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The role of rigid foundation assumption in two-dimensional soil-structure interaction (SSI)
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The role of rigid foundation assumption in two-dimensional soil-structure interaction (SSI)
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Content
THE ROLE OF RIGID FOUNDATION ASSUMPTION IN
TWO-DIMENSIONAL SOIL-STRUCTURE INTERACTION (SSI)
By
Thang H. Le
Ph.D. Advisor: Vincent W . Lee
______________________________________________________________________
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CIVIL ENGINEERING)
December 2017
Copyright 2017 Thang H. Le
i
This Page Intentionally Left Blank
ii
Abstract
Soil-structure interaction (SSI) is a process in which the effects of wave propagation in the
half-space are modified by the response characteristic of a structure and vice versa. SSI is one
of the most dominant subjects in the earthquake engineering research, especially at the interface
of soil and structural dynamics. Classical research has investigated SSI with the assumption of a
rigid foundation with high scattered waves. The structure and soil has also been assumed to be
linear, and the only energy loss in the system is associated with scattered-wave radiation into the
half-space. SSI studies have shown that the dynamic response of a structure supported on a
flexible foundation may differ significantly from the response of the same structure erected on a
rigid foundation. Flexible foundations have been investigated based primarily on numerical
methods. The major drawback of using numerical methods to solve seismic wave problems is
the need for substantial resources and computing time in order to obtain accurate results. In
contrast, analytical methods provide more accurate and relatively simple methods of performing
similar computations on a larger scale, allowing them to be done economically and in a shorter
period of time. Additionally, analytical solutions provide more physical insights into the nature
of the problem and offer benchmarks necessary to verify the other, more approximate, solutions
of the numerical methods. Most importantly, analytical methods contribute to developing
solutions to equations that can be used as the foundation for future research in the field of
earthquake engineering.
The purpose of this thesis is to develop new tools to investigate the SSI for the case of
flexible foundation. The models studied in this thesis present an extension of previous work for
a shear wall on a semi-circular rigid foundation in an isotropic homogeneous and elastic
half-space. Le & Lee (2014) published a new approach and an SSI model to solve for the case
of rigid foundation, an extension of Luco (1969) and Trifunac (1972), using the “big arc
iii
numerical” method, which can be later modified to the case of semi-circular or arbitrarily
flexible foundation. Le et al. (2016) investigated the SSI of a shear wall supported by a rigid
and flexible ring foundation. Chapter 4 of this thesis studies a similar model to Le et al. (2016),
using the big arc approximation developed in Le & Lee (2014). The analytical solutions of these
investigations are for cases in which the foundations are rigid, non-elastic movable foundations.
The solutions serve as an intermediate step toward a goal of solving the SSI of a shear wall on
an elastic foundation. Le et al. (2017) published the analytical solution of a two-dimensional,
moon-shaped alluvial valley embedded in an elastic half-space for incidence plane SH waves
using the wave function expansion, the Discrete Cosine Transform (DCT), and the big arc
approximation. The model studied in the current paper can be recognized as an elastic
foundation without a building on top. Lastly, Chapter 6 of this thesis investigates the SSI of a
shear wall supported by an elastic foundation. This chapter develops a new approach and model
to solve the SSI of a tapered shear wall for all rigid, semi-rigid, and flexible foundations using
an asymptotic of the special functions, the wave function expansion, the Discrete Cosine
transform (DCT) and big-arc approximation as developed in Chapter 5.
iv
Acknowledgements
Firstly, I would like to express my sincere gratitude to my advisor Professor Vincent W.
Lee for the continuous support of my Ph.D. study and related research, for his patience,
motivation, and immense knowledge. His guidance helped me in all the time of research and
writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D.
study. Equally, I would like to thank Professor Mahailo D. Trifunac for his generous review and
valuable suggestions on the manuscript. He imbibed in me many important facets of scientific
research. I realize that it is up to me to carry on those great ideas and dedication in the future.
Besides my advisor, I would like to thank the rest of my thesis committee: Professor Hung
L. Wong, Professor Erik A. Johnson, and Professor Chunming Wang for their insightful
comments and encouragement, but also for the hard question which incented me to widen my
research from various perspectives.
My sincere thanks also goes to Dr. Nazaret Dermendjian, Professor and Chair of the
Department of Civil Engineering and Construction Management at California State University,
Northridge, who recommended me to my advisor Professor Vincent W. Lee and the USC Viterbi
Sonny Astani Department of Civil and Environmental Engineering. Dr. Dermendjian has
provided me encouragement, support, and guidance.
I would like to thank Master Hoang Vu for constantly encouraging, supporting and
inspiring me in completing this dissertation. I sincerely thank my colleagues Duke Huynh and
Jee Hyae Kim for their support and encouragement.
Lastly, I would like to thank my family: my parents and to my brother and sisters for
supporting me spiritually throughout writing this thesis and my life in general.
Last but not the least, I would like to thank my beloved wife and children, whom had been
tolerated my neglectfulness during my Ph.D. research. I dedicate this dissertation to them.
v
TABLE OF CONTENTS
ABSTRACT ...................................................................................................................................... ii
ACKNOWLEDGEMENTS .............................................................................................................. iv
TABLE OF CONTENTS ..................................................................................................................... v
LIST OF FIGURES AND TABLE .................................................................................................... viii
Chapter 1 INTRODUCTION ........................................................................................................ 1
1.1 Motivation ........................................................................................................................ 1
1.2 Review of Methodologies to Solve Wave Propagation Problems .................................... 3
1.3 Literature Review ............................................................................................................. 7
1.3.1 Soil-structure interaction – Analytical methods ................................................... 7
1.3.2 Soil-structure interaction – Numerical methods ................................................... 9
1.3.3 Alluvial valley – Analytical methods ................................................................. 12
1.3.4 Alluvial valley – Numerical methods ................................................................. 14
1.4 Thesis Organization ........................................................................................................ 16
Chapter 2 OUT-OF-PLANE (SH) SOIL-STRUCTURE INTERACTION: SEMI-CIRCULAR
RIGID FOUNDATION REVISITED ......................................................................... 19
2.1 Introduction .................................................................................................................... 19
2.1.1 Brief history ........................................................................................................ 20
2.1.2 Review of Trifunac (1972) paper ....................................................................... 20
2.2 The New Mathematical Model: Tapered-Shape Shear Wall ........................................... 25
2.2.1 The model ........................................................................................................... 25
2.2.2 The Free-Field Waves in the Half-Space ............................................................ 28
2.2.3 The Wave Field within the Structure .................................................................. 29
2.3 The Boundary Conditions .............................................................................................. 30
2.3.1 Displacement continuity ..................................................................................... 30
2.3.2 The Stress-free and stress continuity equations in the building ......................... 30
2.3.3 The Dynamic Equation for the Rigid Foundation .............................................. 32
2.4 Numberical Analysis of the Displacements .................................................................... 34
2.5 Conclusion and Proposed Studies .................................................................................. 38
Chapter 3 OUT-OF-PLANE (SH-WAVES) SOIL-STRUCTURE INTERACTION: A SHEAR
WALL WITH RIGID AND FLEXIBLE RING FOUNDATION ............................... 39
3.1 Introduction .................................................................................................................... 39
3.2 The Mathematic Model .................................................................................................. 40
3.2.1 The free-field wave in the half-Space ................................................................ 42
3.2.2 The wave field within the structure .................................................................... 43
3.2.3 The action of flexible foundation on the rigid foundation .................................. 44
3.3 The Boundary Conditions .............................................................................................. 45
vi
3.3.1 Displacement and stress continuity .................................................................... 45
3.3.2 The dynamic equation for the rigid foundation .................................................. 46
3.4 Numerical Analysis of the Displacement ....................................................................... 49
3.5 Conclusion ...................................................................................................................... 58
Chapter 4 OUT-OF-PLANE (SH-WAVES) SOIL-STRUCTURE INTERACTION: A SHEAR
WALL WITH RIGID AND FLEXIBLE RING FOUNDATION - BIG ARC
NUMERICAL METHOD .......................................................................................... 60
4.1 Introduction .................................................................................................................... 60
4.2 The Mathematical Model ............................................................................................... 60
4.2.1 The free-field wave in the half-Space ................................................................ 63
4.2.2 The wave field within the structure .................................................................... 64
4.2.3 The wave field within the flexible foundation .................................................... 64
4.3 The Boundary Conditions .............................................................................................. 65
4.3.1 At the interface of flexible foundation and free-field ......................................... 65
4.3.2 At the interface of flexible and rigid foundation ................................................ 66
4.3.3 Building boundary conditions ............................................................................ 66
4.3.4 The action of flexible foundation on the rigid foundation .................................. 68
4.3.5 The interaction between rigid foundation and building ...................................... 68
4.3.6 The dynamic equation for the rigid foundation .................................................. 68
4.3.7 The rigid foundation displacement ..................................................................... 69
4.4 Numberical Analysis of the Displacements .................................................................... 72
4.5 Conclusion ...................................................................................................................... 76
Chapter 5 SH WA VES IN A MOON-SHAPED V ALLEY .......................................................... 77
5.1 Introduction .................................................................................................................... 77
5.1.1 Brief history ........................................................................................................ 77
5.1.2 Two examples ..................................................................................................... 78
5.1.2.1 Sherman Oaks ................................................................................ 79
5.1.2.2 Forced vibrations of the Millikan Library ..................................... 81
5.2 The Mathematic Model .................................................................................................. 83
5.2.1 The incident waves in the half-space .................................................................. 85
5.2.2 The waves within the valley ............................................................................... 86
5.2.2.1 The Wave Field within the Outer Semi-Circular Region ............... 86
5.2.2.2 The Wave Field within the Inner Semi-Circular Region ............... 87
5.2.3 Transfering the valley waves between the two coordinate systems ................... 89
5.3 Interface Boundary Conditions of the V alley ................................................................. 91
5.3.1 The outer interface of the valley with a half-space medium .............................. 91
5.3.2 The interface of the inner and the outer regioins of the valley ........................... 91
5.4 New Proposed Solution Verification .............................................................................. 93
5.5 Numerical Examples ...................................................................................................... 94
5.6 Discussion and Conclusions ......................................................................................... 110
vii
Chapter 6 OUT-OF-PLANE (SH-WAVES) SOIL-STRUCTURE INTERACTION: A SHEAR
WALL WITH FLEXIBLE CYLINDRICAL FOUNDA TION ................................. 111
6.1 Introduction .................................................................................................................. 111
6.2 The Mathematic Model ................................................................................................ 111
6.2.1 The free-field waves in the half-space .............................................................. 115
6.2.2 The wave field within the structure .................................................................. 117
6.2.3 The wave field within the flexible foundation .................................................. 1 1 8
6.3.2.1 The wave field within the inner semi-circular flexible foundation.. 118
6.3.2.2 The wave field within the onner semi-circular flexible foundation . 120
6.3 The Interface Boundary Conditions ............................................................................. 120
6.3.1 At the outer interface with half-space medium ................................................ 120
6.3.2 At the interface of the inner and the outer flexible foundations ....................... 121
6.3.3 At the building and the flexible foundatino interface ....................................... 123
6.4 Numerical Implementation: Asymptotic Expansion ..................................................... 124
6.4.1 The Shear Wall Structure Waves ...................................................................... 124
6.4.1.1 The Choice of k
b
R
1
...................................................................... 126
6.4.2 The flexible foundation waves ......................................................................... 126
6.4.3 Normalized strain ............................................................................................. 129
6.4.4 Numerical pa ra m eters ....................................................................................... 130
6.5 Discussion and Conclusions ......................................................................................... 132
Chapter 7 CONCLUSION AND FUTURE WORK ................................................................. 172
7.1 Conclusion .................................................................................................................... 172
7.1.1 Chapter 2 Summary and Conclusion .............................................................. 172
7.1.2 Chapter 3 Summery and Conclusion .............................................................. 173
7.1.3 Chapter 4 Summary and Conclusion .............................................................. 173
7.1.4 Chapter 5 Summary and Conclusion .............................................................. 174
7.1.5 Chapter 6 Summary and Conclusion .............................................................. 174
7.1.6 Continuation of the SSI of a Shear Wall with Flexible Foundation ............... 176
7.1.7 Future Work .................................................................................................... 176
BIBLIOGRAPHY ............................................................................................................................ 177
Appendix A .................................................................................................................................. 189
Clossary .................................................................................................................................. 191
viii
LIST OF FIGURES AND TABLES
Fig. 1.1 Seismic design per building code ................................................................................. 2
Fig. 1.2 Building subjected to seismic waves ............................................................................ 2
Fig. 2.1 Shear wall, foundation, and soil (Trifunac 1972) ....................................................... 23
Fig. 2.2 The Mathematical model of the tapered shear wall with a semi-circular rigid foundation
............................................................................................................................................ 27
Fig. 2.3 The effect of interaction on ∆ the amplitude of foundation vibration ........................ 36
Fig. 2.4 Comparison of the Big Arc Numerical and Trifunac (1972) Plots .............................. 37
Fig. 3.1 The Mathematical model SSI with semi-circular flexible and rigid foundation ......... 41
Fig. 3.2 The effect of interaction on the amplitude of the foundation vibration:
1, 1, 1, 1, 0, 2, 4
BS F S
aa M M M M
............................................... 51
1, 1, 1, 1, 0, 2, 4
BS F S
aa M M M M (Trifunac 1972) .................... 51
Fig. 3.3 The effect of interaction on the amplitude of the foundation vibration:
1.25, 2, 1, 1, 0, 2, 4
BS F S
aa M M M M
........................................ 52
1.25, 2, 2, 1, 0,2,4
BS F S
aa M M M M
....................................... 53
1.25, 2, 4, 1, 0,2,4
BS F S
aa M M M M
....................................... 53
Fig. 3.4 The effect of interaction on the amplitude of the foundation vibration:
1.50, 2, 1, 1, 0, 2, 4
BS F S
aa M M M M
........................................ 54
1.50, 2, 2, 1, 0,2,4
BS F S
aa M M M M
....................................... 54
1.50, 2, 4, 1, 0,2,4
BS F S
aa M M M M
....................................... 55
Fig. 3.5 The base shear force:
1, 4, 1, 2
BS F S
MM M M
...................................................................... 55
1.5, 4, 1, 2
BS F S
MM M M
.................................................................. 56
ix
2, 4, 1, 2
BS F S
MM M M
..................................................................... 56
Fig. 3.6 The dimensionless base shear:
1, 4, 1, 2
BS F S
MM M M
...................................................................... 57
1.5, 4, 1, 2
BS F S
MM M M
.................................................................. 57
2, 4, 1, 2
BS F S
MM M M
..................................................................... 56
Fig. 4.1 The mathematical model SSI with semi-circular flexible and rigid foundation ......... 62
Fig. 4.2a The effect of interaction on the shear wall:
1, 10, 1, 1
BF R F
RH M M M M ............................................................ 73
Fig. 4.2b The effect of interaction on the shear wall:
1, 50, 1 , 1
BF R F
RH M M M M ............................................................ 73
Fig. 4.3a The effect of interaction on the shear wall:
1, 10, 2, 1
BF R F
RH M M M M ........................................................... 74
Fig. 4.3b The effect of interaction on the shear wall:
1, 50, 2, 1
BF R F
RH M M M M .......................................................... 74
Fig. 4.4a The effect of interaction on the shear wall:
1, 10, 4, 1
BF R F
RH M M M M ........................................................... 75
Fig. 4.4b The effect of interaction on the shear wall:
1, 50, 4, 1
BF R F
RH M M M M .......................................................... 75
Fig. 5.1 The San Fernando Valley, northwest of Los Angeles, with a horizontal projection of the
fault (dashed line) that slipped during the 1994 Northridge earthquake. Asterisks show the
aftershocks, which were recorded at accelerograph stations USC#3, USC#6, and USC#53.
The depth to the crystalline basement rocks (from Y erkes et al. 1965), in thousands of feet, is
shown in the southeastern part of the Valley ...................................................................... 80
Fig. 5.2 The San Fernando Valley: “gray zones” (where buildings were damaged) and locations
of reported breaks in the water pipes (black dots) in the 1994 Northridge earthquake.
Locations of “unsafe” buildings following the 1971 San Fernando earthquake are shown by
x
open diamonds.................................................................................................................... 81
Fig. 5.3 (Top) The three trial cross sections A, B, and C through the alluvium in west Pasadena
(west is to the left in this figure). (Bottom) A comparison of the computed (for models A, B,
and C) and normalized measured displacements at 13 points (triangles) ........................... 82
Fig. 5.4 A model of a moon-shaped valley in an elastic half-space ......................................... 84
Fig. 5.5 Surface displacement amplitudes for
0.50, 10, 0.50, 1.50, 0 , 60
oo
vv
ha kk ............................................ 94
Fig. 5.6a Surface displacement amplitudes and phases for
0.50, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
ooo o
vv
ha kk ................. 96
Fig. 5.6b Surface displacement amplitudes and phases for
0.75, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
ooo o
vv
ha kk ................. 96
Fig. 5.6c Surface displacement amplitudes and phases for
1.00, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
oo o o
vv
ha kk ................. 97
Fig. 5.6d Surface displacement amplitudes and phases for
1.50, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
oo o o
vv
ha kk ................. 97
Fig. 5.7 Moon-shaped valleys with different h/a ratios ............................................................ 98
Fig. 5.8a Surface displacement amplitudes and phases for
0.50, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha ........... 99
Fig. 5.8b Surface displacement amplitudes and phases for
1.00, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha ............ 99
Fig. 5.8c Surface displacement amplitudes and phases for
1.50, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha .......... 100
Fig. 5.8d Surface displacement amplitudes and phases for
2.00, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha ......... 100
xi
Fig. 5.9a Surface displacement amplitudes and phases for
0.50, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha ....... 101
Fig. 5.9b Surface displacement amplitudes and phases for
1.00, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha ........ 101
Fig. 5.9c Surface displacement amplitudes and phases for
1.50, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha ........ 102
Fig. 5.9d Surface displacement amplitudes and phases for
2.00, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk ha ....... 102
Fig. 5.10a Top and bottom surface displacement amplitudes and phases for
0.50, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha kk ................................ 103
Fig. 5.10b Surface displacement amplitudes and phases for
1.00, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha kk ................................ 103
Fig. 5.10c Surface displacement amplitudes and phases for
1.50, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha kk ................................ 104
Fig. 5.10d Surface displacement amplitudes and phases for
2.00, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha kk ................................ 104
Fig. 5.11 Transmitted and refracted waves in the valley for 1.00
0.75, 0.50,
v
ha kk 1.50,
v
and 90
o
............................................. 106
Fig. 5.12a The amplitudes of surface relative to the base displacements of the valley versus
ha and , for = 0.5, 1.0, 1.5, and 2.0, and for
v
kk = 0.3 ................................. 108
Fig. 5.12b The amplitudes of surface relative to the base displacements of the valley versus
ha and , for = 0.5, 1.0, 1.5 and 2.0, and for
v
kk = 0.5. The open circles at ha
= 0.75. for = 0° and 90° correspond to the four cases plotted in Figs. 9a through 9d at
= 0.5, 1.0, 1.5, and 2.0 ...................................................................................................... 109
Fig. 6.1 The Model ................................................................................................................. 114
xii
Fig. 6.2 The mathematical model SSI with a semi-circular flexible foundation .................... 114
Fig. 6.3 Surface displacement amplitudes at the interface for 1.0, 0
o
1.5, 0.90, 0.25, 8,16, 32, 50,100 and 1000
bb f f
kk k k ............. 134
Fig. 6.4 Normalized building displacement for 1.0, 0
o
1.5, 0.90, 0.25, 8,16, 32, 50,100 and 1000
bb f f
kk k k ............. 135
Fig. 6.5 Normalized building displacement for 0.25
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
ooo o
bb f f
kk k k ........ 136
Fig. 6.6 Normalized building displacement for 0.50
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 137
Fig. 6.7 Normalized building displacement for 0.75
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 138
Fig. 6.8 Normalized building displacement for 1.00
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 139
Fig. 6.9 Normalized building displacement for 1.25
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 140
Fig. 6.10 Normalized building displacement for 1.50
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 141
Fig. 6.11 Normalized building displacement for 1.75
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 142
Fig. 6.12 Normalized building displacement for 2.00
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 143
Fig. 6.13 Normalized building displacement for 2.25
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 144
xiii
Fig. 6.14 Normalized building displacement for 2.50
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 145
Fig. 6.15 Normalized building displacement for 2.75
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 146
Fig. 6.16 Normalized building displacement for 3.00
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 147
Fig. 6.17 Normalized building strain for 0.25
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 148
Fig. 6.18 Normalized building strain for 0.50
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 149
Fig. 6.19 Normalized building strain for 0.75
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 150
Fig. 6.20 Normalized building strain for 1.00
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 151
Fig. 6.21 Normalized building strain for 1.25
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 152
Fig. 6.22 Normalized building strain for 1.50
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 153
Fig. 6.23 Normalized building strain for 1.75
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 154
Fig. 6.24 Normalized building strain for 2.00
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 155
Fig. 6.25 Normalized building strain for 2.25
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 156
xiv
Fig. 6.26 Normalized building strain for 2.50
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 157
Fig. 6.27 Normalized building strain for 2.75
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 158
Fig. 6.28 Normalized building strain for 3.00
1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb ff
kk k k ......... 159
Fig. 6.29 Surface displacement amplitudes at the interface for 0.25
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 160
Fig. 6.30 Surface displacement amplitudes at the interface for 0.50
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 160
Fig. 6.31 Surface displacement amplitudes at the interface for 0.75
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 161
Fig. 6.32 Surface displacement amplitudes at the interface for 1.00
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 161
Fig. 6.33 Surface displacement amplitudes at the interface for 1.25
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 162
Fig. 6.34 Surface displacement amplitudes at the interface for 1.50
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 162
Fig. 6.35 Surface displacement amplitudes at the interface for 1.75
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 163
Fig. 6.36 Surface displacement amplitudes at the interface for 2.00
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 163
Fig. 6.37 Surface displacement amplitudes at the interface for 2.25
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 164
xv
Fig. 6.38 Surface displacement amplitudes at the interface for 2.50
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 164
Fig. 6.39 Surface displacement amplitudes at the interface for 2.75
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 165
Fig. 6.40 Surface displacement amplitudes at the interface for 3.00
0 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k ... 165
Fig. 6.41 Surface displacement amplitudes at the interface for 0.25
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 166
Fig. 6.42 Surface displacement amplitudes at the interface for 0.50
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 166
Fig. 6.43 Surface displacement amplitudes at the interface for 0.75
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 167
Fig. 6.44 Surface displacement amplitudes at the interface for 1.00
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 167
Fig. 6.45 Surface displacement amplitudes at the interface for 1.25
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 168
Fig. 6.46 Surface displacement amplitudes at the interface for 1.50
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 168
Fig. 6.47 Surface displacement amplitudes at the interface for 1.75
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 169
Fig. 6.48 Surface displacement amplitudes at the interface for 2.00
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 169
Fig. 6.49 Surface displacement amplitudes at the interface for 2.25
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 170
xvi
Fig. 6.50 Surface displacement amplitudes at the interface for 2.50
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 170
Fig. 6.51 Surface displacement amplitudes at the interface for 2.75
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 171
Fig. 6.52 Surface displacement amplitudes at the interface for 3.00
90 , 1.5, 0.90, 0.25, 8,16, 32, 50,100 &1000
o
bb f f
kk k k . 171
1
CHAPTER 1
INTRODUCTION
1.1 Motivation
In an earthquake event, seismic waves propagate up through a structure from the geologic
media underlying the foundation. The response of a shear wall, rigid frame, or structure to the
seismic waves is affected by interactions among the geologic media underlying the foundation
and the structure. For more than a century, procedures in building codes analyzed the response
of structures to an earthquake event by applying virtual lateral (horizontal) forces to each of the
levels, called diaphragms, above the earth’s surface. Previous analyses have typically assumed
that the foundation was fixed to the geologic media at the foundation-soil interface (Figure 1.1).
The virtual lateral forces applied to the structure were obtained based on the lateral-resistant
system and the parameters representing free-field ground motion, which refer to seismic waves
of the half-space without structures in the vicinity of the half-space. However, through
functional analyses, it has become apparent that the motions at the foundation that are
propagated to the structural elements are different from the assumed free-field motion (Figure
1.2). These differences are due to the interaction of the structure foundation and the geologic
media. This phenomenon is understood as soil-structure interaction (SSI), and it is often not
properly understood or accounted for in building design.
2
Figure 1.1. The seismic design per building code.
Figure 1.2. A building subjected to seismic waves.
