Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Soil structure interaction in poroelastic soils
(USC Thesis Other)
Soil structure interaction in poroelastic soils
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
SOIL STRUCTURE INTERACTION IN POROELASTIC SOILS
by
Yousef Saleh Al Rjoub
____________________________________________________________________
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)
May 2007
Copyright 2007 Yousef Saleh Al Rjoub
ii
Acknowledgments
I am indebted to my advisor Prof. Maria Todorovska for her guidance and assistance
through all the stages of this thesis work. I am also grateful to Prof. Mihailo Trifunac for
the useful discussions and suggestions, and to my Ph.D. dissertation committee members:
Prof. Vincent W. Lee, Prof. Hung Leung Wong and Prof. Frank Corsetti for their useful
comments.
The financial support from the University of Southern California I received during the
course of my studies in the form of Teaching Assistantship from the Civil Engineering
Department, Research Assistantship from Prof. Maria Todorovska, and Dissertation
Completion Fellowship from the Graduate School is gratefully appreciated.
Finally, I dedicate this work to my mother, the soul of my father, and my family,
especially my brother Qasem, for their love and support during my study
iii
Table of Contents
Acknowledgments ............................................................................................................. ii
List of Tables ..................................................................................................................... v
List of Figures................................................................................................................... vi
Abstract.............................................................................................................................. x
Chapter 1: Introduction............................................................................................... 1
1.1 Objective and organization of this thesis.................................................... 1
1.2 Literature Review on Wave Propagation in Porous Media ........................ 2
1.3 Literature Review on Soil Structure Interaction in Porous Media.............. 6
Chapter 2: Theoretical Model ................................................................................... 12
2.1 The Soil-Structure Interaction Model ....................................................... 12
2.2 Wave Propagation in Fluid Saturated Poroelastic Medium...................... 16
2.3 Solution for P-waves................................................................................. 18
2.3.1 Solutions for S-waves ................................................................................19
2.3.2 Material Constants for the Mixture............................................................20
2.3.3 Approximate Treatment of Partial Saturation............................................22
2.4 The Soil-Structure Interaction Problem.................................................... 24
2.4.1 Representation of the Scattered Waves......................................................24
2.4.2 Boundary Conditions at the Contact Surface.............................................27
2.4.3 Integral of Stresses along the Contact Surface ..........................................31
2.4.4 Dynamic Equilibrium of the Foundation ...................................................36
2.5 The Free-Field Motion.............................................................................. 39
2.5.1 The Free Field Motion due to Incident P-Wave ........................................40
2.5.2 The Free Field Motion due to Incident SV-Wave .....................................43
2.5.3 The Free Field Motion due to Rayleigh-Wave ..........................................44
2.5.4 Displacements and Stresses due to the Free-Filed Motion ........................48
Chapter 3: Numerical Results and Analysis ............................................................ 52
3.1 Soil Constitutive Properties and Waves Velocities .................................. 53
3.1.1 Input Model Parameters.............................................................................53
3.1.2 Wave Velocities for Full Saturation as Function of the Model
Parameters..................................................................................................55
3.1.3 The Effect of Partial Saturation on the Wave Velocities...........................62
3.2 Foundation Complex Stiffness Matrix...................................................... 64
3.2.1 Foundation Complex Stiffness Matrix for Fully Saturated Soils...............64
3.2.2 Foundation Complex Stiffness Matrix for Partially Saturated Soils .........74
3.3 Free-Field Motion ..................................................................................... 77
iv
3.3.1 Incident Plane Fast P-wave........................................................................78
3.3.2 Incident Plane SV-wave.............................................................................87
3.4 Foundation Input Motion .......................................................................... 95
3.5 Building-Foundation-Soil Response......................................................... 98
3.5.1 Building-Foundation-Soil Response for Incident P-wave .........................98
3.5.2 Building-Foundation-Soil Response for Incident SV-wave ....................104
3.6 Frequency Shift due to Saturation – Millikan Library Case................... 109
3.6.1 Full-scale Observations............................................................................109
3.6.2 Model Parameters for NS Response ........................................................111
3.6.3 Foundation Complex Stiffness and System Response.............................115
Chapter 4: Summary and Conclusions................................................................... 121
References...................................................................................................................... 127
Appendix........................................................................................................................ 133
v
List of Tables
Table 3.1.1 Wave velocities for fully saturated soil for the case of no seepage force...... 56
Table 3.6.1 Wave velocities for fully saturated soil for the model of Millikan library
assuming no seepage force......................................................................... 112
List of Figures
Fig. 2.1.1 The model......................................................................................................... 13
Fig. 2.3.1 The excavation and forces acting on the soil.................................................... 24
Fig. 2.3.2 Dynamic equilibrium of the foundation. .......................................................... 37
Fig. 3.1.3 Normalized wave velocities versus frequency for different values of
permeability. The porosity is ˆ 0.4 n = , and
f
/K µ =0.1 (part a) and 0.01 (part b)......61
Fig. 3.1.4 Modified bulk modulus of the pore fluid versus fraction of air, 1
r
S − ............ 63
Fig. 3.1.5 Modified velocities of fast and slow P-waves versus fraction of air, 1
r
S − ,
for porosity ˆ 0.4 n = and / 0.01
w
K µ = (soft soil).................................................... 63
Fig. 3.2.1 Foundation dynamic stiffness coefficients for different values of skeleton
permeability, and for porosity ˆ 0.4 n = and / 0
f
K .1 µ = . Pat a) shows results for
permeable (open), and part b) for impermeable (sealed) contact surface................. 65
Fig. 3.2.2 Foundation dynamic stiffness coefficients for different values of skeleton
permeability, and for porosity ˆ 0.4 n = and / 0.
f
K 01 µ = . Pat a) shows results
for permeable (open), and part b) for impermeable (sealed) contact surface. .......... 66
Fig. 3.2.3 Comparison of variations of horizontal/vertical foundation complex
stiffness (part a)) and variations of the complex wave velocities (part b)) with
dimensionless frequency eta for different values of permeability, for porosity
ˆ 0.4 n = and / 0
f
K .1 µ = , and for p ermeable (open) contact surface. ................... 70
Fig. 3.2.4 Same as Fig. 3.2.3 but for impermeable (sealed) contact surface. ................... 71
Fig. 3.2.5 Comparison of variations of horizontal/vertical foundation complex
stiffness (part a)) and variations of the complex wave velocities (part b)) with
dimensionless frequency eta for different values of permeability, for porosity
ˆ 0.4 n = and / 0.
f
K 01 µ = , and for permeable (open) contact surface. ................... 72
Fig. 3.2.6 Same as Fig. 3.2.5 but for impermeable (sealed) contact surface. ................... 73
vi
Fig. 3.2.7 Effect of degree of saturation on the foundation complex stiffness for
porosity ˆ 0.4 n = and / 0.1
w
K µ = , and for permeable (open, part a)) and
impermeable (sealed, part b)) contact surface. ......................................................... 75
Fig. 3.3.1.1 Free-field motion due to unit displacement plane fast P-wave versus
incident angle, for different values of permeability. a) Permeable, b)
impermeable half-space. Left: amplitudes of the reflection coefficients. Right:
amplitudes of the surface displacements. The input parameters are: porosity
ˆ 0.4 n = , / 0
f
K .1 µ = , and the frequency is set to 1 f = Hz. ................................... 80
Fig. 3.3.1.2 Same as Fig. 3.3.1.1 but for / 0.
f
K 01 µ = ................................................. 81
Fig. 3.3.1.3 Free-field motion due to unit displacement plane fast P-wave versus
incident angle, for different values of frequency. a) Permeable, b) impermeable
half-space. The input parameters are: porosity ˆ 0.4 n = , / 0.
f
K 01 µ = , and
permeability
7 2
ˆ
10 m k
−
= .......................................................................................... 82
Fig. 3.3.1.4 Same as Fig. 3.3.1.3 but for permeability
10 2
ˆ
10 m k
−
= . ........................... 83
Fig. 3.3.1.5 Free-field motion due to unit displacement plane fast P-wave versus
incident angle, for different levels of saturation. a) Permeable, b) impermeable
half-space. The input parameters are: porosity ˆ 0.4 n = , / 0
f
K .1 µ = , and
frequency 1 f = Hz. The effects of the seepage force are neglected. ..................... 85
Fig. 3.3.1.6 Same as Fig. 3.3.1.5 but for / 0.
f
K 01 µ = .................................................. 86
Fig. 3.3.2.1Free-field motion due to unit displacement plane SV-wave versus incident
angle, for different values of permeability. a) Permeable, b) impermeable half-
space. Left: amplitudes of the reflection coefficients. Right: amplitudes of the
surface displacements. The input parameters are: porosity ˆ 0.4 n = , /0
f
K .1 µ = ,
and the frequency is set to 1 f = Hz......................................................................... 88
Fig. 3.3.2.2 Same as Fig. 3.3.2.1 but for / 0.
f
K 01 µ = ................................................. 89
Fig. 3.3.2.3Free-field motion due to unit displacement plane SV-wave versus incident
angle, for different values of frequency. a) Permeable, b) impermeable half-
space. The input parameters are: porosity ˆ 0.4 n = , / 0.
f
K 01 µ = , and
permeability
7 2
ˆ
10 m k
−
= .......................................................................................... 90
vii
Fig. 3.3.2.4 Same as Fig. 3.3.2.3 but for permeability
10 2
ˆ
10 m k
−
= . ........................... 91
Fig. 3.3.2.5 Free-field motion due to unit displacement plane SV-wave versus
incident angle, for different levels of saturation. a) Permeable, b) impermeable
half-space. The input parameters are: porosity ˆ 0.4 n = , /0
f
K .1 µ = ,
permeability
7 2
ˆ
10 m k
−
= , and frequency 1 f = Hz................................................. 93
Fig. 3.3.2.6 Same as Fig. 3.3.2.5 but for / 0.
f
K 01 µ = ................................................... 94
Fig. 3.4-1 Foundation input motion amplitudes due to an incident plane fast P-wave
at 30
incidence, for / 0
f
K .1 µ = , porosity ˆ 0.4 n = . Part a): permeable half-
space. Part b) impermeable half-space. Left: permeable foundation. Right:
impermeable foundation. .......................................................................................... 96
Fig. 3.4-2 Same as Fig. 3.4.1 but for an incident plane SV-wave at 30
incidence........ 97
Fig. 3.5.1.1 System response due to an incident plane fast P-wave at 30
incidence,
for / 0
f
K .1 µ = , porosity ˆ 0.4 n = , 2 H a = , width W a = , and mass ratios
/2
bf
mm = , / 0
fs
mm =.2
.1
, and for a permeable foundation. Part a): permeable
half-space. Part b) impermeable half-space. The different curves correspond to
different soil permeability....................................................................................... 100
Fig. 3.5.1.2 Same as Fig. 3.5.1.1, but for impermeable foundation............................... 101
Fig. 3.5.1.3 Same as Fig. 3.5.1.1, but the different curves correspond to different
levels of saturation. ................................................................................................. 102
Fig. 3.5.1.4 Same as Fig. 3.5.1.1, but the different curves correspond to different
levels of saturation, and the foundation is impermeable......................................... 103
Fig. 3.5.2.1 System response due to an incident plane SV-wave at 30
incidence, for
/ 0
f
K µ = , porosity ˆ 0.4 n = , 2 H a = , width W a = , and mass ratios
/2
bf
mm = , / 0
fs
mm =.2 , and for a permeable foundation. Part a): permeable
half-space. Part b) impermeable half-space. The different curves correspond to
different soil permeability....................................................................................... 105
Fig. 3.5.2.2 Same as Fig. 3.5.2.1, but for impermeable foundation............................. 106
viii
ix
Fig. 3.5.2.3 Same as Fig. 3.5.2.1, but the different curves correspond to different
levels of saturation. ................................................................................................. 107
Fig. 3.5.2.4 Same as Fig. 3.5.2.1, but the different curves correspond to different
levels of saturation, and the foundation is impermeable......................................... 108
Fig. 3.6.1Wave velocities for the Millikan case. a) Normalized wave velocities for
fully saturated soil as function of inverse permeability. b) Wave velocities of P-
waves as function of the air content in the pores.................................................... 114
Fig. 3.6.2 Foundation complex stiffness matrix coefficients for the model
corresponding to Millikan library. a) Permeable foundation. b) Impermeable
foundation. ................................................................................................................... 116
Fig. 3.6.3 System response for the model corresponding to Millikan library. a)
Permeable foundation. b) Impermeable foundation................................................ 117
Fig. 3.6.4 Enlarged view of the horizontal/vertical foundation complex stiffness
coefficients for the model corresponding to Millikan library. a) Permeable
foundation. b) Impermeable foundation. ................................................................ 118
Fig. 3.6.5Enlarged view of the first peak in the building relative roof response of the
model corresponding to Millikan library. a) Permeable foundation. b)
Impermeable foundation. ........................................................................................ 119
x
Abstract
This thesis presents an investigation of the effects of water saturation on the effective
excitation and system response during building-foundation-soil interaction, using a
simple theoretical model. The model consists of a shear wall supported by a rigid
circular foundation embedded in a homogenous and isotropic poroelastic half-space. The
half-space is fully saturated by a compressible and viscous fluid, and is excited by in-
plane wave motion, consisting of plane P and SV waves, or of surface Rayleigh waves.
Partial saturation is also considered but in a simplified way. The motion in the soil is
described by Biot’s theory of wave propagation in fluid saturated porous media.
According to this theory, two P-waves (one fast and the other one slow) and one S-wave
exist in the medium, which are represent by wave potentials. Helmholtz decomposition
and wave function expansion are used to represent the motion in the soil, and a closed
form solution of the problem is derived in the frequency domain. Numerical results are
presented for the free-field motion, foundation input motion, complex foundation
stiffness matrix, and the foundation and building response to incident plane fast P and SV
waves, as function of the many model parameters. The presented analysis, which is
linear, is of interest for understanding and interpreting the effects of water saturation on
the response of the ground and structures to small amplitude (e.g. ambient noise) and to
some degree earthquake excitation. An attempt is presented to use this model to explain
the observed variation of the apparent frequencies of vibration of Millikan library in
Pasadena, California, with heavy rainfall.
1
Chapter 1: Introduction
1.1 Objective and organization of this thesis
This thesis work presents an analysis of a simple linear model of building-foundation-
soil interaction in poroelastic soil excited by in-plane excitation. The objective of this
study is to gain insight into the effects of the water saturation on both the response of the
soil and of the building and its foundation. Understanding of the effects of water
saturation of soils on the seismic response of soils and structures is useful for interpreting
observed and predicting the features of response of structures and soils to earthquake,
ambient, and forced vibration excitation. It is noted here that, as the model is linear, it
cannot represent true nonlinear response of soils and structures to strong earthquake
shaking (e.g. soil yielding and liquefaction), but can be helpful in understanding the early
smaller amplitude response leading to pore pressure buildup and nonlinear response.
The study is carried out using a simple two-dimensional (2D) model in which the soil
is represented by a poroelastic half-space, and the structure is a shear wall supported by a
cylindrical embedded foundation. Such a soil-structure interaction model has been
considered first for semi-circular foundation embedded in elastic half-space and vertically
incident SH waves by Luco (1969). This model was later generalized to obliquely
incident SH waves by Trifunac (1972), to semi-elliptical foundations by Wong and
Trifunac (1974), and to P, SV and Rayleigh wave excitation by Todorovska and Trifunac
(1990) and Todorovska (1993a,b). Todorovska and Al Rjoub (2006a,b) considered such
a model in which the seepage force was ignored, and the half-space was fully saturated.
In this thesis work, the effects of the seepage force are included, and also of partial
2
saturation, and the emphasis of the analysis is on how the seepage force and the degree of
saturation affect the system response. Also, the free field motion is studied in grater
detail.
The remaining part of this chapter presents literature review on wave propagation
and soil-structure interaction in poroelastic soils. Chapter 2 presents the problem and
method of solution. Chapter 3 presents numerical results for: (1) the wave velocities in
the soil as function of soil permeability, relative stiffness of the skeleton, frequency and
degree of saturation; (2) foundation complex stiffness matrix; (3) the free-field motion
due to incident plane fast P and SV waves; (4) the foundation input motion; (5) the
system response; and (6) shift of the apparent frequency of a model of the NS response of
Millikan library in Pasadena, California, and comparison with its observed frequency
shift during heavy rainfall and recovery days following the rainfall. Finally, Chapter 4
presents a summary and the conclusions.
1.2 Literature Review on Wave Propagation in Porous Media
The theory of wave propagation in a fully saturated poroelastic medium by a viscous
compressible fluid was postulated by Maurice Biot in a series of papers (Biot, 1956a,b;
1962). While in elastic (one phase) medium two waves exist – one dilatational (P) and
one rotational (S) wave, Biot’s theory predicted the existence of an additional P-wave in
a porielastic (two phase) medium, which is a result of the relative motion of the fluid with
respect to the solid. This second P-wave, referred to as the “slow” P wave, is much
slower and is much more attenuated and dispersed than the “true” P wave also referred to
as the “fast” P wave. The existence of the slow P-wave was experimentally confirmed
3
many years later, in the 1980s (Berrymann, 1980). The theory presented in Biot (1956a)
is applicable to lower frequencies for which the flow of the fluid in the pores is
Poiseuille. Biot (1956b) presents an extension of that theory to higher frequencies,
beyond the critical frequency for which the Poiseuille assumption stops to be valid, but
still small enough so that the related wavelengths are still much larger than the size of the
pores. Biot (1962) presents an extension to anisotropic media, and media with solid
dissipation, and other relaxation effects.
The remaining part of this section reviews literature on wave propagation in a
homogeneous or layered poroelastic half-space for incident body and surface waves.
This problem is of interest for the work in this thesis because wave motions in such a
medium are usually used to represents the “free-field” seismic motions exciting structures
on the ground surface or buried at some depth.
The effects of boundaries on wave propagation in fully saturated poroelastic media
were studied by Deresiewicz and coworkers in the 1960s by considering plane body and
surface waves incident onto a traction free poroelastic half-space. Deresiewicz (1960)
considered incident P and SV waves onto a half-space saturated with a nondissipative
liquid, Deresiewicz (1961) considered Love waves in a half-space saturated with a
viscous liquid, and Deresiewicz and Rice (1962) considered incident plane P and SV
waves onto a half-space saturated by a viscous fluid.
Incident plane body waves onto a half-space were considered also by other
investigators, with emphasis on different applications (porous rock and soils). For
example, Sharma and Gogna (1991) considered incident fast P-wave onto a half-space
4
and showed results for water saturated sandstone. Lin et al. (2001, 2005) considered
reflection of plane P and SV waves in water saturated porous half-space (assuming
invicid fluid) for a wide range of skeleton stiffness, from very stiff (porous rock) to very
soft (soft soil), for a range of values of Poisson’s ratio and porosity, and for both drained
and undrained hydraulic boundary condition on the half-space surface. They also
discussed the range of validity of Biot’s theory for different types of soils, and showed
results for the amplitudes of the surface displacements, strains, rotations, and stresses,
and examined the effect of the saturation, and various parameters of the mixture on these
quantities. They found that, for undrained (sealed) half-space surface, the peak
amplitudes of these characteristics of are smaller than the amplitudes for the elastic case.
For a drained (open) half-space surface, they found that these peak amplitudes are smaller
than for the elastic case, with the exception of the peak rotations. Ciarleta and Subatyan
(2003) also studied the refection of plane waves in a fluid saturated poroelastic half-space
but for the general case of a viscous fluid. Their study showed that the refection
coefficients and the vibration amplitude in the saturated half-space are smaller than those
in an elastic half-space.
Liu et al. (2002) studied stress wave propagation in transversely isotropic fluid-
saturated porous medium for plane waves and for surface Rayleigh waves, in particular
the effects of the fluid viscosity and the anisotropy of the solid skeleton. Their study
showed that the fluid viscosity resulted in Rayleigh waves with frequency dependent
phase velocity.
5
Degrande et al. (1998) studied harmonic and transient wave propagation in multi-
layered dry, saturated and unsaturated isotropic poroelastic media, but for small fraction
of gas in the fluid and ignoring the effects of the capillary forces. They examined the
effect of moving ground water table and partial saturation on wave propagation in a
poroelastic layered half-space, and found that air bubbles in the top layer of a saturated
half-space affect the P-wave propagation. Partial saturation was considered in a similar
way by Yang (2000, 2001, 2002) and Yang and Sato (2000a,b) for incident plane waves
in a poroelastic half-space and onto a boundary between two bonded half-spaces, the
lower one being elastic and the upper one being partially saturated poroelastic. Further,
Yang and Sato (2000c) showed that partial saturation in the soil near the surface may
explain the significant amplification of the vertical motion observed by a borehole array
at Port Island, Kobe, during the 1995 Hyogo-Ken Nanbu (Kobe) earthquake, while the
opposite effect was observed for the horizontal motions. In their earlier work, the same
authors studied the effects of the flow condition and viscous coupling (i.e. the effect of
the seepage force) on reflection of waves from an interface between two half-spaces, the
lower one being elastic and the upper one being fully saturated poroelastic (Yung and
Sato, 1998; Yang, 1999).
In contrast to the previously mentioned work dealing with partial saturation, which
used a modified theory for a two phase medium, Carcione et al. (2004) simulated wave
propagation in partially saturated porous rock including capillarity pressure effects.
Their model is based on a Biot type theory for a three-phase medium, which predicted the
existence of a second slow wave.
