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University of Southern California Dissertations and Theses
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Train induced vibrations in poroelastic media
(USC Thesis Other)
Train induced vibrations in poroelastic media
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Content
TRAIN INDUCED VIBRATIONS IN
POROELASTIC MEDIA
By: Mohammad S. Alshaiji
University of Southern California
In Partial Fulfillment of the
Requirements for the Degree
Doctor Of Philosophy
(Civil Engineering)
December, 2015
c
Mohammad Alshaiji
Contents
DEDICATION i
ACKNOWLEDGMENT ii
LIST OF FIGURES iii
LIST OF TABLES xiv
ABSTRACT xv
Chapter 1 Introduction 1
1.1 Motivation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Wave Propagation and Poroelasticity: . . . . . . . . . . . . . . . . 3
1.2.2 Train Induced Vibrations: . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Soil-Structure-Interaction and Trac Induced Vibrations: . . . . . 13
1.2.4 Vibrations and Human Comfort: . . . . . . . . . . . . . . . . . . 14
1.3 Organization of thesis: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 2 Biot's Theory of Poroelasticity 16
2.1 Theory Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 Evaluation of Material Constants for Poroelastic Media: . . . . . 23
2.1.2 Validity of Biot's Theory: . . . . . . . . . . . . . . . . . . . . . . 26
2.1.3 Boundary Conditions for Porous Media: . . . . . . . . . . . . . . 29
2.2 Comparison of Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Medium Constants and Velocities: . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 3 Line Load on top of a Poroelastic Half-space / Layered Media 37
3.1 Introduction: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Mathematical Derivation: . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Problem setup: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 Boundary Conditions: . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2.1 Permeable Surface: . . . . . . . . . . . . . . . . . . . . . 41
1
3.2.2.2 Impermeable Surface: . . . . . . . . . . . . . . . . . . . 41
3.2.2.3 Permeable Intermediate Interface: . . . . . . . . . . . . . 42
3.2.2.4 Permeable Intermediate Interface: . . . . . . . . . . . . . 43
3.2.3 Solution of Wave Amplitudes: . . . . . . . . . . . . . . . . . . . . 44
3.2.4 Surface Displacements: . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Numerical Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 Poroelastic Half-space: . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.2 Poroelastic Layer on Top of a Poroelastic Half-space: . . . . . . . 58
Chapter 4 Concentrated Harmonic Point Load on top of a Poroelastic
Half-space / Layered Media 69
4.1 Introduction: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Mathematical Derivation: . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1 Problem setup: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.2 Boundary Conditions: . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.2.1 Permeable Surface: . . . . . . . . . . . . . . . . . . . . . 73
4.2.2.2 Impermeable Surface: . . . . . . . . . . . . . . . . . . . 73
4.2.2.3 Permeable Intermediate Interface: . . . . . . . . . . . . . 74
4.2.2.4 Permeable Intermediate Interface: . . . . . . . . . . . . . 75
4.2.3 Solution of Wave Amplitudes: . . . . . . . . . . . . . . . . . . . . 76
4.2.4 Surface Displacements: . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Numerical Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 Poroelastic Half-space . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.2 Poroelastic Layer on Top of a Poroelastic Half-space: . . . . . . . 86
4.4 Moving Load Formulation: . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.1 Numerical Results: . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4.1.1 Poroelastic Half-space . . . . . . . . . . . . . . . . . . . 99
4.4.1.2 Poroelastic Layer on Top of a Poroelastic Half-space: . . 103
Chapter 5 The Equivalent Dry Elastic Medium 110
5.1 Introduction: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Mathematical Background: . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 Equivalent Dry Medium Parameters: . . . . . . . . . . . . . . . . . . . . 116
5.4 Numerical Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2
Chapter 6 Soil-Structure Interaction of Buildings Nearby Train Tracks 131
6.1 Introduction: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Superstructure Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.3 A Simplied Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.4 Dynamic Soil-Structure Interaction . . . . . . . . . . . . . . . . . . . . . 140
6.5 Analysis Setup: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.5.1 Soil Proles: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.5.2 Displacement Components: . . . . . . . . . . . . . . . . . . . . . . 142
6.5.3 Distance from Structures: . . . . . . . . . . . . . . . . . . . . . . 143
6.5.4 SAP2000 and Material Denitions: . . . . . . . . . . . . . . . . . 143
6.6 Analysis Models: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.6.1 First Model: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.6.1.1 Results for the First Model: . . . . . . . . . . . . . . . . 145
6.6.2 Second Model: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.6.2.1 Results for the Second Model: . . . . . . . . . . . . . . . 155
6.6.3 Third Model: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.6.3.1 Results for the Third Model: . . . . . . . . . . . . . . . 168
6.7 Discussion: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Chapter 7 Summary and Future work 179
7.1 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.2 Future work: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2.1 Part of a Bigger Model: . . . . . . . . . . . . . . . . . . . . . . . 183
7.2.2 Below Ground Source: . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2.3 Vibration Mitigation: . . . . . . . . . . . . . . . . . . . . . . . . . 185
References 187
Appendix: Mathematical Formulas 196
A.1 Cartesian Coordinate System: . . . . . . . . . . . . . . . . . . . . 196
B.1 Cylindrical Coordinate System: . . . . . . . . . . . . . . . . . . . 201
C.1 Moving Load Formulation: . . . . . . . . . . . . . . . . . . . . . . 206
List of Symbols 210
3
DEDICATION
To my parents who never stopped believing in me.