3
The soil-structure interaction (SSI) is the seismic structural response caused by the seismic
waves of the half-space acting on the foundation, which in turn affects the motion of the soil
caused by the response and vibration of the structure. Recently, the SSI has been recognized and
adopted into building codes to acknowledge that it can affect the response of the building’s
lateral resistant system (ASCE/SEI 7-10). Structural engineers refer to the SSI as the interaction
forces that can change the response of the structure’s foundation in comparison to the free-field
ground motion—i.e., motion recorded on the ground surface without the presence of the
structure. It is believed that the SSI can be beneficial to the structure system response in seismic
design using the response spectrum analysis by reducing the seismic base shear as a result of
lengthening of the structure period and increasing the damping of the overall lateral-resistant
system. However, while these forces occur in every structure, the calculations do not always
reflect the reality of the soil motion. If the structure is analyzed based on a rigid foundation
excited by the free-field motion, then the conclusion is that no SSI occurs even though the
interaction forces, in fact, displace the structure and the same forces impinge on the foundation.
The flexibility of the soil media and the foundation are the main components that affect the
ability of the interaction forces. As this effect has not been recognized in the current codes, it is
crucial to extend the research in this field beyond the assumption of the rigid foundation and
soil media. Moreover, recent testing and study of seismic interaction with structures from past
earthquakes have shown that the vertical component of seismic waves is a critical one in certain
types of structural failure. However, this issue has been completely neglected in analyses
required by most building codes (NIST GCR 12-917-21).
1.2 Review of Methodologies to Solve Wave Propagation Problems
The study of SSI began during the late nineteenth century. In the early part of the twentieth
century, it developed slowly due to research on nuclear power plant design and improvements in
building structure safety due to significant earthquake events around the world. More recently,
4
the sophistication of SSI modeling has grown at a fast pace with computer technology and in
response to the demands of research in the field of nonlinearity of building materials and soil
media. During this period. Many attempts and elaborate computational methods have been
developed to solve the mathematical difficulties and complicated wave propagation problems
(Chen and Chen, 2005). Researchers have developed two major methodologies used to analyze
these problems: (1) numerical methods such as the Finite Difference Method (FDM), the
Boundary Element Method (BEM), and the Finite Element Method (FEM); and (2) analytical
methods such as integral equations, integral transforms, the perturbation method, and the
method of wave function expansion.
The numerical methods developed in large part because of the rapid progress in the
iterative capacity of computer technology. With the availability of computational power, these
methods have been the basis of a number of studies dating from the 1960s to the present. The
finite element methods such as FDM and FEM have been used widely in the SSI field. The
FDM was introduced in the 1930s and further developed in the 1960s (Thomée 2001). A finite
difference approximation was used to solve wave equations and demonstrated the solution
convergence. The FEM, on the other hand, has been discussed since the beginning of the
twentieth century. M. J. Turner, L. C. Topp, J. H. Argyris, R. W. Clough, H. C. Martin, and O. C.
Zienkiewicz developed the FEM in the 1950s using computer technology developed for the
aerospace industry. This new technology was applied to a wide range of engineering
applications including earthquake engineering (Gupta and Meek, 1996). The FEM was
expanded to the field of earthquake engineering as an approach to simulate seismic wave
propagation; however, it suffered from low accuracy in comparison with the FDM. The BEM
began around 1962 and quickly developed into a movement in 1967 when new electronic
computers became more widely available. In 1962, Friedman and Shawn first solved the wave
equation in the time domain for a scattered wave field resulting from seismic waves impinging
on a cylindrical obstacle. Rizzo, Jaswon, Ponter, and Symm elaborated and established the
major development of the BEM in the 1970s (Cheng and Cheng, 2005).
5
The main challenge of the FDM and FEM is to properly simulate seismic wave scattering
and diffraction and the imposition of a radiation boundary condition over infinite or
semi-infinite domains. Some approximations and guesswork have been introduced to solve
seismic wave problems. Truncation is normally used to make an approximation or to cut the
semi-infinite or infinite domains into a finite interval/boundary defined by seismic wavelengths
and wave numbers. This method could introduce inaccuracies into the numerical computations
of wave problems (Wu and Lee, 1994a,b). In addition, the large physical dimensions of the
half-space can create large-sized numerical problems for a seismic wave analysis that could
overwhelm the FDM or FEM. Moreover, to simplify the complexity of the seismic soil behavior,
it is typically assumed to be linearly elastic, homogenous, and isotropic throughout the soil
model, which is not reflected in real-world design problems.
The Boundary Element Method (BEM) can eliminate the disadvantages of the FDM and
the FEM and solve wave problems more efficiently, as it only needs to model the infinite
domain and perform the boundary integrals. However, the BEM has difficulty computing
seismic waves accurately in a heterogeneous medium and addressing singularities in the
numerical integration path. In addition, the BEM can only be applied to general geometries and
media in which the values of the variables are known. Boundary integrals may become singular
depending on the geometry and the distance between the source and the node being integrated
(Cheng and Cheng, 2005).
To conclude, the major drawback of using the FDM, FEM, or BEM to solve seismic wave
problems is the need for substantial resources and computing time in order to obtain accurate
numerical solutions. Special care should be paid to computational cost. Recently, practicing
engineers have advanced at a much lower rate in the numerical methods, because high skill and
knowledge is required to run the finite element computation. In addition, the finite element
results do not present the physical aspect of the seismic wave problems. Finally, the cost to own
and maintain the finite element software has skyrocketed in the past decades, preventing most
small- and medium-sized consulting engineering firms from using and advancing the numerical
6
methods.
In contrast to the numerical methods, which are asymptotic and hindered by necessary
approximations and limitations, the analytical methods provide more accurate and relatively
simple methods of performing similar computations on a larger scale, enabling them to be done
economically and quickly. Frequently, it is possible to solve computations by hand or calculator,
or using simple computer software, such as spreadsheets, Matlab, or Mathematica. Additionally,
analytical solutions provide more physical insights into the nature of the problem and offer
benchmarks necessary to verify the other, more approximate, numerical method solutions.
Moreover, analytical methods contribute to developing solutions to equations that can be used
as the foundation for future research in the field of earthquake engineering.
The purpose of this dissertation is to develop new and more powerful tools to solve for
more or less realistic models. The models studied in this dissertation present a logical extension
of the elastic shear wall with a circular rigid foundation fixed firmly in an isotropic,
homogeneous, and elastic half-space considered only for vertical-incidence SH waves.
Vertical-incidence SH waves were initially studied by Luco (1969) and then formulated to any
angle of incidence SH waves by Trifunac (1972). The more- or less-realistic models should
consider concrete footings, piles, grade beams and mat foundations that, in practice, often
support buildings. These kinds of foundations are not rigid, so the ideal goal should be to solve
for the SSI of a shear wall supported by a flexible or semi-rigid foundation. However, Trifunac’s
(1972) methodology cannot be modified to cases of flexible or semi-rigid foundations. Further,
it is not uncommon for buildings to gradually taper from bottom to top, with the result that the
top of the building is slightly narrower than the base, rather than the same width from the top to
the bottom, which presents further complications. To address both of these complicated realities,
this dissertation develops new approaches and models in order to solve the SSI of a tapered
shear wall for all rigid, flexible, and semi-rigid foundations using the "big-arc approximation."
7
1.3 Literature Review
1.3.1 Soil-structure interaction: Analytical methods
SSI is a fascinating subject in earthquake engineering that has warranted significant
research and documentation. The following overview addresses the literature relevant to the
topic of this dissertation.
Reissner (1936) observed the natural effect of soil inertia and established that it laid in the
inertial properties of the soil media. He also discovered that the radiation damping into the soil
contributed a great deal to the response of the structure, thus marking the beginning of SSI
study using the analytical method. A number of years later, Housner (1957) demonstrated that
the variation of density and elasticity in the soil media caused a change in the wave propagation
velocity that led to the reflection and refraction of incoming seismic energy. This phenomenon
is understood as wave passage or kinematic interaction. Additionally, the weight of the structure
generates inertial forces that impinge on the soil media when responding to seismic waves,
causing additional deformation in the soil known as inertial interaction. Integration of the
dynamic response of a structure and the supporting soil media causes the inertia effects. The
structure deforms to dissipate the energy caused by incoming seismic waves and, in turn, the
waves scatter away from the structure, increasing the soil deformation.
The topic was taken up again in 1969 by Luco (1969), who focused on the diffraction of
normal incidence plane SH waves by an elastic shear wall resting on a rigid semi-circular
foundation embedded in soil media. Trifunac (1972) generalized Luco’s analytical solution into
cases for arbitrary oblique incidence angles of SH waves. Analytical solutions of the dynamic
interaction of shear walls with circular rigid foundations embedded in the half-space were
derived and evaluated. The numerical results demonstrated that waves scattered from a rigid
foundation contribute significantly to the surface ground motion near the shear wall and at
distances at least one order of magnitude greater than the characteristic length of the foundation.
8
Therefore, waves reflected and diffracted by the foundation must not be neglected when the
Fourier amplitude ratios of accelerograms recorded in and around the building are used to study
the SSI. For this reason, when considering low frequencies (long periods) for shear walls
founded on hard soil, it is possible to obtain base displacements and base shear forces higher
than the values computed when neglecting the interaction. For higher frequencies, the effect on
the displacement and base shear force is more important for rigid shear walls founded on soft
soils. In more flexible structures, the effect of interaction depends heavily on the frequency and
stiffness of the soils.
In the early 1970s, Lee and Westley (1973) investigated the SSI influence and its effects on
the seismic response of nuclear reactors using a three-dimensional (3D) model subjected to the
vertical propagation of the SH waves. Normal incidence SH waves used the analytical method
along the two orthogonal directions and the spring-mass models for the structures attached to
the foundations. Luco and Contesse (1973) presented the closed-form analytic solutions for the
two-dimensional (2D), out-of-plane problems of the interaction between elastic shear walls
fixed on rigid circular foundations that are subjected to vertical and oblique angles of incidence
harmonic SH waves. Trifunac and Wong (1974b) extended the analytical solutions for the
incident plane SH waves to shallow or deep elliptical rigid foundations, as well as to multiple
buildings and foundations. The solutions to these problems showed that closely spaced
buildings could affect the fundamental frequencies (or period) of neighboring buildings due to
the SSI.
Lee (1979) studied 3D analytical solutions for the interaction of a single degree-of-freedom
oscillator supported by a semi-spherical foundation for harmonic P-, SV-, and SH waves.
Kobori and Kusakabe (1980) investigated rigid rectangular and circular foundations welded to
the surface of elastic half-spaces and subjected to harmonic seismic waves. Triantafyllidis and
Prange (1988) studied the dynamic interaction of two rigid circular foundations embedded in an
elastic half-space and subjected to Rayleigh waves impinging at an arbitrary angle. These
studies demonstrated that forces react on the foundations perpendicular to the incidence of
9
propagation in addition to forces in the direction of the motion. Todorovska (1993) studied the
in-plane foundation-soil interaction of circular foundations embedded in elastic soil media. The
research mainly focused on the wave passage influences and the depth of foundation below the
half-space for in-plane SSI. In the same year, Hryniewicz (1993) investigated two 2D trip
foundations based on a semi-infinite medium embedded in a homogeneous half-space excited
by anti-plane SH waves.
1.3.2 Soil-structure interaction: Numerical methods
Over the past three decades, advances in computer technology have accelerated the
development of numerical methods in dealing with complicated seismic wave problems. To
obtain more accurate radiation damping of semi-infinite and infinite spaces, the geologic
medium should be modeled using a very large scale. This demands a high consumption of
modeling and running time, as well as a computer with internal memory large enough to
minimize the FEM errors. Some researchers have proposed various boundaries to reduce the
scale. Most of this work has been based on semi-cylindrical or elliptical foundations with
structures modeled by a lumped mass with a single degree of freedom.
Lysmer and Wass (1972) developed the viscous boundary technique, which absorbed
scattering waves effectively. The special feature of the viscous boundary technique is that it can
correctly compute the response of the structure and the outward radiation of wave energy. This
paper presented the SSI of a linear elastic structure supported by two semi-infinite media layers
excited by periodic forces acting perpendicularly to the plane of the structure (SH waves). The
numerical results were compared to the known analytical solutions of the spring and dashpot
soil-foundation system and found that the numerical method led to peak amplitude of footing
displacement significantly lower than the analytical method. The curves obtained from plotting
the frequency of base motion versus the amplitude of footing displacement are quite close to the
analytical results of the circular rigid foundation without a shear wall on top studied by Luco
(1969) and Trifunac (1972). It can be seen that the peak displacement is 1.3 in comparison with
10
1.2 from Luco (1969) and Trifunac (1972). Interestingly, the total computation time was 60
seconds on a CDC 6400 computer in 1972.
Smith (1974) developed a different infinite boundary technique called the superposing
boundary technique, in which solutions from two different conventional boundary conditions
are superimposed. This new development improved the methodology by Lysmer and Wass
(1972) by creating a nonreflecting boundary that prevented damping of the reflections. This
method can be used to solve the SSI with incident SH-, P- and SV waves, but only with plane
boundaries parallel to one of the coordinate axes. This creates a problem with curved boundaries
in that they are concave to the incident wave front and multiple reflections can occur.
To address this, Engquist and Majda (1977) used what is called a paraxial boundary
method that allows wave propagation in only one direction, while the conventional wave
propagation allows it in both positive and negative directions. Thus, waves can propagate only
in the outward direction at the infinite boundary with no reflected waves. The paraxial boundary
is also called the nonreflecting boundary method and is independent of the frequency. It can
solve for curved boundaries, unlike the superposing boundary technique by Smith (1974). Both
viscous and paraxial boundary methods assume the infinite space to be linear elastic.
Liao and Wong (1984) introduced the transmitting boundary, which is intended to absorb
body and surface waves on the lateral infinite boundary, depending on the frequency. The
transmitting boundary method improves on Engquist and Majda’s (1977) methodology by
eliminating round-off errors in an early stage of the computation to prevent the solution from
decaying to the level of numerical noise. The transmitting formula was simplified, which made
it more practical in computing seismic waves. It can solve linear 2D and 3D wave problems,
which can be expressed in the form of an extrapolation formula and implemented in the FEM
and FDM. Luco and Wong (1986a,b) studied and extended the numerical method to the case of
multiple rigid foundations of different shapes embedded in elastic or viscoelastic half-spaces
excited by seismic waves and subjected to external loads. These studies showed that the
discretization of the foundation has a significant effect on the evaluated impedance functions for
11
small separations. Luco and Wong (1986a,b) also found that computer memory is an important
factor when the model is large.
Givoli and Keller (1990) derived a new formulation of the numerical method to solve wave
problems in an unbounded domain called an exact Non-Reflecting Boundary Condition method
(NRBC). The NRBC can reduce large infinite domains to smaller domains and still obtain
accurate and efficient results in comparison with such methods based on approximate local and
artificial boundary conditions. Givoli (2001) extended the NRBC method to construct
high-order local NRBCs with a symmetric structure to solve for both the time-harmonic
Helmholtz equation and time-dependent waves involving any high-order derivatives. This
NRBC is an arbitrary high-order condition that can easily increase accuracy. A user need only
change the input parameter of the code to increase the NRBC order.
Gicev (2005) applied the abovementioned FEM, FDM, and BEM to investigate the
interaction of a soil-flexible foundation and a structure for incident plane SH waves. The
artificial boundaries were also reviewed and illustrated using numerical examples. Soil-structure
interaction with flexible semi-circular foundations that considered only a steady-state analysis
was studied and compared with the results for the special case of a rigid foundation presented in
Trifunac (1972). It was found that the results for the rigid foundation also hold for the flexible
foundation, except that the motion of the flexible foundation depends on the incident angle. The
reason for the difference is that the seismic waves can transmit through the flexible foundation
at the foundation-shear wall interface and generate differential motions. The flexible foundation
amplitudes are found to be larger due to the seismic waves scattering more of the incident
energy from the rigid foundation. Gicev (2009) extended this problem to the case of a
structure-interaction with nonlinear soil. One of the disadvantages of this approach is that short
waves cannot be computed even with very fine grids, and the incidence wave should be filtered
to obtain numerical results more effectively. Gicev and Trifunac (2012a,b) and Gicev et al.
(2012) found that the nonlinear soil response could be a powerful sink of incident seismic
energy. The structure and soil are assumed to be linear and the only energy loss in the system
12
would be associated with radiation of scattered waves into the half space. Ferro (2013) revisited
the nonlinear SSI and applied the integral boundary formulations with the hybrid Laplace-time
domain approach. The results were found to be comparable with the solutions of a full FEM
model of the bounded domain.
Torabi and Rayhani (2014) investigated the seismic soil-structure interaction of flexible
soil foundations using 3D finite element modeling. The numerical results were obtained by
exciting the finite element model with the recorded ground motions of the 1985 Mexico City
and 1989 Loma Prieta earthquakes. Their findings appear to agree with Luco (1969) and
Trifunac (1972) in cases of tall buildings supported by rigid foundations in the following ways.
(1) For more flexible structures, the effect of interaction depends heavily on the frequency
and the stiffness of the soil. This clearly indicates that a flexible and slender building with
low frequency (high period) will interact effectively with the soil and lead to decreasing
base shear force due to dissipation of earthquake-induced energy;
(2) A rigid structure founded on soft soils will ride along with the movement of the
free-field and generate a lower base shear force compared with the flexible structure.
1.3.3 Alluvial valley: Analytical methods
Most of the wave propagation problems were modeled on the flat ground surface in the
half-space, which does not reflect real ground surface topography. Variation in topography and
geological media can affect seismic waves. This problem can be addressed and explained by
solving for a 2D alluvial valley. Trifunac (1971) studied the scattering wave of SH waves in a
semi-cylindrical alluvial valley and derived analytical formulations for the behavior of the
surface motion using a wave function method that was an expansion of Bessel functions.
Trifunac (1973) applied these analytical formulations to the scattering of plane SH waves by a
semi-circular canyon and compared the results to the recorded Pacoima Dam ground motion
data. His comparison suggested that the variation of the surface topography had little effect on
the data recorded by the accelerogram. Wong, Trifunac, and Lo (1976) investigated the SSI of a
13
semi-circular cylindrical rigid foundation embedded in a homogeneous half-space adjacent to a
semi-circular canyon excited by plane SH waves. This study found that the canyon does not
significantly affect the response of the shear wall and rigid foundation.
Trifunac and Wong (1974a,b,c) extended the surface ground motion analysis of a
semi-circular alluvial valley to deep and shallow semi-elliptical alluvial valleys for incident
plane SH waves. Moeen-Varizi and Trifunac (1981) studied the scattering and diffraction of a
semi-cylindrical canal embedded in the elastic, isotropic, and homogeneous half-space when
subjected to plane SH waves. Analytical formations were derived mainly for solving a practical
engineering problem regarding the structural analysis and design of reinforced concrete canals.
By utilizing Hankel wave functions expanded for incident waves, Moeen-Varizi and Trifunac
(1985) modified the semi-cylindrical canal to an arbitrary shape embedded in the elastic,
isotropic, and homogenous half-space to reflect various physical and irregular canal surfaces.
The numerical results of the approximate method were found to be in agreement with the
known exact solution of a semi-circular canal. A few years later, Moeen-Varizi and Trifunac
(1988a,b) further investigated the scattering and diffraction of various shapes of 2D alluvial
canyons with inhomogeneous media subjected to SH-, P-, and SV waves. Again, Bessel and
Hankel wave function expansion was successfully used in the mean square error approximation.
Lee and Cao (1989) applied the incident SV waves to Trifunac and Wong (1974a,b,c)
models of deep and shallow elliptical canyons. This study found that the surface displacements
were amplified to be close to twice as large as the free-field amplitudes, which were similar in
magnitude to the incident P- and SH waves. The same conclusion was drawn for the
displacement amplitudes outside and within the canyon for both P- and SV waves but not for
SH waves. Using integral equation formulations, Clements and Larson (1994) extended the
work of Trifunac and Wong (1974a,b,c) to the case of inhomogeneous alluvial valleys. Yuan and
Liao (1995) extended the investigation to the 2D scattering and diffraction of plane SH waves
impinging on a circular-arc cross-section of a cylindrical alluvial valley in a homogenous
half-space. The analytical formulations were derived using Bessel and Hankel wave function
14
expansion and Graf’s addition theorem. Tsaur and Chang (2008) and Tsaur and Hsu (2014)
extended the alluvial valley problem to include, respectively, partially filled semi-circular
alluvial layers and partially filled semi-elliptic alluvial layers, both subjected to SH waves.
Chang et al. (2014) further investigated the alluvial valley filled with an inclined alluvial layer
excited by SH polarized waves.
Although analytical solutions of the abovementioned models are far too simple to reflect
real-world applications, they were able to help clarify the similar geometries of valleys and
canyons subjected to incident P-, SV- and SH waves. Moreover, these analytical formulations
are important for checking the more approximate solutions of FDM, FEM, and BEM, and they
contributed an enormous amount of understanding to the foundation of seismic wave
computations in dealing with valleys and canyons.
1.3.4 Alluvial valley: Numerical methods
Soil-structure interaction of the building structure in an alluvial valley was an interesting
topic for many decades, and the numerical method for solving it developed at a particularly fast
pace after the 1985 Mexico City earthquake. Scattering and diffraction of elastic waves on the
topographic surface are important factors in studying the behavior of the alluvial valley.
Sanchez-Sesma et al. (1982) first applied the boundary method to investigate the effects of SH
waves on various kinds of ground surfaces. This study was a benchmark for much successful
research on the alluvial valley, canyon, and hill topography surfaces. Bravo et al. (1988) used
the BEM to study the alluvial valley with two different kinds of soil subjected to SH waves. The
numerical results were found to be in agreement with previous studies using the analytical
method as described in Section 1.3.3 and the damages observed from the 1985 Mexico City
earthquake. Luzon et al. (1995) extended the work by Bravo et al. (1988) by simulating the
alluvial basin with recorded ground motions. The numerical results were obtained using the
indirect boundary element method, which combines modal summation and FDM techniques.
This is a powerful method that can predict the ground motion for areas where recorded ground
15
motion is not available. Zuniga et al. (1995) simplified the formulation to the alluvial valley
problem by modeling a rectangular-shaped alluvial valley supported by a rigid base subjected to
SH waves. Unfortunately, this model was too simple to express the alluvial valley phenomenon
in comparison to the exact solutions obtained from the analytical models. Fu and Bouchon
(2004) used the discrete wave number to improve the numerical method in computing seismic
waves in an alluvial valley. The advantages of this method are mainly in eliminating the
singularity problem of Green’s function and in saving computing time and memory.
Practical engineers are not as interested in the specific ground motion behavior of alluvial
valleys as they are in the response of building structures to the ground motion during an
earthquake event. To this end, Bielak et al. (1999) studied the actual structural response of
buildings founded in a shallow alluvial valley in Kirovakan, Armenia, which were subjected to
the 1988 Armenia earthquake. Based on observations from the earthquake, over 50 percent of
the buildings in the deep section of the valley totally collapsed, compared to the shallow section
of the valley, where no buildings collapsed, and despite the fact that the area sits on top of the
seismic fault. A realistic, 2D valley was modeled and analyzed using the FEM and the recorded
ground motion of the 1988 Armenia earthquake. The numerical results indicated that the peak
accelerations within the deep zone were three times higher than the shallow zone. This is in line
with the observations of actual destruction from the earthquake.
Semblat et al. (2002) investigated seismic effects for various depths of alluvial valleys. The
paper focuses on the amplification and frequency of ground motion versus depth in alluvial
valleys. The authors found that the in-depth motion in a shallow valley is decreasing while in a
deep valley it is increasing. Kim et al. (2003) modeled the alluvial basin in the finite element
program called SHAKE. The results concurred with Semblat et al. (2002) that the amplification
of ground motion could be significant in soft alluvial valleys. Panji et al. (2013) revisited the
alluvial valley subjected to SH waves using the half-plane BEM. This method is intended to
reduce the computing time while still obtaining the desired accuracy in comparison with
analytical solutions.
16
The ground motion at the edge of a valley where the soft alluvial material meets the
underlying bedrock was another important area of analysis. The phenomenon of the basin/valley
edge effect was observed and investigated in the aftermath of the 1994 Northridge earthquake.
Adams (2000) studied these effects by employing the FEM to a realistic basin in Lower Hutt
Valley, New Zealand. One of the critical results was that the edge parallel to the basin edge has
the strongest motion and highest displacement due to SH waves. Iyisan and Hasal (2011) did a
similar investigation of the Dinar Basin in Turkey using Quake/W finite element software. The
maximum ground accelerations near the basin edge are amplified to a factor of four in some
cases. These results concur with the analytical results of the investigation of a cylindrical valley
of double circular-arc cross-section in Chapter 6 of this dissertation in which the displacements
can be amplified to six times the free-field amplitudes.