6
Other recent work on wave propagation in fluid saturated poroelastic media includes
that of: Liu and Liu (2004), who analyzed the propagation of Rayleigh waves in
orthotropic fluid-saturated porous media; Sharma (2004), who studied the propagation of
plane harmonic waves in an anisotropic fluid-saturated porous solid; Vashishth and
Khurana (2004), who studied the wave propagation in a multilayered anisotropic
poroelastic medium; Jinting et al. (2004), who studied the refection and refraction of
waves in a multi-layered medium composed of ideal fluid, porous medium, and
underlying elastic solid, and subjected to incident P wave. Other recent work also
includes Edelman (2004a,b), who studied the existence of surface waves along the
interface between vacuum and porous medium in the low frequency range. Finally, Liu
et al. (2005) used the generalized characteristic theory to analyze the stress wave
propagation in anisotropic, in particular orthotropic, fluid-saturated porous media.
1.3 Literature Review on Soil Structure Interaction in Porous
Media
This section presents a literature review of soil-structure interaction in poroelastic
soils. Review of other work on this topic that does not involve poroelasticity is out of the
scope of this thesis.
Halpern and Christiano (1986) present compliance matrices for vertical and rocking
motion of a square rigid plate baring on a water-saturated poroelastic half-space for water
saturated coarse grained sands (with porosity 0.48, and shear modulus of the skeleton 20
times smaller than the bulk modulus of water). Their results indicate smaller (in absolute
value) real and imaginary parts of the compliance (i.e. stiffer soil) for saturated soil as
7
compared to dry soils for both vertical and rocking motions. They also studied the stress
distribution along the contact surface carried separately by the solid and by the fluid, and
concluded that the magnitude of either one of the component stresses can be grater than
the total stress predicted by an equivalent undrained elastic solid model (elastic solid with
Poisson ratio 0.5).
Kassir and Xu (1988) studied interaction of a rigid pervious strip foundation bonded
to a poroelastic half-space for horizontal, vertical, and rocking motions. They concluded
that the influence of the fluid is substantial, and is more pronounced for vertical and
rocking motions. Kassir et al. (1989) studied impedances for vertical motion of circular
footings on a poroelstic half-space. They concluded that, for dense sand, the presence of
ground water affects the magnitude and character of the influence functions and should
be included in dynamic analysis of surface structures to dynamic loading.
Philippacopoulos (1989) present dynamic stiffness for vertical motion of a rigid disk
foundation on a layered poroelastic half-space saturated up to certain depth below the
disk. He concludes “the effect due to saturation on the impedance function is generally
not significant. Specifically, at low dimensionless frequency (i.e. less than 3) this effect is
practically negligible, while at higher dimensionless frequency (i.e. between 3 and 6),
the departure from the dry case was about 30%.” In the discussion of his results, he
states, “the effect of the pore fluid is to generally reduce the stiffness and increase the
damping (compared to the dry case). Furthermore, these effects are more pronounced at
higher dimensionless frequency and at lower saturation depth-to radius ratio. On the other
hand, at low frequencies, the results from both saturated and dry cases agree very well.”
8
This was explained by the fact that “the water has sufficient time to drain and thus avoids
carrying stresses imposed by the skeleton.” It is not clear from the discussion to what
degree the predicted effects are due to the “layer” effect created by the impedance
contrast at the water table level at depth, as compared to the fluid motion.
Bougacha and Tassoulas (1991a,b) developed a finite element technique to solve the
dynamic response of a gravity dam. The sediment is modeled as two-phase medium. It is
found that the partially saturated sediment leads to a significant decrease in the system
fundamental frequency more than fully saturated sediment.
Bougacha et al. (1993a,b) present a computational model and results for dynamic
stiffnesses for rigid strip and circular foundations on fluid filled poroelastic stratum over
a rigid base for horizontal, vertical, rocking and torsional motion, and propose how to
estimate the equivalent properties of an elastic soil. They show results for porosity 0.3,
Poisson ratio 1/3, and shear modulus of the skeleton such that it results in shear wave
velocity of 152 m/s. For torsional loading, they state that the results for a circular disk
obtained for the two-phase medium and the equivalent solid are identical, and explain
that by the fact that the torsional loading for circular footings transmits only shear waves
into the stratum. They conclude that the seepage forces introduce substantial damping at
low frequencies in the case of vertical excitation, while their effect on the rocking, and
especially on the torsional stiffness and damping coefficients were relatively minor.
Rajapakse and Senjuntichai (1995) present a soil-structure interaction model for rigid
strip foundation on a layered half-space, and show results for foundation response to unit
9
vertical and horizontal loads, and vertical impedance for a layered model. They also show
results for the pore pressure distribution with depth.
Kassir et al. (1996) present the impedances of surface circular footing on a
poroelastic half-space, for rocking and horizontal motions. They conclude that for
rocking motion, the presence of pore fluid significantly affects the impedance (both in
magnitude and sign), while the influence is marginal for horizontal motion.
Dargush and Chopra (1996) consider circular footings on a half-space or a layer over
bedrock, for horizontal, vertical, rocking and torsional motion. Their results show that
for surface footing on half-space, and for vertical motions, the compliance is larger for
dry soil than for poroelastic saturated soil, but the difference is small for small
frequencies and high permeability. For low permeability, the compliance is similar for
poroelastic and for undrained solid. For surface footing on layered medium, they note a
significant influence of the soil layer resonance.
Japon et al. (1997) show probably the most comprehensive set of results that shed
light on the effects of the pore water on the foundation stiffness for surface foundations.
They show impedances for strip foundations resting on a half-space, or on a stratum over
rigid or compliant bedrock, for smooth or welded contact, and for horizontal, vertical,
and rocking motions. Their results show that the seepage forces stiffen the foundation
and increase the damping. For a half-space soil model, their results show that the type of
contact condition is only important for the real part of vertical stiffness, which is larger
for a welded contact and for an impervious foundation. Further, the seepage forces
produce an effect of increased stiffness for the whole range of frequencies, and their
10
effect is more pronounced on the imaginary part (i.e. the radiation damping). The added
density (from the coupling mass term) produces increase in stiffness, noticeable only
when there are no seepage forces. For soil represented as a layer, the vertical and
rocking stiffness tend to the half-space values as the layer depth grows. At smaller
frequencies, the foundation stiffness for a layer is larger than that for a half-space, but the
difference is small for depth of layer to half width of foundation > 4. Further, the
foundation stiffness for a layer is oscillatory about the half-space solution, with
increasing frequency and decreasing amplitudes of the oscillations as the depth of layer
increases. For vertical motions, the oscillations are related to resonance of the fast P
waves in the layer, while for horizontal motions – to the resonance of the SH waves in the
layer. Further, they show that the effect of the seepage forces is much more important for
a stratum than for half-space, and finally, that the position of the resonant peaks may
change substantially with the dissipation coefficient b.
Zeng and Rajapakse (1999) studied vertical vibrations of a circular disk on a half-
space, and noted an increase in stiffness and radiation damping due to the poroelastic
effects.
Bo and Hua (1999) present compliances for a circular rigid disk on a half-space for
vertical motions. They conclude that the difference in compliance between pervious and
impervious foundation decreases with increasing seepage forces. Similarly, Jin and Liu
(2000a,b) show such compliances for horizontal and for rocking motions. For the
horizontal motions, they conclude that the permeability of the medium has an important
effect on horizontal vibrations, and that there is a difference between the compliances for
11
elastic and for saturated half-space. The conclusion for the rocking motions is that the
difference in compliance between poroelasic and elastic half-space is < 18% and can be
neglected. However, these three studies show results for a very limited set of parameters.
Senjuntichai et al. (2006) show impedances for axi-symmetric embedded
foundations in a half-space for vertical motions. They study the effects of foundation
depth, soil permeability, and foundation shape. Their results show that for cylindrical
shape, both the stiffness and the damping increase with increasing foundation depth.
Further, there is a notable dependence of the foundation stiffness on the hydraulic
boundary condition especially at higher frequencies and for short cylinders, but this effect
is much smaller for smaller permeability.
Chapter 2: Theoretical Model
2.1 The Soil-Structure Interaction Model
The simple two-dimensional soil-structure interaction model is shown in Fig. 2.1.1.
The structure is represented as a shear beam supported by a circular rigid foundation
embedded in a homogeneous and isotropic poroelastic half-space. The center of
curvature of the foundation is at some point along the z-axis, in general above point
O. The shear beam has height H, width W, and mass per unit length
1
O
b
m . The foundation
has width 2a, depth h, and mass per unit length . The response of the foundation is
described by the horizontal and vertical displacements of point , and V , and the
rotation angle
fnd
m
O ∆
ϕ (positive clockwise). The building moves as a rigid body, with
translations and V , and rotation ∆ ϕ , and also deflects due to elastic deformation (Fig.
2.1.1). The horizontal displacement at the top of the building due to its elastic
deformation is
rel
b
u . The shear wave velocity in the building is , which implies first
mode fixed-base frequency
,b S
V
1,b
/(4 )
S
fV H = . The damping in the building is neglected.
The motion in the half-space is described by the linearized theory of wave
propagation in fluid saturated poroelastic media as described by Biot (1956a). The two-
phase medium is composed of a solid skeleton, formed by the grains, and fluid occupying
completely all voids in the skeleton. The properties of this mixture are defined by the
shear modulus and Poisson’s ratio of the skeleton
s
µ and
s
ν , the bulk modulus of the
12
fluid
f
K , the porosity , the mass density of the grains ˆ n
gr
ρ comprising the skeleton, and
the density of the fluid
f
ρ , both defined per unit volume of pure grain material and pure
Fig. 2.1.1 The model
13
fluid. This implies shear wave velocity of the dry mixture
,dry gr
ˆ /[(1 ) ]
Ss
Vn µ ρ = − . The
skeleton and the foundation are perfectly bonded to each other. The motion of the fluid
along the contact surface relative to that of the solid is constrained by the drainage
condition. It is assumed in this thesis work that the foundation can be either completely
permeable, allowing for free drainage of the pore fluid, or completely impermeable.
These conditions would affect the foundation complex stiffness matrix, and the
foundation driving forces. The half-space surface can also be either perfectly sealed or
unsealed, and this would affect the free-field motion (Lin et al., 2001, 2005).
A closed form solution is obtained by: (1) expanding the scattered waves (a
perturbation to the free-field motion caused by the presence of the foundation) in a series
of outgoing cylindrical waves (represented by Hankel functions in space), (2) expressing
the coefficients of this expansion in terms of the (known) coefficients of expansion of the
free-field motion and the (unknown) motion of the rigid foundation through the
continuity of displacements at the contact surface, and (3) solving for the motion of the
foundation from the dynamic equilibrium conditions. In this process, the zero-stress
condition on the half-space surface, which is automatically satisfied by the free-field
motion, is relaxed for the scattered waves, as in Todorovska and Al Rjoub (2006a). The
zero stress condition for the scattered waves can be imposed numerically along finite
length of the half-space surface adjacent to the foundation, by some point collocation or a
weighted residual method, for example. In the interest of simplicity, and in view of all
the other simplifications in this model (e.g., the restriction of the shape of the foundation,
the assumption that it is absolutely rigid, the assumption of perfect bond at the contact
14
15
5
surface, and finally – the assumption of liner constitutive relations and small
displacements, and of homogeneous and isotropic soil), it was decided to relax this
condition. De Barros and Luco (1995) compared foundation impedances for a semi-
circular foundation in an elastic half-space, when the zero-stress boundary condition is
imposed on the scattered waves, and when it is relaxed. Their results show that for
horizontal and vertical motions the difference is small (the approximate solution
overestimates slightly the damping and the vertical stiffness, while it underestimated
slightly the horizontal stiffness at smaller frequencies and overestimates it slightly for
higher frequencies). The difference is also small for the coupling terms (between
horizontal motion and rocking). The difference is the largest for the rocking motions at
low frequencies, especially for the damping coefficient. The approximate solution
overestimates the rocking stiffness and underestimates the damping at all frequencies in
the range
0
/
S
aaV ω= < , but the difference becomes progressively smaller as the
frequency increases. At
0
0.5 a = , the rocking stiffness is overestimated by as much as
about 28%, and the damping coefficient is underestimated by as much as 38%, but the
shapes of the functions are similar, and the difference rapidly decreases with frequency,
especially for the damping coefficient. However, it turns out that the rocking stiffness is
not noticeably affected by the fluid in the pores, as shown in the companion paper, and
therefore the conclusions of this study are not likely to be affected by this approximation.
2.2 Wave Propagation in Fluid Saturated Poroelastic Medium
The motion in the soil is assumed to be governed by Biot’s theory of wave
propagation in a fully saturated poroelastic medium (1956a), which was postulated based
on the assumption that the motion of the solid matrix is a wave motion, while that of the
fluid relative to the solid is a diffusion process described by Darcy’s law. Biot (1956a)
made the following assumptions:
1. The Reynolds number is less than 2000, which implies that the relative motion of
the fluid in the pores is laminar flow.
2. The size of the unit element of the solid-fluid mixture is much smaller than the
wavelength of the motions considered.
3. The size of the unit element of the mixture is large compared to the size of the
pores.
Then the motion of the solid and that of the fluid is described by the following two
coupled equations of motion
() () ()
[] () ()
2
2
11 12 2
2
11 12 2
ˆ
grad
ˆ
grad
ueQ uUb
t
Qe R u U b u U
tt
µλµε ρρ
ερ ρ
∂∂
⎡⎤ ∇+ + + = + + −
⎣⎦
∂ ∂
∂∂
+= + − −
∂∂
uU
t
(2.2.1)
where
u = displacement vector for the solid-skeleton
U
= displacement vector for the pore fluid
( ) edivu =
16
()
div U ε =
11 12 22
,, ρ ρ ρ = dynamic mass coefficients
ˆ
b = coefficient of dissipation
The coefficient of dissipation depends on the permeability of the skeleton and on
the viscosity of the fluid via the relation
ˆ
b
2
ˆ
ˆ
ˆ
ˆ
bn
k
µ
=
where
ˆ n = porosity
ˆ µ = absolute viscosity in units Pa s=kg/(m s) ⋅ ⋅
ˆ
k = intrinsic permeability (depends only on the properties of the skeleton) in units
2
m
Hence,
ˆ
/ b ω has units of mass density.
Helmholtz decomposition to the displacement vector gives
() ( ) grad curl u φ ψ =+ (2.2.2a)
() ( )
grad curl U=Φ+ Ψ
(2.2.2b)
where φ and are the P-wave potentials, and Φ ψ and Ψ are the S-wave potentials for the
solid and fluid, respectively. Substitution of eqns (2.2.2a,b) into eqn (2.2.1) leads to the
following two sets of equations for the P-wave and S-wave potentials
17
() ()
() ()
2
22
11 12 2
2
22
11 12 2
ˆ
ˆ
PQ b
t
QR b
tt
φρφρ
φρφρ
∂∂
∇+ ∇Φ= + Φ + −Φ
∂ ∂
∂∂
∇+ ∇Φ= + Φ − −Φ
∂∂
t
φ
φ
(2.2.3)
and
() ()
() ()
2
2
11 12 2
2
11 12 2
ˆ
ˆ
0
b
t
b
tt
µψ ρψ ρ ψ
ρψ ρ ψ
∂∂
∇= + Ψ+ −Ψ
∂ ∂
∂∂
=+Ψ− −Ψ
∂∂
t
(2.2.4)
2.3 Solution for P-waves
For harmonic wave motion, the potentials can be represented as
()
1
ikx t
ce
ω
φ
+
= ,
(
2
ikx t
ce
) ω +
Φ= (2.2.5)
()
3
ikx t
ce
ω
ψ
+
= ,
(
4
ikx t
ce
) ω +
Ψ= (2.2.6)
Substitution of eqn (2.2.5) into eqn (2.2.3) leads to the following fourth order differential
equation for the P-wave potential in the solid
42
0 ABC φφ −+ = (2.2.7a)
where
2
APR Q = − (2.2.7b)
11 22 12
2(
ib
2) B RP Q PR Q ρρ ρ
ω
=+ − − ++ (2.2.7c)
2
11 22 12 11 22 12
(2
ib
C ) ρρρ ρ ρ ρ
ω
=− − + + (2.2.7d)
Further, eqn (2.2.7) can be decomposed into the following two equations
( )
22
,
0
jj
k
α
φ ∇+ = , j=1,2 (2.2.8)
18
where
,
,
j
j
k
V
α
α
ω
= , j=1,2 (2.2.9a)
and
()
, 1
2 2
2
4
j
A
V
BB AC
α
=
− ∓
, j=1,2 (2.2.9b)
are the wave numbers and wave velocities of two distinct P-waves (fast and slow) in the
solid.
The wave potential for the fluid can be obtained after substituting eqn (2.2.5) into
eqn (2.2.3), which gives
12 11 22
f f φ φ Φ= Φ + Φ = + (2.2.10)
where
2
,11 12
12 22
(/ )( )
(/ )( )
j
j
AV R Q ib Q R
f
RQ ib QR
α
ρρ ω
ρρ ω
−+ + +
=
−+ +
, j=1,2 (2.2.11)
2.3.1 Solutions for S-waves
Substitutin of eqn (2.2.6) into eqn (2.2.2) leads to the following differential equation
for the S-wave potential of the motion of the skeleton
( )
22
0 k
β
ψ ∇+ = (2.2.12)
where
k
V
β
β
ω
= (2.2.13a)
19
and
22
2
11 22 12 11 22 12
(/)
(2)*
ib
V
ib
β
µρ ω
/ ρρρ ρ ρ ρ ω
−
=
−− + +
(2.2.13b)
are the wave number and wave velocity of the shear waves in the skeleton.
The wave potential for the fluid can be obtained as
3
f ψ Ψ= (2.2.14)
where
12
3
22
( /
(/
ib
f
ib
)
)
ρ ω
ρ ω
+
=−
−
(2.2.15)
2.3.2 Material Constants for the Mixture
The material constants of mixtures can be determined experimentally (Biot and
Willis, 1957), or can be derived from the properties of the components. In the
dimensionless analysis in this work, the set of input parameters consists of the porosity
, the Poisson’s ratio of the skeleton ˆ n
s
ν , the ratio of the bulk modulus of the fluid and
the shear modulus of the skeleton /
f
K
s
µ , and the ratio of the mass density of the fluid
and that of the grains /
f gr
ρ ρ (both per unit volume of “pure” material).
The elastic moduli of the mixture µ , λ , R and are computed using a
simplification (for
Q
R and ) of the formulae proposed by Biot and Willis (1957) based
on the assumption that the compressibility of the mixture is much smaller than that of the
solid skeleton and of the fluid, and can be neglected, which is a common assumption in
soil mechanics (Lin et al., 2005)
Q
20
()
2
/
ˆ 1
ˆ
s
s
f
f
QR
QnK
RnK
µ µ
λλ
=
=+
=−
=
(2.2.16)
where
2
12
s
s s
s
v
v
λ µ =
−
= Lamé constant for the skeleton
For computation of the mass coefficients,
11
ρ ,
22
ρ and
12
ρ , the following relations
proposed by Berryman (1980) are used (as in Lin et al., 2005)
()
()
11 gr 12
22 f 12
12 f
ˆ 1
ˆ
ˆ 1
n
n
n
α
ρ ρρ
ρρ ρ
ρ τρ
=− −
=−
=− −
(2.2.17)
where
ˆ 1
1
ˆ
r
n
n
α
ττ
−
=+ ≥1= dynamic tortuosity (2.2.18a)
Tortuosity is a dimensionless macroscopic parameter characterizing the resistance to
flow of a fluid in porous medium, in particular the effect that, on microscopic scale, the
paths of the fluid particles deviate from a straight line. It depends on the porosity, , as
well as on the shape of the pores, through the parameter
ˆ n
r
τ . It has values 1
α
τ ≤ <∞ . As
(pure fluid) ˆ 1 n → 1
α
τ → , and as (pure solid) ˆ 0 n →
α
τ →∞. For pores formed by
spherical grains, as assumed in this work, 1/ 2
r
τ = , and
21
1 1
1
ˆ 2 n
α
τ
⎛
= +
⎜
⎝⎠
⎞
⎟
(2.2.18b)
It can be seen from eqn (2.2.17) that the dynamic mass coefficients represent physically
mass densities, per unit volume of the mixture. If the coupling term
12
ρ is neglected,
then
11
ρ and
22
ρ represent the mass densities of the solid and fluid phases per unit
volume of the mixture.
2.3.3 Approximate Treatment of Partial Saturation
Partially saturated soil represents a three-phase medium (mixture of solid, fluid and
gas). So far there is no generally accepted theory for wave propagation in such soil
medium. In this work, a simplified approach is followed, in which is the theory for a
two-phase medium is used, but with reduced bulk modulus of the fluid, as in Yang
(2001). Let be the degree of saturation. The relative proportions of the constituent
volumes are defined as
r
S
/
Vt
nV V =
(2.2.19a)
/
rW
SV V =
V
(2.2.19b)
where n
is the porosity of the soil, and and are respectively the volumes of
pores, pore water and the total volume.
,
V W
VV
t
V
In this study, a high degree of saturation is considered > 90%, assuming the
embedded air in the pore water is in the form of bubbles uniformly distributed through
the fluid. In this case, the bulk modulus of fluid
f
K can be written as
22
1
1 1
f
r
Wa
K
S
KP
=
−
+
(2.2.20)
Where is the bulk modulus of pore water and is the absolute fluid pressure.