To my beloved wife and kids who kept me going strong.
To Kuwait, my country, my home.
i
ACKNOWLEDGMENT
I would like to dedicate my heartfelt gratitude to my academic advisor, Dr. Hung
Leung Wong, for his sel
ess support, edication, mentorship, friendship, and guidance.
His valuable knowledge and assistance made the quest of knowledge exciting and a
dream become reality. I am also grateful for the helpful suggestions of my dissertation
committee members, Dr. Vincent Lee, Dr. James Moore, II , and Dr. Felipe de Barros. I
sincerely express my special thanks to Dr. Maria Todorovska for her valuable suggestions
and encouragement in expanding this research.
I would like to acknowledge the Public Authority for Applied Education and Training
in Kuwait for the nancial sponsorship of my studies during the PhD program at USC. I
also like to thank Mr. Khaled Al Awadhi from Kuwait Metro Rapid Transport Company
for the raw data used in this study.
I express my special thanks to my parents for their endless prayers, love, and support.
I would like to show gratitude to my family and friends for their continued encouragement
through the years of this journey. Finally, I want to thank my beloved wife and kids for
their unceasing love and support in this and every endeavor I pursue.
ii
LIST OF FIGURES
1.1 Schematic of the two main options proposed for train tracks (a) on grade,
(b) elevated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Unit size cube with solid stress tensors (
ij
) and
uid stress tensor () . . 17
2.2 The variation in the
uid modulus K
f
with air content (after Santama-
rina et al. (2001)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Boundary conditions between two porous media (Deresiewicz & Skalak,
1963) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Illustration of the boundary conditions in the case of free plane and two
porous media in contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Comparison of re
ected wave amplitudes using material constants result-
ing from Lin et al. (2005) and the ones developed in this study . . . . . . 34
3.1 (a) Line load on top of a poroelastic layered space, (b) 3D view of axis . 38
3.2 Horizontal and vertical displacement values versus distance from the
source for dierent porosity ratios (V
s
= 500 m/s) . . . . . . . . . . . . . 48
3.3 Horizontal and vertical displacement values versus frequency for dierent
distances from the source for = 0:30 and V
s
= 500 m/s . . . . . . . . . 49
3.4 Horizontal displacement versus frequency at dierent distances from the
source with dierent medium shear velocities, = 0:30 (permeable surface) 50
iii
3.5 Vertical displacement versus frequency at dierent distances from the
source with dierent medium shear velocities, = 0:30 (permeable surface) 51
3.6 Horizontal displacement versus frequency at dierent distances from the
source with dierent medium shear velocities, = 0:30 (impermeable
surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 Vertical displacement versus frequency at dierent distances from the
source with dierent medium shear velocities, = 0:30 (impermeable
surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.8 Horizontal displacement versus frequency at dierent distances from the
source with dierent medium porosities V
s
= 500 m/s (permeable surface) 54
3.9 Vertical displacement versus frequency at dierent distances from the
source with dierent medium porosities V
s
= 500 m/s (permeable surface) 55
3.10 Horizontal displacement versus frequency at dierent distances from the
source with dierent medium porosities V
s
= 500 m/s (impermeable
surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.11 Vertical displacement versus frequency at dierent distances from the
source with dierent medium porosities V
s
= 500 m/s (impermeable
surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.12 Horizontal and vertical displacement values versus distance from the
source for dierent porosity ratios and a layer thickness of 1 m with
a stiness ratio V
s
= 0:50 . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.13 Horizontal and vertical displacement values versus distance from the
source for dierent porosity ratios and a layer thickness of 10 m with
a stiness ratio V
s