1.4 Thesis Organization
The models studied in this thesis are 2D, elastic rectangular, and tapered shear wall
(structure) supported by a semi-circular rigid or flexible foundation and fixed firmly on a
half-space considered for the incident SH waves. All materials here are homogeneous, elastic,
and isotropic. The interaction between the shear wall, foundation, and soil media is investigated.
The thesis is organized into chapters as follows:
1. The first chapter includes motivation and a brief history and literature review of prior
research work done on the SSI.
2. The second chapter presents the extension of the shear wall supported by a
semi-circular rigid foundation fixed firmly in an isotropic, homogeneous, and elastic
half-space, considered only for vertical-incidence SH waves by Luco (1969) and then
formulated to any angle of the incidence SH waves by Trifunac (1972). A new
approach and model are developed to solve for a rigid foundation using the big arc
approximation, which can be extended to solve for flexible or semi-rigid foundations.
17
The analytical expression of the interaction of a tapered shear wall, rigid foundation,
and elastic half-space is derived. The results of the numerical analysis are then
compared with Trifunac (1972). This is a reproduction of the 2014 paper by Le and
Lee.
3. The third chapter presents the interaction of an elastic shear wall, flexible-rigid
foundation, and the elastic half-space for incident plane SH waves. The foundation
consists of a semi-circular rigid foundation that is wrapped with an elastic semi-circular
flexible one. In this chapter, the displacement of the shear wall structure, flexible-rigid
foundation, and the ground motion close to the subjected building are investigated and
discussed in a 2016 published paper by Le, Lee, and Luo. The results are compared
with Trifunac (1972), who studied the interaction of a shear wall, rigid foundation and
the half-space for incident SH waves.
4. The fourth chapter revisits the wave propagation problem in Chapter 3 using the big arc
approximation method developed in Chapter 2. The displacement of the shear wall
structure, flexible-rigid foundation, and the ground motion close to the subjected
building are investigated and compared with the results in Chapter 3.
5. The fifth chapter studies the analytical solution of a 2D, moon-shaped alluvial valley
embedded in an elastic half-space that is analyzed for incident plane SH waves using
the wave function expansion and the Discrete Cosine Transform (DCT). A series of
solutions with different depth-to-radius ratios have been computed, analyzed, and
discussed in a 2017 published paper by Le, Lee, and Trifunac.
6. The sixth chapter extends the shear wall with a circular rigid foundation to a flexible
foundation embedded in an isotropic, homogeneous, and elastic half-space considered
for incident plane SH waves. The displacement of the shear wall, foundation, and nature
of the ground motion near the foundation are studied. The results of the special case
where the flexible foundation is approaching the limiting case of a rigid foundation will
be compared with the results of previous work using the numerical methods.
18
7. The seventh chapter presents the conclusion of various models described in previous
chapters, and a plan for future work.
19
CHAPTER 2
OUT-OF-PLANE (SH) SOIL-STRUCTURE
INTERACTION: SEMI-CIRCULAR RIGID
FOUNDATION REVISITED
2.1 Introduction
The model studied in this paper presents a logical extension of the shear wall on top of a
semi-circular rigid foundation in an isotropic homogeneous and elastic half-space. Luco (1969)
considered only the case of vertical-incidence SH waves. Trifunac (1972) formulated and
solved the problem for the same model subjected to SH waves of arbitrary angles of the
incidence. A new approach and model is developed here to solve for the same semi-circular
rigid foundation with a tapered-shape (instead of rectangular-shape) superstructure on top. The
analytical expression for the deformation of the semi-circular rigid foundation below a tapered
shear wall (structure) with soil-structure interaction in an isotropic homogeneous and elastic
half-space is derived. Results are then compared with that of Trifunac (1972). The formulation
of this problem can later be extended to the case of flexible foundation, semi-circular or
arbitrarily shaped.
2.1.1 Brief history
Soil-structure interaction (SSI) is a process in which the effects of wave propagation in the
half-space are modified by the response characteristic of a structure and vice versa. SSI
continues to be an active area of research especially at the interface of soil and structural
dynamics. During the first half of the twentieth century, engineered structures were designed
with the assumption that a foundation was fixed to a rigid underlying medium; soil inertia was
not considered and at best the soil was modeled as a spring element in structural models.
Reissner (1936) studied soil inertia and discovered that the material damping in the soil
could modify to the response of the structure. In 1969, Luco (1969) took up the topic and solved
a two-dimensional (2D) interaction of a shear wall for an incident plane SH waves. Trifunac
(1972) generalized Luco’s solution for arbitrary incidence of SH waves. Trifunac and Wong
20
(1974b) then presented solutions for shallow and deep elliptical-rigid foundations, as well as for
multiple buildings and foundations (Wong and Trifunac 1975). Wong and Trifunac (1976) also
studied the effects of nearby canyons on soil-structure interaction and Abdel-Ghaffar and
Trifunac (1977) investigated the interaction of a simple two-dimensional (2D) bridge excited by
SH waves. Lee (1979) presented the first three-dimensional (3D) analytical solution of
interaction for a single degree-of-freedom oscillator resting on a semi-spherical foundation for
the incidence of harmonic P-, SV-, and SH waves. Todorovska (1993a, 1993b) described in-
plane foundation-soil interaction for an embedded circular foundation and the effect of wave
passage and embedment depth for in-plane building-soil interaction.
The purpose of this investigation is to develop new tools to solve more realistic models.
The model studied in this paper presents an extension of the elastic shear wall with a circular
rigid foundation studied by Luco (1969) and by Trifunac (1972). However, our work will
continue to be limited to a plane-wave representation of incident waves that was recently shown
to be a good approximation (Kara and Trifunac 2013) and to the homogeneous half-space—
hence, any effects of the local soil layers will be neglected (Liang et al. 2013). Finally, the
structure and soil will be assumed to be linear and the only energy loss in the system will be
associated with radiation of scattered waves into the half-space. It is known that the nonlinear
soil response can be a powerful sink of incident seismic energy (Gicev and Trifunac 2012a,b;
Gicev et al. 2012c; Trifunac and Todorovska 1994, 1997, 1998, 1999). We will study the related
energy loss when we generalize the analysis presented here to the case of flexible foundation.
In practice, concrete footings, piles and grade beams, or mat foundations support buildings
and such support structures do not behave as rigid bodies when excited by seismic waves
(Trifunac et al. 1999). As these foundations are not rigid, the ideal objective would be to solve
for the soil-structure interaction of a shear wall with a flexible foundation. The methodology of
Trifunac (1972) cannot be modified to the case of a flexible or semi-rigid foundation, thus a
new approach and model are developed in this paper to solve the SSI of a tapered shear wall
(structure) for rigid, flexible, and semi-rigid foundations using a “big arc numerical” method.
Further, it is not uncommon for high-rise buildings to gradually taper from the bottom to the top
so that the top of the building is slightly narrower than the bottom.
2.1.2 Review of Trifunac (1972) paper
21
The model studied in the Trifunac (1972) paper is a 2D, infinitely long elastic shear wall
resting on a semi-circular rigid foundation of radius a embedded in a half-space. It is subjected
to plane incident SH waves with harmonic frequency ω. All materials here are homogeneous,
elastic, and isotropic. Contact between the soil, the foundation, and the shear wall is assumed to
be welded, and no slippage exists. The material constants, namely the shear modulus and wave
speed of the half-space soils and shear wall, are denoted by ,C
and ,
b
b
C
, respectively.
Since Trifunac’s model used exp( ) it as the time-dependent term and a polar-coordinate
system that measured the angle with respect to the vertical y-axis, and our model uses
exp( ) it as the time-dependent term and a polar-coordinate system that measures the angle
with respect to the horizontal x-axis (Fig. 2.1), we will first re-derive the Trifunac (1972) results
with respect to the notation used in this paper. For the coordinate system shown in Fig. 1, the
analytical expression for the out-of-plane (SH) deformation of the rigid foundation, ∆ will now
take the form
1
11
1
1
1
1
tan
2
f b
b
ss b
o
o
o
o
M kH
M ka
MM kH
Jka
Jka H ka a
Hka
H ka
H ka
, (2.1)
which is independent of the angle of incidence SH waves. Here,
s
M is the mass per unit length
of soil to be replaced by the rigid foundation, and
f
M is the mass per unit length of the rigid
foundation.
A train of plane harmonic incident SH waves impinge on the model from deep earth with
an incidence angle with respect to the horizontal axis. A Cartesian coordinate system , x y
and a corresponding polar coordinate system , r
have been defined with the origin at the
center of the semi-circular foundation.
The incident wave field consists of a train of plane SH waves of unit amplitude with
harmonic frequency ω, wave speedC
, shear wave number kk
C
, and incidence
angle γ. The incident
() i
w and reflected
() r
w waves can be expressed together as follows:
22
cos ysin cos ysin
cos cos
,
,
ir ikx it ikx it
ikr ikr ir
it
ir
ir
ww e e
ww e e e
wxy
wr
, (2.2)
where the out-of-plane motions are all in the z direction and perpendicular to the x, y plane (Fig.
1). Equation (2.2) represents waves that propagate in the positive x direction with phase velocity
cos
C
c
. From here on, the harmonic term exp( ) it will be understood and omitted in
all subsequent equations.
The free-field incident and reflected waves
ir
w
given by Eq. (2.2) can be expanded into a
Fourier-Bessel series as
00
, 2 cos cos cos
n
nn nn
nn
ir
r iJkr n n aJkr n w
, (2.3)
where 01 2 ,, n ,
n
Jkr is the Bessel function of the first kind with argument kr
and
order n , and 2cos
n
nn
ai n . For n = 0, 1, 2, 3….
n
a are the coefficients of the free-field
waves: 1
o
and 2
n
for 0 n , so that 2
o
a . The scattered and diffracted wave from
the foundation
R
w must satisfy the Helmholz wave equation for harmonic waves with
frequency :
2
22
222
0
11
w
ww w
k
rrr r
, (2.4)
for ra and
2
, it must satisfy the boundary conditions given by
1
0
z
z
w
r
at 0, and ra (2.5)
and
ir R
ww
at 0 and ra . (2.6)
23
Figure 2.1. The shear wall, foundation, and soil (Trifunac 1972).
Here is the unknown movement of the rigid foundation. The motion
R
w represents an
outgoing wave from the cylindrical foundation. It must also satisfy Eq. (2.4) and boundary
condition (2.6). This wave can be represented as follows:
1
0
,cos
nn
n
R
rAHkrn w
, (2.7)
24
where
n
A are the unknown complex numbers to be determined by boundary conditions and the
wave functions and
1
p
Hkr are the Hankel functions of the first kind with argument kr and
order p .
The displacement of the shear wall, also in the z-direction (out-of-plane), has the same
harmonic frequency , and
b
w and must satisfy the Helmholtz wave equation with y the axis
pointing vertically down (Fig. 2.1):
2
2
2
0
b
b
b
kw
w
y
for 0 H y , (2.8)
with
b
b
k
C
being the building shear wave number, and
b
C
the wave speed in the shear
wall. The shear wall must satisfy the boundary conditions of
at topof shear wall
at
w
, 0
0
b
yz b
bit
w
yH
y
ey
. (2.9)
Dependence on x in the shear wall is eliminated in Eq. (2.10) by the assumption that the
foundation is rigid. The solution of Eqs. (2.8) and (2.9) is then given by
cos tan sin
bit
bb b
we ky kH ky
. (2.10)
The base shear force per unit length of the shear wall
b
z
f can be expressed as
2
tan
bit
b
zb
b
kH
f Me
kH
, (2.11)
where 2
bb
M aH is the mass of the shear wall per unit length and the natural frequencies
of the shear wall on the fixed foundation are
21
2
b
kH n
. (2.12)
25
To find the displacement in terms of
z
u , it is necessary to write the dynamic equilibrium
equation for the rigid foundation. This equation, assuming no slippage exists between the soil
and the foundation, is
2 it s b
zz f
Me f f
, (2.13)
where
f
M is the mass per unit length of the foundation. The
b
z
f , the action force per unit length
of the shear wall on the foundation, is given by Eq. (2.11) as shown above. The
s
z
f , the action
force of the soil on the foundation, is obtained by integrating the forces from wave stresses in
the half-space at the surface of the rigid foundation. From Eqs. (2.2) and (2.5), we obtain the
following:
1
11
2
2
nn
s it
zrz
ra
akaJ ka kAH ka fad e
. (2.14)
The movement of the rigid foundation can then be expressed using Eqs. (2.11), (2.13), and
(2.14) (Trifunac 1972):
1
11
1
1
1
1
tan
2
f b
b
ss b
o
o
o
o
M kH
M ka
MM kH
Jka
Jka H ka a
Hka
H ka
H ka
, (2.15)
where
s
M is the mass per unit length of soil to be replaced by the rigid foundation, and
f
M is
the mass per unit length of the rigid foundation.
2.2 The New Model: Tapered Shear Wall
2.2.1 The model
The model studied in this paper is a 2D, elastic, tapered shear wall supported by a semi-
circular rigid foundation of radius a attached to the elastic half-space, as illustrated in Figure
2.2. All materials are homogeneous, elastic, and isotropic. The material constants—i.e., shear
modulus and wave speed of the elastic half-space soil and shear wall—are denoted by ,C
and
26
,
b
b
C
, respectively. Contact between the soil-to-foundation and foundation-to-shear wall is
assumed to be bonded so that no slippage can occur between the contact surfaces. The structure
on top of the foundation is an elastic shear wall of which a section is a circular sector
0 of large radius R . The center of the circular sector is at O
, a point high above the
structure, so that the base of the shear wall in contact with the foundation is at a radius R and of
width 2a . The shear wall has height H above the foundation, thus the top of the shear wall is a
circular arc with radius
1
R RH . Here, the radius R is assumed to be very large compared
with its half width, R a , so that the full width of the shear wall, which is also the diameter of
the semi-circular foundation, is 2aR or 2 Ra .
27
Figure 2.2. The mathematical model of the tapered shear wall with a semi-circular
rigid foundation.
28
2.2.2 Free-field waves in the half-space
The excitation consists of a series of plane SH waves incident onto the rigid foundation
from a half-space at an incidence angle with respect to the horizontal axis. A Cartesian
coordinate system , x y and a corresponding polar-coordinate system , r have been defined
with the origin at the center of the semi-circular foundation. These waves are identical to the
free-field waves expressed in the last section and are now considered in more detail. The
incident free-field wave consists of plane waves with unit amplitude, harmonic frequency
,
wave speedC
, and wave number kk C
. The incident waves can be expressed in
both the rectangular and polar coordinates as follows:
cos ysin
cos cos sin sin cos
,
,
iikx
iikr ikr
wxy e
wr e e
, (2.16)
and the reflected plane waves can be written as
cos ysin
cos cos sin sin cos
,
,
ik x r
ikr ikr r
wxy e
wr e e
, (2.17)
where is the angle of incidence or reflection with respect to the horizontal axis; cos
x
kk
and sin
y
kk represent the components of the SH wave number along the x- and y-axes,
respectively. The
it
e
harmonic time factor is understood and omitted from all wave equations.
Applying the Jacobi-Anger Expansion (Pao and Mow 1973), we have
cos
0
cos
0
0
cos
cos
cos cos sin sin
n
ikr
nn
n
ikr n
nn
n
n
nn
n
eiJkrn
eiJkrn
iJ kr n n n n
, (2.18)
29
where 1 i is the imaginary complex unit, .
n
J is the Bessel function of the first kind
with order n, and the two expressions in polar coordinate , r of Eqs. (2.16) and (2.17) can be
expanded into an infinite series. The free-field wave field is then given by their sum as follows:
cos cos
00
,
2 cos cos cos
ff i ikr ikr r
n
nn nn
nn
wr w w e e
iJ kr n n aJ kr n
, (2.19)
where 2cos
n
nn
ai n , exactly as in Eq. (2.3) above, which represents the free-field waves
that are finite everywhere in the half-space for n = 0, 1, 2, 3…, and
n
a
are the coefficients of
the free-field waves; 1
o
and 2
n
for 0 n , so that 2
o
a .
The free-field waves will arrive towards the foundation, resulting in scattered and diffracted
waves in the half-space. The wave field in the half-space scattered from the rigid foundation for
ra and 0 is given (as in the last section) by:
1
0
,cos
S
nn
n
wr AH kr n
, (2.20)
where
n
A are the unknown complex numbers to be determined by boundary conditions and the
wave functions and
1
.
it
n
He
represent outgoing waves satisfying Summerfield’s radiation
condition. The foundation is assumed to be rigid and thus the particles anywhere on and inside
the foundation must have the same out-of-plane motion.
2.2.3 The wave field within the structure
Since the building structure on top is a shear wall that is defined as a circular sector with
center at O , a point above the structure (Figure 2.2), the building waves will be defined using
the polar-coordinate system
, r with an origin at O . The out-of-plane motion is
independent of coordinate x and can be represented as, for R Hr R and 0 :
11 2 2
0
, cos
BB
nn n n bb
n
r
n
w w B H kr B H kr
,
(2.21)
30
where
1
n b
Hkr
and
2
n b
Hkr
are the Hankel functions of the first or second kind with
argument
b
kr and order n ;
1
n
B and
2
n
B are the unknown complex numbers to be
determined by boundary conditions and the wave functions.
2.3 The Boundary Conditions
The boundary conditions on the flat ground surface are automatically satisfied by the free-
field waves
ff
w and the scattered waves
S
w . The stress and displacement continuity
equations along the semi-circular interface of the rigid foundation at ra and 0 will
be expressed as in the following.
2.3.1 Displacement continuity
for 0
ff S
ra
ww
, (2.22)
is the displacement continuity requirement, where is the amplitude of the rigid foundation
displacement.
The substitution of Eqs. (2.19) and (2.20) with Eq. (2.22) leads to the following two boundary
conditions:
for
for
for
for
1
1
1
0
0
0
0
0
nn n n
oo
o
n
nn
n
aJ ka AH ka
aJ ka
Hka
A
aJ ka
Hka
n
n
n
n
. (2.23)
2.3.2 The stress-free and stress-continuity equations in the building
The stress at the top of the building for 0,1 , 2, 3,... n
is
31
1
1
11 2 2
11
11 2 2
11
1
21
1
2
1
0
0
B
zr n n n n bbb b b
rR
rR
nn n n bb
n b
nn
n b
w
kB H kR B H kR
r
B H kR B H kR
HkR
BB
HkR
.
(2.24)
Substituting Eq. (2.24) with (2.21), the out-of-plane motion of the shear wall is simplified to the
following:
12
11 1
2
0
1
12 1 2
1 11
2
0
1
1
1
0
cos
cos
ˆ
,cos
n
B nb n b
nn b
n
nb
B n b nb nb n b
n
n
nb
B
nb b
n
HkRH kr
n
wBHkr
HkR
HkrH kR H kRH kr
n
wB
HkR
n
wBHkRkr
,
(25)
where
12 1 2
11
2
1
1
ˆˆ
,
nb n b n b nb
nn
nb
bbb
H kr H k R H k R H k r
HkR
Hkr H kRkr
is a
“scaled” Hankel function defined as a linear combination of Hankel functions of the first and
second kind. The boundary condition at the interface of shear wall and rigid foundation
B
rR
w
, gives
for
for
1
0
00
ˆ
,
nn bb
n
R
n
BH kR k
, (2.26a)
resulting in
1
ˆ
,
o
o bb
B
HkR kR
and 0
n
B for n > 0, so that the building wave
1
1
ˆ
,
ˆ
,
ob b
b
ob b
BB
HkRkr
kr
HkRkR
ww , (2.26b)
32
becomes a one-term expression.
2.3.3 The dynamic equation for the rigid foundation,
fit
we
As pointed out by Luco (1969) and Trifunac (1972), the displacement of the foundation
can be determined by applying the dynamic equilibrium equation for the rigid foundation as
follows:
2
2
f
fit it
ff sb
sb
M
Mw M e f f e
ff
, (2.27)
where
2
1
2
f f
a M is the mass of the rigid foundation per unit length in the z-axis;
f
is
the mass density of the foundation;
b
f is the force of the shear wall acting on the foundation per
unit length atrR and 0 ; and
S
f denotes the force due to total (free-field +
scattered) waves at ra and 0 . Thus, we have
1
11
1
,
ˆ
ˆ
,
ˆ
,
bb o o b b
B
bb
o
rR
bbb
bb
obb
kR kR kR
w
fRd BH
r
fkR
HkRkR
HkRkR
, (2.28a)
where
111
ˆˆ
,,
o bb bb
H k R k R H kR kR
, and
1
11
1
0
0
1
0
cos
oo
rz
nn n n
n
oo o o
n
s
o ra
s
s
s
ka aJ ka AH ka
a J ka A H ka n d
k a a J ka A H ka
fad
fka
f
f
,
(2.28b)
33
where the integral:
0
for 0
0for 0
cos
n
n
nd
.
The displacement of the rigid foundation can be determined by solving Eqs. (2.28a), (2.28b),
and (2.27) for 0 n . Hence,
1
11
1
1
2
11 1
1
1
ˆ
,
ˆ
,
o
o
o
fbb
bb
ob b o
Jka
Jka H ka a
Hka
M H kR kR H ka
kR
ka ka
HkRkR H ka
. (2.29a)
Let
2
1
2
s
a M again (as in Section I above) be defined as the mass per unit length of the
soil to be replaced by the rigid foundation, , the mass density of the soil. For
1
,2, R RH aR a so 2
B B
aH M is again the mass of the building per unit
length, where
B
here is the mass density of the building (with the tapered shear wall
approaching the shape of a rectangular shear wall with width 2a and height H). Thus
2
1
2
bb b
sb
kR M ka
ka M kH
, Eq. (2.29a) can further be simplified to
11
1
1
11
1
1
1
1
,
1
2
,
ˆ
ˆ
f bb
b
ss b
ob b
o
o
o
o
M kR kR
M ka
MM kH
kR kR
Jka
Jka H ka a
Hka
H ka H
H H ka
. (2.29b)
Using the asymptotic approximation
11
1
tan
ˆ
,
ˆ
,
b
bb
o
bb
kH
HkR kR
HkRkR
as , RaR
(see Eq. (A16), Appendix A), the displacement of the rigid foundation Eq. (2.29b) will be
identical to Eq. (2.1) above in Trifunac (1972):
34
1
11
1
1
1
1
tan
2
f b
b
ss b
o
o
o
o
M kH
M ka
MM kH
Jka
Jka H ka a
Hka
H ka
H ka
. (2.30)
2.4 Numerical Analysis of the Displacements
As pointed out by Trifunac (1972), the envelope of the rigid foundation displacement is
given by
22
1
11
1
11
ooo
e o
oo
o
Jka J ka Y ka
Jka H ka a
Y kaJ ka Y kaJ ka
Hka
.
(2.31)
The backbone curve of could be understood as the displacement of the rigid foundation
whose density is identical to that of the surrounding soil by setting 0
b
s
M
M
.
1
11
1
1
1
1
2
o
f
s
o
o
o
o
M
ka
M
Jka
Jka H ka a
Hka
Hka
Hka
(2.32)
To characterize the problem in terms of dimensionless parameters, we define
b
b
kH H
ka a
where
1
HR R . The shape of the tapered shear wall is characterized by the ratio
R
H
, the
ratio of the circular radius of the sector to its height. We plot the amplitude of the foundation vs.
the dimensionless
a
for 10 and 100
R
H
, 10
H
a
, 1
f
s
M
M
and 1, 2, 4
b
s
M
M
with
0,2,4
(Figs. 2.3(i), 2.3(ii), 2.3(iii), and 2.4) and compare them with Trifunac (1972). The
results agree well when 10
R
H
and are found to be almost identical when 100
R
H
.
35
The two figures show that the new model of the tapered-shaped shear wall is a
legitimate and realistic model and the results show good agreement with the model studied by
Luco (1969) and Trifunac (1972). The existing model has one advantage over the models
studied by Luco (1969) and Trifunac (1972). While the previous models allow an explicit
analytic expression to be derived for the displacement of the semi-circular rigid foundation
below a rectangular shear wall, it has the limitation that the derivation works only for a rigid
foundation and cannot be extended to the case of a flexible foundation. The present model also
allows explicit analytic expressions for the displacements of the same semi-circular foundation
below a tapered-shaped shear wall. When the shape of the shear wall is close enough to the
rectangular one, the expressions for the displacement amplitudes agree at all angles of incidence
and at all frequencies (Figs. 2.3 & 2.4). Further, results for the present model can and will be
extended to cases in which the foundation is flexible and elastic with different elastic properties,
which will be discussed in Chapter 4. This new model is thus formulated for future work on SSI
by flexible, elastic foundations.
36
Fig. 2.3(i)
R/H = 10,
M
b
/M
s
= 1,
M
f
/M
s
= 1,
= 0, 2, 4
Fig. 2.3(ii)
R/H = 100,
M
b
/M
s
= 1,
M
f
/M
s
= 1,
= 0, 2, 4
Fig. 2.3(iii)
M
b
/M
s
= 1,
M
f
/M
s
= 1,
= 0, 2, 4
(Trifunac,
1972)
Figure 2.3. The effect of interaction on the amplitude of foundation vibration.