W
K
a
P
23
2.4 The Soil-Structure Interaction Problem
2.4.1 Representation of the Scattered Waves
The scattered waves are represented by a triplet of potentials,
1
R
φ ,
2
R
φ , and
R
ψ , each
expanded in Fourier-Bessel series with period 2 π , representing outgoing cylindrical
waves with origin at point (see Fig. 2.3.1 showing an excavation in the soil where
the foundation is embedded)
1
O
(1)
11, 11, 1 11
0
(1)
21, 11, 1 21
0
(1)
1, 1 1, 1 1
0
(cos sin ) ( )
(cos sin ) ( )
(sin cos ) ( )
R it
nn nP
n
R it
nn nP
n
R it
nn nS
n
An B nH kre
En F nH kre
Cn D nH kre
ω
ω
ω
φθ θ
φθ θ
ψ θθ
∞
−
=
∞
−
=
∞
−
=
=+
=+
=+
∑
∑
∑
(2.3.1)
Fig. 2.4.1 The excavation and forces acting on the soil.
24
The radial and tangential components of the displacements of the skeleton due to
these waves are
()
1
1
1,
(3) (3) (3)
11 1 12 1 13 1
11 1,
(3) (3) (3)
0
1 21 1 22 1 23 1
1,
(3) (3) (3)
11 1 12 1 13 1
(3)
1 21 1 2
cos cos cos 1
,
sin sin sin
sin sin sin 1
cos
R
n
r
n
n
n
A
u
Dn Dn D n
rE
u rDn Dn D n
C
Dn Dn D n
rDn D
θ
θθ θ
θ
θθ θ
θθ θ
θ
+
∞
++
=
−
−
⎧
⎧⎫
⎧⎫
⎡⎤
⎪ ⎪⎪ ⎪ ⎪
=
⎨⎬ ⎨ ⎨ ⎬ ⎢⎥
⎪⎪⎣⎦ ⎪⎪
⎩⎭
⎩⎭ ⎩
+
∑
1,
1,
(3) (3)
21 23 1
1,
cos cos
n
it
n
n
B
Fe
nD n
D
⎪
ω
θθ
−
−
⎫
⎧⎫
⎡⎤
⎪ ⎪⎪
⎨⎬⎬ ⎢⎥
⎣⎦⎪⎪⎪
⎩⎭
⎭
(2.3.2a)
and the same components of the displacement of the fluid are
()
1
1
1,
(,3) (,3) (,3)
11 1 12 1 13 1
11 1,
( ,3) (,3) (,3)
0 1 21 1 22 1 23 1
1,
( ,3) (,3) (,3)
11 1 12 1 13
1
cos cos cos 1
,
sin sin sin
sin sin sin 1
R
n
ff f
r
n
ff f
n
n
ff f
A
U
Dn Dn D n
rE
U rDnD n D n
C
Dn Dn D
r
θ
θθ θ
θ
θθ θ
θθ
+
∞
++
=
−
⎧ ⎧⎫
⎧⎫
⎡⎤
⎪ ⎪⎪ ⎪ ⎪
=
⎨⎬ ⎨ ⎨ ⎬ ⎢⎥
⎪⎪⎣⎦ ⎪⎪
⎩⎭
⎩⎭ ⎩
+
∑
1,
1
1,
(,3) (,3) (,3)
21 1 22 1 23 1
1,
cos cos cos
n
it
n
ff f
n
B
n
Fe
DnDn D n
D
⎪
ω
θ
θθ θ
−
−−
⎫ ⎧⎫
⎡⎤
⎪ ⎪⎪
⎨⎬⎬ ⎢⎥
⎣⎦⎪⎪⎪
⎩⎭ ⎭
(2.3.2b)
The radial and tangential components of the stresses in the skeleton due to these waves
are
()
11
11
1,
(3) (3) (3)
11 1 12 1 13 1
11 1, 2 (3) (3) (3)
0
1 21 1 22 1 23 1
1,
(3) (3)
11 1 11
cos cos cos 2
,
sin sin sin
sin
R
n
rr
n
n r
n
A
En En E n
rE
r En En E n
C
En E
θ
τ
θθ θ µ
θ
τ θθ θ
θ
+
∞
++
=
⎧
⎧⎫
⎧⎫
⎡⎤
⎪ ⎪⎪ ⎪ ⎪
=
⎨⎬ ⎨ ⎨ ⎬ ⎢⎥
⎪⎪⎣⎦ ⎪⎪ ⎩⎭
⎩⎭ ⎩
+
∑
1,
(3)
113 1
1,
(3) (3) (3)
21 1 22 1 23 1
1,
sin sin
cos cos cos
n
it
n
n
B
nE n
Fe
En En E n
D
⎪
ω
θθ
θθ θ
−
−
−−
⎫
⎧⎫
⎡⎤
⎪ ⎪⎪
⎨⎬⎬ ⎢⎥
⎣⎦⎪⎪⎪
⎩⎭
⎭
(2.3.3a)
and the stress in the fluid is
25
()
1,
(,3) (,3)
11 1 1 2 1
0 1,
1,
( ,3) ( ,3)
112 1
1,
,cos cos
sin sin
n
Rf f
n n
n
f fi
n
A
sr E n E n
E
B
t
E nE n e
F
ω
θθ θ
θθ
∞
=
−
⎧⎫
⎡⎤ =
⎨⎬
⎣⎦
⎩⎭
⎫
⎧⎫
⎪
⎡⎤ +
⎨⎬⎬
⎣⎦
⎪ ⎩⎭
⎭
∑
(2.3.3b)
where
()
()
()
()
()
()
()
11 11 11
()
22 12 12
()
13
()
1 21
()
2 22
()
1 23
() ()
() ()
()
()
()
() ()
l
nP P n P
l
nP P n P
l
nS
l
nP
l
nP
l
nS n S
Dr nCkr krC kr
Dr nCk r k rC k r
Dr nCkr
Dr nCkr
Dr nCkr
Dr nCkr krC kr
β
−
−
±
±
±
−
=− +
=− +
=±
=
=
=−
∓
∓
(2.3.4a)
() ()
(,) ()
, 1,2; 1,2,3
fl l
ij j ij
Dr fD r i i = == (2.3.4b)
(1)
(2)
(), 1
(), 2
()
(), 3
(), 4
n
n
n
n
n
Jl
Yl
C
Hl
Hl
⋅= ⎧
⎪
⋅=
⎪
⋅=
⎨
⋅ =
⎪
⎪
⋅=
⎩
(2.3.5)
()
n
J ⋅ and are the Bessel functions of first and second kind, and ()
n
Y ⋅
(1)
()
n
H ⋅ and
are the Hankel functions of the first and second kind, and
(2)
()
n
H ⋅
j
f is as defined in eqn
(2.2.1).
26
()
()
()
() 2 2 2
11 111 11
() 2 2 2
22 2 2 21 12
()
1 13
11
() () ( ) 1 ( ) ( )
22 2
11
() () ()1 ( ) (
22 2
(1) ( ) ( )
l
PnP f nf Pn P
l
PnP P nP Pn P
l
nS S n
E r n n k r Ck r k r Q Ck r k rC k r
E r n n k r Ck r k r Q Ck r k rC k r
Er n n Ckr krC kr
β
λ
λ
−
−
±
−
⎡⎤ ⎛ ⎞
=+− − + + −
⎜⎟
⎢⎥
⎣⎦ ⎝ ⎠
⎡⎤ ⎛ ⎞
=+− − + + −
⎜⎟
⎢⎥
⎣⎦ ⎝ ⎠
⎡⎤ =− + +
⎣
∓
()[]
()[]
()
2
)
()
11 11 21
()
22 12 22
() 2 2
1 23
(1) ( ) ( )
(1) ( ) ( )
1
() () ()
2
l
nP P n P
l
nP P n P
l
SnS Sn S
Er n n Ckr krC kr
Er n n Ckr krC kr
Er n n kr Ckr krC kr
±
−
±
−
−
⎦
=− + +
=− + +
⎡⎤
=− + − −
⎢⎥
⎣⎦
∓
∓
(2.3.6a)
()
()
()
(,) 2
11 1 1 1
(,) 2
12 2 2 2
(,)
13
1
() ()
2
1
() (
2
0
fl
PnP
fl
PnP
fl
Er SkrCkr
Er SkrCkr
Er
=
=
=
) (2.3.6b)
where
()
/, 1,2
jj
SQ fR j µ =+ = (2.3.7)
2.4.2 Boundary Conditions at the Contact Surface
The motion of the rigid foundation for incident monochromatic waves is harmonic,
and can be written as
0
0
0
e
i t
VV
aa
ω
ϕϕ
−
⎧⎫ ⎧ ⎫
⎪⎪ ⎪ ⎪
∆= ∆
⎨⎬ ⎨ ⎬
⎪⎪ ⎪ ⎪
⎩⎭ ⎩ ⎭
(2.3.8)
Along the contact surface
10 0
:, rb θ θ θ Σ =− ≤ ≤
1
0
sin ( / ) ab θ
−
= , (Fig. 2.3.1), the
displacements of the skeleton are constrained by the displacements of the foundation, and
the motion of the fluid is constrained by the drainage condition. Perfect bond between
27
the skeleton and the foundation, and perfectly sealed contact (i.e. no drainage of the pore
fluid) imply
() ()
11
11
11 11
11 1 0
11 1 0
0
cos sin ( / )sin
sin cos / ( / )cos e
00 0
ff R
rr
i t
rr rr
uu
da V
uu bada
a
uU uU
ω
θθ
θθ θ
θθ θ
ϕ
−
ΣΣ
⎧⎫⎧⎫
⎡⎤
⎪⎪⎪⎪
⎪⎪⎪⎪ ⎪⎪
⎢⎥
+=− −+ ∆
⎨⎬⎨⎬ ⎨⎬
⎢⎥
⎪⎪⎪⎪ ⎪⎪
⎢⎥
⎣⎦
−−
⎪⎪⎪⎪
⎩⎭⎩⎭
⎧⎫
⎩⎭
(2.3.9a)
Where the matrix on the right-hand side is the foundation influence matrix.
Similarly, perfect bond between the skeleton and the foundation, and unsealed contact
(i.e. free drainage of the pore fluid) imply
11
11
11 1 0
11 1 0
0
cos sin ( / )sin
sin cos / ( / )cos e
00 0
ff R
rr
i t
uu
da V
uu bada
a
ss
ω
θθ
θθ θ
θθ θ
ϕ
−
ΣΣ
⎧⎫ ⎧⎫
⎡⎤
⎪⎪ ⎪⎪ ⎪⎪
⎢⎥
+=− − + ∆
⎨⎬ ⎨⎬ ⎨ ⎬
⎢⎥
⎪⎪ ⎪⎪ ⎪ ⎪
⎢⎥
⎣⎦
⎩⎭ ⎩⎭
⎧⎫
⎩⎭
(2.3.9b)
The application of these conditions enables expressing the unknown coefficients of
expansion of the scattered waves in terms of the known free-field displacements, and the
displacements of the foundation. However, this requires expansion of the free-field
displacements at in Fourier series of
1
r b =
1
θ with period 2 π . This can be done by
expanding the potentials in Fourier-Bessel series, and then computing the displacements,
similarly as for the scattered waves, but such series converge only for the plane waves,
and diverge for the surface waves (Lee and Cao, 1989). Hence, for the surface waves, we
expand the displacements at
1
r b = in Finite Fourier series of
1
θ , up to , which is
the truncation index for the expansion of the scattered and plane free-field waves (Lee
n N =
28
and Cao, 1989). Let us assume that such expansions are available, with
n
A
i
and
being the Fourier coefficients for the symmetric and anti-symmetric terms. Then
n
B
i
()
() ()
1
1
11
11
1
0
11
cos sin
1
sin cos
cos sin
rr
rr rr
ff
uu
r
nn
N
u u
n n
n
uU u U
nn nn
rr
u
An B n
uAn B
b
AA n B B n
uU
θ θ
θ
θθ
θ
θθ
=
Σ
⎧⎫
⎧⎫
⎧⎫⎧
⎪⎪
⎪⎪
⎪⎪⎪
⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪
= +
⎨ ⎬ ⎨⎨ ⎬ ⎨ ⎬⎬
⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪
−−
−
⎪⎪⎪ ⎪⎪ ⎪
⎩⎭⎩
⎩⎭
⎩⎭
∑ 1
nθ
⎫
⎪
⎪⎪
⎭
(2.3.10a)
and
1
1
11
1
0
11
cos sin
1
sin cos
cos sin
rr
ff
uu
r
nn
N
u u
n n
n
ss
nn
u
An B n
uAnB
b
An B n
s
θ θ
θ
θθ
θ
θθ
=
Σ
⎧⎫ ⎧⎫ ⎧⎫⎧
⎪⎪ ⎪⎪ ⎪⎪⎪
= +
⎨ ⎬ ⎨⎨ ⎬ ⎨ ⎬⎬
⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪
⎩⎭⎩ ⎩⎭⎩⎭
∑ 1
nθ
⎫
⎪
⎭
(2.3.10b)
For the scattered waves
()
1
1
11
(3) (3) (3)
11 1 12 1 13 1 1,
(3) (3) (3)
21 122 1 23 11,
0
( ,3) ( ,3) ( ,3)
11 1 12 1 13 1 1,
cos cos cos
1
sin sin sin
cos cos cos
R
r
n
N
n
n
rel rel rel
n
rr
u
Dn Dn Dn A
u D nD nD n E
b
DnDnDn C
uU
θ
θθθ
θθθ
θθθ
=
Σ
Σ
⎧⎫
⎧
⎡⎤⎧⎫
⎪⎪
⎪
⎢⎥ ⎪⎪ ⎪
=
⎨⎬ ⎨ ⎨
⎢⎥
⎪⎪ ⎪ ⎪
⎢⎥
⎩ ⎭ ⎣⎦ −
⎩ ⎪⎪
⎩⎭
∑
(3) (3) (3)
11 1 12 1 13 1 1,
(3) (3) (3)
21 1 22 1 23 1 1,
(,3) (,3) (,3)
11 1 12 1 13 1 1,
cos cos cos
sin sin sin
cos cos cos
n
n
rel rel rel
n
Dn Dn Dn B
Dn Dn Dn F
DnD nDn D
θθθ
θθθ
θθθ
Σ
⎫
⎡⎤⎧⎫
⎪
⎢⎥⎪⎪
+
⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎩⎭ ⎣⎦
⎭
⎪
⎬
⎪
⎬
⎪
(2.3.11a)
where
( ,3) ( ,3) (3) rel f
ij ij ij
D D D = − (2.3.11b)
and
29
1
1
(3) (3) (3)
11 1 12 1 13 1 1,
(3) (3) (3)
21 1 22 1 23 1 1,
0
( ,3) ( ,3) ( ,3)
11 1 12 1 13 1 1,
(3)
11
cos cos cos
1
sin sin sin
cos cos cos
c
R
r
n
N
n
n
ff f
n
u
DDD A
uD D D
b
EEE C
s
D
θ
θθθ
θθθ
θθθ
=
Σ Σ
⎧ ⎧⎫⎡⎤
E
⎧ ⎫
⎪
⎪⎪⎢⎥ ⎪ ⎪
=
⎨⎬ ⎨ ⎨ ⎬
⎢⎥
⎪⎪ ⎪ ⎪ ⎪
⎢⎥
⎩⎭ ⎣⎦ ⎩⎭ ⎩
+
∑
(3) (3)
112 1 13 1 1,
(3) (3) (3)
21 1 22 1 23 1 1,
(,3) (,3) (,3)
11 1 12 1 13 1 1,
os cos cos
sin sin sin
cos cos cos
n
n
ff f
n
DD B
DDD F
EEE D
θθθ
θθθ
θθθ
Σ
⎫
⎡⎤ ⎧ ⎫
⎪
⎢⎥ ⎪ ⎪
⎨ ⎬⎬
⎢⎥
⎪ ⎪⎪
⎢⎥
⎩⎭ ⎣⎦
⎭
(2.3.11c)
After substitution for the appropriate expansions in eqn’s (2.3.11a) and (2.3.11b),
and matching the terms multiplying the same basis functions (due to the orthogonality of
Fourier series), the coefficients of expansion of the scattered field can be expressed in
terms of the coefficients of expansion of the free-field motion and the displacements of
the foundation. Then for sealed boundary
1,
0
1
(3)
1, ,sealed 0
0
1,
1,
0
1
(3)
1, ,sealed
1,
() , 0,...,
()
r
rr
r
rr
u
n
n
u
nn
uU
nn
n
u
n
n
u
nn
uU
nn
n
A
AV
ED A Xn n
AA a
C
B
BV
FD B Xn
BB
D
θ
θ
ϕ
−
++
Σ
−
−−
Σ
⎧⎫ ⎧⎫ ⎧⎫
⎧⎫
⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪
⎡⎤ ⎡ ⎤ =− + ∆ =
⎨⎬ ⎨⎨ ⎬ ⎨ ⎬⎬
⎣⎦ ⎣ ⎦
⎪⎪ ⎪⎪ ⎪ ⎪ ⎪⎪
−
⎩⎭
⎩⎭ ⎩⎭⎩
⎧⎫ ⎧⎫
⎪⎪ ⎪⎪
⎡⎤ ⎡ ⎤ =− + ∆
⎨⎬ ⎨ ⎬
⎣⎦ ⎣ ⎦
⎪⎪ ⎪ ⎪
−
⎩⎭ ⎩⎭
0
0
,0,..., nN
a ϕ
⎧⎫
⎧⎫
⎪⎪
⎪⎪
=
⎨⎨⎬⎬
⎪⎪⎪⎪
⎩⎭
⎩⎭
N
⎭
N
(2.3.12a)
and for an unsealed boundary
1,
0
1
(3)
1, ,unsealed 0
0
1,
1,
0
1
(3)
1, ,unsealed 0
0
1,
() , 0,...,
()
r
r
u
n
n
u
nn
s
n
n
u
n
n
u
nn
s
n
n
A
AV
ED A Xn n
Aa
C
B
BV
FD B Xn
Ba
D
θ
θ
ϕ
ϕ
−
++
Σ
−
+−
Σ
⎧⎫ ⎧⎫ ⎧⎫
⎧⎫
⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪
⎡⎤ ⎡ ⎤ =−+ ∆ =
⎨⎬ ⎨⎨ ⎬ ⎨ ⎬⎬
⎣⎦ ⎣ ⎦
⎪⎪ ⎪⎪ ⎪ ⎪ ⎪⎪
⎩⎭
⎩⎭ ⎩⎭ ⎩ ⎭
⎧⎫ ⎧⎫
⎧
⎪⎪ ⎪⎪ ⎪
⎡⎤ ⎡ ⎤ =−+ ∆
⎨⎬ ⎨ ⎬ ⎨
⎣⎦ ⎣ ⎦
⎪⎪ ⎪ ⎪
⎩
⎩⎭ ⎩⎭
,0,..., nN
⎧⎫
⎫
⎪⎪
⎪
=
⎨⎬⎬
⎪⎪⎪⎪
⎭
⎩⎭
(2.3.12b)
30
where
31
nb
⎦
=
=
()
()
(3) (3) (3)
11 12 13
(3) (3) (3) (3)
,sealed 21 22 23
(,3) ( ,3) (,3)
11 12 13
(3) (3) (3)
11 12 13
(3) (3) (3) (3)
,unsealed 21 22 23
(,3) (,3) (,3)
11 12 13
,
,
rel rel rel
fff
DDD
DD D D
DDD
DDD
D DDD nb
EEE
±
±
Σ
±
±
Σ
⎡⎤
⎢⎥
⎡⎤ =
⎢⎥ ⎣⎦
⎢⎥
⎣
⎡⎤
⎢⎥
⎡⎤ =
⎢⎥ ⎣⎦
⎢⎥
⎣⎦
(2.3.13)
10 0
() 0 0 0 , 1
00 0
00 0
() 0 0 0 , 1
00 0
Xn n
Xn n
+
+
−⎡⎤
⎢⎥
⎡⎤==
⎣⎦
⎢⎥
⎢⎥
⎣ ⎦
⎡⎤
⎢⎥
⎡⎤=≠
⎣⎦
⎢⎥
⎢⎥
⎣⎦
(2.3.14a)
00 0
() 0 0 / , 0
00 0
01 /
() 0 1 / , 1
00 0
000
() 0 0 0 , 1
000
Xn ba n
da
Xn d a n
Xn n
−
−
−
⎡⎤
⎢⎥
⎡⎤=−
⎣⎦
⎢⎥
⎢⎥
⎣⎦
⎡⎤
⎢ ⎥
⎡⎤=
⎣⎦
⎢ ⎥
⎢⎥
⎣⎦
⎡⎤
⎢⎥
⎡⎤=>
⎣⎦
⎢⎥
⎢⎥
⎣⎦
(2.3.14b)
2.4.3 Integral of Stresses along the Contact Surface
Next we compute vertical and horizontal forces
( ) s
z
f and
( ) s
x
f , and moment about O ,
( )
0
s
M , which result from all stresses in the soil along the contact surface , and have
signs as shown in Fig. 2.3.1. We also introduce a generalized force vector notation for
Σ
this triplet of forces and moment
{ }
0
,, /
T
zx
f fM a = F and refer to it as the force, and
generalized displacement vector { } ,,
T
V a ϕ =∆ ∆ and refer to it as the displacement. For
harmonic excitation, is also harmonic and can be written as ∆
0
i t
e
ω −
= ∆∆ (2.3.15)
where is its complex amplitude.