= 0:50 . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.14 Horizontal and vertical displacement values versus distance from the
source for dierent top layer thicknesses with
= 1.20 and V
s
= 0:50 . . 62
iv
3.15 Horizontal and vertical displacements values versus frequency for dier-
ent distances from the source with a layer thickness of 1 m,
= 1.20,
and V
s
= 0:50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.16 Horizontal and vertical displacements values versus frequency for dier-
ent distances from the source with a layer thickness of 5 m,
=1.20, and
V
s
= 0:50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.17 Horizontal and vertical displacements values versus frequency for dier-
ent distances from the source with a layer thickness of 10 m,
=1.20,
and V
s
= 0:50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.18 Vertical displacement values versus frequency for dierent porosity ratios
for a layer thickness of 1 m and V
s
= 0:50 (permeable surface) . . . . . . 66
3.19 Vertical displacement values versus frequency for dierent porosity ratios
for a layer thickness of 10 m and V
s
= 0:50 (permeable surface) . . . . . 67
3.20 Vertical displacement values versus frequency for dierent top layer thick-
nesses for
=1.50 and V
s
= 0:50 (permeable surface) . . . . . . . . . . . 68
4.1 (a) Concentrated harmonic point load on top of a poroelastic layered
space, (b) 3D view of axis . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Horizontal and vertical displacement values versus distance from the
source for dierent medium porosities for both permeable and imper-
meable surface (V
s
= 500 m/s) . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Horizontal and vertical displacement values versus distance from the
source for dierent medium shear wave velocities for both permeable
and impermeable surface, = 0:20 . . . . . . . . . . . . . . . . . . . . . 80
v
4.4 Horizontal displacement values versus frequency at dierent medium
shear wave velocities at dierent distances from the source and a porosity
of = 0:20 (permeable surface) . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Vertical displacement values versus frequency at dierent medium shear
wave velocities at dierent distances from the source and a porosity of
= 0:20 (permeable surface) . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Horizontal displacement values versus frequency for dierent medium
porosity values at dierent distances from the source for V
s
= 500 m/s
(permeable surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.7 Vertical displacement values versus frequency for dierent medium poros-
ity values at dierent distances from the source for V
s
= 500 m/s (per-
meable surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Vertical displacement values versus frequency for dierent medium poros-
ity values at dierent distances from the source for V
s
= 500 m/s (im-
permeable surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.9 Horizontal and vertical displacements values versus distance from the
source for dierent porosity ratios with a layer thickness of 1 m and V
s
= 0.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.10 Horizontal and vertical displacements values versus distance from the
source for dierent porosity ratios with a layer thickness of 5 m and V
s
= 0.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.11 Horizontal and vertical displacement values versus distance from the
source for dierent top layer thicknesses with
= 1.50 . . . . . . . . . . 90
4.12 Horizontal displacement values versus frequency at dierent distances
from the source for dierent porosity ratios and a layer thickness h = 5
m (permeable surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
vi
4.13 Horizontal displacement values versus frequency at dierent distances
from the source for dierent porosity ratios and a layer thickness h = 10
m (permeable surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.14 Vertical displacement values versus frequency at dierent distances from
the source for dierent porosity ratios and a layer thickness h = 5 m
(permeable surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.15 Vertical displacement values versus frequency at dierent distances from
the source for dierent porosity ratios and a layer thickness h = 10 m