37
38
2.5 Conclusion and Proposed Studies
In the majority of papers on soil-structure interaction (similar to the papers by Luco (1969)
and Trifunac (1972)), the foundation is simplified as a rigid body. Almost every building has a
foundation that transfers upper loads to the soil evenly. With the exception of some towers and
mast structures, most engineered structures cannot to be mathematically simplified to a single
degree-of-freedom dynamic system. Therefore, a mathematical model with a flexible foundation
is of considerable significance in SSI research.
From the results of the numerical analyses of the proposed model in Figure 2.2, it is
concluded that the tapered-shape structure methodology works well with large-enough radius R
. This methodology can be extended to solve for the soil-structure interaction of a shear wall
supported by a flexible foundation as part of future studies.
As seen in Trifunac (1972), when a foundation is assumed to be rigid, every particle at any
horizontal cross-section of the structure parallel to the half-space surface must have the same
out-of-plane motion of the shear wall and is independent of coordinate y . Thus, dependence on
yin the shear wall is eliminated in Eq. (2.10). This simplicity in the dependence of the
displacement solution does not permit an extension of the solution for a flexible foundation. The
wave field within the structure of the tapered-shape methodology only depends on the polar
coordinate system, thus the methodology can be expanded to solve for both rigid and flexible
elastic foundations.
39
CHAPTER 3
OUT-OF-PLANE (SH) SOIL-STRUCTURE
INTERACTION: A SHEAR WALL WITH
RIGID AND FLEXIBLE RING FOUNDATION
3.1 Introduction
The soil-structure interaction (SSI) of a building and shear wall above a foundation in an
elastic half-space has long been an important research subject for earthquake engineers and
strong-motion seismologists. While numerous papers have been published on the topic since the
early 1970s, very few have had analytic closed-form solutions available.
The soil-structure interaction problem is one of the most classic problems connecting the
two disciplines of earthquake engineering and civil engineering. The interaction effect
represents the mechanism of energy transfer and dissipation among the elements of the dynamic
system, namely the soil subgrade, foundation, and superstructure. This effect is important across
many structure, foundation, and subgrade types; however, it is most pronounced when a rigid
superstructure is founded on a relatively soft lower foundation and subgrade. The interaction
effect may only be ignored when the subgrade is much harder than a flexible superstructure—
for instance, a flexible moment frame superstructure founded on a thin compacted soil layer on
top of very stiff bedrock below.
Figure 3.1 is the realization of the two-dimensional (2D) mathematical model in this paper
and presents the interaction of an elastic shear wall flexible-rigid foundation and the elastic half-
space for incident plane SH waves. The foundation consists of a semi-circular rigid foundation
wrapped with an elastic semi-circular flexible foundation. The lower semi-circular section can
be modeled as flexible for soft soil or as rigid for hard soil, such as bedrock, which provides a
more accurate mathematical model for the shear wall or structure interacting with the soil
media. This accuracy is important in modeling an SSI with a stiffer layer of soil overlaying a
more flexible layer. An engineer goes through a decision-making process when selecting the
optimum type of foundation system for soil deposits that are soft and not suitable to support the
superstructure. Selecting an optimum system is based on the principle that cost-effective
40
alternatives such as soft ground improvements must be sought first before considering relatively
costly foundation alternatives (such as mat or pile deep foundation). Soil treatment and
stabilization are techniques to enhance some aspects of soil behavior and improve the strength
and bearing capacity of soft ground conditions. Through the process of soil treatment, the
property of soil supporting the superstructure foundation is modified and improved in compared
with the surrounding strata. This layer of modified soil will alter the seismic wave propagating
from the half-space to the superstructure foundation. The model in this chapter will accurately
analyze this interaction and assist in the selection of optimum foundation systems.
The analytical solution of the interaction of a shear wall, flexible-rigid foundation, and an
elastic half-space is derived for incident SH waves with various angles of incidence. The
displacement of the shear wall structure and flexible/rigid foundation and the ground motion
close to the subjected structure are investigated and compared with the results of Luco (1969)
and Trifunac (1972), who studied the interaction of a shear wall, rigid foundation, and the half-
space for incident SH waves. Luo (2008) also studied this problem but the derivation and final
equations here are much simplified with improved convergency and accuracy of numerical
results. Moreover, the emphasis in the current study is different: the graphs computed here
confirm the results of Trifunac (1972) for only a rigid foundation case and the effect of shear
wall structure response caused by the flexible ring is studied. The base shear force of the wall
structure is also studied with various thickness and stiffness of the flexible ring.
3.2 The Mathematical Model
The model studied in this paper is a 2D rectangular building resting on a semi-circular rigid
foundation of radius a wrapped with an elastic semi-circular flexible foundation of radius a
embedded in a half-space, as illustrated in Figure 1. All materials here are assumed to be
homogeneous, elastic, and isotropic. The material constants, namely the shear modulus and
wave speed of the half-space soil, building, and flexible foundation are denoted by ,C
and
,
b
b
C
, and ,
f
f
C
. Contacts between the soil, foundation and building are assumed to be
fixed with no slippage between them and with the assumption that the foundation is removable.
A train of parallel harmonic incident SH-waves impinges on the foundation from the half-
space at an incidence angle with respect to the horizontal axis. The width and height of the
41
structure are 2a and H , respectively. A Cartesian coordinate system , x y and a
corresponding polar coordinate system , r have been defined with the origin at the center of
the semi-circular foundation.
Figure 3.1. The mathematical model SSI with a semi-circular flexible and rigid
foundation.
ݓ ሺ ሻ
42
3.2.1 The free-field wave in the half-space
The incident wave field consists of a train of plane waves of unit amplitude with harmonic
frequency , wave speed C
, and wave number kk C
. The incident waves can be
expressed in both the rectangular and polar coordinates as follows:
cos sin
cos cos sin sin cos
,
,
xy i
i
ikx k y ix y
ikr ikr
wxy e e
wr e e
, (3.1)
and the reflected plane waves can be written as
cos sin
cos cos sin sin cos
,
,
xy r
r
ikx ky ix y
ikr ikr
wxye e
wr e e
. (3.2)
The
it
e
harmonic time factor is present in all wave equations, and will be understood to be
omitted from all equations. Here γ is the angle of incidence or reflection with respect to the
horizontal axis; cos
x
kk and sin
y
kk represent the components of the SH wave
number k along the x- and y-axes, respectively. Applying the Jacobi-Anger Expansion (Pao and
Mow 1973),
cos
0
cos
n
ikr
nn
n
eiJkrn
, (3.3)
where 1 i is the imaginary complex unit, and .
n
J is the Bessel function of the first kind
with order n as shown:
cos
0
cos
0
cos
cos cos sin sin
ikr n
nn
n
ikr n
nn
n
eiJkrn
eiJkrnnnn
. (3.4)
The two formulas in Eqs. (3.1) and (3.2) can be expanded into infinite series by using polar
coordinates , r . The free-field wave field is then given by the sum of the represented waves
that are finite everywhere in the half-space for n = 0, 1, 2, 3….
43
cos cos
00
,
,2 coscos cos
ff i r ikr ikr
ff n
nn nn
nn
wr w w e e
wr iJkr n n aJkr n
, (3.5)
where 2cos
n
nn
ai n are the coefficients of the free-field waves, and 1, 2
on
for
0 n .
The wave field in the half-space scattered from the flexible foundation is written as
1
0
, cos for
S
nn
n
wr AH kr n na
, (3.6)
where
n
A are the unknown complex numbers to be determined by boundary conditions and the
wave function and
1 it
n
He
represent outgoing waves toward infinity satisfying Sommerfeld’s
radiation condition.
The expression of the wave inside the flexible foundation is
11 2 2
0
,cos
F
nn f n n f
n
w r B H kr B H kr n
, (3.7)
for 0 ar aand where
1
n f
Hkr and
2
n f
Hkr are the Hankel functions of
the first or second kind with argument
f
kr and order n ;
1
n
B and
1
n
B are the unknown
complex numbers to be determined by boundary conditions and the wave functions.
3.2.2 The wave field within the structure
As pointed out by Trifunac (1972), the displacement of the shear wall, in the z-direction
(out-of-plane), has the same harmonic frequency
as the rigid foundation, and
B
w must
satisfy the Helmholtz wave equation with y , the axis pointing vertically down (Figure 1):
for
2
2
00
B
B
b
w
kw H y
y
, (3.8)
44
with
b
b
k C
being the building shear wave number, and
b
C
the wave speed in the shear
wall. The shear wall must satisfy the boundary conditions of
at at the top of shear wall
at at the bottom of shear wall
0,
0,
B
yz b
B it
w
yH
y
we y
. (3.9)
Dependence on x in the shear wall is eliminated in Eq. (3.8) by the assumption that the
foundation is rigid. The solution of Eqs. (3.8) and (3.9) is then given by
cos tan sin
B it
bb b
wy e ky kH ky
. (3.10)
The shear stress along the interface of building and foundation could be derived as
0
0
tan
B
yz b b b b
y
y
w
kkH
y
. (3.11)
The base shear force per unit length of the shear wall
b
z
f can be expressed as
2
tan
2tan
b
bb b B
b
b
z
kH
ka kH M
kH
f
. (3.12a)
From Eq. (12a), a dimensionless function proportional to the base shear force acting on the
shear wall can be expressed as
2
tan
b
Bb
b
z
kH
MkH
f
. (3.12b)
3.2.3 The action of flexible foundation on the rigid foundation
The action of flexible foundation on the rigid foundation,
f
z
f , can be expressed in term of
stress as follows:
0
11 2 2
0
0
cos
rz
ra
ff n n f n n f
n
f
z
f
z
ad
ka B H ka B H ka n d
f
f
, (3.13)
45
where
0
,0
cos
0, 0
n
nd
n
.
Thus,
f
z
f can be rewritten as
11 2 2
11 ff o f o f
f
z
ka B H ka B H ka f
. (3.14)
3.3 The Boundary Conditions
3.3.1 Displacement and stress continuity
The free-stress boundary conditions of the ground surface should be satisfied by the free-
field waves
ff
w and the scattered waves
S
w . The displacement and stress continuity
equations along the semi-circular interface at 0 and ra , respectively, are:
for 0
ff S F
ra ra
ww w
(3.15)
for 0
ff S F
f
ra ra
ww w
rr
. (3.16)
Substitution of Eqs. (3.5), (3.6) and (3.7) into Eqs. (3.15) and (3.16) leads to the following two
boundary condition equations, for n = 0, 1, 2, 3, …:
111 22
nn n n n n f n n f
a J ka A H ka B H k a B H k a (3.17)
111 22
nn n n n n f n n f
a J ka A H ka B H k a B H k a
, (3.18)
where
f f
k
k
is the material property ratio in the equation for the stress continuity
boundary condition.
The boundary condition at the interface of rigid and flexible foundations can be expressed as
below:
FR it
ra ra
ww e
. (3.19)
46
Substituting Eq. (3.7) into Eq. (3.19) to solve for
2
n
B ,
for
for
11 2 2
11 2 2
0
00
oo f o o f
nn f n n f
B H ka B H ka n
B H ka B H ka n
. (3.20)
2
n
B
can be written in terms of
1
n
B and as,
for
for
11
2
2
11
2
0
0
oo f
of
n
nn f
nf
BH ka
n
Hka
B
BH ka
n
Hka
. (3.21)
3.3.2 The dynamic equation for the rigid foundation
As pointed out by Luco (1969) and Trifunac (1972), displacement of the foundation
can
be determined by the kinetic equation for the rigid foundation as
R fb it
Rzz
Mw ff e
, (3.22)
where
2 R it
we
,
R
M
is the mass of the rigid foundation per unit depth in the z-axis
and
R
w represents the displacement function of the rigid foundation in terms of time factor t
as described in Eq. (3.19).
The foundation displacement
can be solved from Eqs. (3.12), (3.14) and (3.22),
1
12
11 2
1
2
1
2
2
2tan
of
ff f f
of
o
f
Rbb b ff
of
Hka
kaH ka H ka
Hka
B
Hka
Mka kH ka
Hka
. (3.23)
Equation (3.23) can be further simplified as,
47
2
11
2 2
1
2
4
tan
2
fo f
ooo
f
f
b RB
FF b
of
i
kaH ka
BB
Hka ka
kH MM
aM M kH
Hka
, (3.24)
where
2
2 2
1
2
4
tan
2
fo f
o
f f
b RB
FF b
of
i
kaH ka
Hka
ka
kH MM
aM M kH
Hka
. (3.25)
B
M ,
R
M , and
F
M are the masses of the building, rigid foundation, and flexible ring,
respectively,
b
,
r
,
f
stand for the density of those three media, sequentially, and the
Wronskian:
12 1 2 1 2
4
,
n f nf n f nf n f n f
f
i
WH ka Hka H kaHka H kaH ka
ka
.
Other terms can be found in Appendix I.
As pointed out by Luco (1968), the natural frequencies of a shear wall on a fixed
foundation correspond to 21
2
b
kH n
. Thus, becomes zero at the values of
b
kH in
the abovementioned equation.
Substituting Eq. (3.24) into Eq. 3.21),
1
n
B can be derived explicitly in terms of
2
n
B ,
for
for
for
11 1
2
2
11
2
1
21
2
0
0
0, 1, 2, 3, 4
oo o o f
of
n
nn f
nf
no n f
nn
nf
BBH ka
n
Hka
B
BH ka
n
Hka
Hka
BBn
Hka
, (3.26)
48
where
1for 0
0for 0
n
n
n
.
By substituting Eq. (3.26) into Eqs. (3.17) and (3.18), we can solve for wave function
coefficients
1
and
nn
A B explicitly as:
1
11 2 1
2
no n f
nn n n n f n f n
nf
Hka
a J ka A H ka H k a H k a B
Hka
(3.27)
1
11 2 1
2
no n f
nn n n n f n f n
nf
Hka
a J ka A H ka H k a H k a B
Hka
(3.28)
From Eq. (3.27),
n
A can be derived and expressed in terms of
1
and
nn
aB as
1
1
12 1
n
n
nn n
nnf n
Jka
G
A Ba
HkaH ka Hka
, (3.29)
where
11 2 1 2 2
nn f n f n f n f non f
G H ka H k a H ka H k a H ka .
Substituting Eq. (29) into Eq. (28),
n
B can be solved explicitly as
11 2
1
21 1 1
nn n n nfn
n
nn n n
J kaH ka J kaH ka H kaa
B
GH ka G H ka
,
(3.30)
where
21 2 1 2 2 2
4
n n f n f n f n f n on f n on f
f
i
G H ka H k a H ka H k a H ka H k a
ka
Solving for
1
n
B from Eq. (3.30),
2
1
11 2 1
2
nf
nn
nn n n
i
Hka
ka
Ba
G H ka G H ka
, (3.31)
49
where Wronskian
11 1
2
,
nn n n n n
i
W J ka H ka J ka H ka J ka H ka
ka
.
Deriving an equation for
1
o
B (n = 0) from Eq. (3.31) presents as,
22
1
11 2 1
11 2 1
1
22
of o o f o
o
ooo
oo o o
ii
Hkaa Hkaa
ka ka
B
GH ka G H ka
G H ka G H ka
.
(3.32)
By combining Eqs. (3.28), (3.29), and (3.31), we can derive the expressions for wave function
coefficients and
nn
A C as the following:
1
1
1
11 2 1
2
n
n
n
n n
n
nn n n
G i
ka
Hka
Jka
A a
Hka
GH ka G H ka
(3.33)
1
2
11 2 1
2
no n f
nn
nn n n
i
Hka
ka
Ba
GH ka G H ka
. (3.34)
3.4 Numerical Analysis of the Displacement
As pointed out by Trifunac (1972), the condition for the envelope of the rigid foundation
displacement, , corresponding to the case of which 1,aa , is given by
22
1
11
1
11
ooo
e o
oo
o
Jka J ka Y ka
Jka H ka a
YkaJ ka Y kaJ ka
Hka
.
(3.35)
The backbone curve of could be understood as the displacement of the rigid foundation
whose density is identical to that of the surrounding soil by setting 0
B
F
M
M
and 1
R
F
M
M
, as
shown:
50
2
2
1
2
4
2
o
oo
R
F
o
i
kaH ka
a
Hka
ka M
M
Hka
. (3.36)
To characterize the problem, the dimensionless parameter is defined as
f
f
b
b
H
kH
ka a
. It is
seen that the flexible, slender, and tall shear walls are described by large values of .
First, the correctness of the numerical results can be verified by comparing the results from
the rigid semi-circular foundation case. This is done by setting 1,
ff
k
aa
k
. Figure
2 represents the plots of the displacement on a Cartesian coordinate system with the x-axis
being the “wave number” and the y-axis being the “displacement” and the initial conditions
shown in the legends. The abscissa in these figures is the dimensionless frequency a
and
the ordinate is the foundation displacement . The displacement would be equal to one if
the movement of the rigid foundation does not depend on the shear wall. This serves as a
reference value, which shows the influence of the SSI on the movement of the foundation and,
consequently, on the base shear force. The results and plots are in line with Luco (1969) and
Trifunac (1972).
Figures 3.3 and 3.4 represent the displacement curves for 1.25 aa and 1.50 aa ,
respectively. The zeros correspond to the fixed-base natural frequencies of the shear wall. It can
be seen that the flexible ring thickness has a strong effect on the displacement amplitude of the
rigid foundation. As the ratio of aa becomes large, peaks of the displacement amplitude
increase significantly. The displacement amplitude increases greatly for low frequencies and is
dependent on the wave amplitude in the flexible ring coefficient
1
o
B . It is also noticed that the
structure response is independent of the angle of incidence and only depending on the
foundation displacement .
51
(i) 1, 1, 1, 1, 0, 2, 4
BS F S
aa M M M M
(ii) 1, 1, 1, 1, 0, 2, 4 Trifunac 1972
BS F S
aa M M M M
Figure 3.2. The effect of interaction on the amplitude of the foundation
vibration.
52
Figure 3.5 represents the base shear force acting on the flexible shear wall for 2 . When
there is no interaction, the base shear force is infinite for the fixed-base natural frequencies of
the shear wall. When the interaction is considered, the base shear force is bounded for all
frequencies. It can be seen that the thickness of the flexible ring has little effect on the peaks the
base shear force.
Figure 3.6 describes the dimensionless base shear coefficient of the flexible wall structure.
For the lower frequencies range which is of special importance for earthquake engineering, the
peaks of the base shear forces are increased in proportion to the ratio of
f f
k
k
and
decreased for larger aa . It is also indicated that the base shear force is bounded faster for a
higher value of material property ratio.
(i) 1.25, 2, 1, 1, 0, 2, 4
BS F S
aa M M M M
53
(ii) 1.25, 2, 2, 1, 0, 2, 4
BS F S
aa M M M M
(iii) 1.25, 2, 4, 1, 0, 2, 4
BS F S
aa M M M M
Figure 3.3. The effect of interaction on the amplitude of the foundation
vibration.
54
(i) 1.50, 2, 1, 1, 0, 2, 4
BS F S
aa M M M M
(ii) 1.50, 2, 2, 1, 0, 2, 4
BS F S
aa M M M M
55
(iii) 1.50, 2, 4, 1, 0, 2, 4
BS F S
aa M M M M
Figure 3.4. The effect of interaction on the amplitude of foundation vibration.
(i) 1, 4, 1, 2
BS F S
MM M M
56
(ii) 1.5, 4, 1, 2
BS F S
MM M M
(iii) 2, 4, 1, 2
BS F S
MM M M
Figure 3.5. The base shear force.
57
(i) 1, 4, 1, 2
BS F S
MM M M
(ii) 1.5, 4, 1, 2
BS F S
MM M M
58
(iii) 2, 4, 1, 2
BS F S
MM M M
Figure 3.6. The dimensionless base shear.
3.5 Conclusion
The analytical solution of the interaction of a shear wall, flexible-rigid foundation and an
elastic half-space is derived for incident SH waves with various angles of incidence.
Comparison of these displacement amplitude plots and the ones in which the superstructure sits
only on the rigid foundation shows that the flexible ring has the effect of diminishing the ground
displacement amplitude as the building absorbs wave energy and scatters it back into the half-
space. This effect results in increased displacement amplitude of the foundation and decreased
amplitude of ground displacement, and is an important phenomenon that differs from the typical
SSI models in the absence of the flexible ring. It was also found that the flexible ring (soft
layer) could not be used as an isolation mechanism to decouple a superstructure from its
substructure resting on a shaking half-space because waves are transmitted into and scattered
from the flexible foundation.
59
Interesting results are shown in the displacement amplitude graphs. The foundation
displacement is zero for a rigid foundation with a fixed-base excited at the natural frequency of
the shear wall. This concurs with the results of Luco (1969) and Trifunac (1972). However, in a
heavy and flexible structure, the peaks of displacement increase for low-frequency waves as the
thickness of the flexible foundation layer increases.
Compared with the surrounding soil medium, the base shear forces increase as the rigidity of the
flexible foundation layer increases. Interestingly, for high-frequency waves, peaks of the base
shear force are the same for all ratios of aa . For shear walls founded on soft soil and for low
frequencies, the base shear forces are higher than values computed for shear walls supported on
hard soil. The structure response is independent of the angle of incidence and only depend on
the foundation displacement.
60
CHAPTER 4
OUT-OF-PLANE (SH) SOIL-STRUCTURE
INTERACTION: A SHEAR WALL WITH
RIGID AND FLEXIBLE RING FOUNDATION
– BIG ARC NUMERICAL METHOD
4.1 Introduction
Chapter 3 presents the interaction of an elastic shear wall, flexible-rigid foundation, and the
elastic half-space for incident plane SH waves. The analytical solutions presented here are
possible for cases of rigid and non-elastic movable foundation, because this type of rigid
foundation is characterized by the only one parameter. In addition, the solutions demonstrated
in this chapter, although analytic, cannot be directly adapted to solutions representing a fully
flexible foundation. A more realistic assumption would be to allow the foundation to be elastic.
As such, analytical solutions for the soil-structure interaction (SSI) of a building on an elastic
foundation deserve investigation.
This chapter serves as an intermediate step for such a goal. The model consists of the soil-
structure interaction of a tapered shear wall on a semi-circular rigid foundation and wrapped
with an elastic semi-circular flexible foundation. The analytical solutions are derived using the
“big arc numerical method” developed in Chapter 2. The displacement of the shear wall
structure and flexible-rigid foundation, as well as the ground motion close to the subjected
building are investigated. Results are then compared with those of Trifunac (1972) and Le et al.
(2016).
4.2 The Mathematical Model
The model studied in this chapter is a two-dimensional (2D), tapered-shaped building
resting on a semi-circular rigid foundation of radius a, and wrapped with an elastic semi-
circular flexible foundation radius a outside embedded on a half-space, as illustrated in Figure
4.1. All materials here are assumed to be homogeneous, elastic, and isotropic. The material
constants, namely shear modulus and wave speed of the half-space soils, building, and flexible
61
foundation, are denoted by ,C
and ,
b
b
C
, and ,
f
f
C
. The contacts between the soils
and foundation and building are assumed to be fixed with no slippage between them and the
foundation is removable.
A train of parallel harmonic incident SH waves impinge on the foundation from the half-
space at an incidence angle with respect to the horizontal axis. A Cartesian coordinate system
, x y and a corresponding polar coordinate system
, r have been defined with the origin at
the center of the semi-circular foundation. The structure on top of the foundation is an elastic
shear wall of which a section is a circular sector 0 of large radius R . The center of
the circular sector is at O , a point high above the structure, thus the base of the shear wall in
contact with the foundation is at a radius R and of width 2a . The shear wall has height H above
the foundation, thus the top of the shear wall is a circular arc with radius
1
R RH . Here the
radius R is assumed to be very large compared with its half width, R a , thus the full width of
the shear wall, which is also the diameter of the semi-circular foundation is 2aR or
2 Ra .
62
Figure 4.1. The mathematical model SSI with a semi-circular flexible and rigid
foundation.