0
∆
The resultant force vector,
( ) s
F , is the sum of the force vectors due to the free-field
motion and due to the scattered waves,
( ) s
ff
F and
( ) s
R
F , and can be computed as follows
0
11 1 1
11 1 1 0
( ) () ()
11
11
11
cos sin
sin cos
(/ )sin / ( / )cos
ss s
ff R
ff R
rr rr
rr
s s
bd
da b a da
θ
θθ θ
θθ
ττ
1
θθθ
ττ
θθ
−
ΣΣ
=+
−+ ⎡ ⎤
⎧⎫
+ + ⎧⎫ ⎧⎫
⎪⎪ ⎪ ⎪ ⎪ ⎪
⎢ ⎥
=− − +
⎨⎨ ⎬ ⎨ ⎬ ⎬
⎢⎥
⎪⎪ ⎪⎪
⎪⎪ ⎩⎭ ⎩⎭
⎢⎥⎩⎭ −−
⎣⎦
∫
FF F
(2.3.16)
Eqn (2.36) holds for both sealed and unsealed conditions. We note however that, for
an unsealed boundary, the total stress in the pore fluid,
ff
s s
R
+ , is actually zero on the
boundary, as preset by the drainage condition (see eqn (2.29b)).
Similarly as in Section 2.3.1, we expand the stresses of the free-field motion along
the contact surface in Fourier series of
1
θ with period 2 π
11
11
1
2
0
1 1
()cos ( )sin
2
sin cos
rr rr
rr
ff
ss
N
rr
nn nn it
n r
n n
s
AA n B B n
e
b
An B n
θθ
ττ
1 ω
ττ
θ
τ
θθ
µ
τ
θ θ
−
=
Σ
⎧⎫ + ⎧⎫ ⎧⎫⎧ ++
⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪
=+
⎨ ⎬ ⎨⎨ ⎬ ⎨ ⎬⎬
⎪⎪⎪ ⎪⎪⎪⎪ ⎩⎭⎩ ⎩⎭⎩⎭
∑
⎫
⎪
⎭
(2.3.17)
and substitute in eqn (2.3.16). Then, for the forces due to the free-field motion we get
32
()
0
2
() ()
rr rr
rr
ss
N
nn nn s i t
ff
n
nn
AA BB
Ln Ln e
b
AB
θθ
ττ
ω
τ τ
µ
+−
=
⎧⎫ ⎧⎫ ⎧⎫ ++
⎪⎪ ⎪ ⎪ ⎪⎪
⎡⎤ ⎡ ⎤ = +
⎨⎨ ⎬ ⎨ ⎬⎬
⎣⎦ ⎣ ⎦
⎪⎪ ⎪⎪ ⎪⎪ ⎩⎭ ⎩⎭
⎩⎭
∑
F
−
1
(2.3.18)
where
[]
[]
0
0
11 1 1
11 11
11 1 1
11 1 1
11 1
11 1
cos cos sin sin
() sin cos cos sin
(/ )sin cos / (/ )cos sin
cos sin sin cos
() sin sin cos cos
(/ )sin sin / (/ )cos cos
n
Ln n d
da n b a d a n
n
Ln n
da n b a d a n
θ
θ
θθ θ θ
θθθθ
θθ θ θ
θθ θ θ
θθ θ θ
θθ θ
+
−
−
⎡⎤ −
⎢⎥
⎡⎤=− −
⎣⎦⎢⎥
⎢⎥
−−
⎣⎦
−
⎡⎤=− −
⎣⎦
−−
∫
0
0
1
1
d
θ
θ
θ
θ
θ
−
⎡⎤
⎢⎥
⎢⎥
⎢⎥
⎣⎦
∫
(2.3.19)
Some of the terms of matrices () L n
+
⎡ ⎤
⎣ ⎦
and () L n
−
⎡ ⎤
⎣ ⎦
are automatically zero (when
the integrand is an odd function), and the nonzero ones can be evaluated analytically,
which gives
14
41
45
() ()
() 0 0
00
00
() () ()
(/ ) ( ) ( / ) ( ) (/ ) ( )
In I n
Ln
Ln I n I n
daI n b aI n daI n
+
−
−⎡⎤
⎢⎥
⎡⎤ =
⎣⎦
⎢⎥
⎢⎥
⎣⎦
⎡⎤
⎢⎥
⎡⎤=− −
⎣⎦
⎢⎥
⎢⎥ −−
⎣⎦1
(2.3.20)
The expressions for the integrals
1
( ) I n ,
4
( ) I n , and
5
( ) I n are given in Appendix.
Similarly, for the forces from the scattered waves we get
1, 1,
() (3) (3)
1, 1,
0
1, 1,
2
() () () ()
nn
N
s i t
Rn
n
nn
AB
Ln E n E L n E n F e
b
CD
n
ω
µ
++ − −
Σ Σ
=
⎧⎫ ⎧⎫ ⎧ ⎫
⎪⎪ ⎪⎪ ⎪ ⎪
⎡⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ = +
⎨⎨⎬ ⎨
⎣⎦⎣ ⎦ ⎣ ⎦⎣ ⎦
⎪⎪⎪ ⎪
⎩⎭ ⎩ ⎭
⎩⎭
∑
F
−
⎬⎬
⎪⎪
(2.3.21)
33
where
34
) (
(3) ( ,3) (3) ( ,3) (3) ( ,3)
(3) 11 11 12 12 13 13
(3) (3) (3)
21 22 23
() ,
ff f
EE EE EE
Enn
EEE
±
±
Σ
⎡⎤ +++
⎡⎤ =
⎢⎥
⎣⎦
⎣⎦
b (2.3.22)
Further, substituting in eqn (2.3.16) for the coefficients of expansion of the scattered
waves from eqn (2.3.12a,b) it follows that for a sealed boundary
0
1
() (3) (3)
R ,sealed 0
0
0
1
(3) (3)
,sealed
2
() () ()
( ) ( )
r
rr
r
r
u
n
N
u s
n
n
uU
nn
u
n
u
n
u
n
AV
Ln E n D A b X n
b
AA a
B
Ln E n D B
B
θ
θ
µ
ϕ
−
++ + +
ΣΣ
=
−
−− −
ΣΣ
⎧
⎧⎫ ⎧⎫
⎧⎫
⎪ ⎪⎪
⎪⎪ ⎪⎪
⎡⎤⎡ ⎤⎡ ⎤ ⎡ ⎤ =−+
⎨⎨⎨⎬
⎣⎦⎣ ⎦⎣ ⎦ ⎣ ⎦
⎪⎪⎪⎪
−
⎩⎭
⎩⎭
⎩⎭ ⎩
⎡⎤⎡ ⎤⎡ ⎤ +−
⎣⎦⎣ ⎦⎣ ⎦
−
∑
F
0
0
0
()
r
it
U
n
V
bX n e
Ba
∆
⎨⎬⎬
⎪⎪⎪
ω
ϕ
−−
⎫
⎧⎫
⎧⎫
⎧⎫
⎪
⎪⎪ ⎪⎪ ⎪⎪
⎡⎤ +∆
⎨ ⎨ ⎬ ⎨ ⎬⎬⎬
⎣⎦
⎪ ⎪ ⎪ ⎪ ⎪⎪⎪
⎩⎭
⎩⎭
⎩⎭
⎭
(2.3.23)
It is seen that
( ) s
R
F depends both on the displacements from the free-field motion and
that of the foundation, and can be written as
() () ( )
R scat
s s
∆
= + FF F
s
s
)n⎤
⎦
(2.3.24)
where
() ()
2
s
K µ
∆
⎡ ⎤ =
⎣ ⎦
F ∆ (2.3.25)
with
1
() (3) (3)
,sealed
0
1
(3) (3)
,sealed
2() () (
( ) ( ) ( )
N
s
n
KLnEnD X
Ln E n D X n
µ
−
++ + +
ΣΣ
=
−
−− − −
ΣΣ
⎡
⎡⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ =
⎣⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣
⎢
⎣
⎤
⎡⎤⎡ ⎤⎡ ⎤⎡ ⎤ +
⎣⎦⎣ ⎦⎣ ⎦⎣ ⎦
⎥
⎦
∑
(2.3.26)
and
1
() (3) (3)
scat ,sealed
0
1
(3) (3)
,sealed
2
() ()
( ) ( )
r
rr
r
rr
u
n
N
u s
n
n
uU
nn
u
n
u it
n
uU
nn
A
Ln E n D A
b
AA
B
Ln E n D B e
BB
θ
θ
ω
µ
−
++ +
ΣΣ
=
−
−− −
ΣΣ
⎧
⎧⎫
⎪
⎪⎪
⎡⎤⎡ ⎤⎡ ⎤ =−
⎨⎨
⎣⎦⎣ ⎦⎣ ⎦
⎪⎪
−
⎩ ⎭
⎩
⎫
⎧⎫
⎪
⎪⎪
⎡⎤⎡ ⎤⎡ ⎤ +
⎨⎬⎬
⎣⎦⎣ ⎦⎣ ⎦
⎪⎪⎪
−
⎩⎭
⎭
∑
F
−
⎬
⎪
(2.3.27)
For an unsealed boundary
1
() (3) (3)
,unsealed
0
1
(3) (3)
,unsealed
() () ()
( ) ( ) ( )
N
s
n
K LnE nD X n
LnE nD X n
−
++ + +
ΣΣ
=
−
−− − −
ΣΣ
⎡
⎡⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ =
⎣⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣
⎢
⎣
⎤
⎡⎤⎡ ⎤⎡ ⎤⎡ ⎤ +
⎣⎦⎣ ⎦⎣ ⎦⎣ ⎦
⎤
⎦
⎥
⎦
∑
(2.3.28)
and
1
() (3) (3)
scat ,unsealed
0
1
(3) (3)
,unsealed
2
() ()
( ) ( )
r
r
u
n
N
u s
n
n
s
n
u
n
u it
n
s
n
A
Ln E n D A
b
A
B
Ln E n D B e
B
θ
θ
ω
µ
−
++ +
ΣΣ
=
−
−− −
ΣΣ
⎧
⎧⎫
⎪
⎪⎪
⎡⎤⎡ ⎤⎡ ⎤ =−
⎨⎨
⎣⎦⎣ ⎦⎣ ⎦
⎪⎪
⎩⎭
⎩
⎫
⎧⎫
⎪
⎪⎪
⎡⎤⎡ ⎤⎡ ⎤ +
⎨⎬⎬
⎣⎦⎣ ⎦⎣ ⎦
⎪⎪⎪
⎩⎭
⎭
∑
F
−
⎬
⎪
(2.3.29)
Then the integral of all stresses in the soil along the contact surface is
( ) () () ( )
ff scat
() ( )
driv
s ss
ss
∆
∆
=+ +
=+
FF F F
FF
s
(2.3.30)
where
() () ()
driv ff scat
s s
=+ FF F
s
(2.3.31)
35
The interpretation of these forces and of matrix
() s
K ⎡ ⎤
⎣ ⎦
is as follows.
( ) s
∆
F defined by
eqn (2.3.25) is an external force required to move the foundation by displacement
when there is no free-field motion, and the matrix elating them, , is the
foundation stiffness matrix. This matrix is complex, with its real part representing the
stiffness of the foundation, and its imaginary part the radiation damping.
∆
()
2
s
K µ⎡
⎣
⎤
⎦
()
driv
s
F defined by
eqn (2.3.31) is the external force required to hold the foundation in place when it is
subjected to the action of the free-field waves. Its reaction is the force with which the
free-field motion effectively drives the foundation, and is the generalized foundation
driving force. It is different from force
( )
ff
s
F , which is the integral of the stresses of the
free-field motion, because of the scattering of waves from the foundation.
2.4.4 Dynamic Equilibrium of the Foundation
The only remaining unknown is the foundation displacement vector, , which can
be determined from the dynamic equilibrium condition for the foundation. Fig. 2.3.2
shows a free-body diagram of the foundation, which is subjected to the forces from the
building, , and the forces from the soil,
∆
( ) b
F
( ) () ()
driv
2
ss s
K µ
⎡ ⎤
=+
⎣ ⎦
FF ∆ .
For small amplitudes of the response, the forces from the building can be represented
in terms of the displacement vector, ∆ , as follows (Todorovska, 1993b)
[ ]
() 2 b
bb
m M ω = F ∆
(2.3.32)
36
Fig. 2.4.2 Dynamic equilibrium of the foundation.
where [
b
]
M is a dimensionless matrix that depends on the building model and
characteristics. For a shear beam model, and neglecting the effect of the gravity forces,
its entries are
()
()
() ()
()
,
,11
,
,
,22
,
,23 ,32 ,
2
,
2
,
2
,33
2
, ,
tan /
/
tan /
/
1
11/cos /
(/)
tan /
11
1(/
/12 (/ )
Pb
b
Pb
Sb
b
Sb
bb Sb
Sb
Sb
b
Sb Sb
HV
M
HV
HV
M
HV
H
MM HV
a HV
HV
H
MW
HV a HV
ω
ω
ω
ω
ω
ω
ω
ω ω
=
=
−
== −
⎡⎤ ⎛⎞
⎛⎞
⎢⎥ ⎜⎟ =− +
⎜⎟
⎜⎟
⎢⎥⎝⎠
⎝⎠ ⎣⎦
)H
(2.3.33)
37
In eqns (2.3.32) and (2.3.33), is the mass per unit length of the beam, and
b
m H and
are its height and width, and and W
, Sb
V
, P b
V are its S and P wave velocities. The
dynamic equilibrium of forces acting onto the foundation implies
[ ] [ ]
2(s) ()2
fndfnd bb driv
2
s
mM K m M ωµω
⎡⎤
−− + =
⎣⎦
∆ F ∆ 0∆ (2.3.34)
where is the mass of the foundation,
fnd
m [ ]
fnd
M is the foundation dimensionless mass
matrix
[]
fnd
2
fnd,0 fnd
10 0
01 0
00 /( )
M
I ma
⎡⎤
⎢⎥
=
⎢
⎢⎥
⎢⎥
⎣⎦
⎥
(2.3.35)
where
()
2
2 fnd 3 1
fnd,0 0 0 0 0
2 2
00 0
cos sin cos
sin cos
bm
I θθθ
θθ θ
⎡⎤
=+ −
⎣⎦
−
θ (2.3.36)
is the mass moment of inertia of the foundation relative to point O, and is the mass
per unit length of the foundation. Finally, one can solve for by inverting a
fnd
m
∆ 33 ×
matrix
[] []
1
22
() (s) fnd b
fnd b driv
2
22
s
mm
MMK
ωω
µ
µµ
−
⎡⎤
⎡⎤
=+ −
⎢
⎣⎦
⎢⎥
⎣⎦
∆ F
⎥
(2.3.37)
For the purpose of brevity, and without loss of generality, the dynamic moments of the
gravity forces were neglected in the above derivations. The expressions including these
moments can be found in Todorovska (1993a,b).
38
2.5 The Free-Field Motion
This section deals with the representation of the free-field motion for incident plane
P-wave, incident plane SV-wave, and for a Rayleigh wave in a fully saturated porous
half-space, considering the effects of the seepage force (which leads to complex valued
and frequency dependent wave velocity in the mixture). The coefficients of the reflected
waves of the P- and S-wave potentials of the Rayleigh waves are derived for open
(permeable) and sealed (impermeable) hydraulic condition on the half-space boundary,
for which the zero stress condition on the half-space surface is Deresiewicz (1960)
0
0
0
0
yy
xy
y
τ
τ
σ
=
⎧⎫ ⎧⎫
⎪⎪ ⎪⎪
=
⎨⎬ ⎨⎬
⎪⎪ ⎪⎪
⎩⎭ ⎩⎭
open (permeable) boundary (2.4.1)
and
0
0
0
0
yy
xy
yy
y
uU
τσ
τ
+
=
⎧⎫
⎧ ⎫
⎪⎪ ⎪ ⎪
=
⎨⎬ ⎨
⎪⎪ ⎪
−
⎩⎭
⎩⎭
⎬
⎪
sealed (impermeable) boundary (2.4.2)
For incident plane waves (P and SV), the derivation presented in this section, which
is for the genral dissipative case, follows Deresiewicz and Rice (1962). For the Rayleigh
waves, and nondissipative case, the derivation follows Lin et al. (2005).
39
2.5.1 The Free Field Motion due to Incident P-Wave
Let the excitation consist of a plane fast P-wave incident onto the half-space free
surface (Fig. 2.4.1), represented by its potential
] ) cos sin ( exp[ t i z x i
i
ω γ γ κ φ
α
− − = (2.4.3)
Its interaction with the free surface will generate a triplet of reflected waves, consisting of
a fast and slow P-wave and an S-wave, represented by their wave potentials
] ) cos sin ( exp[
1
t i z x ik K
f f f f
r
f
ω θ θ ϕ
α α
− + = (2.4.4a)
] ) cos sin ( exp[
1
t i z x ik K
s s s s
r
s
ω θ θ ϕ
α α
− + = (2.4.4b)
] ) cos sin ( exp[
2
t i z x ik K
r
ω θ θ ψ
β β β
− + = (2.4.4c)
40
Fig. 2.5.1 Fluid-saturated porous half-space subjected to incident P-wave.
where
f
k the complex wave number of the fast plane P-wave
s
k the complex wave number of the slow plane P-wave
β
k the complex wave number of the shear plane wave
We recall that the wave numbers are in general complex, when the dissipation forces due
to the seepage force are considered. These wave numbers can be written in terms of
their real and imaginary parts as
41
ki jkr
ki η η =− , , , j fs β = (2.4.5)
To ensure dissipation at infinity
ki
η in eqn (2.4.3) should be positive. In this work, the
angles of incident wave and of the reflected waves are taken as in H. Deresiewicz and
Rice (1962)
() ( ) ()
sin
arc tan
sgn
k kr k
j kk
lr l li l l
k
p qn
η θ
θ
ηη
=
−
(2.4.6)
where
k
j
θ is the reflection angle,
k
θ the incident angle, and ,and
kr li
η η are the real and imaginary
parts of the wave numbers.