63
4.2.1 The free-field wave in the half-space
The incident wave field consists of a train of plane waves of unit amplitude with harmonic
frequency , wave speed C
, and wave number kk C
. The incident waves can be
expressed in both the rectangular and polar coordinates as follows:
cos sin
cos cos sin sin cos
,
,
xy i
i
ikx k y ix y
ikr ikr
wxy e e
wr e e
, (4.1)
and the reflected plane waves can be written as
cos sin
cos cos sin sin cos
,
,
xy r
r
ikx k y ix y
ikr ikr
wxy e e
wr e e
. (4.2)
The
it
e
harmonic time factor is present in all wave equations and will be understood and
omitted from all equations. Here γ is the angle of incidence or reflection with respect to the
horizontal axis, cos
x
kk , and sin
y
kk represent the components of the SH wave
number k along the x- and y-axes, respectively. Applying the Jacobi-Anger Expansion (Pao
and Mow 1973) we have,
cos
0
cos
n
ikr
nn
n
eiJkrn
, (4.3)
where 1 i is the imaginary complex unit, and
.
n
J is the Bessel function of the first kind
with order n as follows:
cos
0
cos
0
cos
cos cos sin sin
ikr n
nn
n
ikr n
nn
n
eiJkrn
eiJkrnnnn
. (4.4)
The two formulas in polar coordinate
, r of Eqs. (4.1) and (4.2) can be expanded into an
infinite series. The free-field wave field is then given by the sum of represented waves that are
that are finite everywhere in the half-space for n = 0, 1, 2, 3…. as:
64
cos cos
00
,
, 2 cos cos cos
ff i r ikr ikr
ff n
nn nn
nn
wr w w e e
wr iJkr n n aJkr n
, (4.5)
where 2cos
n
nn
ai n are the coefficients of the free-field waves and 1, 2
on
for
0 n .
The wave field in the half-space scattered from the flexible foundation is written as
1
0
,cosfor
S
nn
n
wr AH kr n na
, (4.6)
where
n
A are the unknown complex numbers to be determined by boundary conditions and the
wave function and
1 it
n
H e
represent outgoing waves toward infinity, thus satisfying
Sommerfeld’s radiation condition.
4.2.2 The wave field within the structure
Since the building structure on top is a shear wall that is defined as a circular sector with
center at O , a point above the structure (Fig. 4.1), the building waves will be defined using the
polar-coordinate system
, r with an origin at O . The out-of-plane motion is independent of
coordinate x and can be represented as, for R Hr R and 0 :
11 2 2
0
, cos
BB
nn n n bb
n
r
n
w w B H kr B H kr
, (4.7)
where
1
n b
Hkr
and
2
n b
Hkr
are the Hankel functions of the first or second kind with
argument
b
kr and order n ; and
1
n
B and
2
n
B are the unknown complex numbers to be
determined by boundary conditions and the wave functions.
65
4.2.3 The wave field within the flexible foundation
The expression of the wave inside the flexible foundation is
11 2 2
0
,cos
F
nn f n n f
n
w r C H kr C H kr n
, (4.8)
for 0 ar aand where
1
n f
Hkr and
2
n f
Hkr are the Hankel functions of
the first or second kind with argument
b
kr and order n, and
1
n
C and
1
n
C are the unknown
complex numbers to be determined by boundary conditions and the wave functions.
4.3 The Boundary Conditions
4.3.1 At the interface of the flexible foundation and free-field
The free-stress boundary conditions of the ground surface should be satisfied by the free-
field waves
ff
w and the scattered waves
S
w . The stress and displacement continuity
equations along the semi-circular interface at ra and 0 include:
displacement continuity:
for 0
ff S F
ra ra
ww w
; and (4.9)
stress continuity:
for 0
ff S F
f
ra r a
ww w
yy
. (4.10)
Substituting Eqs (3.5), (3.6) and (3.7) into Eqs (3.15) and (3.16) to solve for the boundary
equations, for n = 0, 1, 2, 3,… we have
111 22
nn n n n n f n n f
a J ka A H ka C H k a C H k a , (4.11)
111 22
nn n n n n f n n f
a J ka A H ka C H k a C H k a
(4.12)
66
where
f f
k
k
is the material property ratio in the equation for the stress continuity
boundary condition.
4.3.2 At the flexible and rigid foundation interface
As the rigid and flexible foundations are assumed to be welded, they have the same
displacement. The boundary condition at the interface of rigid and flexible foundations can be
expressed as follows:
FR it
ra ra
ww e
. (4.13)
Substituting Eq. (4.8) into Eq. (4.13) to solve for
2
n
C is,
for
for
11 2 2
11 2 2
0
00
oo f o o f
nn f n n f
CH ka C H ka n
CH ka C H ka n
. (4.14)
2
n
C
can be expressed in terms of
1
n
C and as:
for
for
11
2
2
11
2
0
0
oo f
of
n
nn f
nf
CH ka
n
Hka
C
CH ka
n
Hka
. (4.15)
4.3.3 Building boundary conditions
Stress free at top of the building for n = 0, 1, 2, 3,… is
1
1
11 2 2
11
0
zr b b n n b n n b
rR
rR
B
B H kR B H kR
y
w
. (4.16)
Expressing
2
n
B in terms of
1
n
B is
67
1
1
21
2
1
nb
nn
nb
HkR
B B
HkR
. (4.17)
Substituting Eq. (4.17) into (4.7), the wave within the building can be simplified to
12
1
11 1
2
0
1
12 1 2
11
1
2
0
1
1
00
cos
cos
ˆˆ
,cos cos
nn n n
nb n b
B
nn b n
n
nb
nb n b n b n b
B
n
n
nb
B
bb b
nn
HkRH kr
n
wBHkr B
HkR
HkrH kR H kRH kr
n
wB
HkR
nn
wBHkRkr BHkr
,
(4.18)
where
1
ˆˆ
,
nb nb b
Hkr H kRkr
is the modified Hankel function defined as a linear
combination of Hankel function of the first and second kind, and
12 1 2
11
1
2
1
ˆ
,
nb n b n b n b
nb b
nb
H kr H kR H kR H kr
HkRkr
HkR
.
(4.19)
The boundary condition at the interface of building and rigid foundation can be expressed
as
BR
rR r R
ww
.
(4.20)
Substituting Eq. (4.18) into Eq. (4.20) we have
for
for
and for
1
1
ˆ
0
ˆ
00
00
,
,
oo
nn
n
bb
bb
Hn
Hn
n
BkRkR
BkRkR
B
. (4.21)
Therefore, the wave field within the structure can be further simplified as
68
1
ˆ
,
B
oobb
H w B kR kR . (4.22)
4.3.4 The flexible foundation action on the rigid foundation
The building action on the rigid foundation,
f
z
f , can be expressed in terms of stress as
follows:
0
11 2 2
0
0
11 2 2
11
cos
f
zrz
ra
f
zff nn f nn f
n
f
zff o f o f
fad
f ka C H ka C H ka n d
fkaCHkaCHka
, (4.23)
where
0
,0
cos
0, 0
n
nd
n
.
4.3.5 The rigid foundation and building interaction
The building action on the rigid foundation,
b
z
f , can be expressed in term of stress as
follows:
1
0
11
ˆ
,
ˆ
,
B
b
zb bb oob b
rR
b
zbb o b b
w
f Rd kR BH kR kR
r
fkR BHkRkR
, (4.24)
where
111
ˆˆ
,,
o bb bb
HH kR kR kR kR
.
4.3.6 The dynamic equation for the rigid foundation
As pointed out by Luco (1969) and Trifunac (1972), displacement of the foundation
can
be determined by the kinetic equation for the rigid foundation as,
R fb it
Rzz
Mw ff e
, (4.25)
69
where
2 R it
we
,
R
M
is the mass of the rigid foundation per unit depth in the z-axis,
f
z
f denotes the action of flexible foundation on the rigid foundation,
b
z
f is the force of the
building acting on the foundation per unit length, and
R
w represents the displacement function
of the rigid foundation in terms of time factor t as described in Eq. (4.26):
R it
we
.
(4.26)
4.3.7 The rigid foundation displacement
The rigid foundation displacement is solvable by plugging Eqs. (4.23), (4.24) and (4.26)
into Eq. (4.25), as shown:
11 2 2 2
11
11
11
11 2 2
11
2
11
1
ˆ
,
ˆ
,
ˆ
,
Rff o f o f
bb o b b
oo f
Rff o f f
of
bb
bb
obb
MkaCHkaCHka
kR BH kR kR
CH ka
MkaCHka Hka
Hka
HkRkR
kR
H kR kR
(4.27)
12 1 2
11
2
1
2
1 11 2
2
1
ˆ
,
ˆ
,
fo f o f f
ff
of
o
f bb
Rbb ff
o of bb
HkaH ka HkaH ka
ka
Hka
C
Hka HkRkR
MkR ka
H Hka kR kR
.
(4.28)
The displacement of the rigid foundation can be further simplified as,
2
11
2 2
1
11
2
1
4
ˆ
,
ˆ
2 ,
fo f
ooo
f
fbb
RB
FF o of bb
i
kaH ka
CC
Hka ka H kR kR
MM
aM M H Hka kR kR
,
(4.29)
70
where
2
2 2
1 11
2
1
4
ˆ
,
ˆ
2 ,
fo f
o
f fbb
RB
FF o of bb
i
kaH ka
H ka
ka H kR kR
MM
aM M H H ka kR kR
;
(4.30)
R
M ,
B
M , and
F
M are the masses of the building, rigid foundation, and flexible foundation,
respectively;
b
,
r
,
f
stands for the density of those three media sequentially, and
12 1 2 1 2
4
,
nf n f n f n f n f n f
f
i
W H ka H k a H ka H k a H ka H k a
ka
is
the Wronskian.
By substituting Eq. (4.9) into Eqs. (4.15),
2
n
C can be solved in terms of
1
n
C as
for
for
for
11 1
2
2
11
2
1
21
2
0
0
0, 1, 2, 3, 4
oo o o f
of
n
nn f
nf
no n f
nn
nf
CCH ka
n
Hka
C
CH ka
n
Hka
Hka
CCn
Hka
, (4.31)
where
1for 0
0for 0
n
n
n
.
By substituting Eq. (4.31) into Eqs. (4.11) and (4.12), the expressions for wave function
coefficients
n
A and
1
n
C can be solved explicitly as:
71
1
111 2 1
2
12 1 2 2
1
12 1
1
1
12 1
no n f
nn n n n n f n f n
nf
n f n f n f nf nonf
nn
n n
nnf n
nn
n
nn
nnf n
Hka
a J ka A H ka C H k a H k a C
Hka
H ka H ka H ka H ka H ka
aJ ka
AC
H ka H k a H ka
aJ ka
G
AC
HkaH ka Hka
(4.32)
where
11 2 1 2 2
nn f n f n f n f non f
G H ka H k a H ka H k a H ka .
From Eq. (4.12),
1
n
C can be found as
2
1
21 1 1
2
nf
nn
nn n n
i
Hka
ka
Ca
G H ka G H ka
, (4.33)
where
21 2 1 2 2
nn f n f n f n f non f
G H ka H k a H ka H k a H ka
and the
Wronskian
11 1
2
,
nn n n n n
i
W J ka H ka J ka H ka J ka H ka
ka
.
Deriving the equation for
1
o
C (n = 0) from Eq. (4.33), we have
2
1
21 1 1
1
2
of
oo
oo o
i
Hka
ka
Ca
G H ka G H ka
. (4.34)
By combining Eqs. (4.31), (4.32) and (4.33), the expressions for wave function coefficients
n
A
and
2
n
C can be solved simultaneously as shown:
72
1
21 1 1 1
2
nn
n n
nn n n n
i
GJka
ka
A a
GH ka G H ka H ka
(4.35)
1
2
21 1 1
2
no n f
nn
nn n n
i
Hka
ka
Ca
GH ka G H ka
. (4.36)
4.4 Numerical Analysis of the Displacements
First, the correctness of the numerical results can be verified by comparing the results from
the rigid semi-circular foundation case (Trifunac 1972) and the results in Chapter 3 by setting
1 and aa . Figures 4.2 to 4.4 represent the plots of the displacement versus the wave
number and radius a with the initial condition shown in the legends. The abscissa in these
figures is the dimensionless frequency a
and the ordinate is the foundation displacement
amplitude . The results are in excellent agreement with the results of Trifunac and Le et al.
(2016).
73
Figure 4.2a. The effect of interaction on the shear wall:
1, 10, 1, 1
BF R F
RH M M M M .
Figure 4.2b. The effect of interaction on the shear wall
1, 50, 1, 1
BF R F
RH M M M M .
74
Figure 4.3a. The effect of interaction on the shear wall
1, 10, 2, 1
BF R F
RH M M M M .
Figure 4.3b. The effect of interaction on the shear wall
1, 50, 2, 1
BF R F
RH M M M M .
75
Figure 4.4a. The effect of interaction on the shear wall
1, 10, 4, 1
BF R F
RH M M M M .
Figure 4.4b. The effect of interaction on the shear wall
1, 50, 4, 1
BF R F
RH M M M M .
76
4.5 Conclusions
The analytical solution of the interaction of a shear wall, flexible-rigid foundation and an
elastic half-space is derived for incident SH waves with varies angle of incidence. From results
of the numerical analysis of the proposed model in Figure 4.1, it is concluded that the big arc
numerical methodology works well with a reasonably large radius R . This methodology can be
used to solve for the SSI of a shear wall supported by the flexible foundation as discussed in
Chapter 6.
77
CHAPTER 5
SH WAVES IN A MOON-SHAPED VALLEY
5.1 Introduction
The analytical solution of a two-dimensional (2D), moon-shaped, alluvial valley embedded
in an elastic half-space is analyzed for incident plane SH waves, using the wave function
expansion and the Discrete Cosine Transform (DCT). A series of solutions with different depth-
to-radius ratios have been computed, analyzed, and discussed. It is shown that amplification of
incident motions along the thinning valley segment can be significant. The phenomena of
combined action of the waves resulting from (a) turning (reversing the direction of
propagation), (b) focusing, and (c) diffraction from the half space into the valley have been
examined with an emphasis on the significance for surface-motion amplification and the power
to damage man-made structures.
5.1.1 Brief history
Amplification of strong earthquake ground motion has been studied for simplified site
characterizations (in terms of categorical variables: basement rocks versus sedimentary
deposits, for example; Trifunac and Brady 1975); site characterization in terms of horizontal
layers with various degrees that include the number and depth of the near-surface layers (Lee
and Trifunac 2010; Trifunac 1989, 2016); regional variations that include the entire sedimentary
basins (Todorovska and Trifunac 1997a,b; Trifunac et al. 1994; Trifunac and Todorovska
2013); the relationship of site conditions to the observed damage (Trifunac and Todorovska
2004; Trifunac et al. 1999); the relationship of site conditions to the measured microtremors
(Trifunac and Todorovska 2000; Udwadia and Trifunac 1973); and in terms of numerous
analytical models involving 2D and 3D inclusions (Sanchez-Sesma et al. 2002).
Amplification of seismic waves along the sloping alluvial deposits belongs to a more
advanced form of site modeling, because it includes complex superposition of amplification
within the sloping layer, as well as amplification of refracted and diffracted waves from below
the layer, which are amplified by propagation—i.e., from harder into softer materials. The
complexity of this problem in a realistic setting is further increased by irregular topography,
78
irregular shape of the bottom of the layer (Lee et al. 2014a,b), and the 3D geology and
geometric complexities. In this chapter, we examine waves in a simple 2D version of such a
problem for a sedimentary valley that is bounded by two circular surfaces. By varying the two
radii, our model of the valley can be used to study waves in the sedimentary inclusion with
shapes that can range between semi-circular and slender moon-shapes.
Many researchers have investigated elastic-wave propagation in wedge-shaped layers, in
the case of a single wedge, and with either stress-free or rigid boundaries (Hudson 1963).
Knopoff (1969) reviewed the Kantorovich-Lebedev transformation solution method. Ishii and
Ellis (1970a,b), Hong and Helmberger (1977), Wojcik (1979), Pao and Ziegler (1982), and
Sanchez-Sesma and Velazquez (1987) produced studies on a wedge bonded to an elastic half-
space. The subject is of considerable interest for earthquake engineering, because the rays of the
waves inside the wedge, which propagate towards its apex, are progressively turned to a point
where they start to propagate backwards. The focusing of wave energy at the turning point can
lead to a large amplification of surface motions, and, as such, becomes important in analyses
that aim to interpret the causes of damage to man-made structures.
We introduce the problem of propagation-direction reversal and the associated focusing
and amplification of incident wave energy by reviewing two examples that we believe are the
governing mechanisms of the observed surface displacement motions.
5.1.2 Two examples
Many settlements are built in valleys surrounded by mountains where the sediment
thickness gradually increases from higher ground toward the valley centers. Typically, such
sediments are formed by water, wind erosion, and the associated transport of fine particles
downhill toward the valley’s flat central areas. During strong earthquake shaking, higher-than-
average destruction to man-made structures often occurs along the edges of these sedimentary
valleys, when wave energy within the sediments is amplified as it progresses toward decreasing
layer thickness (Papageorgiou and Kim 1991). In this chapter we illustrate such amplification
for the Sherman Oaks area in the California San Fernando Valley. We also cite the results of a
full-scale shaking experiment during which monochromatic SH waves are amplified by
focusing on the shallowing section along a sedimentary layer in eastern Pasadena, California.
79
5.1.2.1 Sherman Oaks
The California earthquakes in San Fernando (1971) and Northridge (1994) caused
extensive damage throughout the San Fernando Valley, a northwestern segment of the Los
Angeles metropolitan area. Figure 5.1 shows a horizontal projection of the fault (dashed line)
that slipped during the main event on January 17, 1994. The aftershocks, which were recorded
by strong-motion accelerographs, are shown by asterisks. Strong-motion accelerograph stations
of the Los Angeles strong-motion array (Anderson et al. 1981), which recorded the main event
and the aftershocks, are shown by triangles.
The areas in which buildings were damaged by the San Fernando earthquakes and
Northridge are shown in Figure 5.2 by gray zones. The location of breaks in the water pipes is
shown by black dots (these indicate a large strain in the surface soils). This figure shows that the
areas in which buildings were severely damaged (the gray zones) do not overlap with the areas
of “high” strain in the soil (areas with high concentration of pipe breaks), where motions were
moderate to large. However, in the areas of very severe ground motion, for example, in
Sherman Oaks (south of the 101 freeway), buildings were damaged and pipes were broken in
the same area, implying a very large nonlinear soil response resulting from excessive input
motions and large nonlinearity in the site response (Trifunac and Todorovska 1998; 1999).
In the Sherman Oaks area, the damage was severe during both earthquakes. We interpret
this to imply that the wave energy from the deep sediments in the central and northern San
Fernando Valley propagated southwards into progressively shallower surface layers where it
was focused and amplified. This amplification was further increased by the waves arriving from
below, which then further amplified by propagating into softer sediments. Together,
superimposed, these two amplified motions resulted in excessive peak ground velocity that may
have reached 150 cm/s.
The San Fernando Valley is an asymmetric basin that deepens northward from Sherman
Oaks (Fig. 5.1), with the crystalline basin composed of metamorphic and plutonic rocks of the
Precambrian and pre-Cretaceous age (Yerkes 1995; Yerkes et al. 1996). These rocks are
exposed in the San Gabriel, Verdugo, and Santa Monica Mountains. Near the surface, the
Matilija and Milbank seismic reflection profiles (Figs. 5.1 and 5.2) show Plio-Pleistocene
deposits deepening towards the north, with the bottom sloping between 15 and 22 degrees and
reaching a depth of about 300 m at the end of the Matilija seismic line (Stephenson et al. 2000).
80
For the purposes of this chapter, we will view this complex picture as one in which the overall
geology underlying the Sherman Oaks area may be represented approximately by a model
consisting of a sedimentary layer that is progressively deepening northwards.
Figure 5.1. The San Fernando Valley, northwest of Los Angeles, with a horizontal
projection of the fault (dashed line) that slipped during the 1994 Northridge
earthquake. Asterisks show the aftershocks, which were recorded at accelerograph
stations USC#3, USC#6, and USC#53. The depth to the crystalline basement rocks
(from Yerkes et al. 1965), in thousands of feet, is shown in the southeastern part of
the Valley.
81
Figure 5.2. The San Fernando Valley: “gray zones” (where buildings were
damaged) and locations of reported breaks in the water pipes (black dots) in the
1994 Northridge earthquake. Locations of “unsafe” buildings following the 1971
San Fernando earthquake are shown by open diamonds.
5.1.2.2 Forced vibrations at the Millikan Library
Wong et al. (1977a) describe a forced vibration experiment during which the Millikan
Library, a nine-story reinforced concrete (RC) building in Pasadena, California, was
shaken in a north-south (NS) direction. During this experiment, shaker baskets were
fully loaded with lead weights (Hudson 1962) and, at 1.8 Hz, generated periodic force
acting on the ground surface in a NS direction with a maximum amplitude of about
2750 lbs. The resulting horizontal force and moment on the soil were approximately
2.8×
5
10 lb and 2.8×
7
10 lb-ft (Wong et al. 1977b). This type of wave source generated
motions whose NS components in the layered half-space consisted of SH and Love
waves west of Millikan Library. These motions occurred because the radiation pattern
of SH and Love waves has a maximum in eastward and westward directions for a point
source consisting of a harmonic NS force and a harmonic rocking moment acting in the
same direction (Luco et al. 1975).
82
Figure 5.3. (Top) The three trial cross sections A, B, and C through the alluvium in
west Pasadena (west is to the left in this figure). (Bottom) A comparison of the
computed (for models A, B, and C) and normalized measured displacements at 13
points (triangles).
The central and western portions of Pasadena are underlain by alluvium with depth that
ranges from zero to about 1200 ft (Gutenberg 1957). The depth of the alluvium underneath the
Millikan Library is approximately 900 ft and becomes shallower toward the west and southwest.
About 4.5 km west of the Library and north of the Raymond fault, the crystalline basement
rocks and deep tertiary become exposed on the ground surface. The variations of measured
ground displacements (triangles in Fig. 5.3) agree with the computed variations for model A.
The characteristic pattern of the observed variations of displacement amplitudes versus distance
is virtually identical with the pattern predicted by the theoretical model for waves propagating
toward a decreasing alluvium depth.
83
One of the most important elements in numerous engineering studies that have dealt with
the effects of alluvium layers on the variations of strong earthquake ground motions is
associated with the assumption concerning the direction and the manner in which the waves
enter into the model. Whether seismic energy arrives from below, the side, outside, or within,
the body of alluvium layer can lead to remarkably different surface displacement amplitudes
even for the same model. Patterns of constructive and destructive interference, as well as the
fraction of energy scattered by the material discontinuities in the model, can change
dramatically for these different inputs. An initial buildup followed by a rapid decay of observed
displacement amplitudes described in Wong et al. (1977a) results from the focusing as wave
energy propagates toward shallower alluvium. The sloping bottom of the layer causes the
incident rays to turn and reflect off the free surface with progressively larger angles of
incidence. When the angle exceeds 90
o
instead of propagating forward and away from the
source, the rays turn back toward the source. For an ensemble of different rays, the net effect is
to focus the energy in the vicinity of the turning points. The input energy here is partly refracted
through the bottom of the layer, radiating away into the underlying half-space, and partly
reflected back toward the source. Only a fraction of total input energy, which is associated with
essentially horizontal rays, is transmitted farther, past the focusing points. Behind the turning
points, where all energy is carried inside the layer, one would expect to find relatively low
displacement amplitudes and then a gradual buildup as the depth of alluvium reaches zero. This
gradual buildup of amplitudes results mainly from the waves traveling through the underlying
half-space. On the other hand, if all input energy arrives from below, through the underlying
half-space—as is commonly assumed for many simple models of local site conditions—the
focusing of incident waves would be diminished or eliminated altogether.
5.2 The Mathematical Model
The model studied in this chapter is a 2D, elastic, moon-shaped, alluvial valley embedded
in an elastic half-space, as illustrated in Figure 5.4. All materials are homogeneous, elastic, and
isotropic. The material constants, shear modulus, and wave speed of the elastic half-space and
the alluvial valley are denoted by , C
and ,
v
v
C
. The contact between the half-space and
the alluvial valley is assumed to be welded. Two Cartesian coordinate systems, (, ) x y and
(, ) x y , with the two corresponding polar coordinate systems (, ) r and (, ) r used in this
chapter are shown in Figure 5.4. The origin of the coordinate system (, ) x y is located at O at
84
the center of the circular arc, and the coordinate system (, ) x y has its origin O at the center of
the top surface of the alluvial valley. The vertical distance between the two origins is h .
Figure 5.4. A model of a moon-shaped valley in an elastic half-space.
The waves considered in this chapter are out-of-plane SH waves, with motions in the z-
direction, perpendicular to the x-y plane. A Cartesian coordinate system , x y and a
corresponding polar-coordinate system , r are defined with the origin at the center of the
circular arc of the bottom surface of the alluvial valley. All SH waves in the half-space are
assumed to be harmonic, with frequency , wave number k , and particle motions in the z-
direction, of the form (, , ) (, )
it
wx y t w x y e
. These motions satisfy the Helmholtz wave
equation:
22
0 wkw . (5.1)
85
The harmonic frequency term
it
e
in all the waves is present but will be omitted in all of the
following equations.
5.2.1 Incident waves in the half-space
The excitation is assumed to consist of a series of plane SH waves with incidence angle
with respect to the horizontal axis of the Cartesian system , x y . The incident wave consists of
a plane wave with unit amplitude, harmonic frequency , wave speed C
, and wave number
kk C
. The incident waves can be expressed in both the rectangular and polar
coordinates as
cos ysin
cos cos sin sin cos
,
,
iikx
iikr ikr
wxy e
wr e e
. (5.2)
The plane waves reflected from the horizontal half space boundary can be written as
cos ysin
cos cos sin sin cos
,
,
ik x r
ikr ikr r
wxy e
wr e e
, (5.3)
where cos
x
kk and sin
y
kk are the wave numbers along the x- and y-axes. The
it
e
harmonic time factor is understood and omitted. Applying the Jacobi-Anger Expansion (Mow
and Pao 1971), we have
cos
0
cos
0
0
cos
cos
cos cos sin sin
n
ikr
nn
n
ikr n
nn
n
n
nn
n
eiJkrn
eiJkrn
iJ kr n n n n
, (5.4)
where 1 i is the imaginary complex unit, .
n
J is the Bessel function of the first kind
and order n , and the two expressions in the polar coordinate , r can be expanded into an
infinite series. The free-field wave field is then given by their sum:
86
cos cos
00
,
2 cos cos cos
ikr ikr ff i r
n
nn onn
nn
wr w w e e
iJ kr n n a J kr n
, (5.5)
where 2cos
n
on n
ai n , as in Eq. (5.5) shown above, which represents the free-field waves
that are finite everywhere in the half-space for 1 , 2, 3, ... n ;
on
a are the coefficients of the
free-field waves; and 1
o
and 2
n
for 0 n , so that 2
oo
a .
The free-field waves will arrive toward the valley and result in scattered and diffracted
waves. The wave field in the half-space, scattered from the valley forra and 0 , can
be represented as
1
0
,cos
S
nn
n
wr AH kr n
, (5.6)
where
n
A are the unknown complex numbers to be determined by boundary conditions and the
wave functions
1
.
it
n
He
represent outgoing waves satisfying Summerfield’s radiation
condition.
The total displacement of the half-space is then the combination of the incident wave
,
ff
wr and the scattered waves
,
S
wr . Both the incident waves and the scattered
waves satisfy the wave Eq. (5.1) and the stress-free boundary conditions at the surface. For
,
S
ww r and ra :
()
0,
1
0,
0
0 sin
S
nn
n
w nA H kr n
. (5.7)
5.2.2 The Waves within the valley
The valley will first be divided into two regions, the inner semi-circular, 0 ra , and
the outer semi-circular ring,
1
ar a , regions:
5.2.2.1 The wave field within the outer semi-circular region,
1
ar a :
87
The wave field within the outer semi-circular valley can be formulated with waves
expressed in the polar coordinate , r with respect to the origin O at the surface of the half-
space surface, which is also the origin for the scattered waves
S
w . With respect to this
coordinate system, the wave field for this region can be expressed as
(1) (2)
11 1 2
0
,cos
vv
nn v n n v
n
ww r CH kr CH kr n
, (5.8)
for 0 and
1
ar a where
1n
C and
2n
C are an unknown complex number to be
determined by boundary conditions and the wave functions of the half-space.
1
nv
Hkr and
2
nv
Hkr are the Hankel functions of the first and second kind with argument
v
kr and order
n . As for the scattered waves
S
w (Eq. 5.6) in the half-space medium, the valley waves
1
v
w
here in the outer ring also satisfy the zero-stress boundary condition at the flat surface of the
foundation. For
11 1
,,
v
ww r a r a
,
12
11 2
0,
0,
0
0 sin
v
vvnnvnnv
n
wCHkrCHkr nn
. (5.9)
5.2.2.2 The wave field within the inner semi-circular region, 0 ra
The wave field within the inner semi-circular valley of the same form as the scattered
waves outside with respect to
, r can be expressed as
12 (1) (2)
0
,cos
vv
nn v n n v
n
n
w w r CHkr C Hkr
, (5.10)
for ra and 0 within in the valley where
1
n
C
and
2
n
C
are the unknown complex
numbers to be determined by boundary conditions and the wave functions of the half-space, and
1
nv
H kr
and
2
nv
H kr
are the Hankel functions of the first and second kind with argument
v
kr and order n . Hankel functions of this order are chosen since the arc of the moon-shaped
88
valley is within the range of 0 , thus the cosine functions are complete and
orthogonal, making n the eigenvalues of the trigonometric functions.
Hankel functions of the first and second kind are both used to represent the waves in the
valley as they represent outgoing and incoming waves from the top surface of the moon-shaped
valley. Together the waves are to satisfy the stress-free boundary condition at the top surface.
The valley waves
,
vv
ww r with respect to the coordinate system
, r with origin at
O (Eq. 5.10) can first be simplified by applying the zero stress boundary condition at ra :
0
v
zr v
ra
w
r
.
(5.11)
This gives
12 (1) (2)
0
cos 0
vv n n v n n v
n
n
k CHka CHka
, (5.12)
for 0 . The orthogonality of the
cos
n
functions for 1 , 2, 3, ... n gives
12 (1) (2)
0
nn v n n v
C H ka C H ka
(5.13)
or
(1)
21
(2)
nv
nn
nv
Hka
CC
Hka
, (5.14)
for 1 , 2, 3, ... n . Thus
2
n
C
can be expressed in terms of
1
n
C
and the valley waves become
0
ˆ ˆ
,cos
vv
nn v
n
n
ww r CH kr
, (5.15)
where
89
(1)
(1)
(1) (2)
(2)
ˆ
ˆ
nn
nv v
nv v n v v nv v
nv v
CC
Hka
H kr H k r H kr
Hka
. (5.16)
5.2.3. Transferring the valley waves between the two coordinate systems
The wave field within the valley can also be formulated with waves expressed in the polar
coordinates , r with respect to the origin O at the surface of the half-space surface, which is
the same origin for the scattered waves
S
w . With respect to that coordinate system, writing
,, , rrr r , the waves in the valley can be expressed from Eq (5.15) as:
0
ˆ ˆ
,,cos,
vv
nn v
n
n
ww r CH krr r
.
(5.17a)
However, ,
vv
ww r with respect to the coordinate system , r with origin at O also
satisfies the Helmholtz wave equation for 0 in the valley , rar a and is finite
everywhere. Using separation of variables of the wave equation, the waves can thus be chosen
to be expressed in terms of the orthogonal set of trigonometric cosine functions
cos , 0,1,2 nn in 0 , the waves thus take the form:
0
,() cos
vv
vnv
n
ww r Rkr R kr n
. (5.17b)
The trigonometric cosine functions are chosen over the sine functions to match with the
scattered waves outside the valley. Here it is known that the radial function
nv
R kr for each
0,1,2 n satisfies the Bessel differential equation of order n, with the Bessel functions as
the solution. Since the waves are finite everywhere, the Bessel functions of the 1
st
kind,
()
nv
Jkr , are chosen. The waves thus take the form:
0
,()cos
vv
nn v
n
ww r CJ kr n
, (5.17c)
90
referring to both coordinate systems, where
n
C for 1, 2 , 3 , ... n are the unknown complex
coefficients to be determined by boundary conditions and the wave functions of the half space.
Next, consider the waves given in abovementioned Eqs. (5.15), (5.17a) and (5.17c) with respect
to the coordinate systems
, r
at O and , r at O , respectively,
Eq. (5.15)
Eq. (5.17a)
Eq. (5.17c)
0
0
0
ˆ ˆ
,cos
ˆ ˆ
,,cos,
,cos
vv
nn v
n
vv
nn v
n
vv
nn v
n
n
ww r CH kr
n
ww r CH krr r
ww r CJ kr n
. (5.18)
If the waves are known everywhere around the valley from Eq. (5.A17c), they can be evaluated
at equally spaced points along the upper surface of the moon-shaped valley at ra and
0 as
00
ˆˆ ˆ
ˆ ,cos cos
v
nn v n
ra
nn
nn
fwr CHka c
, (5.19)
from which the coefficients
ˆ ˆ
ˆ
nnn v
cCH ka
for 1, 2, 3 , ... n can then be computed using
the Discrete Cosine Transform (DCT). The DCT algorithm and software is readily available in
Fortran 90 of Numerical Recipes (DCT-II) (Fortran 90, 1991) or in the built-in library of Matlab
(Matlab R2013b, 2013). Knowing ˆ
n
c , the coefficients
ˆ
n
C will be known.
Conversely, if the waves are known in the valley using Eq. (5.15), they can also then be
evaluated at equally spaced points along the surface ra and 0 , and the bottom
surface of the moon-shaped valley is as shown below:
00
,cos cos
v
nn v n
ra
nn
f wa CJ ka n c n
, (5.20)
from which the coefficients,
n
c for 1 , 2, 3, ... n can also be evaluated using DCT. Once
n
c is
known,
n
C can be calculated.
91
5.3 Interface Boundary Conditions of the Valley
5.3.1 The outer interface of the valley with a half-space medium
The displacements and stresses of the moon-shaped valley and the half space at
1
ra
must satisfy the continuity condition at interface for 0 as follows:
1
1
1
1 (1) (2)
111 12 1
ff S v
ra
ra
on n n n n n v n n v
ww w
a J ka A H ka C H k a C H k a
(5.21)
1
1
1
1 (1) (2)
11 1 12 1
ff S v
v
ra
ra
on n n n v n n v n n v
ww w
rr
a J ka A H ka C H k a C H k a
, (5.22)
where
vv
v
k
k
. Equations (5.21) and (5.22) in matrix form are
1 (1) (2)
11 1
1 1
1 (1) (2)
1 2
11 1
nv n v n
n n
on n
n n
vn v v n v nn
Hka H ka Hka
Jka C
aA
Jka C
Hka H ka Hka
. (5.23)
5.3.2 The interface of the inner and the outer valley regions
The displacements and stresses of the inner (Eq. 5.17c) and outer (Eq. 5.8) regions of the
moon-shaped valley at ra must satisfy the continuity at the interface for 0
, thus,
1
(1) (2)
12
vv
ra r a
nn v n n v n n v
ww
C H ka C H ka C J ka
(5.24)
1
(1) (2)
12
vv
vv
ra r a
nn v n n v n n v
ww
rr
C H ka C H ka C J ka
A
, (5.25)
and in matrix form as
(1) (2)
1
(1) (2)
2
nv n v
nv n
n
nv n
nv n v
Hka H ka
Jka C
C
Jka C
Hka H ka
, (5.26)
92
from which
11
22
nn
n
nn
C
C
C
, where, with Wronskian given by , Wf g fg gf
(Abramowitz and Stegun 1972), so that
(1) (2)
(1) (2)
4
,
22
,, ,
nv n v
v
nv n v nv n v
vv
i
WH ka H ka
ka
ii
W J ka H k a W J k a H ka
ka ka
(5.27)
(2) (1) (2)
1
(1) (1) (2)
2
1
,,
2
1
,,
2
n v nv n v nv
n
n
nv n v n v n v
W J ka H k a W H k a H ka
WJ ka Hka WHka H ka
, (5.28)
and
1
2
1
2
1
2
n
n
n
C
C
C
. (5.29)
The wave field within the outer semi-circular valley Eq. (5.8) then becomes
(1) (2)
11
0
11
0
1
,cos
2
,cos
vv
nn v n v
n
vv
nn v
n
ww r C H krH kr n
ww r CJ kr n
, (5.30)
where
(1) (2)
1
2
nv n v n v
Jkr H kr H kr
. This indicates that the waves in the outer
semi-circular region
1
ar a given by Eq. (5.8) are the same as the waves in the inner
semi-circular region 0 ra given by Eq. (5.17c). Therefore, the waves within the valley
can be evaluated for 0 ra and 0 . Equation (5.23) at the outer interface for
ra then becomes
1 (1) (2)
1
1 (1) (2)
2
nv n v n
n nv n
on n n
n vn v n
vn v v n v nn
Hka H ka Hka
Jka J ka C
aA C
Jka J ka C
Hka H ka Hka
or
93
1
1
nnv n n
on
n n
nvnv
Hka J ka
Jka A
a
Jka C
Hka J ka
. (5.31)
Recall
vv
v
k
k
. Here
n
A and
n
C can be solved explicitly as
1
1
1
11
nnv
n n
on
n n
nvnv
vn n v n v n
on
n
nn n n
Hka J ka
Jka A
a
Jka C
Hka J ka
JkaJ ka J kaJ ka
a
H kaJ ka H kaJ ka
,
(5.32)
where
11
n n vn vn vn
J ka H ka J ka H ka
.
5.4 New Proposed Solution Verification
The accuracy of the new proposed solution can be verified by comparing the results with
the published solution of the surface motion of a semi-circular alluvial valley for incident plane
SH waves (Trifunac 1971) by setting the height-to-width ratio ha to be a large value. Figure
5.5 illustrates the surface displacements for
0.50, 0 .5 0 , 1.50
vv
kk
and the incidence
angles
0,60
oo
of plane SH waves whose amplitude is unity. The abscissa is the
dimensionless distance x a
and the ordinate is the surface displacement amplitude. The points
1 xa correspond to the edges of the valley and 0 xa is the center of the valley. It can be
seen from Figure 5.5 that the results for
4 0 N
and 10 ha of our new solution are in
excellent agreement with the results of Trifunac (1971).
94
Figure 5.5. Surface displacement amplitudes for
0.50, 10, 0.50, 1.50, 0 , 60
oo
vv
ha k k
.
5.5 Numerical Examples
The waves in the half-space and within the valley were calculated numerically by
truncating the infinite sums to a finite number of terms for 1 , 2, 3, ..., nN , and the radius of
the outer valley is taken to be equal to the radius of the inner valley for all cases—i.e.,
1
aa
(Figure 5.4) as described in Section 5.3.2. The dimensionless frequency is chosen to be the
ratio of the width of the valley 2a and the incident wavelength
CT
that is
22
CC
aa a
T
, (5.33)
where the period of motion is 2/ T .
The amplitudes
w
and phases
of all waves are
95
12
22
1
Re Im
Im
tan
Re
ww w
w
w
, (2)
where Re . and Im . are the real and image parts of a complex number. Both
w
and
depend on the frequency of the incident waves, the physical dimensions of the valley, and the
material constants—namely
,,
v
C
and are the material densities, and shear wave
velocities—in the half-space and inside the valley, respectively.
Figures 5.6a through 6d show the amplitudes of surface displacements versus
dimensionless distance x a for dimensionless frequencies 0.50 , 0 .75 , 1.00 , and1.50 of
the softer material in the valley and the harder material in the half-space corresponding to the
material constant ratios
1.50
v
and
0.50
v
kk
, and the ratio of wave numbers in the
half space and inside the valley. In each figure, the valley with a dimensionless alluvium
thickness of 0.75 ha is subjected to the plane SH waves with incidence angles
0
o
,30
o
,60
o
and 90
o
. The following figures show that for the waves arriving from the left,
the amplification tends to be concentrated toward the right side of the valley.
v
C
96
Figure 5.6a. Surface displacement amplitudes and phases for
0.50, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
oo o o
vv
ha k k .
Figure 5.6b. Surface displacement amplitudes and phases for
0.75, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
oo o o
vv
ha k k .
97
Figure 5.6c. Surface displacement amplitudes and phases for
1.00, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
oo o o
vv
ha k k .
Figure 5.6d. Surface displacement amplitudes and phases for
1.50, 0.75, 0.50, 1.50, 0 , 30 , 60 and 90
oo o o
vv
ha k k .
98
For waves incident from the left and propagating rightwards, the corresponding phases
have a positive slope versus distance / x a . The segments of phases with a negative slope
indicate a leftward propagation—i.e., a propagation reversal caused by the converging top and
bottom boundaries of the valley.
The surface displacement amplitudes for various valley thicknesses (Figure 5.7) are shown
for incidence angles
0 a nd 60
oo
in Figures 5.8a–5.8d and Figures 5.9a–5.9d, respectively.
For shallow valleys, the surface wave amplitudes increase slowly relative to the deeper valleys.
For incidence angle
60
o
, shown in Figures 8a–8d, the peak displacement amplitudes in the
shallow valley ascend with increasing frequency, and are larger than in the deep valley.
An analysis of the motion of the top surface relative to the motion of the bottom of the
valley (interface with half-space) can help to explain where the large amplifications take place.
Figures 5.10a–5.10d illustrate these displacements and phases for both surfaces and for a
dimensionless alluvium thickness of 0.75 ha excited by the plane SH waves with incidence
angles
0 a nd 90
oo
.
Figure 5.7. Moon-shaped valleys with different h/a ratios.
99
Figure 5.8a. Surface displacement amplitudes and phases for
0.50, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
Figure 5.8b. Surface displacement amplitudes and phases for
1.00, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
100
Figure 5.8c. Surface displacement amplitudes and phases for
1.50, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
Figure 5.8d. Surface displacement amplitudes and phases for
2.00, 0.50, 1.50, 0 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
101
Figure 5.9a. Surface displacement amplitudes and phases for
0.50, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
Figure 5.9b. Surface displacement amplitudes and phases for
1.00, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
102
Figure 5.9c. Surface displacement amplitudes and phases for
1.50, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
Figure 5.9d. Surface displacement amplitudes and phases for
2.00, 0.50, 1.50, 60 , 0.50, 1.00, 1.50 and 2.00
o
vv
kk h a .
103
Figure 5.10a. Top and bottom surface displacement amplitudes and phases for
0.50, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha k k .
Figure 5.10b. Top and bottom surface displacement amplitudes and phases for
1.00, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha k k .
104
Figure 5.10c. Top and bottom surface displacement amplitudes and phases for
1.50, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha k k .
Figure 5.10d. Top and bottom surface displacement amplitudes and phases for
2.00, 0.75, 0.50, 1.50, 0 and 90
oo
vv
ha k k .
105
Figure 5.11 illustrates the waves in the moon-shaped valley for vertically incident SH
waves. The standing wave occurs between / x a = 0.34 and -0.34 because of the refractions in
the opposite direction from the bottom interface. For / x a greater than 0.34 and smaller than –
0.34, but less than about 0.75 and greater than –0.75, the waves inside the valley propagate
toward / x a = 0. For / x a greater than 0.75 and smaller than –0.75, the waves inside the valley
become larger, and have essentially the same phase as the incident motions.
To study the amplification of relative motions between the valley surface and its interface
with the half-space, we calculated the largest displacement amplitudes within the moon-shaped
valley (the difference between the surface and interface displacements) for /
v
kk = 0.3 and 0.5,
for /
v
= 1.5, between 0.5 and 2.0, / ha between 0.25 and 2.00, and for between 0°
and 90°. In Figures 5.12a and 5.12b, we show the contours of these displacement amplitudes
versus / ha and . Figure 11a shows the contours of the largest relative displacement
amplitudes for /
v
kk = 0.3 while Figure 11b shows it for /
v
kk = 0.5.
106
Figure 5.11. Transmitted and refracted waves in the valley for 1.00,
0.75, 0.50,
v
ha k k 1.50,
v
and 90
o
.
In all examples shown in this chapter, the excitation consists of harmonic SH waves whose
amplitude is equal to 1. The contours in Figures 5.12a and 5.12b are given with the same units;
thus, for example, 8 means that the displacement amplitude is eight times the incident
displacement amplitude. Since the displacement on the surface of the homogeneous half-space
would be 2 for all incidence angles, an amplitude of 8 represents the amplification of surface
motions (relative to the base of the valley) by a factor of 4.
107
Figure 5.12b shows large amplitudes that exceed 13 (for = 1.5, and / ha = 2.0). These
correspond to the amplification of surface relative to the base motions in the valley by 6.5.
Figures 5.12a and 5.12b show that large amplifications can occur for essentially all incident
angles and all frequencies of incident waves. In addition, the location of turning points and
focusing is a complex function of the ray and valley geometries. As such, the location cannot be
associated in a simple way with any parameter that describes the physical and geometrical
properties of the model we study in this chapter. The complexity of amplifications increases as
the wavelength of incident motions decreases (i.e., as increases). For the range of parameters
illustrated in Figure 5.12a ( /
v
kk = 0.3), the largest amplifications occur at / ha = 0.5, = 90°
for = 0.5; between / ha = 1.0 and 1.5, for = 45° and = 1.0; at / ha = 2.0, for = 30°
and 60°; at = 1.5; and for = 2.0 at / ha = 1.25 for = 45°, and at / ha = 1.5, and =
90°. For /
v
kk = 0.5, in Figure 11b, the large amplitudes occur at / ha = 0.50 and = 60°, and
/ ha = 2.0 and = 0° for = 1.0; at / ha = 0.25 for = 60° and = 90°; at / ha = 2.0 and
= 0° for = 1.5; and at / ha = 0.5 and = 60°; at / ha = 1.0 and = 30°; and at / ha =
2.0 and = 0° for = 2.0.
Figures 5.12a and 5.12b show that for the range of / ha and , amplifications do not
occur everywhere, and where the contours have amplitudes equal to 2, the amplitudes are the
same as for the homogeneous half-space. In some instances, the presence of the moon-shaped
valley also leads to the reduction of surface displacement amplitudes. For example, in Figure
5.12b, for = 0.5, at / ha = 0.75 for = 90°, the computed amplitude is 0.54. This
corresponds to “amplification” by a factor equal to 0.27. It is associated with a long wave ( =
0.5), arriving horizontally and then scattered by the “canyon” in the half-space, while inside the
valley, no opportunity exists for interference and focusing due to the long wavelength.
108
Figure 5.12a. The amplitudes of surface relative to the base displacements of the
valley versus ha and , for = 0.5, 1.0, 1.5, and 2.0, and for
v
kk = 0.3.
109
Figure 5.12b. The amplitudes of surface relative to the base displacements of the
valley versus ha and , for = 0.5, 1.0, 1.5 and 2.0, and for
v
kk = 0.5. The
open circles at ha = 0.75. for = 0° and 90° correspond to the four cases plotted
in Figures 9a through 9d at = 0.5, 1.0, 1.5, and 2.0.
110
5.6 Discussion and Conclusions
The subject of focusing and amplification of seismic waves in sedimentary layers with
progressively decreasing thickness is of considerable interest for studies that aim to understand
and interpret larger-than-average destruction of man-made structures. The problem is complex
because of simultaneous contributions from (a) progressive turning of incident rays by sloping
layer interfaces; (b) by amplification of waves propagating from hard-to-soft geological media;
and (c) by diffraction of incident waves from basement rocks into the soft sediments. The model
we describe in this chapter is a simplified 2D representation of the dipping layers, but it offers
the advantage that it can be solved analytically, thus enabling a straightforward examination of
how its behavior depends on material and geometrical properties.
In the range of parameters we examined, we found that amplification of surface motions of
the moon-shaped valley can be as large as 6.5. We also found the reduction of surface motions
by a factor as small as 0.27. Of course, considering excitation by shorter waves (we presented
results only up to = 2.0), the complexities and peak amplifications should increase relative to
those we have illustrated here.
As can be seen from the figures in this chapter, the amplification (as well as the
deamplification) of the surface displacement amplitudes is localized to within a fraction of the
wavelength of the exciting motion. The location where the amplification occurs depends on the
local geometry of the valley, the impedance jump from the basement rocks into the valley, the
geometry of the valley, and the angle at which the waves arrive from below. Since the waves
can arrive with a broad range of directions, it is clear that the amplification pattern on the
ground surface will not only be very complex, but also essentially unpredictable.
With reference to the example we presented in the Sherman Oaks area, it can be concluded
that the spatial variability and complexity of the amplification of ground motions was further
increased by the variation of arrival azimuths and vertical incidence angles from the moving
dislocation of the Northridge earthquake (Figures 5.1 and 5.2). As such, it should be clear that
simplified engineering representations based on the models of vertically incident waves that
amplified as they propagated into a horizontal softer layer are not capable of describing the
average amplification that can be expected from future earthquakes, and hence should not be
used.