()
22
sin
k kr li ki lr
l
lr li
n
k
ηηηη
θ
ηη
−
=
+
(2.4.7)
The reflection angle sine and cosine for the case of l k ≠ can be written as
() () ( )
1
sin
kk
ll
m in = +
k
l
(2.4.8)
42
k ( ) () () ( )
3
cos sgn
kk k
ll l l
p iq n = (2.4.9)
where
()
22
sin
k kr lr ki li
l
lr li
m
k
ηηηη
θ
ηη
+
=
+
(2.4.10)
() ( )
1
2
22
() () () () ()
1
,
2
kk k k k
ll l l l
pq K L K
⎡
=+ ±
⎢
⎣⎦
⎤
⎥
k
l
k
l
⎧⎫
⎪
⎬
⎪
⎩⎭
⎧⎫
, (2.4.11)
() ( )
22
() ( ) ()
1
kk
ll
Km n =− + (2.4.12a)
and
() () ( )
2
k k
l l
Lmn = (2.4.12b)
To determine the reflection coefficients both boundary conditions at the free surface are
taken (open boundary and sealed boundary condition):
For open (permeable) half-space surface, the zero stress condition implies
11,1 11,2 12 1 11,1
21,1 21,2 22 1 0 21,1
61,1 61,2 2 61,1
0
f
s
GG G K G
GG G K aG
GG K G
⎡⎤ − ⎧⎫
⎪⎪ ⎪ ⎢ ⎥
−− =−
⎨⎬ ⎨
⎢ ⎥
⎪⎪ ⎪
⎢⎥
⎩⎭
⎣⎦
(2.4.13)
and for sealed (impermeable) half-space surface
11,1 61,1 11,2 61,2 12 1 11,1 61,1
21,1 21,2 22 1 0 21,1
141,1 2 41,2 3 42 2 141,1
(1 ) (1 ) (1 ) (1 )
f
s
GG G G G K GG
GG G K aG
fG f G f G K f G
⎡⎤ ++ − + ⎧⎫
⎪ ⎪⎪ ⎢ ⎥
−− =−
⎪
⎨ ⎬⎨
⎢ ⎥
⎬
⎪ ⎪⎪
⎢⎥
−− −− − −
⎩⎭
⎣⎦
⎪
⎩⎭
(2.4.14)
where
) sin 2 (
2
1
2
1 , 11 f f
M k G
α α
θ − − = (2.4.15a)
) sin 2 (
2
2
2
2 , 11 s s
M k G
α α
θ − − = (2.4.15b)
β β
θ 2 sin
2
12
k G = (2.4.15c)
f f
k G
α α
θ 2 sin
2
1 , 21
− = (2.4.15d)
s s
k G
α α
θ 2 sin
2
2 , 21
− = (2.4.15e)
β β
θ 2 cos
2
22
k G − = (2.4.15f)
f f
ik G
α α
θ cos
1 , 41
= (2.4.15g)
s s
ik G
α α
θ cos
2 , 41
= (2.4.15h)
β β
θ sin
42
ik G − = (2.4.15i)
1
2
1 , 61
S k G
f α
− = (2.4.15j)
2
2
2 , 61
S k G
s α
− = (2.4.15k)
2.5.2 The Free Field Motion due to Incident SV-Wave
Similarly as for the fast incident P-wave, an incident SV wave can be represented by
its potential
exp[ ( sin cos ) ]
i
ix z it
α
ψ κγ γ = − ω− (2.4.16)
and the interaction with the free-surface will generate the reflected fast and slow P-wave
and an SV-wave, represented by their potentials
] ) cos sin ( exp[
1
t i z x ik K
f f f f
r
f
ω θ θ ϕ
α α
− + = (2.4.17)
] ) cos sin ( exp[
1
t i z x ik K
s s s s
r
s
ω θ θ ϕ
α α
− + = (2.4.18)
43
] ) cos sin ( exp[
2
t i z x ik K
r
ω θ θ ψ
β β β
− + = (2.4.19)
The critical angle for the fast P-wave is
1
sin / 2
cr
PQ R θµ
−
= ++ (2.4.20)
The zero stress condition for open (permeable) half-space surface implies
11,1 11,2 12 1 12
21,1 21,2 22 1 0 22
61,1 61,2 2
00
f
s
GG G K G
GG G K bG
GG K
∗
∗
⎧ ⎫ ⎡⎤ − ⎧⎫
⎪ ⎪ ⎪⎪ ⎢ ⎥
−− =−
⎨⎬ ⎨
⎢ ⎥
⎪⎪ ⎪
⎢⎥
⎩⎭
⎣⎦ ⎩⎭
⎬
⎪
(2.4.21)
and for sealed (impermeable) half-space surface it implies
11,1 61,1 11,2 61,2 12 1 12
21,1 21,2 22 1 0 22
141,1 2 41,2 3 42 2 3 42
(1 ) (1 ) (1 ) (1 )
f
s
GG G G G K G
GG G K bG
fG f G f G K f G
∗
∗
∗
⎧ ⎫ ⎡⎤ ++ − ⎧⎫
⎪ ⎪ ⎪⎪ ⎢ ⎥
−− =−
⎨ ⎬⎨
⎢ ⎥
⎬
⎪ ⎪⎪
⎢⎥
−− −− − −
⎩⎭
⎣⎦ ⎩⎭
⎪
(2.4.22)
2.5.3 The Free Field Motion due to Rayleigh-Wave
A Rayleigh in the half-space propagating in the positive x- direction can be
represented by its potentials
( ) 1 f
by ikx t
ff
ce e
ω
φ
− −
=
( )
1s
ikx t by
ss
ce e
ω
φ
− −
= (2.4.23)
( )
2
ikx t by
De e
ω
ψ
− −
=
where
2
1 2
1
f
f
c
bk
V
α
=− (2.4.24a)
44
2
1 2
1
s
s
c
bk
V
α
= − (2.4.24b)
2
2 2
1
c
bk
V
β
= − (2.4.24c)
The zero stress condition for open (permeable) half-space surface implies
11,1 11,2 12 1
21,1 21,2 22 1
61,1 61,2
0
0
00
f
s
GG G b
GG G b
GG D
∗∗ ∗∗ ∗∗
∗∗ ∗∗ ∗∗
∗∗ ∗ ∗
⎡⎤ − ⎧ ⎫⎧ ⎫
⎢ ⎥ ⎪ ⎪⎪ ⎪
−− =
⎨ ⎬⎨ ⎬
⎢ ⎥
⎪ ⎪⎪ ⎪
⎢⎥
⎩⎭ ⎩⎭
⎣⎦
(2.4.25)
and for sealed (impermeable) half-space surface it implies
11,1 61,1 11,2 61,2 12 1
21,1 21,2 22 1
141,1 2 41,2 3 42
0
0
(1 ) (1 ) (1 ) 0
f
s
GG G G G b
GG G b
fG f G f G D
∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗
∗∗ ∗∗ ∗∗
∗∗ ∗∗ ∗∗
⎡⎤ ++ − ⎧ ⎫⎧ ⎫
⎢⎥ ⎪ ⎪⎪ ⎪
−− =
⎨ ⎬⎨ ⎬
⎢⎥
⎪ ⎪⎪ ⎪
⎢⎥
−− −− −
⎩⎭ ⎩⎭
⎣⎦
(2.4.26)
where
22
2 1
11,1 2
2
f
kc M
Gk
V
α
∗∗
=− (2.4.27a)
22
2 2
11,2 2
2
s
kc M
Gk
V
α
∗∗
=− (2.4.27b)
2
2
12 2
21
c
Gik
V
β
∗∗
=− − (2.4.27c)
2
2
21,1 2
21
f
c
Gik
V
α
∗∗
=− (2.4.27d)
45
2
2
21,2 2
21
s
c
Gik
V
α
∗∗
= − (2.4.27e)
22
2
22 2
2
kc
Gk
V
β
∗∗
=− (2.4.27f)
22
1
61,1 2
f
Sk c
G
V
α
∗∗
=− (2.4.27g)
22
2
61,2 2
s
Sk c
G
V
α
∗∗
=− (2.4.27h)
For a nontrivial solution, the determinate of the matrices in eqns (2.4.25) and (2.4.26)
have to be equal to zero, which gives for open (permeable) half-space surface
() ( ) ( )
()
222 2 22 22
21 1 2 12 21
22 2 2
422 22
12 2 1 2 1 22 2 2
det 4 2 2
41 1 4 1 1
fs s f
fs
fs s s
SV SV V SV S V c SM S M V c
cc c c
MS M S c SV V SV V
VV V V
αα β α α β
αβ α β
αα α α
=− + − + −
+− − − − + − − 0=
(4.2.28)
and for sealed (impermeable) half-space surface
46
()()()
()()()()
()()()
22 2 2 2 2
1
22 2 2 2 2
2
31 2
22 2 4 2
1
2
22 2 2 2 2 42
31 2 1 1
2
22 2 2 22
32 1
42
14 2 1
2 det 1
12 1 2
4
14 2 1
fs s
fs s
fs s
s
sss
fs
fs f
VV V cMV V
f V VV cMVV f
c
cV V c MV
V
fScVV f ScVV ScV
VV
fVVV cMVV f
αα β α β
αα β α β
αα α
α
αβ αβ α
αα
αα β α β
⎡⎤ ⎛⎞ −−
⎢⎥ ⎜⎟ −− − + −
⎜⎟
+ ⎢⎥ =−
⎝⎠
⎢⎥
+− + − − + ⎢⎥
⎣⎦
−− +−
+
()()()()
()
22 2 2 2
2
2
22 2 4 2
2
2
22 2 2 2 2 4 2
32 1 2 2
22 2
22 2
21 22 2
2
2 1
12 1 2
41 1 1 0
f
fs f
f
fsf
fs
fs
VcMVV
c
cV V c M V
V
f S cV V f ScV V ScV
cc c
ff VVV
VV V
βαβ
αα α
α
αβ α β α
αα β
αα β
⎡⎤ ⎛⎞ −−
⎢⎥ ⎜⎟
⎜⎟
+ ⎢⎥ −
⎝⎠
⎢⎥
−− − − − + ⎢⎥
⎣⎦
⎡⎤ +− − − − =
⎣⎦
(2.4.29)
Equations (2.4.28) and (2.4.29) are the characteristic equations the solution of which
gives the allowable phase velocities of Rayleigh waves, , for both conditions. Then by
substituting the corresponding velocity in eqns (4.2.25) and (4.2.26), the coefficients of
the wave potentials can be determined.
c
47
2.5.4 Displacements and Stresses due to the Free-Filed Motion
2.5.4.1 Rectangular Coordinates
Once the amplitude coefficients of the wave potentials have been determined, the
displacements and stresses in the half-space can be computed. For a given triplet of
potentials, the displacements and stresses in rectangular coordinates can be computed
using the following relationships
48
) ( 31,1 31,2 32
41,1 41,2 42
0
f
x ikx t
s
y
y
C
u GG G
Ce
u GG G
D
ω
∗∗ ∗∗ ∗∗
−
∗∗ ∗ ∗ ∗∗
=
⎛⎞
⎡⎤ − ⎛⎞
⎜⎟
=
⎢ ⎥ ⎜⎟
⎜⎟
−−
⎝⎠⎣⎦⎜⎟
⎝⎠
(2.4.30)
and
(
11,1 11,2 12
21,1 21,2 22
61,1 61,2
0
0
yy f
ikx t
xy s
y
GG G C
GG G Ce
GG D
) ω
τ
τµ
σ
∗∗ ∗ ∗ ∗∗
− ∗ ∗ ∗∗ ∗∗
∗∗ ∗∗
=
⎡⎤ − ⎛⎞ ⎛ ⎞
⎢ ⎥ ⎜⎟ ⎜ ⎟
=− −
⎢ ⎥ ⎜⎟ ⎜ ⎟
⎜⎟ ⎜ ⎟
⎢⎥
⎝⎠ ⎝ ⎠
⎣⎦
(2.4.31)
where
fs
C ,C and D are the known coefficients of the wave potentials.
31,1
Gi
∗∗
=k
k
31,2
Gi
∗∗
=
32 2
Gb
∗∗
=
41,1 1 f
Gb
∗∗
=
41,2 1s
Gb
∗∗
=
42
Gi
∗∗
=−k
2.5.4.2 Cylindrical Coordinates and Series Expansion along the
Contact Surface
To match the displacements and stresses along the contact surface, they have to be
represented in Fourier-Bessel series. For the plane waves, the representation can be done
easily after expansion of the corresponding wave potentials in Fourier-Bessel series using
the addition theorem. For surface waves, however, these series diverge, and the addition
theorem cannot be used. This problem can be avoided by computing the displacements
and stresses along the contact surface,
1
r b = , and then expanding them in finite Fourier
series of
1
θ , as proposed by Lee and Cao (1989).
The derivations for the inhomogeneous waves are greatly simplified by representing
them as plane waves but for a complex reflection angles. These angles are
2
ff
i
α α
π
θ φ =−
2
s s
i
α α
π
θ φ =− (2.4.32)
2
i
β β
π
θ φ =−
where
f α
φ ,
s α
φ ,
β
φ are real quantities such that
cosh /
ff
Vc
α α
φ =
cosh /
ss
Vc
α α
φ =
cosh / Vc
β β
φ =
49
Then the potentials in terms of these complex angles are
( ) ( ) 11
1
cos
ff
f
ik r i t
bd
ff
Ce e
αα
θθω
φ
−−
=
( ) ( ) 11
1
cos
s s
s
ik r i t
bd
ss
Ce e
α α
θθ ω
φ
−−
= (2.4.33)
( ) ( ) 11
2
cos ik r i t
bd
De e
ββ
θθ ω
ψ
−−
=
Displacements and stresses can be computed in cylindrical coordinates from the
potentials using the following relationships
1
11
1
j
r
u
rr
1
φ
ψ
θ
∂
∂
=+
∂∂
(2.4.34a)
1
11 1
1
j
u
rr
θ
φ
ψ
θ
∂
∂
= −
∂∂
(2.4.34b)
and
11 1 1 1 1
11
1
11 1 1 1 1 1 1
1 1
2
rr r r
rr
r
uu u U U U u
Q
rr r r r r
θ θ
τλ µ
θθ
⎛⎞ ∂∂ ∂ ∂ ⎛⎞
=++ + + + + ⎜ ⎟
⎜⎟
⎜ ⎟
∂∂ ∂ ∂
⎝⎠
⎝⎠
1
r
∂
∂
(2.4.35a)
11 1
11
11 1 1
1
r
r
uu u
rr r
θθ
θ
τµ
θ
∂∂ ⎛⎞
=++
⎜
∂∂
⎝⎠
⎟
(2.4.35b)
This gives for the displacements along the canyon rim
1
r b = and for
00
θ θ θ −≤ ≤
()
() ()
()
1
1 1
1
1
cos cos
1
,
sin
f
s
A
A
ff f s s s
it
r
A
Cik b e Cik b e
ub e
b
Dik b e
β
αα α α
ω
ββ
θθ θθ
θ
θθ
−
⎡⎤
−+ − −
⎢⎥
=
⎢⎥
−
⎣⎦
(2.4.36a)
()
() ()
()
1
1 1
1
1
sin sin
1
,
cos
f
s
A
A
ff f s s s
i t
A
Cik b e Cik b e
ub e
b
Dik b e
β
αα α α
ω
θ
ββ
θθ θθ
θ
θθ
−
⎡⎤
−− − −
⎢⎥
=
⎢⎥
−
⎣⎦
−
(2.4.36b)
for the solid,
50
()
() ()
()
1
11 2 1
1
31
cos cos
1
,
cos
f
s
A
A
ff f s s s
it
r
A
fC ik b e f C ik b e
Ub e
b
fDik b e
β
αα α α
ω
ββ
θθ θθ
θ
θθ
−
⎡⎤
−+ − −
⎢⎥
=
⎢⎥
−
⎣⎦
(2.4.37a)
()
( ) ()
()
1
11 2 1
1
31
sin sin
1
,
cos
f
s
A
A
ff f s s s
i t
A
f C ik b e f C ik b e
Ub e
b
fDik b e
β
αα α α
ω
θ
ββ
θθ θθ
θ
θθ
−
⎡⎤
−− − −
⎢⎥
=
⎢⎥
−
⎣⎦
−
(2.4.37b)
for the fluid. Similarly, the stresses acting on the solid and on the fluid along the canyon
rim are
()
() ()
() ()
() ( ) ( )
11
2
2 1
1 2
2
2 2
1 1 2
2
11 2
2
1sin
22
2
,1 sin
22
2
sin cos
f
s
A
ff f
A i t
rr s s s
A
fQ
Ck b e
b
fQ
bCkb e
b
Dk b e
b
β
αα
e
ω
α α
ββ β
µλ
θθ
µλ
τθ θ θ
µ
θθ θθ
−
⎡⎤ − ⎡⎤
++ − − −
⎢⎥ ⎢⎥
⎣⎦
⎢⎥
⎢⎥
⎡ ⎤
=++−− +
⎢⎥
⎢ ⎥
⎣⎦
⎢⎥
⎢⎥
−−
⎢⎥
⎣⎦
(2.4.38a)
()
() ( ) () ( )
() ()
11
2
2
11
1 2 2
2
1
sin sin
2
,
1
sin
2
f
s
A
A
ff f s s s
i t
r
A
Ck b e C k b e
be
b
Dk b e
β
αα α α
ω
θ
ββ
θθ θθ
µ
τθ
θθ
−
⎡⎤
−+ − +
⎢⎥
=
⎢⎥
⎡⎤
−−
⎢⎥
⎢⎥
⎣⎦ ⎣⎦
(2.4.38b)
and
() () ()
2
2
11 2 2
1
,
f
s
A
A i t
ff s s
bSCkbeSCkbe
b
e
ω
α α
σθ
−
⎡
=− −
⎢
⎣
⎤
⎥
⎦
(2.4.38c)
where
()
11 1
cos
ff f f
Abd ik r
αα
θ θ =+ −
()
11 1
cos
s ss
Abd ikr
ααs
θ θ =+ −
()
21 1
cos Abd ikr
β ββ
θ θ =+ −
51
Chapter 3: Numerical Results and Analysis
This chapter presents numerical results for: the wave velocities, free-field motion,
foundation input motion, foundation stiffness and damping, and system response in the
form of a parametric study, with the objective of understanding the effects of the many
model parameters. The following Chapter 6 shows results for a model with properties
similar to those for Millikan library, attempting to explain the observed effects.
The results were computed using a FORTRAN computer program SSI_POROUS
(Todorovska and Al Rjoub, 2006a), which was generalized to work for a dissipative
medium (i.e. for a mixture that has finite permeability and is saturated with viscous
fluid), and for partially saturated soils. This generalization required computation of
Bessel functions of complex arguments, and subroutines from Zhang and Jin (1996) were
used. The computer program was written in terms of dimensionless parameters, defined
using as reference: length a , material modulus
s
µ , and mass density
gr
ρ . Then, the
system response is a function of the following dimensionless parameters: stiffness of the
fluid relative to the skeleton, defined through the ratio /
f s
K µ and the Poisson’s ratio
s
ν ;
mass density of the skeleton relative to the fluid, defined through the ratio
gr
/
f
ρ ρ and
the porosity ; mass of the building relative to the mass of the foundation, and mass of
the foundation relative to the mass of the replaced soil, through the ratios
ˆ n
b fnd
/ mm and
, where,
fnd gr
/ m m
gr fnd gr
mA ρ =
)
is the mass of the excavated soil (per unit length) if there
were no voids, and is the area of the foundation; the flexibility of the building
relative to that of the soil, through the ratio
fnd
A
ref ,b
()/(
S
VH V a ε =
52
[ ]
,b ref
()/ /( )/
S
HV a V ω ω ⎡⎤ =
⎣⎦
= ratio of the number of wavelength in the shear beam in
length H and the number of reference wavelengths in the soil in length a; dimensionless
frequency
ref
/( ) aV η ωπ = , where
ref gr
/
s
V µ ρ = is a reference velocity; foundation
shape, through the ratio ; and on the type, amplitude and angle of the incident waves. / h a
3.1 Soil Constitutive Properties and Waves Velocities
The wave velocities in the soil and the frequency of motion are fundamental for
significance of the effects of the soil-structure interaction. When the effects of the
seepage force are considered, the wave velocities depend themselves on the degree of
dissipation and also on frequency. Hence, the analysis begins with understanding how
the wave velocities depend on frequency and on the soil permeability (for fixed value of
viscosity of the fluid), which will help later on, interpret the results for the foundation
stiffness and damping.
3.1.1 Input Model Parameters
The following range of the input parameters was considered in the analysis. The pore
fluid in this study is water, which has mass density kg/m
3
10
w
ρ =
3
and bulk modulus
Pa, giving bulk wave velocity
9
2.2 10
w
K =× /
w w
K ρ =1,483 m/s. For full saturation,
f w
K K = and
f w
ρ ρ = . For the mass density of the material of which the grain is made
value was used. The ratio
3
gr
2.7 10 kg/m ρ =×
3
/
f
K µ , where
f w
K K = describes the
stiffness of the skeleton, which is varied so that /
f
K µ = 0.01, 0.1, 1, and 10. The value
/
f
K µ = 0.01 corresponds to soft soil, /
f
K µ = 0.1 to stiff soil, and /
f
K µ = 1 and 10 to
53
porous rock (Lin et al., 2001). Only one value of Poisson ratio is considered,
s
ν =0.3, and
two values of porosity, n
=0.3 and 0.4, which are representative for soils. The
dissipation depends on the ratio
ˆ
ˆ / k µ , where ˆ µ is the absolute viscosity of the fluid, and
is the intrinsic permeability of the skeleton. The absolute viscosity of water at about
25 C is
ˆ
k
° ˆ
w
µ =
-3
0.89 10 × Pa s ⋅ (1 Pa s ⋅ =1 ). In this work, a rounded value
is used. The intrinsic permeability of the skeleton depends on the
type of geologic material. Pervious consolidated geo materials, such as highly fractured
rock, and pervious unconsolidated geo materials such as well-sorted gravel and well-
sorted sand and gravel, have intrinsic permeability in the range 10 .
Semi-pervious consolidated geo materials, such as oil reservoir rocks and fresh
sandstone, and semi-pervious unconsolidated geo materials, such as very find sand, silt,
loess and loam, have in the range
2
N s/m
3
ˆˆ 10
w
µµ
−
==
2
N s/m
ˆ
k
6 10
10
−−
−
2
m
ˆ
k
11 14
10 10
− −
−
2
m (Bear, 1972). Geo materials with
in the range are considered to be impervious. In this work, results are
shown for , and also for the case when the effects of the dissipative force
are neglected, and the wave velocities are real values (this case is referred to as the “no
seepage force” or the “ ” case). For dry soil (pores filled with air), the following
values are taken:
ˆ
k
15 19
10 10
− − 2
m −
− 12 6
ˆ
10 10 k
−
≤≤
ˆ
k →∞
gr
/
f
ρ ρ =0.001 and /
f
K
s
µ =0.001, and the dry shear wave velocity is
computed as
,dry gr
ˆ /[(1 ) ]
Ss
V n µ ρ = − . Another variable of the model is the frequency
of motion.
54
The upper bound of frequency is constrained by the requirements that: (1) the flow of
the fluid in the pores is laminar, and (2) the wavelengths are much larger than the size of
the pores. The first requirement implies frequency bound
2
ˆ
4
t
f
d
π ν
= , where d is the
diameter of the pores, and ˆ ν is the kinematic viscosity, which is related to the dynamic
viscosity ˆ µ and fluid density
f
ρ by ˆ ˆ /
f
ν µρ = (Biot, 1956a). This implies maximum
frequencies Hz for 10,000
t
f = 0.01 d = mm (silt), 100
t
f = Hz for mm (sands),
and Hz for mm (coarse sands to gravel). We consider frequencies not
larger than 100 Hz in presenting results for the wave velocities and free-field motions,
and dimensionless frequency
0.1 d =
10
t
f = 1 d =
5 η ≤ , where
ref ref
/( ) (2 ) / aV afV η ωπ = = , and
, and is the characteristic dimension for the wave scattering problem and
radiation problem. Then for the softest soils considered (
ref S,dry
V V = 2a
/
f
K µ =0.01), for which the
smallest wave velocity is of the order of 100 m/s, and for =24 m, which is the value
used for the case study (Millikan library), the maximum value of
2a
5 η = implies highest
frequency Hz, which is low enough for the flow to be laminar. 20 f =
3.1.2 Wave Velocities for Full Saturation as Function of the Model
Parameters
For very small
ˆˆ 1
ˆ
n
k
µ
ω
, which has dimension of mass density, the seepage force is much
smaller than the inertial forces, and its effects are negligible (see eqns (2.2.7c,d)). In that
case, the wave velocities are real valued, do not depend on frequency, and are used in this
55
section as reference in showing how much the wave velocities change as result of the
seepage force. Table 3.1.1 shows these frequency independent velocities for different
combinations of all other input parameters. The S and P –wave velocity of the dry solid
is also shown, used as reference in studying the effects of saturation.