111
CHAPTER 6
OUT-OF-PLANE (SH) SOIL-STRUCTURE
INTERACTION: A SHEAR WALL WITH
FLEXIBLE CYLINDRICAL FOUNDATION
6.1 Introduction
The model studied in this chapter presents a logical extension of the elastic shear wall with
a circular rigid foundation fixed firmly in an isotropic, homogeneous, and elastic half-space
considered only for vertical-incidence SH waves developed by Luco (1969) and then formulated
to any angle of incidence SH waves by Trifunac (1972). In practice, buildings are often
supported by concrete footings, piles, grade beams, or mat foundations. As these types of
foundations are not rigid, the ideal goal should be to solve for the soil-structure interaction (SSI)
of shear walls with a flexible or semi-rigid foundation. Todorovska, Hayir, and Trifunac (2001)
investigated the response of an elastic circular wedge (a dike) on a flexible foundation
embedded into a half-space in the frequency domain for incident plane SH waves. The problem
was solved by an expansion of the dike structure motion in cylindrical wave functions that
accounted for both the differential ground motion and soil-structure integration in which a rigid
foundation ignores the differential ground motion. However, both methodologies by Trifunac
(1972) and Todorovska, Hayir, and Trifunac (2001) cannot be modified to the case of a shear
wall structure supported by flexible or semi-rigid foundation because of the expansion
limitation of cylindrical wave functions. To address both of these complicated realities, this
chapter develops a new approach and model in order to solve the SSI of a tapered shear wall for
all rigid, flexible, and semi-rigid foundations using the "big arc numerical method" developed in
Chapter 2 and the Discrete Cosine Transform (DCT). Further, it is not uncommon for buildings
to gradually taper from the bottom to the top, with the result that the top of the building is
slightly narrower than the base (rather than the same width from the top to the bottom), which
presents further complications that must be addressed.
112
6.2 The Mathematical Model
Figure 6.1 illustrates the model of the problem studied in this chapter. As in Trifunac
(1972), this is a two-dimensional (2D) elastic shear wall structure of height H, width 2a erected
on a semi-cylindrical foundation of radius
1
aa that is same or wider than the structure above.
All materials here are homogeneous, elastic, and isotropic. The elastic properties of the half-
space medium, foundation, and shear wall structure are , , ,
ff
, and ,
bb
,
respectively. The corresponding wave velocities at the half-space medium, foundation,
and shear wall structures are respectively C
,
f
ff
C
, and
b
bb
C
. At frequency , the corresponding wave numbers k ,
f
k , and
B
k are
respectively given by . As illustrated in the figure, the
semi-circular flexible foundation is divided into two regions: (1) the inner semi-circular region
with radius a supporting the shear wall structure above, and (2) the outer ring region with
radius
1
ar a embedded on a half-space. The contact surfaces between the soils–foundation
and foundation–building are assumed to be fixed and no slippage exists between them. This
model is similar to and an extension to that in Le et al. (2016),of the case of SSI of a shear wall
on a rigid foundation enclosed by a flexible semi-circular ring.
The model to be solved, as illustrated in Figure 6.2, is a 2D, simulated tapered shear wall
structure resting on the same wider, flexible foundation. This is the same model studied in Le
and Lee (2014), that is, the case of a tapered shear wall structure on a semi-circular rigid
foundation. Imagine that high above the middle of the structure is the point O , which is the
origin of a polar coordinate system
, r such that the right side of the structure is along
0 , and for some very, very small angle , the left side of the shear wall is along
. In other words, the building is modeled as an elastic shear wall with a circular sector
0 of radius
1
R rR , where the arc is at the surface of the foundation with width
of 2a, and the bottom of the shear wall corresponds to the arc at
1
rR R H . When the
radius R is very large, the width of the shear wall is approximately 2aR or 2 Ra .
113
At the base of the center of the structure is the point O, which is the origin of a Cartesian
coordinate system , x y and a corresponding polar coordinate system , r .
Figure 6.1. The Model.
114
Figure 6.2. The mathematical model SSI with a semi-circular flexible foundation.
115
All waves studied here are out-of-plane SH waves, with motions in the z-direction,
perpendicular to the x-y plane, a Cartesian coordinate system
, x y , and a corresponding
polar-coordinate system
, r , which have been defined with the origin at the center of the
semi-circular foundation. All SH waves in the half-space are of a given harmonic frequency ,
wave number k and motions in the z-direction of the form (, , ) (, )
it
xyt w xy e
W with
motions (, ) wx y satisfying the Helmholtz wave equation:
22
0 wkw . (6.1)
The harmonic frequency term is present in all the waves and will be understood and
omitted from all equations here on.
6.2.1 The free-field waves in the half-space
The excitation consists of a series of plane and out-of-plane SH waves incident onto the
flexible foundation from a half-space at an incidence angle with respect to the horizontal axis.
It has unit amplitude and harmonic frequency and wave number k. The incident waves can
be expressed in both the rectangular and polar coordinates as follows:
cos ysin
cos cos sin sin cos
,
,
iikx
iikr ikr
wxy e
wr e e
, (6.2a)
and the reflected plane waves from the half-space surface can be written as
cos ysin
cos cos sin sin cos
,
,
ik x r
ikr ikr r
wxy e
wr e e
, (6.2b)
where is the angle of incidence or reflection with respect to the horizontal axis, and
cos
x
kk and sin
y
kk represent the components of the SH wave number along the x-
and y-axes, respectively. Applying the Jacobi-Anger Expansion (Pao and Mow 1973), we have
116
cos
0
cos
0
0
cos
cos
cos cos sin sin
n
ikr
nn
n
ikr n
nn
n
n
nn
n
eiJkrn
eiJkrn
iJ kr n n n n
, (6.3)
where 1 i is the imaginary complex unit,
.
n
J is the Bessel function of the first kind
with order n, and the two expressions in polar coordinate
, r of Eqs. (6.2a) and (6.2b) can be
expanded into the above infinite wave function series. The free-field wave is then given by their
sum as follows:
cos cos
00
,
2 cos cos cos
ikr ikr ff i r
n
nn onn
nn
wr w w e e
i J kr n n a J kr n
,
(6.4)
where 2cos
n
on n
ai n are the coefficients of the free-field waves exactly as in Eq. (3)
above, which represents the free-field waves that are finite everywhere in the half-space for
1, 2, 3 , ... n , and are the coefficients of the free-field waves; 1
o
and 2
n
for
0 n , so that 2
oo
a .
The free-field waves will arrive toward the foundation, resulting in scattered and diffracted
waves in the half-space. The wave field in the half-space scattered from the flexible foundation
for ra and 0 is given by:
1
0
,cos
S
nn
n
wr AH kr n
, (6.5a)
where
n
A are the unknown complex numbers to be determined by boundary conditions and the
wave functions and
1
.
n
H represent outgoing waves satisfying Summerfield’s radiation
condition.
on
a
117
The total displacement of the half-space is the combination of the free-field wave
,
ff
wr and the scattered wavefield
,
S
wr . Both the free-field waves and the
scattered waves satisfy the wave Eq. (6.1) and the stress-free boundary conditions at the free-
field surface. For
,
1
,
S
ww r r a :
()
0,
1
0,
0
0 sin
S
nn
n
w nA H kr n
. (6.5b)
6.2.2 The wave field within the structure
The shear wall building structure is modeled as an elastic shear wall defined as a circular
sector with center at O , a point above the structure (Fig. 6.2). The building waves are defined
using the polar-coordinate system
, r with an origin at O . The wave field within the
structure can be represented for
1
RRH r R and 0 as:
11 2 2
0
, cos
nn
BB
nb n b
n
r
n
w w B H kr B H kr
,
(6.6a)
where
1
n b
Hkr
and
2
n b
Hkr
are the Hankel functions of the first or second kind with
argument
b
kr and order n ; and
1
n
B and
2
n
B are the unknown complex numbers to be
determined by boundary conditions and the wave functions. As in Le and Lee (2014), the waves
satisfy the zero-shear stress boundary condition at the left
and right
0 side of
the structure. For
1
R RH r R :
,
,
()
0
0
11 2 2
0
0 sin
B
BB n n nnbb
n
n
w
n
B H kr B H kr
(6.6b)
The building waves can be further simplified by applying this zero-stress boundary
condition at the top of the building,
1
rR for 1, 2 ,3 ,... n
118
1
1
()
11 2 2
11
1
21 1
2
1
0
0
nn
n
n
B
zr b
rR
rR
bb n b n b
b
nn
b
w
r
kB H kR B H kR
HkR
BB
HkR
. (6.7)
Substitute (6.7) into (6.6a), the building motion is simplified to a single complex coefficient .
Hence,
12
11 1
2
0
1
12 1 2
1 11
2
0
1
1
00
cos
cos
ˆˆˆ
cos cos
nn
n
n
nn n n
n
nnn
B bb
nb
n
b
B bb b b
n
n
b
B
nb b
nn
HkRH kr
n
wBHkr
HkR
HkrH kR H kRH kr
n
wB
HkR
nn
wBHkr BHkr
(6.8a)
where
(1)
ˆ
nn
BB and
12 1 2
11
2
1
ˆ
n b nb nb n b
n
nb
b
H kr H kR H kR H kr
HkR
Hkr
, (6.8b)
is the “reshaped” Hankel functions for the shear wall defined as a linear combination of the
Hankel function of the first and second kinds.
6.2.3 The wave field within the flexible foundation
6.2.3.1 The wave field within the inner semi-circular flexible foundation
The wave field within the inner semi-circular flexible foundation is be expressed both with
respect to the
, r
coordinate system with origin at O and the , r
coordinate system
with an origin at O . With respect to
, r
it takes the form of
ˆ
n
B
119
12 (1) (2)
0
,cos
FF
nn f n n f
n
n
w w r CHkr CHkr
, (6.9a)
for rR and 0 where
1
n
C
and
2
n
C
are the unknown complex coefficients to be
determined by boundary conditions and the wave functions of the half space;
1
nf
Hkr
and
2
nf
H kr
are the Hankel functions of the first or second kind with argument
f
kr and order
n . We chose Hankel functions of this order since the arc of the inner region flexible
foundation is within the range of 0 , which makes the cosine functions complete and
orthogonal with respect to n , the eigenvalues of the trigonometric functions. The Hankel
functions thus have the order n , the same as the eigenvalues. Hankel functions of the first
and second kind are both used to represent the incoming and outgoing waves, respectively, from
the top surface of the enclosed inner-region flexible foundation.
The wave field within the inner semi-circular flexible foundation can also be formulated
with waves expressed in the polar coordinate , r
with respect to the origin O at the surface
of the outer flexible foundation, which is the same origin for the scattered waves
S
w . With
respect to this coordinate system, the wave equation for the inner flexible foundation can be
expressed as
0
,cos
FF
nn f
n
ww r CJkr n
, (6.9b)
for 0 in the flexible foundation , rRr a
, referring to both coordinate systems,
where
n
C
is unknown complex number to be determined by boundary conditions and the wave
functions of the half space. .
n
J
is the Bessel function of the first kind with order n , which is
used because the wave field is finite everywhere.
6.2.3.2 The wave field within the outer semi-circular flexible foundation
The wave field within the outer semi-circular flexible foundation can be formulated with
waves expressed in the polar coordinate , r
with respect to the origin O at the surface of the
120
half-space surface, which is also the same origin for the scattered waves
S
w . With respect to
this coordinate system, the wave equation for the flexible foundation can be expressed as
(1) (2)
11 1 2
0
,cos
FF
nn f n n f
n
ww r CH kr CH kr n
, (6.10a)
for 0 and
1
ara where
1n
C
and
2n
C
are an unknown complex number to be
determined by boundary conditions and the wave functions of the half space.
1
nf
H kr
and
2
nf
Hkr
are the Hankel functions of the first or second kind with argument
f
kr and order n
. As in the scattered waves
S
w (Eq. (6.5a) ) in the half-space medium, the foundation waves
1
S
w here in the outer ring also satisfy the zero-stress boundary condition at the flat surface of
the foundation. For
,
11
,
F
ww r a r a
,
()
1
0,
12
12
0,
0
0
0 sin
F
f
fnnf nnf
n
w
C H kr C H kr nn
.
(6.10b)
6.3 The Interface Boundary Conditions
6.3.1 At the outer interface with a half-space medium
The displacement and stress of the flexible foundation and the half space at
1
ra
must
satisfy the continuity at the interface for 0 :
11
1
1 (1) (2)
111 12 1
ff S F
ra ra
on n n n n n f n n f
ww w
a J ka A H ka C H k a C H k a
(6.11a)
121
1
1
1
1 (1) (2)
11 1 12 1
ff S F
f
ra
ra
on n n n f n n f n n f
ww w
rr
aJ ka AH ka C H ka C H ka
,
(6.11b)
where
ff
f
k
k
. Equations (6.11a) and (6.11b) in matrix form are
(1) (2)
1
11
1
1 1
0
1
(1) (2)
1 2
1 11
nf n f
n
n n
nn
n n
n fn f f n f
Hka H ka
Hka
Jka C
aA
Jka C
Hka Hka H ka
. (6.11c)
6.3.2 At the interface of the inner and outer flexible foundations
The displacement and stress of the inner and outer regions of the flexible foundation at
ra must satisfy the continuity at the interface for 0 , as follows:
1
(1) (2)
12
FF
ra r a
nn f n n f n n f
ww
C H ka C H ka CJ ka
(6.12a)
1
(1) (2)
12
FF
ff
ra r a
nn f n n f n n f
ww
rr
C H ka C H ka CJ ka
, (6.12b)
in matrix form
(1) (2)
1
(1) (2)
2
nf n f nf
n
n
n
nf nf n f
Hka H ka Jka
C
C
C
Jka Hka H ka
, (6.12c)
from which
11
22
nn
n
nn
C
C
C
, where, with Wronskian given by W(f, g) = f g ′ – g f ′, so that
122
(1) (2)
(1) (2)
4
,
22
,, ,
nf n f
f
nf n f nf n f
ff
i
WH ka H ka
ka
ii
W J ka H ka W J ka H ka
ka ka
(6.12d)
and
(2) (1) (2)
1
(1) (1) (2)
2
1
,,
2
1
,,
2
nf n f n f n f
n
n
n f nf nf n f
WJ ka H ka W H ka H ka
WJ kaHka WHkaH ka
(6.12e)
or
(1) (2) (1) (2)
12
1
2
nn f n n f n n f n f nn f
C H kr C H kr C H kr H kr CJ kr ,
which means the waves in the ring region
1
ara given by Eq. (6.10a) are the same as the
waves in the inner semi-circular region 0 ra given by Eq. (6.9b). Equation (6.11c) at the
outer interface becomes
(1) (2)
1
11 1
1
1 1
0
1
(1) (2)
1 2
1 1 11
nf n f nf
n
n n
nn n
n n
fn f n fn f f n f
Hka H ka Jka
Hka
Jka C
aA C
Jka C
Jka Hka Hka H ka
or
(6.13)
1
11
1
1
1
11
nnf
n n
on
n n
nfnf
Hka J ka
Jka A
a
Jka C
Hka Jka
.
Recall
ff
f
k
k
. Here
n
A and
n
C can be solved explicitly as shown:
1
1
11
1
1
1
11
11 1 1
11
11 11
nnf
n n
on
n n
nfnf
fn nf nf n
on
n
nn n n
Hka J ka
Jka A
a
Jka C
Hka Jka
Jka J ka Jka J ka
a
H kaJ ka H kaJ ka
,
(6.14a)
123
where
11
11 1 1 n nfn fnfn
Jka H ka Jka H ka
. (6.14b)
6.3.3 At the building and the flexible foundation interface
The displacement and stress at the interface between the building and the flexible
foundation at rR must satisfy the continuity at the interface for . Hence,
12 (1) (2)
ˆˆ
BF
rR r R
nn b n n f n n f
ww
BH kR CHkR CHkR
(6.15a)
12 (1) (2)
ˆˆ
BF
v
rR
rR
nn b n n f n n f
ww
rr
BH kR C H kR C H kR
(6.15b)
where
bb
ff
k
k
, and
12 1 2
11
2
1
12 1 2
11
2
1
ˆ
ˆ
n b nb nb n b
n
nb
nb nb n b nb
n
nb
b
b
H kR H kR H kR H kR
HkR
H kRHkR H kRHkR
HkR
HkR
HkR
. (6.16)
Combining Eqs (6.15) and (6.16), we have
(1) (2)
1
2
(1) (2)
ˆ
ˆ
ˆ
nv n f nb
n
n
n nv n f n b
HkR H kR HkR
C
B
C HkR H kR HkR
.
(6.17a)
Now when solving the
1
n
C and
2
n
C in terms of
ˆ
n
B from (6.17a):
0
124
(1) (2)
1
2
(1) (2)
(2) (2)
1
2
(1) (1)
ˆ
ˆ
ˆ
ˆ
1
ˆ
ˆ
nf n f nb
n
n
n nf n f n b
nf n f nb
n
n
n nb nf n f
HkR H kR HkR
C
B
C HkR H kR H kR
HkR H kR HkR
C
B
W
C HkR HkR H kR
,
(6.17b)
of the form
1
(1)
(2)
2
ˆ
n n
n
n
n
C
B
C
, where
(2) (2)
(1)
(2)
(1) (1)
ˆˆ
4
ˆˆ
nf n b n f n b
f n
n
nf n b n f n b
H kRH kR H kRH kR
kR
i
H kRH kR H kRH kR
, (6.17c)
with the determinant of the matrix in Eq. (6.17a), namely the Wronskian involved, and given by
12 1 2
4
W
n f nf nf n f
f
i
H kRH kR H kRH kR
kR
. (6.17d)
Therefore, the wave field within the flexible foundation in Eq. (6.9) can be expressed in terms
of the building coefficient, with
12 (1) (2)
ˆˆ
,
nnn n n n
CBC B ;
(1) (1) (2) (2)
0
0
ˆ
,cos
ˆ
,cos
FF
nn n f n n f
n
FF
nn f
n
n
ww r B H kr H kr
n
ww r BH kr
(6.17e)
where
(1)(1) (2)(2)
nf n n f n n f
H kr H kr H kr
(6.17f)
is the “Revamped” Hankel functions of the flexible foundation.
125
6.4 Numerical Implementation: Asymptotic Expansion
6.4.1 The Shear Wall Structure Waves
When 1
b
kr , corresponding to large arguments for Hankel functions
1
nb
H kr
and
2
nb
H kr
. Consider their 1st-term asymptotic expansion in http://dlmf.nist.gov/10.17:
12
12
111
12
222
1
24
1
24
21
~ cos
24
2
~ ~
2
~ ~
n
nnn
nnn
n
iz
n
iz
n
Jz z
z
HzHziHz
z
HzHziHz
z
e
e
. (6.18a)
Apply this to
1
1
b
zkR , and
1
1
1
11
11
2
2
1
1
1
1
24
2
24
1
24
~~ ~
b
b
b
nb n b
nb
nb
n
ikR
n
ikR
n
ikR
HkR H kR
HkR
HkR
e
e
e
.
(6.18b)
If
1 b
kR is carefully and freely chosen to be such that
1
14 1
b
kR M , for some large
integer M and 01 (say around 0.1), then
1
1
24
b
n
NkR
would be a large
positive integer, so that
1
1
1
2
24
1
1
2
1
2
1
2
24
~1
1
b
b
n
ikR
nb
nb
iN
n
ikR
e
HkR
HkR
e
e
, (6.18c)
and
ˆ
n b
Hkr
in Eq. (6.8b) together with its derivative take the form:
126
12
12
~2
~2
ˆ
ˆ
nnb nb nb
nnb nb nb
b
b
H kr H kr J kr
H kr H kr J kr
Hkr
Hkr
. (6.18d)
In other words, when 1
b
kr , these “reshaped” Hankel functions
ˆ
n b
Hkr
behave as
Bessel functions of the first kind of the same order.
6.4.1.1 The choice of
1 B
kR
For example, when 1 , so ka in the half-space and 0.25 4
bb
CCkaka then
44
b
ka ka . Let
1
25 to 50 1 16 Ra , so
1
100 to 200 1 4
b
kR . Table 1 has
many other cases.
Table 1. Choice of with
ka
0.25
b
CC
1
50 Ra 1 b
kR
0.25
0.25
50 1 4
50 1 4
0.5
0.5 2 50 1 8
100 1 4
0.75
0.75 3 50 1 12
150 1 4
1
4 50 1 16
200 1 4
2
2 8 50 1 32
400 1 4
3
3 12 50 1 48
600 1 4
4
4 16 50 1 64
800 1 4
5
5 20 50 1 80
1000 1 4
7.5
7.5 30 50 1 120
1500 1 4
10
10 40 50 1 160
2000 1 4
15
15 60 50 1 240
3000 1 4
6.4.2 The flexible foundation waves
Again with 1
ff
kr k R , Eq. (6.17c) takes the asymptotic form of
1 b
kR 0.1
127
1
12
24
(1)
(2)
1
24
2
~
2
f
f
n
ik R
nb nb
f
n
n
ik R
n
nb nb
eiJkRJkR
kR
i
eiJkRJkR
. (6.19a)
With 1
f
kr , the “revamped” Hankel functions of the flexible foundation in Eq. (6.17f) has
the asymptotic expansion of the form:
12
11
24 24 (1) (2)
1
12 24 12
2
~
2
2
~
2
ff
f
nn
ikr i kr
nf n n
f
n
ik R
nb nb
f
f
Hkr e e
kr
iJ k R J k R e
kR
kr i
1
24
f
n
ikr
e
1
24
f
n
ikR
nb nb
iJ k R J k R e
1
24
f
n
ikr
e
(6.20a)
or
and in particular, at ,
12
~2 cos sin
~2
nf n b f n b f
nf n b
R r
R
H kr J kR kr R J kR kr R
r
rR H kr J kR
,
(6.20b)
so that the “revamped Hankel functions of the flexible foundation waves are also like the Bessel
functions but with a wave number of the shear wall structure. Also,
and in particular, at
12
~2 sin cos
,2
bb
nf n b f n b f
ff
nf nf n b
rR
k
k
R
H kr J kR kr R J kR kr R
r
rR H kr H kR J kR
.
(6.20c)
Note that the wave number
f
k in
nf
H kR
is replaced by
b
k , that of the foundation, in the
asymptotic term
nb
J kR
. Equations (6.20b,c) will ensure that the waves in asymptotic forms
still satisfy the boundary conditions of the continuity of stress and displacement at the
foundation and shear wall interface (Eqs. 6.15a,b, 6.17e), namely:
128
00
00
ˆˆ ˆ
cos ~ 2 cos
ˆˆ
cos ~ 2 cos
B
nn b n n b
nn
F
nn f n n b
nn
rR
rR
nn
wBHkR BJkR
nn
wBHkR BJkR
, (6.20d)
00
00
ˆˆ ˆ
2~2
ˆˆ
2~2
B
bbbnnbbbnnb
nn
F
fffnnfffnnb
nn
rR
rR
wkBHkR kBJkR
r
wkBHkR kBJkR
r
, (6.20e)
so that both the displacement and stress are continuous at the interface.
Next consider the waves given in Eq. (6.10) and (6.17e) above with respect to
, r
and
, r , respectively with origins at O and O,
(6.10)
(6.17e)
0
0
,cos
ˆ
,cos
FF
nn f
n
FF
nn f
n
ww r CJkr n
n
ww r BH kr
. (6.21)
If the waves are known around the flexible foundation using Eq. (6.10), they can be evaluated at
equally spaced points along the upper surface of the flexible foundation at rR and
0 as follows:
00
ˆˆ ˆ
,coscos
F
nn f n
rR rR
nn
nn
fwr BHkR b
, (6.22)
from which the coefficients
ˆ ˆ
nnn f
bBH kR
for 1, 2, 3,... n can be computed using the
Discrete Cosine Transform (DCT) method. The DCT algorithm and software is readily
available in Fortran 90 of Numerical Recipes (DCT-II) (Fortran 90, 1991) or in the built-in
library of Matlab (Matlab R2013b, 2013). Knowing
ˆ
n
b
, the coefficients
ˆ
n
B
will be known.
The displacement at any point on the building, from Eq. (6.8a–d) can then be used as,
129
00
ˆ
ˆ ˆˆ
cos cos
B
nn n
n b
b
nn
n f
w
Hkr
nn
BH kr b
HkR
, (6.23a)
to be evaluated asymptotically using Eqs. (6.8d) and (6.18c):
~ ~
~
12
12
12
21
cos
ˆ
24
21
cos
24
1
cos
ˆ
24
1
cos
24
b
nb n b b
nb nf
b
b
b
nb
nf
b
n
kr
Hkr J kr kr
JkR HkR
n
kR
kR
n
kr
Hkr
R
r n HkR
kR
, (6.23b)
in particular, when
ˆ
, ~1
nb
nf
HkR
rR
HkR
, which is the result from imposing the boundary
conditions of the continuity of stress and displacement at the foundation and shear wall
interface, using Eqs. (6.8a–d, 6.18a–d).
6.4.3 Normalized strain
From Eq. (6.2a), the incident wave is
cos ysin
cos cos sin sin cos
,
,
iikx
iikr ikr
wxy e
wr e e
,
(6.24)
so that the strain component due to the incident wave is
()
() ()
cos
,cos
i
ii
zr zr
ikr w
rik
r
e
. (6.25)
At the half-space surface, where the foundation is situated and interfaced with the base of
the building, the strain amplitude of the incident waves, taking 0 , is
()
cos
i
zr
k so that
()
max
i
zr
k , (6.26)
130
which would be max when 0
o
(horizontal incidence) and zero when 90
o
(vertical
incidence). We will take the max incident strain to be the normalizing factor:
()
max
i
zr
k
.