Table 3.1.1 Wave velocities for fully saturated soil for the case of no seepage force.
/
f
K µ
s
ν n
S,dry
V
[m/s]
P,dry
V
[m/s]
S
V
[m/s]
Pf
V
[m/s]
Ps
V
[m/s]
0.01 0.3 0.3 107.9 201.9 103.6 1,922.4 101.5
0.1 0.3 0.3 314.2 638.5 327.5 1,986.8 310.7
1.0 0.3 0.3 1,078.9 2,018.9 1,033.6 2,620.5 745.0
10.0 0.3 0.3 3,411.8 6,384.6 3,274.7 6,344.7 973.0
0.01 0.3 0.4 116.5 218.1 110.8 1,763.5 131.8
0.1 0.3 0.4 368.5 689.5 350.4 1,831.1 401.5
1.0 0.3 0.4 1,165.3 2,180.4 1,108.2 2,561.0 907.7
10.0 0.3 0.4 3,685.1 6,894.9 3,504.4 6,697.9 1,097.5
Figure 3.1.1 shows the variation of the normalized wave velocities with inverse
permeability , which is proportional to the seepage force. In this one and all other
figures, , , the wave velocities are normalized by their respective value for no
seepage force (
1/ K
ˆ
Kk ≡ ˆ n n ≡
ˆˆ 1
ˆ
n
k
µ
ω
=0, see Table 3.1.1), and the real parts of the wave velocities are
shown on the left hand side, and the imaginary parts are shown on the right hand. In this
figure, the different curves correspond to different values of /
f
K µ , and the porosity is
56
ˆ 0.4 n = . Parts a) and b) differ in that in part a) all the results are for same absolute
frequency, set to 1 f = Hz, while in part b), all the results are for same relative
frequency, which is the dimensionless frequency set to
,dry
/( )
S
aV η ωπ = set to 1 η = ,
where the reference length a = 12 m. Because is different for each
,dry S
V /
f
K µ (see
Table 3.1.1), the absolute frequency is different for the different values of /
f
K µ ,
having values f = 3.76, 11.9, 37.6 and 118.9 Hz for /
f
K µ = 0.01, 0.1, 1 and 10. Part b)
is shown because the effects of scattering and radiation depend on the dimensionless
frequency rather than on the absolute frequency. The results in Fig. 3.1.1 show that the
velocities vary significantly only within a band of values of permeability. Outside this
band, they rapidly approach their asymptotic values, which are real valued. The velocity
of the fast P-wave is affected little by the seepage force, decreasing only by up to about
5%. The velocity of the slow P-wave is affected most by the seepage force, decreasing to
zero for very large seepage force. The velocity of the S-waves reduces by up to about
40% of its value for zero seepage force. A comparison of the results in parts a) and b)
shows that, for fixed permeability and absolute frequency f , the change of the wave
velocities with permeability does not depend on the relative stiffness of the skeleton,
although their absolute values are very much dependent on the relative stiffness of the
skeleton (see Table 3.1.1). For fixed relative frequency,
,dry
/( )
S
aV η ωπ = , the change
of the wave velocities is similar but occurs within a different range of values of 1/ .
The change starts to become significant for smaller permeability for materials with stiffer
skeleton.
K
57
Fig. 3.1.1 Normalized wave velocities of the mixture versus inverse permeability for
different values of /
f
K µ , and for porosity ˆ n = 0.4. In part a), the absolute frequency is
set to 1 f = Hz, and in part b) the relative frequency is set to
,dry
/( )
S
aV η ωπ = =1, where
the reference length 12 m. a =
58
Figure 3.1.2 also shows the variation of the normalized wave velocities with inverse
permeability , but the different curves correspond to different values of frequency
, 1, 10 and 100 Hz. In part a),
1/ K
0.1 f = /
f
K µ =0.1 (stiff soil), in part b) /
f
K µ =0.01
(soft soil), and in both parts the porosity is ˆ n = 0.4. It can be seen from this figure that,
for smaller frequencies of motion, the seepage force starts to affect the wave velocities at
smaller 1/ , i.e. at higher permeability. This trend is as expected, because the effect of
the seepage force is through the combination
K
ˆ
ˆˆ /( ) nk µ ω .
Fig. 3.1.3 shows the normalized wave velocities versus frequency f in Hz on a
logarithmic scale for different values of inverse permeability 1/ . As in Fig. 3.1.2, in
part a),
K
/
f
K µ =0.1 (stiff soil), in part b) /
f
K µ =0.01 (soft soil), and in both parts the
porosity is 0.4. It can be seen that the trend of the variation of the normalized wave
velocities with increasing frequency is the same as the trend with increasing permeability
(see Fig. 3.1.2), which is due to the fact that in Biot’s theory, the effect of the seepage
force is through the ratio
ˆ n =
ˆ
ˆˆ /( ) nk µ ω .
59
Fig. 3.1.2 Normalized wave velocities of the mixture versus inverse permeability for
different values of frequency, 0.1 f = , 1, 10 and 100 Hz. The porosity is , and ˆ 0.4 n =
f
/K µ =0.1 (part a) and 0.01 (part b).
60
Fig. 3.1-3 Normalized wave velocities versus frequency for different values of
permeability. The porosity is ˆ 0.4 n = , and
f
/K µ =0.1 (part a) and 0.01 (part b).
61
3.1.3 The Effect of Partial Saturation on the Wave Velocities
This section illustrates the variation of the wave velocities of the mixture as function
of the degree of saturation, accounted for as described in Section 2.2.3. Fig. 3.1.4 shows
the variation of the adjusted bulk modulus of the fluid,
f w
K K ≤ , plotted on a
logarithmic scale, as function of the fraction of air in the pores, 1 , where is the
saturation (ratio of the volume of the pore water and the volume of the pores). The
different curves correspond to different values of the absolute pore pressure ( in eqn
(2.2.20)), which takes values 0.2, 0.6 and 1 MPa. It can be seen that the modified bulk
modulus reduces rapidly as the saturation becomes partial even for very small fraction of
air. Fig. 3.1.5 shows the variation of the modulus of the (complex) velocities of the fast
and slow P-wave versus 1 , for porosity
r
S −
r
S
a
P
r
S − ˆ 0.4 n = , and / 0.01
w
K µ = (soft soil). It can
be seen that both wave velocities decrease rapidly with increasing fraction of air content,
the velocity of the fast P-wave approaching the P-was velocity in the dry solid (about 200
m/s, see Table 3.1.1), and the velocity of the slow P-wave approaching zero. In this
model, which is used only for very high degrees of saturation ( ), the effects of
the saturation on the mass densities is ignored, and hence, the shear wave velocity does
not depend on the degree of saturation.
90%
r
S >
62
Fig. 3.1.4 Modified bulk modulus of the pore fluid versus fraction of air, 1
r
S − .
Fig. 3.1.5 Modified velocities of fast and slow P-waves versus fraction of air, 1
r
S − ,
for porosity and ˆ 0.4 n = / 0.01
w
K µ = (soft soil).
63
3.2 Foundation Complex Stiffness Matrix
This section shows results for the real and imaginary part of the foundation
impedance matrix, where the real part describes the foundation stiffness, and the
imaginary part is related to the damping due to radiation of energy from a vibrating
foundation. Section 3.2.1 shows results for fully saturated soils, and Section 3.2.2 for
partially saturated soils. In all figures, the Poisson’s ratio is 0.3 ν = , the porosity is
, and the dimensionless frequency is defined with respect to the shear wave
velocity of the dry solid,
ˆ 0.4 n =
, s dry
a
V
ω
η
π
= , which is independent of the state of saturation.
Results are shown only for “stiff” and “soft” soils, i.e. for / 0.1
f
K µ = and 0.01, because
the effects of the pore water are more significant for solids with relatively soft skeleton.
3.2.1 Foundation Complex Stiffness Matrix for Fully Saturated Soils
Figure 3.2.1 shows results for / 0.1
f
K µ = , for different values of permeability. Part
a) shows results for permeable (open) foundation-soil interface, and part b) for
impermeable (sealed) interface. It is noted here that the hydraulic condition on the half-
space surface does not affect the results, as the effect of the free surface on the scattered
waves from the foundation was neglected in the development of the model. In each part,
the plots on the left show the real part and those on the right show the imaginary part of
the corresponding foundation stiffness matrix coefficient, and the three rows of plots
show respectively (horizontal and vertical stiffness which are equal for this
model for a semi-circular foundation), (rocking stiffness), and (coupling term
11 22
= K K
33
K
23
K
64
Fig. 3.2.1 Foundation dynamic stiffness coefficients for different values of skeleton
permeability, and for porosity ˆ 0.4 n = and / 0
f
K .1 µ = . Pat a) shows results for
permeable (open), and part b) for impermeable (sealed) contact surface.
65
Fig. 3.2.2 Foundation dynamic stiffness coefficients for different values of skeleton
permeability, and for porosity ˆ 0.4 n = and / 0.
f
K 01 µ = . Pat a) shows results for
permeable (open), and part b) for impermeable (sealed) contact surface.
66
67
01
between horizontal and rocking motions). Similarly, Fig. 3.2.2 shows results for
/ 0.
f
K µ = (soft soil). The results are discussed in what follows.
A noted in Todorovska and Al Rjoub (2006b) the effect of saturation for this shape
of embedded foundation is such that it affects significantly both the horizontal and the
vertical stiffness, while the effect on the rocking and coupling stiffness coefficients is
very small. This can be explained by the fact that the rocking motion of the foundation
results only in shear deformations in the soil and motion of soil tangent to the contact
surface, which does not cause flow of fluid perpendicular to the foundation-soil interface
(hence pressure from the fluid onto the foundation). What is different in this thesis from
the study in Todorovska and Al Rjoub (2006b) is that the effects of the seepage force are
considered, which, as noted in Section 3.1 lead to complex valued wave velocities in the
soil, and also the effects of partial saturation. Hence, in this thesis work, the emphasis is
on analyzing the effects of finite permeability and partial saturation.
Todorovska and Al Rjoub (2006b) explained the trend of the effect of the pore water
as increasing the stiffness of the foundation for small frequencies for which the water
moves in phase with the solid, but the effects reverses for high enough frequency, when
the pore water moves in the opposite direction of the solid, and it reduces the foundation
stiffness. For very stiff skeleton (e.g. rock) the effect is very small, while for some very
soft soils, the window of frequencies where there would be an increase of stiffness may
be very small for the increase to be noticeable. The presence of a dissipative force is
expected to increase the dynamic stiffness of the foundation. Figures 3.2.1 and 3.2.2
shows that the horizontal and vertical stiffness (real part of ) do increase with
11 22
= K K
decreasing permeability (i.e. with increasing seepage force), but only up to a certain value
of permeability. For smaller permeability than that value, the foundation stiffness
decreases with further decrease of permeability, but only for smaller η , and for large
enough η it exceeds the stiffness for larger values of permeability. This change in the
trend can be explained by the dependency of the wave velocities both on frequency and
permeability.
To observe better and explain these effects, the next four figures, 3.2.3 through 3.2.6,
show in part a) enlarged plots of versus
11 22
= K K η from Figs 3.2.1 and 3.2.2, and in part
b) they show the variation of the wave velocities with dimensionless frequency η .
Again, the real parts of both complex stiffness coefficients and velocities are shown on
the left hand side, and the imaginary parts are shown on the right hand side. Figures 2.2.3
and 2.2.4 show results for / 0.1
f
K µ = , respectively for permeable and impermeable soil-
foundation interface. Figures 2.2.5 and 2.2.6 show the same cases and quantities but for
/ 0.
f
K 01 µ = . It can be seen from these figures that, for fixed frequency, the wave
velocities are larger for more permeable materials. For given permeability, the wave
velocities increase with increasing η , and their value and rate of the increase are different
in different frequency intervals. For higher permeability, the wave velocities reach their
high frequency asymptote, which is their value for zero seepage force. Hence, for
saturated realistic soils (with finite permeability and nonzero viscosity) and for the
frequencies of interest in earthquake engineering, the variation of the foundation stiffness
with frequency is governed by two competing mechanisms, one through the flow of fluid
68
through the pores which affects the wave velocities, and the other one via the associated
wave phenomena (scattering and diffraction), which depend on the relative size of the
foundation and the wavelength of the incident waves.
A comparison of the foundation stiffness for different hydraulic condition at the
interface shows that, for smaller η , when the pore water moves in phase or nearly in
phase with the skeleton, it’s the stiffening is larger for an impermeable foundation than
for permeable one. For the foundation damping it is the opposite – the damping is larger
a permeable foundation, and it is larger for less permeable soils. The foundation
damping is also larger for softer soils than for stiff soil.
69
Fig. 3.2.3 Comparison of variations of horizontal/vertical foundation complex
stiffness (part a)) and variations of the complex wave velocities (part b)) with
dimensionless frequency eta for different values of permeability, for porosity ˆ 0.4 n =
and / 0
f
K .1 µ = , and for p ermeable (open) contact surface.
70
Fig. 3.2.4 Same as Fig. 3.2.3 but for impermeable (sealed) contact surface.
71
Fig. 3.2.5 Comparison of variations of horizontal/vertical foundation complex
stiffness (part a)) and variations of the complex wave velocities (part b)) with
dimensionless frequency eta for different values of permeability, for porosity ˆ 0.4 n =
and / 0.
f
K 01 µ = , and for permeable (open) contact surface.
72
Fig. 3.2.6 Same as Fig. 3.2.5 but for impermeable (sealed) contact surface.
73
3.2.2 Foundation Complex Stiffness Matrix for Partially Saturated
Soils
Figures 3.2.7 and 3.2.8 show results for partially saturated stiff and soft soil
respectively ( / 0.1
f
K µ = and 0.01). The different curves correspond to dry soil, fully
saturated soil and partially saturated soil with saturation 99
r
S = % and 90%. It can be
seen that the results for the partially saturate soils are very close to those for dry soil even
for such high saturation ratios.
74
Fig. 3.2.7 Effect of degree of saturation on the foundation complex stiffness for
porosity and ˆ 0.4 n = / 0.1
w
K µ = , and for permeable (open, part a)) and
impermeable (sealed, part b)) contact surface.
75
Fig. 3.2.8 Same as Fig. 3.2.7 but for ˆ 0.4 n = and / 0.01
w
K µ = .
76
3.3 Free-Field Motion
Free-field motion is the motion of the half-space not affected by the presence of
structures. It is of interest for the problem analyzed in this thesis because it represents the
excitation of the soil-structure system. This motion is modified by the scattering of
waves from the soil-foundation interface, and radiation of waves by the vibrating
foundation. Its modification due to scattering only is referred to as foundation input
motion, which asymptotically approaches the free-field motion for wavelengths much
longer than the size of the foundation. Hence, understanding of the free-field motion
helps understand the foundation input motion. In what follows, results are shown for the
free-field motion on the surface of a porous half-space that is fully or partially saturated,
and due to incident plane fast P-wave and incident plane SV-wave. Incident slow P-wave
is not considered because it attenuates very fast, and hence is not likely to be a carrier of
any significant energy from the earthquake source. Locally generated slow P-waves,
however, are considered. The results are presented in the form of the magnitudes of the
(complex) coefficients of the reflected waves from the half-space surface, and the
magnitudes of the (complex) horizontal and vertical displacements on the surface of the
half-space. In all the results presented in this section, the Poisson’s ratio is 0.3 ν = and
the porosity is . ˆ 0.4 n =
77
3.3.1 Incident Plane Fast P-wave
3.3.1.1 Incident P-wave and Fully Saturated Soil
Figures 3.3.1.1 and 3.3.1.2 show variations of the reflection coefficients’ amplitudes
(left) and surface displacement amplitudes (right) due to an incident plane fast P-wave
with unit displacement amplitude, versus the angle of incidence, for different values of
permeability , and
6
ˆ
10 k
−
=
8
10
− 10
10
− 2
m . The case when the effects of the seepage force
are neglected is also shown, as the case , as well as the dry soil case. These two
figures differ in the stiffness of the skeleton. In Fig. 3.3.1.1
ˆ
k →∞
/ 0
f
K .1 µ = (stiff soil) and
in Fig. 3.3.1.2 / 0.
f
K 01 µ = (soft soil). Parts a) and b) differ in the hydraulic boundary
condition at the surface. Part a) corresponds to a permeable half-space and part b) to an
impermeable half-space. The frequency is set to 1 f = Hz.
The results in Figs 3.3.1.1 and 3.3.1.2 show that, at 1 f = Hz, the horizontal
displacements of the saturated soil decrease with decreasing permeability, are affected
very little by the type of hydraulic boundary conditions, and are always and significantly
smaller than those of the dry soil. The vertical displacements for permeable half-space
are practically not affected by the saturation for all values of permeability considered.
For impermeable half-space, vertical displacements are larger for saturated soil, and
decrease with decreasing permeability, approaching the dry soil displacements. The
difference is the largest for vertical incidence.
78
Figs 3.3.1.3 and 3.3.1.4 also show variations of the reflection coefficients’ amplitudes
and surface displacement amplitudes due to an incident plane fast P-wave with unit
displacement amplitude, versus the angle of incidence, but for different values
79
01
of frequency , 1, 10 and 100 Hz. The case when there is no seepage force is also
shown. In both figures,
0.1 f =
/ 0.
f
K µ = (soft soil), and they differ only in the value of
permeability. In Fig. 3.3.1.3
7 2
ˆ
10 m k
−
= and in Fig. 3.3.1.4 . Figure
3.3.1.3 shows that, for (larger) permeability of
10 2
ˆ
10 m k
−
=
7 2
ˆ
10 m k
−
= , the horizontal surface
displacements are the smallest for 0.1 f = , and they approach their values for no seepage
force as the frequency increases. Figure 3.3.1.4 shows that, for less permeable soils, with
permeability of , the horizontal displacements are considerably smaller for
the cases with finite permeability, then for the “no seepage force” case, even for
10 2
ˆ
10 m k
−
=
100 f =
Hz. The vertical displacements, for permeable half-space, vary insignificantly with
frequency (for the range considered) and are practically the same as for the “no seepage
force” case even for the less permeable soil (
10 2
ˆ
10 m k
−
= ).
Fig. 3.3.1.1 Free-field motion due to unit displacement plane fast P-wave versus
incident angle, for different values of permeability. a) Permeable, b) impermeable
half-space. Left: amplitudes of the reflection coefficients. Right: amplitudes of the
surface displacements. The input parameters are: porosity ˆ 0.4 n = , / 0.1
f
K µ = , and
the frequency is set to 1 f = Hz.
80
Fig. 3.3.1.2 Same as Fig. 3.3.1.1 but for / 0.01
f
K µ = .
81
Fig. 3.3.1.3 Free-field motion due to unit displacement plane fast P-wave versus
incident angle, for different values of frequency. a) Permeable, b) impermeable half-
space. The input parameters are: porosity ˆ 0.4 n = , / 0.01
f
K µ = , and permeability
.
72
ˆ
10 m k
−
=
82
Fig. 3.3.1.4 Same as Fig. 3.3.1.3 but for permeability .
10 2
ˆ
10 m k
−
=
83
3.3.1.2 Incident P-wave and Partially Saturated Soil
Similarly as the previous figures, Figs 3.3.1.5 and 3.3.1.6 show variations of the
reflection coefficients’ amplitudes (left) and surface displacement amplitudes (right) due
to an incident plane fast P-wave with unit displacement amplitude, versus the angle of
incidence, for different values of saturation 100
r
S = , 99 and 90%. The effects of the
seepage force are neglected, and hence, the results are not dependent on frequency.
These two figures differ only in the stiffness of the skeleton. In Fig. 3.3.1.5 /0
f
K .1 µ =
(stiff soil) and in Fig. 3.3.1.6 / 0.
f
K 01 µ = (soft soil). Parts a) and b) differ in the
hydraulic boundary condition at the surface. Part a) corresponds to a permeable half-
space and part b) to an impermeable half-space. Fig. 3.3.1.5 shows that, for stiff soil, the
horizontal displacements are very similar for 90 and 99% saturation, and are significantly
larger than for 100% saturation, for both permeable and impermeable half-space. The
vertical displacements, for permeable half-space are practically the same for all three
levels of saturation, but for impermeable half-space are significantly larger for 100%
saturation than for the partial saturation, the difference being the largest for vertical
incidence. Fig. 3.3.1.6 shows that, for stiff soil, as far as the effect of the degree of
saturation is concerned, the results differ from those for stiff soil only in that the
horizontal motions are much more sensitive to the degree of saturation, being almost an
order of magnitude larger for 90% saturation than for 100% saturation.
84
Fig. 3.3.1.5 Free-field motion due to unit displacement plane fast P-wave versus incident
angle, for different levels of saturation. a) Permeable, b) impermeable half-space. The
input parameters are: porosity ˆ 0.4 n = , / 0.1
f
K µ = , and frequency 1 f = Hz. The
effects of the seepage force are neglected.
85
Fig. 3.3.1.6 Same as Fig. 3.3.1.5 but for / 0.01
f
K µ = .