The waves in the foundation interface from the building and foundation are respectively,
from Eqs. (6.8a), (6.17c) and (6.20d):
00
00
ˆˆ ˆ
cos ~ cos
ˆˆ
cos ~ cos
B
nn b n n b
nn
F
nn f n n b
nn
rR
rR
nn
wBHkR BJkR
nn
wBHkR BJkR
, (6.27)
from which the strain from the foundation waves at the interface is,
()
() ()
00
,
ˆˆ
cos ~ cos
F
FF
zr zr
rR
fnn f fnnb
nn
w
r
r
nn
kBHkr k BJkR
,
(6.28)
so that the strain amplitude normalized with that of the incident wave is,
()
()
() 0
max
()
0
ˆ
cos
ˆ
~cos
F
zr
f F
rR
zr n n f
i n
zr
f F
zr n n b
n
k
n
BH k R
k
k
n
BJ kR
k
. (6.29)
This indicates that that strain amplitude normalized with a factor of
f
kk .
6.4.4 Numerical parameters
HALF-SPACE – VERY DENSE SOIL-SOFT ROCK SOIL PROFILE
2
120
1800 /
389 6
pcf
C fts
CE
FOUNDATION - CONCRETE
131
2
22
150
6500 /
6338 6
1
60
2
f
f
fff
f
pcf
Cfts
CE
Ma a
BUILDING – CONCRETE MOMENT FRAME
2
2
150
2000 /
600 6
2 3000
b
b
bbb
bb
pcf
Cfts
CE
Mah a
BUILDING – CONCRETE SHEAR WALL
2
2
150
2660 /
1061 6
2 3000
b
b
bbb
bb
pcf
Cfts
CE
Mah a
RATIOS - CONCRETE MOMENT FRAME BUILDING
2
2
6334
16
389
600 1
0.1
6334 10
600
1.5
389
3000
16
60
1800
0.25
6500
1800
0.90
2000
f
b
f
b
b
f
f
f
b
b
M a
Ma
k
C
kC
k C
kC
132
RATIOS - CONCRETE SHEAR WALL BUILDING
2
2
6334
16
389
1061 1
6334 6
600
1.5
389
3000
16
60
1800
0.25
6500
1800 2
0.67
2660 3
f
b
f
b
b
f
f
f
b
b
M a
Ma
k
C
kC
k C
kC
6.5 Discussion and Conclusions
For more than a century, analyses procedures in building codes that examined the response
of structures to an earthquake event typically assumed that the foundation was fixed to the
geologic media at the foundation-soil interface. There is a common assumption in SSI analyses
that the foundation is rigid, thus ignoring differential ground motion and its effects on the
structure. Relative to the soil, what level of foundation stiffness is required to validate the rigid
foundation assumption? Todorovska, Hayir, and Trifunac (2001) suggest that for 50
f
,
the foundation behaves as rigid. The model studied in this chapter concludes that the foundation
can assume to be rigid for 1000
f
, as shown in Figures 6.3 and 6.4.
In the case of rigid foundation, the building response eliminates the torsional phenomena
in the building due to wave passage effects—i.e., the building rotating about the y-axis in Figure
6.2. Our results for the case of flexible foundation indicate that the wave passage along the base
deforms the building as the wave propagates along the foundation width. It is also well
understood that the soil media flexibility and the foundation are the main components that affect
the interaction forces. In the presence of the SSI, foundations will scatter considerable incident
seismic wave energy (Todorovska (2001) and Gicev (2012)).
133
As can be seen from Figures 6.5–6.28, in a foundation that is less rigid or more flexible,
scattered enegery is relatively small and more energy is transmitted (leaked) up into the
building. The excitation of the building is the combination of translation (out-of-plane) and
torsion of the base for large dimensionless frequency 1.0 . The torsion becomes small and
approaches to zero as the dimensionless frequency decreases 0 . It is also seen that a
translational response is generally excited at the base of the building for vertical incidence,
90
o
due to the lateral heterogeneity at the base when its corners become secondary sources
of cylindrical waves. The results of our solution are in excellent agreement with those of Gicev
and Trifunac (2012).
Normalized strain amplitudes are also plotted versus the height of the building in Figures
6.17–6.28. It can be seen that the wave passage along the base of the building does increase the
horizontal strains at the building-foundation interface, particularly near the corners at 1 x .
Strain amplitude in the building becomes larger as the dimensionless frequency decreases,
eventually reducing to the point of approaching zero as the dimensionless frequency increases.
134
Figure 6.3. Surface displacement amplitudes at the interface for
1.0, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
135
Figure 6.4. Normalized building displacement for
1.0, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 200 and 1000
o
bb f f
kk k k .
136
Figure 6.5. Normalized building displacement for
0.25, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
137
Figure 6.6. Normalized building displacement for
0.50, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k
138
Figure 6.7. Normalized building displacement for
0.75, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
139
Figure 6.8. Normalized building displacement for
1.00, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
140
Figure 6.9. Normalized building displacement for
1.25, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
141
Figure 6.10. Normalized building displacement for
1.50, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
142
Figure 6.11. Normalized building displacement for
1.75, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
143
Figure 6.12. Normalized building displacement for
2.00, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
144
Figure 6.13: Normalized building displacement for
2.25, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
145
Figure 6.14. Normalized building displacement for
2.50, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
146
Figure 6.15: Normalized building displacement for
2.75, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
147
Figure 6.16. Normalized building displacement for
3.00, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
148
Figure 6.17. Normalized building strain for
0.25, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
149
Figure 6.18. Normalized building strain for
0.50, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
150
Figure 6.19. Normalized building strain for
0.75, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
151
Figure 6.20. Normalized Building strain for
1.00, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
152
Figure 6.21. Normalized building strain for
1.25, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
153
Figure 6.22. Normalized building strain for
1.50, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
154
Figure 6.23. Normalized Building strain for
1.75, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
155
Figure 6.24. Normalized building strain for
2.00, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
156
Figure 6.25. Normalized building strain for
2.25, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
157
Figure 6.26: Normalized building strain for
2.50, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
158
Figure 6.27. Normalized Building strain for
2.75, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
159
Figure 6.28: Normalized building strain for
3.00, 1.5, 0.90, 16, 0.25, 0 , 30 , 60 and 90
oo o o
bb f f
kk k k .
160
Figure 6.29. Surface displacement amplitudes at the interface for
0.25, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
Figure 6.30. Surface displacement amplitudes at the interface for
0.50, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
161
Figure 6.31: Surface displacement amplitudes at the interface for
0.75, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
Figure 6.32. Surface displacement amplitudes at the interface for
1.00, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
162
Figure 6.33: Surface displacement amplitudes at the interface for
1.25, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
Figure 6.34. Surface displacement amplitudes at the interface for
1.50, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
163
Figure 6.35. Surface displacement amplitudes at the interface for
1.75, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
Figure 6.36. Surface displacement amplitudes at the interface for
2.00, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
164
Figure 6.37: Surface displacement amplitudes at the interface for
2.25, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
Figure 6.38. Surface displacement amplitudes at the interface for
2.50, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
165
Figure 6.39. Surface displacement amplitudes at the interface for
2.75, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
Figure 6.40. Surface displacement amplitudes at interface for
3.00, 0 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 and 1000
o
bb f f
kk k k .
166
Figure 6.41. Surface displacement amplitudes at the interface for
0.25, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
Figure 6.42. Surface displacement amplitudes at the interface for
0.50, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
167
Figure 6.43. Surface displacement amplitudes at the interface for
0.75, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
Figure 6.44. Surface displacement amplitudes at the interface for
1.00, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
168
Figure 6.45. Surface displacement amplitudes at the interface for
1.25, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
Figure 6.46. Surface displacement amplitudes at the interface for
1.50, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
169
Figure 6.47. Surface displacement amplitudes at the interface for
1.75, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
Figure 6.48. Surface displacement amplitudes at the interface for
2.00, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
170
Figure 6.49. Surface displacement amplitudes at the interface for
2.25, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
Figure 6.50. Surface displacement amplitudes at the interface for
2.50, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
171
Figure 6.51. Surface displacement amplitudes at the interface for
2.75, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
Figure 6.52. Surface displacement amplitudes at the interface for
3.00, 90 , 1.5, 0.90, 0.25, 8, 16, 32, 50, 100 &1000
o
bb f f
kk k k .
172
CHAPTER 7
CONCLUSIONS AND FUTURE WORK
7.1 Conclusion
In this dissertation, various two-dimensional (2D) models have been studied. Superficially,
these models may seem to be different, however, they are mathematically similar. Although the
solutions are based on an analytical approach, some require numerical calculations. In addition,
with a simplified scheme, such as the double-arc, moon-shaped valley, the solutions may create
unexpected errors when we consider the wave motion at particular locations. However, an
advantage of analytical methods over numerical methods is that an error in an analytical
solution can be quantified. Finally, it is worth mentioning that a variety of models involved in this
dissertation focus on methodologies and mathematical derivations based on wave function
expansion. Such solutions provide additional physical insights into both the nature of the
problems and reference criteria, and offer necessary benchmarks to verify other, more
approximate, numerical method solutions. Moreover, analytical methods contribute to developing
solutions to equations that can be used as the foundation for future research in the field of
earthquake engineering.
This dissertation presents numerical results and case studies. The following subsections
summarize the findings from Chapters 2 to 6.
7.1.1 Chapter 2 Summary and Conclusion
As seen in Trifunac (1972), when a foundation is assumed to be rigid, every particle at any
horizontal cross-section of the structure parallel to the half-space surface must have the same
out-of-plane motion of the shear wall, and is independent of coordinate y . Thus, dependence on
y in the shear wall is eliminated. This simplicity in the dependence of the displacement solution
does not permit an extension of the solution for a flexible foundation.
173
From the results of the numerical analyses of the proposed model, it is concluded that the
tapered-shape structure methodology works well with sufficiently large radius R . This
methodology can be extended to solve the soil-structure interaction (SSI) of a shear wall
supported by a flexible foundation since the wave field within the structure of the tapered-shape
methodology only depends on the polar coordinate system.
7.1.2 Chapter 3 Summary and Conclusion
The model studied in this chapter is a 2D rectangular building resting on a semi-circular
rigid foundation wrapped with an elastic semi-circular flexible foundation. The analytical
solution of the interaction of a shear wall, flexible-rigid foundation and an elastic half-space is
derived for incident SH waves with various angles of incidence. Results show that the flexible
ring has the effect of diminishing the ground displacement amplitude as the building absorbs
wave energy and scatters it back into the half-space compared with the case of a rigid
foundation. It was also found that the flexible ring cannot be used as an isolation mechanism to
decouple a superstructure from its substructure resting on a shaking half-space, as waves are
transmitted into and scattered from the flexible ring.
Shear force at the base of the shear wall was investigated. It can be seen that base shear
forces increase as the rigidity of the flexible foundation layer increases compared with the
surrounding soil medium. For shear walls founded on soft soil and for low frequencies, the base
shear forces are higher than values computed for shear walls supported on hard soil.
7.1.3 Chapter 4 Summary and Conclusion
The model studied in this chapter consists of the SSI of a tapered shear wall on a
semi-circular rigid foundation wrapped with an elastic semi-circular flexible foundation. The
analytical solutions are derived using the “big arc numerical method” developed in Chapter 2.
The displacement of the shear wall structure and flexible-rigid foundation, as well as the ground
motion close to the subjected building, are investigated.
174
From the numerical analysis results of the proposed model, it is concluded that the big arc
numerical method works well with a reasonably large radius. The numerical examples are in
excellent agreement with the Chapter 3 results. This methodology can then be used to solve for
the SSI of a shear wall supported by the flexible foundation as discussed in Chapter 6.
7.1.4 Chapter 5 Summary and Conclusion
In this chapter, the analytical solution of a 2D, moon-shaped, alluvial valley embedded in
an elastic half-space is analyzed for incident plane SH waves using the wave function expansion
and the Discrete Cosine Transform (DCT). A series of solutions with different depth-to-radius
ratios have been computed, analyzed, and discussed.
In the range of parameters examined, it can be seen that amplification of surface motions
of the moon-shaped valley can be as large as 6.5. It was also found that the surface motions can
be reduced by a factor as small as 0.27. As can be seen from the figures in this chapter, the
amplification (as well as attenuation) of the surface displacement is localized to within a
fraction of the wavelength of the exciting motion; further, the amplification pattern on the
ground surface is not only very complex, but also essentially unpredictable.
With reference to the example we presented in the Sherman Oaks area, it can be concluded
that the spatial variability and complexity of the amplification of ground motions were further
increased by the variation of arrival azimuths and vertical incidence angles from the moving
dislocation of the Northridge earthquake. As such, it should be clear that simplified engineering
representations, based on the models of vertically incident waves amplified, should not be used.
7.1.5 Chapter 6 Summary and Conclusion
In this chapter, the model studied presents a logical extension of the elastic shear wall with
a circular rigid foundation fixed firmly in an isotropic, homogeneous, and elastic half-space.
Luco (1969) developed this model for this case with only vertical-incidence SH waves, which
was then formulated to any angle of incidence SH waves by Trifunac (1972). The analytical
175
solutions of the SSI of a tapered shear wall with rigid, flexible, and semi-rigid foundations are
derived using the big arc numerical method developed in Chapter 2 and the Discrete Cosine
Transform (DCT).
From the numerical results, the model studied in this chapter concludes that the foundation
can be assumed to be rigid for 1000
f
. For the case of rigid foundation, the building
response eliminates the torsional phenomena due to wave passage effects—i.e., the building
rotating about the y-axis. Our results for the case of flexible foundation indicate that the wave
passage along the base deforms the building as the wave propagates along the foundation width.
It is also well understood that the soil media flexibility and the foundation are the main
components that affect the interaction forces. In the presence of the SSI, foundations scatter
considerable energy of incident seismic waves (Todorovska (2001) and Gicev (2012)).
As can be seen from numerical examples, in foundations with less rigidity, energy
scattered by the foundation is relatively small; thus more energy is transmitted up into the
building. The excitation of the building is the combination of translation (out-of-plane) and
torsion of the base for large dimensionless frequency, 1.0 , and torsion becomes small and
approaches zero as the dimensionless frequency decreases, 0 . It can also be seen that a
translational response is generally excited at the base of the building for vertical incidence,
90
o
, due to the lateral heterogeneity at the base when the corners of the base become
secondary sources of cylindrical waves.
Normalized strain amplitudes are also computed and plotted versus the height of the
building. It can be seen that the wave passage along the base of the building does increase the
horizontal strains at the building-foundation interface, particularly near the corners at 1 x .
Strain amplitude in the building becomes larger as the dimensionless frequency decreases,
eventually reducing to the point of approaching zero as the dimensionless frequency increases.
176
7.1.6 Continuation of the SSI of a Shear Wall with Flexible Foundation
Good results are obtained from Chapter 6, which explored the SSI of a shear wall with
flexible foundation. However, this derivation can be extended to cases of out-of-plane SSI of
flexible foundation to non-homogeneous elastic layered media in which the shear-modulus is
constant but different in each layer or a function of surface depth. Although the waves remain as
SH waves in each layer, a formulation will be complicated and challenging. As a postdoctoral
researcher, I would like to study these cases.
7.1.7 Future work
To date, no one has derived wave function series analytic solutions to SSI of rigid
foundations in an elastic half-space for in-plane P and SV waves. Luco and Wong (1986b)
Green's function integral solutions are the theoretical solutions that come the closest to analytic
solutions. Nevertheless, a more sophisticated numerical approach is needed to evaluate the
integral solutions. Most existing approaches are numerical solutions using finite element, finite
difference, and/or boundary element methods (BEM). The difficulties for in-plane P and SV
waves lies in the fact that the free-stress boundary conditions at the half-space surface are
difficult to satisfy analytically because solutions involve wave functions of both P and SV
waves of different wave numbers, and hence are non-orthogonal, unlike SH waves. The
objective of future work in solving SSI problems for P and SV waves would be to formulate
solutions to satisfy the zero-stress boundary conditions at the half-space surfaces, in addition to
the stress and displacement continuity at the flexible foundation interface. This problem is
currently an active research subject considered by many earthquake engineers and
strong-motion seismologists. As a postdoctoral researcher, my future objective is to solve these
problems.
177
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189
APPENDIX A
Asymptotic Approximation for
1
R RH a
The mass of the soils foundation,
s
M per unit length is
2
22 22
22
22 2
s
aak ak
M
Ck
(A1)
The mass of the building,
s
M per unit length is
2
2
22
22
2
bb bb
bb
bb
aH k k aH
MaH
Ck
(A2)
22
2
bb b b bb bb b
bb b
kR H k aH kR kR kH
ka ka kH kakH kakH
(A3)
2
2
22
1
2 2
bb b b
sb s
b
kR M ka M
ka M kH M
kH
ka
(A4)
12 1 2
11 1 1
1
2
11
ˆ
,
ob b b o b
ob b
b
HkRH kR HkRH kR
HkRkR
HkR
(A5)
12 1 2
11 1 1
1
2
11
ˆ
,
ob b b o b
ob b
b
H kR H kR H kR H kR
HkRkR
HkR
(A6)
Note that
1
ˆ
,
ob b
H kR kR
with respect to R .
12 1 2
11 1 1 11
11
2
11
ˆ
,
bb b b
bb
b
HkRH kR HkRH kR
HkRkR
HkR
(A7)
12 1 2
11 1 1 1 1 1 1
12 1 2
1 11 1 1
ˆ
,
ˆ
,
bb b b b b
ob b ob b b o b
H kR kR H kR H kR H kR H kR
HkRkR HkRH kR HkRH kR
(A8)
190
where
1
4
2
ix
o
Hx e
x
(A9)
2
4
2
ix
o
Hx e
x
(A10)
3
1
4
1
2
ix
Hx e
x
(A11)
3
2
4
1
2
ix
Hx e
x
(A12)
11
11
11
1
33 3 3
44 4 4
33
44 44
ˆ
,
ˆ
,
bb b b
bb bb
RR
bb
RR
obb
ikR ik ik ikR
ik ik ikR ikR
HkRkR
HkRkR
ee e e
ee e e
(A13)
11
11
11
1
ˆ
,
ˆ
,
bb
bb
bb
obb
ik R R ik R R
ik R R ik R R
i
HkRkR
HkRkR
ee
ie e
(A14)
11
11
11
1
ˆ
,
ˆ
,
bb
bb
ik R R ik R R
bb
o
bb
ik R R ik R R
ee
HkRkR
HkRkR
ee
i
(A15)
where
1
HR R . Equation (A15) can be simplified to
11
1
tan
ˆ
,
ˆ
,
bb
bb
b
ik H ik H
bb
o
bb
ik H ik H
kH
ee
HkRkR
HkRkR
ee
i
(A16)
The displacement of the foundation, , Equation (2.29) can be expressed to Equation (2.1)
(Trifunac 1972).
1
11
1
1
1
1
tan
2
f b b
ss b
o
o
o
o
M kH ka M
MM kH
Jka
J ka H ka a
Hka
Hka
Hka
(A17)
191
GLOSSARY
a Radius of the semi-circular rigid foundation
a Radius of the semi-circular flexible foundation
n
a Coefficients of the free-field waves
B Width of building
n
A Complex constants
12
,
nn
BB Complex constants
C
Shear wave velocity in the soil
b
C
Shear wave velocity in the building
f
C
Shear wave velocity in the flexible foundation
f
f Force per unit length acting on the rigid foundation from the flexible foundation
b
f Force per unit length acting on the building
1
n
Hx Hankel function of the first kind with argument x and order n
2
n
Hx Hankel function of the second kind with argument x and order n
i Imaginary unit
n Subscripts used for sequence number
n
J x Bessel function of the first kind with argument x and order n
k Wave number in the soil, ݇ൌ ߱ ଶ ܥ ఉ ൗ
b
k Wave number in the building, ݇ ൌ߱
ଶ ܥ ఉ ್ ൗ
B
M Mass of shear wall per unit length
F
M Mass of rigid foundation per unit length
S
M Mass per unit length of soil to be replaced by the rigid foundation
192
Angle of incidence for SH-waves
Amplitude of the displacement of the foundation
w Amplitude of the displacement of the total wave field in the soil
ff
w Amplitude of the displacement of the free-field wave in the half-space soil
B
w Amplitude of the displacement of the wave field in the building
R
w Amplitude of the displacement of the wave field in the rigid foundation
S
w Amplitude of the displacement of the scattered wave field in the soil
i
w Amplitude of the displacement of the incident plane wave in the soil
r
w Amplitude of the displacement of the reflected plane wave in the soil
Shear modulus of the soil
b
Shear modulus of the shear wall
Density of the soil
b
Density of the shear wall
Circular frequency of the incident SH waves
n
Unit impulse function
Dimensionless parameters
Abstract (if available)
Abstract
Soil-structure interaction (SSI) is a process in which the effects of wave propagation in the half-space are modified by the response characteristic of a structure and vice versa. SSI is one of the most dominant subjects in the earthquake engineering research, especially at the interface of soil and structural dynamics. Classical research has investigated SSI with the assumption of a rigid foundation with high scattered waves. The structure and soil has also been assumed to be linear, and the only energy loss in the system is associated with scattered-wave radiation into the half-space. SSI studies have shown that the dynamic response of a structure supported on a flexible foundation may differ significantly from the response of the same structure erected on a rigid foundation. Flexible foundations have been investigated based primarily on numerical methods. The major drawback of using numerical methods to solve seismic wave problems is the need for substantial resources and computing time in order to obtain accurate results. In contrast, analytical methods provide more accurate and relatively simple methods of performing similar computations on a larger scale, allowing them to be done economically and in a shorter period of time. Additionally, analytical solutions provide more physical insights into the nature of the problem and offer benchmarks necessary to verify the other, more approximate, solutions of the numerical methods. Most importantly, analytical methods contribute to developing solutions to equations that can be used as the foundation for future research in the field of earthquake engineering. ❧ The purpose of this thesis is to develop new tools to investigate the SSI for the case of flexible foundation. The models studied in this thesis present an extension of previous work for a shear wall on a semi-circular rigid foundation in an isotropic homogeneous and elastic half-space. Le & Lee (2014) published a new approach and an SSI model to solve for the case of rigid foundation, an extension of Luco (1969) and Trifunac (1972), using the “big arc numerical” method, which can be later modified to the case of semi-circular or arbitrarily flexible foundation. Le et al. (2016) investigated the SSI of a shear wall supported by a rigid and flexible ring foundation. Chapter 4 of this thesis studies a similar model to Le et al. (2016), using the big arc approximation developed in Le & Lee (2014). The analytical solutions of these investigations are for cases in which the foundations are rigid, non-elastic movable foundations. The solutions serve as an intermediate step toward a goal of solving the SSI of a shear wall on an elastic foundation. Le et al. (2017) published the analytical solution of a two-dimensional, moon-shaped alluvial valley embedded in an elastic half-space for incidence plane SH waves using the wave function expansion, the Discrete Cosine Transform (DCT), and the big arc approximation. The model studied in the current paper can be recognized as an elastic foundation without a building on top. Lastly, Chapter 6 of this thesis investigates the SSI of a shear wall supported by an elastic foundation. This chapter develops a new approach and model to solve the SSI of a tapered shear wall for all rigid, semi-rigid, and flexible foundations using an asymptotic of the special functions, the wave function expansion, the Discrete Cosine transform (DCT) and big-arc approximation as developed in Chapter 5.
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Asset Metadata
Creator
Le, Thang Huu
(author)
Core Title
The role of rigid foundation assumption in two-dimensional soil-structure interaction (SSI)
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Civil Engineering
Publication Date
11/06/2017
Defense Date
09/14/2017
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
big arc numerical method,closed-form analytic solution,flexible foundation,flexible/semi-circular rigid foundation,focusing by non-parallel interfaces,Fourier-Bessel series,OAI-PMH Harvest,rigid-flexible foundation,SH waves,SH waves in a moon-shaped valley,SH waves in cylindrical coordinates, discrete cosine transform (DCT),soil-structure interaction,tapered shear-wall structure
Language
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(provenance)
Advisor
Lee, Vincent W. (
committee chair
), Johnson, Erik A. (
committee member
), Trifunac, Mihailo D. (
committee member
), Wang, Chunming (
committee member
), Wong, Hung Leung (
committee member
)
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thang@thanglese.com,thangle@usc.edu
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Tags
big arc numerical method
closed-form analytic solution
flexible foundation
flexible/semi-circular rigid foundation
focusing by non-parallel interfaces
Fourier-Bessel series
rigid-flexible foundation
SH waves
SH waves in a moon-shaped valley
SH waves in cylindrical coordinates, discrete cosine transform (DCT)
soil-structure interaction
tapered shear-wall structure