86
3.3.2 Incident Plane SV-wave
3.3.2.1 Incident SV-wave and Fully Saturated Soil
Figures 3.3.2.1 through 3.3.2.4 show results for the free-field motion due to an
incident plane SV-wave onto a half-space for the same parameters as Figs 3.3.1.1 through
3.3.1.4. It can be seen that for vertical incidence, the horizontal displacement on the
surface always approaches 2, and the vertical displacement approaches zero. As the
incident angle approaches 90 , both the vertical and horizontal displacements approach
zero. For incidence, and for permeable half-space, the horizontal displacements are
always zero, while for impermeable half-space, they are not necessarily zero but are
small. It can be seen that the surface displacements for incident SV wave are not very
sensitive to the hydraulic boundary condition. A comparison with the results for incident
P-wave shows that, for incident SV-wave, both the horizontal and vertical surface
displacements are more sensitive to the variations of permeability and frequency, i.e. are
affected more by the seepage force, than for incident P-wave. For frequency set to
45
1 f =
Hz, the maximum displacement amplitudes (over all incident angles) are significantly
larger for larger seepage force than for “no seepage force” (see Figs 3.3.2.1 and 3.3.2.2),
but as the frequency increases, they decrease and approach the values for “no seepage
force”. A comparison of the fully saturated and dry soil cases shows that the maximum
horizontal displacements (over all incident angles) are larger for the dry soil, while the
vertical displacements are larger for the saturated soil, and for smaller permeability.
87
Fig. 3.3.2.1Free-field motion due to unit displacement plane SV-wave versus incident
angle, for different values of permeability. a) Permeable, b) impermeable half-space.
Left: amplitudes of the reflection coefficients. Right: amplitudes of the surface
displacements. The input parameters are: porosity ˆ 0.4 n = , / 0.1
f
K µ = , and the
frequency is set to 1 f = Hz.
88
Fig. 3.3.2.2 Same as Fig. 3.3.2.1 but for / 0.01
f
K µ = .
89
Fig. 3.3.2.3 Free-field motion due to unit displacement plane SV-wave versus incident
angle, for different values of frequency. a) Permeable, b) impermeable half-space. The
input parameters are: porosity ˆ 0.4 n = , / 0.01
f
K µ = , and permeability .
72
ˆ
10 m k
−
=
90
Fig. 3.3.2.4 Same as Fig. 3.3.2.3 but for permeability .
10 2
ˆ
10 m k
−
=
91
92
3.3.2.2 Incident SV-wave and Partially Saturated Soil
Figs 3.3.2.5 and 3.3.2.6 show results for the free-field motion due to an incident SV-
wave for partial saturation, also for the same soil properties as in Figs 3.3.1.5 and 3.3.1.6.
It can be seen from these results that the vertical displacements are not very sensitive to
the degree of saturation, but the maximum (over all incident angles) horizontal
amplitudes are much larger for the partially saturated soils than for the fully saturated
soil.
Fig. 3.3.2.5 Free-field motion due to unit displacement plane SV-wave versus incident
angle, for different levels of saturation. a) Permeable, b) impermeable half-space. The
input parameters are: porosity ˆ 0.4 n = , / 0.1
f
K µ = , permeability , and
frequency
7 2
ˆ
10 m k
−
=
1 f = Hz.
93
Fig. 3.3.2.6 Same as Fig. 3.3.2.5 but for / 0.01
f
K µ = .
94
3.4 Foundation Input Motion
Foundation input motion is, by definition, the response of a massless foundation to
the incident waves. The interaction of the incident waves with the massless foundation is
also referred to as kinematic interaction. Hence, the foundation input motion is
essentially the free-field motion plus some perturbation due to scattering of the waves
from the foundation. This perturbation is very small for very long incident waves
compared to the size of the foundation. While the free-field motion was studied in detail
in Section 3.3, this section focuses on the effect of the saturation as function of
dimensionless frequency η , showing results only for one incident angle and for stiff soil.
Figure 3.4.1 and 3.4.2 show the amplitudes of the vertical displacements V ,
horizontal displacements ∆ and rocking amplitudes a ϕ , versus dimensionless
frequency
S,dry
a
V
ω
η
π
= , respectively for an incident fast P- and an incident SV-wave,
both at 30 incidence. The results are for stiff soil, with
/ 0
f
K .1 µ = , porosity ˆ 0.4 n = ,
and full saturation. Results for dry soil are also shown. In part a), the half-space is
permeable, and in part b) it is impermeable. In both parts a) and b), the pots on the left
correspond to permeable soil-foundation interface, and the plots on the right to
impermeable interface. It can be seen that the nature of the variation of the amplitudes of
the three components of the foundation input motion with frequency is similar for
saturated soil as for dry soil, which has been studies previously in detail.
95
Fig. 3.4.1 Foundation input motion amplitudes due to an incident plane fast P-wave at
incidence, for 30
/ 0
f
K .1 µ = , porosity ˆ 0.4 n = . Part a): permeable half-space. Part b)
impermeable half-space. Left: permeable foundation. Right: impermeable foundation.
96
Fig. 3.4.2 Same as Fig. 3.4.1 but for an incident plane SV-wave at incidence. 30
97
3.5 Building-Foundation-Soil Response
98
.1
This section shows results for the amplitudes of the foundation displacements and
building relative displacement (due to elastic deformation of the building only), for stiff
soil with / 0
f
K µ = and porosity ˆ 0.4 n = , and for a building with height 2 Ha = ,
width , and mass ratios W a = / 2
bf
mm = and / 0.2
fs
mm = . The excitation is an
incident plane fast P-wave or a plane SV-wave, both at incidence, and with unit
displacement of the incident wave.
30
3.5.1 Building-Foundation-Soil Response for Incident P-wave
Figures 3.5.1.1 through 3.5.1.4 show results for incident P-wave at 30 incidence. In
all figures,
V is the amplitudes of the vertical displacements, ∆ is the amplitudes of the
horizontal displacements, a ϕ is the rocking angle multiplied by the characteristic length
a,
rel
b
u is the relative building response at the top, and the dimensionless frequency is
S,dry
a
V
ω
η
π
= . In each figure, part a) corresponds to permeable half-space, and part b) to
impermeable half-space.
Figures 3.5.1.1 and 3.5.1.2 show the variation of the system response with η for fully
saturated soil with different permeability, respectively for permeable and for
impermeable foundation. The response for dry soil is also shown. Figures 3.5.1.3 and
3.5.1.4 show the variation of the system response with η for different levels of
saturation. These figures show that both the foundation and the building relative
response exhibit large variations near the building fixed-base frequencies. The
99
“backbone” curve of the foundation displacement amplitudes is governed by the
foundation input motion, shown in Section 3.4, which also affects the amplitudes of the
peaks of the building relative response. The building relative response has peaks at the
system frequencies. Due to the interaction with the foil, the first system frequency is
lower than the fundamental fixed-base frequency, and the amplitude of the peak is
generally reduced due to the radiation of energy in the soil. It is well known that the
amount of the shift and the reduction of amplitude depend on the relative flexibility of the
soil, which in case of porous soil depends on the water content, soil permeability, and
also frequency. The effect of the flexibility of the soil is described through the
foundation complex stiffness matrix (See Section 3.2). In the “coarse” resolution plots in
these figures, no significant shift of the peaks can be seen as function of the water content
and permeability. Hence, if there is a change, it is small. Investigation of such changes
for one case study building is presented in the following section.
Fig. 3.5.1.1 System response due to an incident plane fast P-wave at 30 incidence, for
/ 0
f
K .1 µ = , porosity , ˆ 0.4 n = 2 H a = , width W a = , and mass ratios / 2
bf
mm = ,
, and for a permeable foundation. Part a): permeable half-space. Part b)
impermeable half-space. The different curves correspond to different soil permeability.
/ 0
fs
mm =.2
100
Fig. 3.5.1.2 Same as Fig. 3.5.1.1, but for impermeable foundation.
101
Fig. 3.5.1.3 Same as Fig. 3.5.1.1, but the different curves correspond to different levels
of saturation.
102
Fig. 3.5.1.4 Same as Fig. 3.5.1.1, but the different curves correspond to different levels
of saturation, and the foundation is impermeable.
103
3.5.2 Building-Foundation-Soil Response for Incident SV-wave
Similarly as in Section 3.5.1, Figures 3.5.2.1 through 3.5.2.4 show results for
incident SV-wave at 30 incidence. In all figures,
V is the amplitudes of the vertical
displacements, ∆ is the amplitudes of the horizontal displacements, a ϕ is the rocking
angle multiplied by the characteristic length a,
rel
b
u is the relative building response at
the top, and the dimensionless frequency is
S,dry
a
V
ω
η
π
= . In each figure, part a)
corresponds to permeable half-space, and part b) to impermeable half-space. The
observations in these figures are the same as in Section 3.5.1 for incident P-wave.
104
Fig. 3.5.2.1 System response due to an incident plane SV-wave at 30 incidence, for
/ 0
f
K .1 µ = , porosity , ˆ 0.4 n = 2 H a = , width W a = , and mass ratios / 2
bf
mm = ,
, and for a permeable foundation. Part a): permeable half-space. Part b)
impermeable half-space. The different curves correspond to different soil permeability.
/ 0
fs
mm =.2
105
Fig. 3.5.2.2 Same as Fig. 3.5.2.1, but for impermeable foundation.
106
Fig. 3.5.2.3 Same as Fig. 3.5.2.1, but the different curves correspond to different levels of
saturation.
107
Fig. 3.5.2.4 Same as Fig. 3.5.2.1, but the different curves correspond to different levels of
saturation, and the foundation is impermeable.
108
109
3.6 Frequency Shift due to Saturation – Millikan Library Case
This section shows results for the model response for parameters chosen so that it
corresponds approximately to the NS response of Millikan Library in Pasadena,
California. This building has been chosen because changes of its frequencies of vibration
have been reported such that can be correlated with heavy rainfall (Clinton et al., 2006).
Todorovska and Al Rjoub (2006b) attempted to explain these changes as resulting from
changes in the soil due to saturation. They used a soil-structure interaction model in
poroelastic soils, presented in Todorovska and Al Rjoub (2006a), and showed that
changes in the soil due to saturation produced the same trend and order of magnitude of
the shift as observed. The model of Todorovska and Al Rjoub (2006a) is an earlier
version of the model analyzed in this thesis that did not include the effects of the seepage
force. Hence, this section focuses on how different assumptions on the soil permeability
would affect the frequency shift. For the purpose of completeness of the analysis in this
thesis work, a summary of the full-scale observations, as well as of the choice of model
parameters in Todorovska and Al Rjoub (2006b) is included.
3.6.1 Full-scale Observations
The frequencies of vibration of Millikan Library have been monitored since 1967
using different excitations such as ambient noise, forced vibrations, and earthquakes.
Permanent and temporary changes in its frequencies have been observed, most recently
summarized in Clinton et al. (2006). The permanent change is decrease with time, from
1.45 to 1.19 Hz for EW vibrations, and from 1.9 to 1.72 Hz for NS vibrations, as
measured during small amplitude vibrations. The lowest measured values occurred
110
during the strong earthquake shaking from the 1971 San Fernando earthquake (Udwadia
and Trifunac, 1974), and 1994 Northridge earthquake (0.94 Hz for EW vibrations, and
1.33 Hz for NS vibrations; Cinton et al., 2006). Since February of 2001, continuous data
steams of the 9
th
floor response have been recorded by a tri-axial 24-bit accelerometer,
which is one of the stations of the California Integrated Seismic Network. This has
enabled monitoring of changes of the building apparent frequencies on different time
scales (Clinton et al., 2006). Further, weather data from the nearby JPL Weather station,
located about 8.5 km north of the building, has been available for a period of about 2.5
years following the installation of this sensor, which enabled to study possible
correlations of the changes of its frequencies with weather (temperature, wind and rain).
The reported observation of the changes of the building frequencies with rainfall is
as follows. The building first and second apparent frequencies for EW and NS motions,
and the first frequency for torsional motions increase during heavy rainfall (above 40 mm
per day) in a matter of hours, and recover in about a week following the rain (Clinton,
2004; Clinton et al., 2006). For example, in early February of 2003, when over 100 mm
rain fell over a period of two days, an increase of about 3% was measured for the EW and
the torsional frequencies, followed by a slow decay over a period of about 10 days (see
Fig. 9 in Clinton et al., 2006). The measured change of the NS frequency was smaller
(slightly less than 1%). Clinton et al. (2006) note that this increase in frequencies
occurred in spite of the fact that strong winds that often accompany heavy rainfall tend to
decrease the system frequencies by exciting larger amplitudes of response (by up to 3%
in dry weather). Further, they note an increase of the system frequencies with
temperature (about 1 −2 % on very hot days with temperature near 40
o
C). For wind and
temperature, the recovery is practically instantaneous, while for rainfall it is slow, and
can take about a week.
3.6.2 Model Parameters for NS Response
Todorovska and Al Rjoub (2006b) chose the system parameters guided by the
information provided in Luco et al. (1986), who specified: building weight =
N, foundation weight = N, building height 44 m, foundation depth 4 m, and
building in plan dimensions
8
1.05 10 ×
8
0.14 10 ×
21m 23 m × . Further, they classify the local soil as alluvium
with depth to “bedrock” about 275 m, and mention apparent frequencies of vibration
measured during forced vibration tests that are about
1
1.8 f = Hz for NS vibrations, and
Hz for EW vibrations. Clinton et al. (2005) provided further history on the
variations of these frequencies, which shows strong correlation with amplitudes of
response, and overall trend of decrease with time, from initial values Hz for NS
vibrations, and Hz for EW vibrations during forced vibrations in 1967, to
Hz for NS vibrations, and
1
1.4 f =
1
1.9 f =
1
1.45 f =
1
1.54 f =
1
1.19 f = Hz for EW vibrations during ambient
vibrations. For the soil in their homogeneous half-space model, Todorovska and Al
Rjoub (2006b) chose values that would correspond approximately to the soil at Millikan
Library near the surface. They use soil porosity = 0.4, Poisson’s ratio ˆ n
s
ν = 0.3,
kg/m
3
gr
2.7 10 ρ =×
3
, and
s
µ that correspond to dry shear wave velocity
,dry gr
ˆ /[(1 ) ]
Ss
V n µ ρ = − = 300 m/s. For the water, kg/m
3
10
f
ρ =
3
and
9
2.2 10
f
K =×
111
Pa (which gives bulk wave velocity 1,483 m/s). For dry soil, they set
gr
/
f
ρ ρ =0.001,
and /
f
K
s
µ =0.001. These parameters imply /
f
K
s
µ = 15.089. For the building, they
assumed fundamental fixed-base frequency
1
2.5 f = Hz. Further, based on the foundation
plan dimensions of , they chose reference length a = 12 m. They also
choose , i.e. a semi-circular. They noted that a rigid foundation model might be
acceptable for the NS vibrations, for which experiments have shown that the base moves
as a rigid body (Foutch et al., 1975), and as much as 30% of the roof response is due to
rigid body rocking (Luco et al., 1986). In contrast, in the EW direction, the building
behaves as one on a flexible foundation with a stiff central core (Foutch et al., 1975).
23.3 m 25.1m ×
/ ba =1
Table 3.6.2.1 shows the model wave velocities for dry and fully saturated soil with no
seepage force. It can be seen that the saturated soil has slightly smaller shear wave
velocity than the dry solid and more than 3 times larger velocity of the fast P-wave
velocity.
Table 3.6.1 Wave velocities for fully saturated soil for the model of Millikan library
assuming no seepage force
/
fs
K µ /
fgr
ρ ρ
S
V
1 P
V
2 P
V
Dry 0.001 0.001 300 561.3 175.6
Saturated 15.089 0.371 285.3 1,805.3 331.5
112
The variation of the wave velocities with permeability and degree of saturation is shown
in Fig. 3.6.1 Part a) shows the variations of the normalized wave velocities versus inverse
permeability at frequency 1 η = ( 12.5 f = Hz). As in Section 3.1, the velocities are
normalized by their values for zero seepage force. It can be seen that the wave velocities
decrease for permeability smaller than
7
10
−
. Part b) shows variation of the velocities of
the fast P-wave (when there is no seepage force) versus the amount fraction of air in the
pores for different values of the pore pressure. It can be seen that both velocities
decrease rapidly with increasing air content, with the velocity of the fast P-wave
approaching the value for “dry” soil even for very small percentage of air such as 1%.
This implies that to observe any significant changes in the system response due to
rainfall, the level of saturation has to be very high.
113
Fig. 3.6.1Wave velocities for the Millikan case. a) Normalized wave velocities for fully
saturated soil as function of inverse permeability. b) Wave velocities of P-waves as
function of the air content in the pores.
114
3.6.3 Foundation Complex Stiffness and System Response
Fig. 4.6.2 shows plots of the foundation complex stiffness matrix versus frequency
between 0 and 60 Hz. Part a) shows results for a permeable foundation, and part b)
for an impermeable foundation, and the real parts are shown of the left and the imaginary
parts are shown on the right. The different types of lines correspond to dry soil and to
fully saturated soil with different permeability. Similarly, Fig. 4.6.3 shows results of the
system response (foundation translations, foundation rotation, and building relative
response at the top), for vertically incident SV wave with unit displacement amplitude.
f =
To help measure and understand the shifts of the first system frequency, enlarged
plots of the horizontal/vertical stiffness coefficients and of the amplitudes of the first
peak in the relative response are shown respectively in Figs. 3.6.4 and 3.6.5. Similarly
as in the previous two figures, parts a) show results for a permeable foundation, and parts
b) for an impermeable foundation. Figs. 3.6.4 shows that for small frequencies (i.e. near
the frequency of the first mode of vibration of the building), the foundation stiffness is
larger for saturated soil than for dry soil, and this effect is more significant for
impermeable foundation. Also, it is larger for smaller permeability but only up to some
value beyond which the effect reverses.
As it can be seen from Fig. 3.6.5, the shift in frequency for a permeable foundation is
insignificant. For an impermeable foundation, Todorovska and Al Rjoub (2006b)
reported increase of about by 2% (from 1.44 to 1.47 Hz) for fully saturated soil relative to
dry, using their model which neglected the effects of the seepage force, which agreed
115
Fig. 3.6.2 Foundation complex stiffness matrix coefficients for the model corresponding
to Millikan library. a) Permeable foundation. b) Impermeable foundation.
116
Fig. 3.6.3 System response for the model corresponding to Millikan library. a) Permeable
foundation. b) Impermeable foundation.
117
Fig. 3.6.4 Enlarged view of the horizontal/vertical foundation complex stiffness
coefficients for the model corresponding to Millikan library. a) Permeable foundation. b)
Impermeable foundation.
118
Fig. 3.6.5 Enlarged view of the first peak in the building relative roof response of the
model corresponding to Millikan library. a) Permeable foundation. b) Impermeable
foundation.
119
approximately with the observations. Fig. 3.6.5 (part b)) shows that this effect is smaller
for less permeable soils and may reverse for small enough permeability of the soil.
It is noted here that the observed effect of finite permeability on the shift of frequency
predicted by the model is due to the strong dependency of the wave velocities in Biot’s
original theory (Biot, 1956a) on permeability and frequency, i.e. for large enough
ˆˆ 1
ˆ
n
k
µ
ω
.
This dependency is most significant for the shear wave velocity, which can be reduces to
up to 60-70% of its value for zero seepage force. This reduction is due to increased
effective mass (due to the seepage force) in the computation of the wave velocities. It is
also noted that Biot’s theory does not consider the molecular forces between the different
phases (solid, water and air in the case of partial saturation).
120
121
Chapter 4: Summary and Conclusions
This thesis presented an investigation of the effects of water saturation on the
effective input motion and system response during building-foundation-soil interaction
using a simple two-dimensional model. In this model, the building is represented as a
shear wall supported by a semi-cylindrical foundation imbedded in a homogeneous and
isotropic poroelastic half-space, and the excitation is a plane P, plane SV or a Rayleigh
wave. Biot’s theory of wave propagation in fully saturated poroelastic medium was used
to describe the motion in the soil. By relaxing the zero stress condition at the free surface
for the scattered waves, a closed form solution was derived using the wave function
expansion method. The boundary conditions along the contact surface between the soil
and the foundation are perfect bond for the skeleton, and either drained or undrained
hydraulic condition for the fluid (i.e. permeable or impermeable boundary). The effect of
partial saturation, versus full saturation for which the Biot’s theory has been derived, is
accounted for approximately, by changing the effective bulk modulus of the fluid.
Numerical results are shown for variations of the wave velocities, free-field motion
amplitudes, foundation input motion amplitudes, foundation complex stiffness, and the
system response in the frequency domain for different values of the model parameters,
and for incident plane P- and SV-waves. Also, the effects of the saturation on the
building apparent frequency are analyzed for a model that approximately corresponds to
the NS response Millikan library in Pasadena, California, for which shift in frequency
and recovery have been observed due to heavy rainfall. The numerical results were
computed using a computer program written in Fortran. The following summarizes the
results of the analysis and the conclusions.
The solution of the problem was expressed entirely in terms of dimensionless
parameters, which were defined using reference: length (half with of the foundation),
material modulus
a
s
µ (the shear modulus of the skeleton), and mass density
gr
ρ (the
density of the material of the grains without pores). The dimensionless parameters that
control the system response are the following: stiffness of the skeleton relative to the bulk
modulus of the fluid, through the ratio /
s f
K µ and the Poisson’s ratio
s
ν ; mass density
of the skeleton relative to the fluid, defined through the ratio
gr
/
f
ρ ρ and the porosity ;
mass of the building relative to the mass of the foundation, and mass of the foundation
relative to the mass of the replaced soil, through the ratios
ˆ n
b fnd
/ mm and ,
where,
fnd gr
/ mm
gr fnd gr
mA ρ = is the mass of the excavated soil (per unit length) if there were no
voids, and is the area of the foundation; the flexibility of the building relative to that
of the soil, through the ratio
fnd
A
ref ,b
()/( )
[ ]
,b ref
()/ /( )/
S
HV a V ω ω ⎡⎤ =
⎣⎦
S
VH V a ε = = ratio of
the number of wavelength in the shear beam in length H and the number of reference
wavelengths in the soil in length a; dimensionless frequency
ref
/( ) aV η ωπ = , where
ref gr
/
s
V µ ρ = is a reference velocity; foundation shape, through the ratio ; and on
the type, amplitude and angle of the incident waves. An additional parameter for viscous
fluids and finite permeability is the coefficient of dissipation
/ h a
2
ˆˆ
ˆˆ / bn k µ = , were ˆ µ is the
absolute viscosity, and is the coefficient of permeability.
ˆ
k
122
The wave velocities in the soil medium are the most fundamental quantities that
affect the system response. Therefore, the effects of the model parameters on the wave
velocities were first analyzed. Biot’s theory predicts the existence of two P-waves (one
fast and the other one slow) and one S-wave in poroelastic soil medium. The results
show that, when the seepage force is considered, the velocities of these waves are
frequency dependent. The wave velocities in the soil decrease with increasing seepage
force, the effects of which are more pronounced for smaller frequency of motion, larger
fluid viscosity, larger porosity and smaller skeleton permeability, i.e. for larger
ˆˆ 1
ˆ
n
k
µ
ω
.
For large enough
ˆˆ 1
ˆ
n
k
µ
ω
, the velocities of the SV and fast P-wave approach their
asymptotic value, which is real valued, and the velocity of the slow P-wave approaches
zero. The variation is very small for the fast P-waves (about 5%), is about up to 40% for
the S-wave, and is up to 100% for the slow P-wave. The percentage change depends very
little on the relative stiffness of the skeleton (i.e. /
s f
K µ ). Both P-wave velocities
decrease with increasing air content in the pores. The velocity of the fast P-wave
approaches the P-wave velocity of the dry soil, and that of the slow P-wave approaches
zero.
The results show that the amplitudes of the free-field motion, which excites the
structure, depend strongly on the angle of incidence, and also on the properties of the soil
model: relative stiffness of the skeleton (i.e. /
s f
K µ ), porosity and permeability (for
given viscosity and Poisson’s ratio), and are frequency dependent when
ˆˆ 1
ˆ
n
k
µ
ω
is not
123
small. The effect of the water is more significant for mixtures with softer skeleton (i.e.
smaller /
s f
K µ ), e.g. soft soils, as compared to porous rock. For incident P-wave, the
amplitudes of the horizontal motion at the surface are larger for dry soil than for saturated
soil, but the amplitudes of the vertical motion are larger for saturated soil and
impermeable boundary. For incident SV wave, the peak horizontal amplitude is also
larger for dry soil, and but the vertical amplitudes are smaller regardless of the hydraulic
boundary condition on the surface.
The amplitudes of the free-field displacements at the ground surface depend most on
the incident angle. For incident P waves, the amplitude of the horizontal displacement is
zero for vertically and horizontally incident waves, and is the largest near 45 degrees
incidence. The amplitudes of the vertical displacement is the maximum for vertical
incidence, and it decreases monotonically to zero for horizontal incidence. For incident
SV waves and Poison ratio ν =0.3, the horizontal amplitude equals 2 for vertical
incidence, is zero at 45 degrees incidence, exhibiting a sharp peak for incidence below 45
degrees, beyond which it increases a little and decreases again to zero for horizontal
incidence. The amplitudes for the vertical motion are zero for vertical and horizontal
incidence, and exhibit a peak near 30 degrees incidence. The frequency dependence is
more significant for incident SV waves, but it is still small. The amplitudes decrease
with increasing frequency, approaching those for zero seepage force.
The foundation input motion is the effective input motion exciting the structure. It is
defined as the response of a massless foundation, without the structure, to the incident
waves, and represents the free field motion that has been modified by the scattering of
124
waves from the foundation. As the dimensionless frequency 0 η → (i.e. for very long
incident waves compared to the size of the foundation), the horizontal and vertical
amplitudes of the foundation input motion approach those of the free-field motion, and
the rotation approaches zero. For very short incident wavelengths, the amplitudes
approach zero, due to the ironing effect of the foundation.
The building response is affected by the dynamic soil-structure interaction through
the foundation complex stiffness matrix. The real part of this matrix is referred to as
“foundation stiffness”, and the imaginary part is related to the loss of energy due to
radiation which effect is related to the “radiation damping”. The results show that the
presence of water in the pores affects significantly the foundation impedance matrix for
soft and stiff soils. The rocking impedance is affected negligibly by the presence of
water, which can be explained by the fact that the rocking motions produce mostly a
shearing deformation of the soil near the contact surface. The conclusion is similar for
the coupling term. The horizontal impedance is significantly affected by the hydraulic
condition along the contact surface, which permits or stops the flow of water through the
contact surface. As it can be expected, the foundation stiffness is larger when the pores
are filled with water and the contact surface is impermeable (sealed contact). The
imaginary part, however, i.e. the radiation damping, is larger for permeable foundation.
For relatively small seepage force, the foundation stiffness increases with increasing
seepage force, but for large enough seepage force the effect becomes the opposite. In
that case, the decreasing foundation stiffness with increasing seepage force can be
explained by the smaller wave velocities for larger seepage force.
125
126
For the case corresponding to NS response of Millikan library, the apparent frequency
increases by about 2% for fully saturated soil, impermeable foundation, and negligible
seepage force (very high permeability). For decreasing permeability, however, the
increase becomes smaller and the trend reverses for small enough permeability. Because
of this, and the fact that the wave velocities as well as foundation stiffness are strongly
dependent on frequency, conclusions from analysis of one model or particular
observation for a specific soil site and structure cannot be automatically generalized to
any structure and type of soil.
127
References
[1] Berryman JG (1980). Confirmation of Biot’s theory, Appl. Physics Letter, 34(4),
382-384.
[2] Biot MA (1956a). Theory of propagation of elastic waves in a fluid-saturated
porous solid: I low-frequency range, J. of Acoustical Society of America, 28(2),
168-178.
[3] Biot MA (1956b). Theory of propagation of elastic waves in a fluid-saturated
porous solid: II higher-frequency range, J. of Acoustical Society of America,
28(2), 179-191.
[4] Biot MA (1962). Mechanics of deformation and acoustic propagation in porous
media, J. of Applied Physics, 33(4), 1482-1498.
[5] Biot MA, Willis DG (1957). The elastic coefficients of a theory of consolidation",
Journal of Applied Mechanics, ASME, 29, 594-601.
[6] Bo J, Hua L (1999). Vertical dynamic response of a disk on a saturated
poroelastic half-space, Soil Dynamics and Earthquake Engineering, 18, 437–443.
[7] Bougacha S, Tassoulas JL (1991a). Seismic analysis of gravity dams. I: modeling
of sediments, J. of Engineering Mechanics, ASCE, 117(8), 1826-1837.
[8] Bougacha S, Tassoulas JL (1991b). Seismic response of gravity dams. II: effects
of sediments, J. of Engineering Mechanics, ASCE, 117(8), 1839-1850.
[9] Bougacha S, Tassoulas JL, Roesset JM (1993a). Analysis of foundations on
fluid-filled poroelastic stratum, J. of Engineering Mechanics, ASCE, 119(8),
1632-1648.
[10] Bougacha S, Roesset JM, Tassoulas JL (1993b). Dynamic stiffness of foundations
on fluid filled poroelastic stratum, J. of Engineering Mechanics, ASCE, 119(8),
1649-1662.
128
[11] Carcione JM, Cavallini F, Santos JE, Ravazzoli CL, Gauzellino PM (2004). Wave
propagation in partially saturated porous media: simulation of a second slow
wave, Wave Motion, 39(3), 227-240.
[12] Ciarletta M, Sumbatyan MA (2003). Reflection of plane waves by the free
boundary of a porous elastic half-space, J. of Sound and Vibration, 259(2), 253-
264.
[13] Clinton JF, Bradford SK, Heaton TH, Favela J (2006). The observed wander of
the natural frequencies in a structure, Bull. Seismological Society of America, 96
(1), 237-257.
[14] Dargush GF, Chopra MB (1996). Dynamic analysis of axisymmetric foundations
on poroelastic media, J. of Engineering Mechanics, ASCE, 122(7), 623-632.
[15] Degrande G, De Roeck G, Broeck PVD, Smeulders D (1998). Wave propagation
in layered dry, saturated and unsaturated poroelastic media, Int.l J. of Solids and
Structures, 35(34-35), 4753-4778.
[16] Deresiewicz H (1960). The effect of boundaries on wave propagation in a liquid-
filled porous solid: I. reflection of plane waves at a free plane boundary (non-
dissipative case), Bull. Seismological Society of America, 50(4), 599-607.
[17] Deresiewicz H (1961). The effect of boundaries on wave propagation in a liquid-
filled porous solid: II. Love waves in a porous layer, Bull. Seismological Society
of America, 51(1), 51-59.
[18] Deresiewicz H, Rice JT (1962). The effect of boundaries on wave propagation in
a liquid-filled porous solid: III. Reflection of plane waves at a free plane boundary
(general case), Bull. Seismological Society of America, 52(3), 595-625.
[19] Edelman I (2004a). On the existence of the low-frequency surface waves in a
porous medium, Comptes Rendus Mecanique, 332(1), 43-49.
[20] Edelman I (2004b). Surface waves at vacuum/porous medium interface: low
frequency range, Wave Motion, 39(2), 111-127.
129
[21] Halpern MR, Christiano P (1986). Steady-state harmonic response of a rigid plate
bearing on a liquid-saturated poroelastic half space, Earthquake Engineering and
Structural Dynamics, 14, 439-454.
[22] Japon BR, Gallego R, Dominguez J (1997). Dynamic stiffness of foundations on
saturated poroelastic soils, J. Engineering Mechanics, ASCE, 123(11), 1121-1129.
[23] Jin B, Liu H (2000a). Horizontal vibrations of a disk on a poroelastic half-space,
Soil Dynamics and Earthquake Engineering, 19, 269–275
[24] Jin B, Liu H (2000b). Rocking vibrations of rigid disk on saturated poroelastic
medium, Soil Dynamics and Earthquake Engineering, 19, 469-472.
[25] Jinting W, Chuhan Z, Feng J (2004). Analytical solutions for dynamic pressures
of coupling fluid-solid-porous medium due to P wave incidence, Earthquake
Engineering and Engineering Vibration, 3(2), 263-273.
[26] Kassir MK, Xu JM (1988). Interaction functions of a rigid strip bonded to
saturated elastic half-space, Int. J. of Solids and Structures, 24(9), 915-936.
[27] Kassir MK, Bandyopadhyay KK, Xu J (1989). Vertical vibration of a circular
footing on a saturated half-space, Int. J. Engineering Science, 27(4), 353-361.
[28] Kassir MK, Xu J, Bandyopadyay KK (1996). Rotary and horizontal vibrations of
a circular surface footing on a saturated elastic half-space, Int. J. of Solids and
Structures, 33(2), 265-281.
[29] Lin C-H, Lee VW, Trifunac MD (2001). On the reflection of elastic waves in a
poroelastic half-space saturated with non-viscous fluid. Report CE 01-04, Dept. of
Civil Eng, Univ. of Southern California.
[30] Lin C-H, Lee VW, Trifunac MD (2005). The reflection of plane waves in a
poroelastic half-space saturated with inviscid fluid, Soil Dynamics and
Earthquake Engineering, 25, 205–223.
130
[31] Liu K, Liu Y (2004). Propagation characteristic of Rayleigh waves in orthotropic
fluid-saturated porous media, Journal of Sound and Vibration, 271(1-2), 1-13.
[32] Liu Y, Liu K, Tanimura S (2002). Wave propagation in tansversley isotropic
fluid-saturated poroelastic media, JSME International Journal, Series A, 45(3),
348-355.
[33] Liu Y, Liu K, Gao L, Yu TX (2005). Characteristic analysis of wave propagation
in anisotropic fluid-saturated porous media, Journal of Sound and Vibration,
282(3-5), 863-880.
[34] Luco JE (1969). Dynamic interaction of a shear wall with the soil, J. Engineering
mechanics, ASCE, 95:333-346.
[35] Philippacopoulos AJ (1989). Axisymmetric vibration of disk resting on saturated
layered half space. J. Eng. Mech. Div. ASCE, 115(10), 2301-2322.
[36] Rajapakse RKND, Senjuntichai T (1995). Dynamic response of a multi-layered
poroelastic medium, Earthquake Engineering and Structural Dynamics, 24, 703-
722.
[37] Senjuntichai T, Mani S, Rajapakse RKND (2006). Vertical vibration of an
embedded rigid foundation in a poroelastic soil, Soil Dynamics and Earthquake
Engineering, 26, (Special issue on Biot Centennial – Earthquake Engineering).
[38] Sharma MD, Gogna ML (1991). Seismic wave propagation in a viscoelastic
porous solid saturated by viscous liquid, Pure and Applied Geophysics, 35(3),
383-400.
[39] Sharma MD (2004). 3-D Wave Propagation in a general anisotropic poroelastic
medium: reflection and refraction at an interface with fluid, Geophysics J. Int.
157, 947-958.
[40] Todorovska MI, Trifunac MD (1990). Analytical Model for in Plane
Building-Foundation-Soil Interaction: Incident P-, SV- and Raleigh Waves,
Report No. 90-01, Dept. of Civil Engrg, Univ. of Southern California. pp.122.
131
[41] Todorovska MI (1993a). Effects of the wave passage and the embedment depth
during building-soil interaction, Soil Dynamics and Earthquake Engineering
12(6):343-355.
[42] Todorovska MI (1993b). In-plane foundation-soil interaction for embedded
circular foundations, Soil Dynamics and Earthquake Engineering 12(5):283-297.
[43] Todorovska, M.I. & Y. Al Rjoub (2006a). Effects of rainfall on soil-structure
system frequency: examples based on poroelasticity and a comparison with full-
scale measurements, Soil Dynamics and Earthquake Engrg, 26(6-7), 708-717.
(Special issue on Biot Centennial – Earthquake Engineering).
[44] Todorovska, M.I. & Y. Al Rjoub (2006b). Plain strain soil-structure interaction
model for a building supported by a circular foundation embedded in a poroelastic
half-space, Soil Dynamics and Earthquake Engrg, 26(6-7), 694-707. (Special
issue on Biot Centennial – Earthquake Engineering).
[45] Trifunac MD (1972). Interaction of a shear wall with the soil for incident plane
SH-waves. Bull. Seismological Society America, 62, 63-83.
[46] Vashishth K, Khurana P (2004). Waves in stratified anisotropic poroelastic media:
a transfer matrix approach, J. Sound and Vibration, 277(1-2), 239-275.
[47] Wong HL, Trifunac MD (1974). Interaction of shear wall with the soil for
incident plane SH waves: elliptical rigid foundation, Bulletin of Seismological
Society of America, 64, 1825-1842.
[48] Yang J (1999). Importance of flow condition on seismic waves at a saturated
porous solid boundary, Journal of Sound and Vibration, 221(2), 391-413.
[49] Yang J (2000). Influence of water saturation on horizontal and vertical motion at
a porous soil interface induced by incident P wave, Soil Dynamics and
Earthquake Engineering, 19(8), 575–581.
132
[50] Yang J (2001). Saturation effects on horizontal and vertical motions in a layered
soil–bedrock system due to inclined SV waves, Soil Dynamics and Earthquake
Engineering, 21(6), 527–536.
[51] Yang J (2002). Saturation effects of soils on ground motion at free surface due to
incident SV waves, Journal of Engineering Mechanics, 128(12), 1295-1303.
[52] Yang J and Sato T. (1998). Influence of viscous coupling on seismic reflection
and transmission in saturated porous media, Bulletin of the Seismological Society
of America, 88 (5), 1289-1299.
[53] Yang J and Sato T. (2000a). Effects of Pore-Water Saturation on Seismic
Reflection and Transmission from a Boundary of Porous Soils, Bulletin of the
Seismological Society of America, 90 (5), 1313-1317.
[54] Yang J and Sato T. (2000b). Influence of water saturation on horizontal and
vertical motion at a porous soil interface induced by incident SV wave, Soil
Dynamics and Earthquake Engineering, 19(5), 339–346.
[55] Yang J and Sato T. (2000c). Interpretation of Seismic Vertical Amplification
Observed at an Array Site, Bulletin of the Seismological Society of America, 90
(2), 275-285.
[56] Zeng X, Rajapakse RKND. (1999). Vertical vibrations of a rigid disk embedded
in a poroelastic medium. Int. J. Numerical and Analytical Methods in
Geomechanics, 23, 2075-2095.
Appendix
∫
− ⎪
⎩
⎪
⎨
⎧
= +
≠
−
−
+
+
+
= =
0
0 1 ........ .......... ,.........
2
2 sin
1 ,....
1
) 1 sin(
1
) 1 sin(
cos cos ) (
0
0
0 0
1 1 1 1
θ
θ
θ
θ
θ θ
θ θ θ
n
n
n
n
n
n
d n n I (1)
∫
− ⎪
⎩
⎪
⎨
⎧
= +
−
≠
−
−
+
+
+ −
= =
0
0 1 ........ .......... ,.........
2
2 sin
1 ,....
1
) 1 sin(
1
) 1 sin(
sin sin ) (
0
0
0 0
1 1 1 4
θ
θ
θ
θ
θ θ
θ θ θ
n
n
n
n
n
n
d n n I (2)
∫
− ⎪
⎩
⎪
⎨
⎧
=
≠
= =
0
0 0 ........ .......... ,......... 2
1 .......... ,.........
sin 2
cos ) (
0
0
1 1 5
θ
θ
θ
θ
θ θ
n
n
n
n
d n n I (3)
133
Abstract (if available)
Abstract
This thesis presents an investigation of the effects of water saturation on the effective excitation and system response during building-foundation-soil interaction, using a simple theoretical model. The model consists of a shear wall supported by a rigid circular foundation embedded in a homogenous and isotropic poroelastic half-space. The half-space is fully saturated by a compressible and viscous fluid, and is excited by in-plane wave motion, consisting of plane P and SV waves, or of surface Rayleigh waves. Partial saturation is also considered but in a simplified way. The motion in the soil is described by Biot's theory of wave propagation in fluid saturated porous media. According to this theory, two P-waves (one fast and the other one slow) and one S-wave exist in the medium, which are represent by wave potentials. Helmholtz decomposition and wave function expansion are used to represent the motion in the soil, and a closed form solution of the problem is derived in the frequency domain. Numerical results are presented for the free-field motion, foundation input motion, complex foundation stiffness matrix, and the foundation and building response to incident plane fast P and SV waves, as function of the many model parameters. The presented analysis, which is linear, is of interest for understanding and interpreting the effects of water saturation on the response of the ground and structures to small amplitude (e.g. ambient noise) and to some degree earthquake excitation. An attempt is presented to use this model to explain the observed variation of the apparent frequencies of vibration of Millikan library in Pasadena, California, with heavy rainfall.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
The role of rigid foundation assumption in two-dimensional soil-structure interaction (SSI)
PDF
Train induced vibrations in poroelastic media
PDF
Three applications of the reciprocal theorem in soil-structure interaction
PDF
Diffraction of SH-waves by surface or sub-surface topographies with application to soil-structure interaction on shallow foundations
PDF
In-plane soil structure interaction excited by plane P/SV waves
PDF
Wave method for structural system identification and health monitoring of buildings based on layered shear beam model
PDF
Diffraction of elastic waves around layered strata with irregular topography
PDF
Settlement of dry cohesionless soil deposits under earthquake induced loading
PDF
Numerical simulation of seismic site amplification effects
PDF
A complete time-harmonic radiation boundary for discrete elastodynamic models
PDF
Three-dimensional diffraction and scattering of elastic waves around hemispherical surface topographies
PDF
Three-dimensional nonlinear seismic soil-abutment-foundation-structure interaction analysis of skewed bridges
PDF
Structural system identification and health monitoring of buildings by the wave method based on the Timoshenko beam model
PDF
Characterization of environmental variability in identified dynamic properties of a soil-foundation-structure system
PDF
Scattering of elastic waves in multilayered media with irregular interfaces
PDF
Scattering of a plane harmonic SH-wave and dynamic stress concentration for multiple multilayered inclusions embedded in an elastic half-space
PDF
Transient scattering of a plane SH wave by a rough cavity of arbitrary shape embedded in an elastic half-space
PDF
Numerical study of shock-wave/turbulent boundary layer interactions on flexible and rigid panels with wall-modeled large-eddy simulations
PDF
Situated proxemics and multimodal communication: space, speech, and gesture in human-robot interaction
PDF
Numerical study of flow characteristics of controlled vortex induced vibrations in cylinders
Asset Metadata
Creator
Al Rjoub, Yousef Saleh
(author)
Core Title
Soil structure interaction in poroelastic soils
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Civil Engineering
Publication Date
02/14/2007
Defense Date
12/14/2006
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
OAI-PMH Harvest,water saturation
Language
English
Advisor
Todorovska, Maria I. (
committee chair
), Corsetti, Frank A. (
committee member
), Lee, Vincent W. (
committee member
), Trifunac, Mihailo D. (
committee member
), Wong, Hung Leung (
committee member
)
Creator Email
alrjoub@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m247
Unique identifier
UC1126720
Identifier
etd-AlRjoub-20070214 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-160129 (legacy record id),usctheses-m247 (legacy record id)
Legacy Identifier
etd-AlRjoub-20070214.pdf
Dmrecord
160129
Document Type
Dissertation
Rights
Al Rjoub, Yousef Saleh
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
water saturation