(permeable surface) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.16 Horizontal displacement values versus frequency at dierent distances
from the source for dierent layer thicknesses,
=1.20 (permeable surface) 95
4.17 Vertical displacement values versus frequency at dierent distances from
the source for dierent layer thicknesses,
=1.20 (permeable surface) . . 96
4.18 Schematic of a moving harmonic point load along the x-axis and an
observer at distance D along the y-axis. . . . . . . . . . . . . . . . . . . . 97
4.19 Schematic showing the moving loads at x
1
, x
2
, x
3
, and x
n
. . . . . . . . . 98
4.20 Vertical displacement for dierent values of half-space porosities as a
function of distance from the source for train speeds of (a) 120 km/hr,
(b) 60 km/hr, and (c) 20 km/hr . . . . . . . . . . . . . . . . . . . . . . . 101
4.21 Vertical displacement for dierent values of half-space porosities as a
function of train speed for away distances from the tracks of (a) 20 m,
(b) 50 m, and (c) 100 m . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.22 Vertical displacement for dierent values of layer porosity ratios as a
function of distance from train track speed for layer thickness of (a) 1 m,
(b) 5 m, and (c) 10 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
vii
4.23 Vertical displacement for dierent values of layer thicknesses as a function
of train speeds for distance from train track of (a) 20 m, (b) 50 m, (c)
100 m, and (d) 150 m (lower frequency) . . . . . . . . . . . . . . . . . . . 106
4.24 Vertical displacement for dierent values of layer thicknesses as a function
of train speeds for distance from train track of (a) 20 m, (b) 50 m, (c)
100 m, and (d) 150 m (higher frequency) . . . . . . . . . . . . . . . . . . 107
4.25 Vertical displacement for dierent values of layer thickness as a function
of distance from train tracks for speeds of (a) 120 km/hr, (b) 60 km/hr,
and (c) 20 km/hr (lower frequency) . . . . . . . . . . . . . . . . . . . . . 108
4.26 Vertical displacement for dierent values of layer thickness as a function
of distance from train tracks for speeds of (a) 120 km/hr, (b) 120 km/hr,
and (c) 120 km/hr (higher frequency) . . . . . . . . . . . . . . . . . . . . 109
5.1 Dimensionless frequency () vs. Wave velocities of the medium for dif-
ferent in-situ S-wave velocities (a) Fast P-wave (b) Slow P-wave (c) S-wave 115
5.2 Porosity () vs. Characteristic frequency (!
o
) for dierent values of
medium permeability () . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3 Porosity () vs. Characteristic frequency (!
o
) for permeability =
110
04
m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4 Equivalent wave velocity vs. porosity of the medium for dierent in-situ
S-wave velocities (a) P-wave (b) S-wave . . . . . . . . . . . . . . . . . . . 124
5.5 Calculated and equivalent P and S-wave velocities vs. Porosity of the
medium for (a) V
s
= 250 m/s (b) V
s
= 500 m/s (c) V
s
= 1000 m/s . . . . 125
5.6 Percentage dierence between calculated P-wave velocities and equiva-
lent P-wave velocities for dierent medium porosities . . . . . . . . . . . 126
viii
5.7 Percentage dierence between calculated S-wave velocities and equivalent
S-wave velocities for dierent medium porosities . . . . . . . . . . . . . . 126
5.8 Calculated / Equivalent / Dry P-wave velocities vs. Porosity of medium
at (a) V
s
= 250 m/s (b) V
s
= 500 m/s (c) V
s
= 1000 m/s . . . . . . . . . 127
5.9 Equivalent Poisson's ratio vs. Porosity of medium at dierent in-situ
S-wave velocity of equivalent dry medium . . . . . . . . . . . . . . . . . . 128
5.10 Dry / Saturated / Equivalent shear modulus of the medium vs. Porosity
of the medium for dierent values of in-situ V
s
. . . . . . . . . . . . . . . 128
5.11 Equivalent wave velocity vs. porosity of the medium for dierent in-situ
S-wave velocities (a) P-wave (b) S-wave . . . . . . . . . . . . . . . . . . . 129
5.12 Dry / Equivalent (saturated) density of the medium vs. Porosity of the
medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.13 Percentage dierence between dry and equivalent (saturated) density of
the medium vs. Porosity of the medium . . . . . . . . . . . . . . . . . . 130
6.1 Schematic of how the buildings are in a close proximity of the train tracks 133
6.2 Soft and hard cases soil proles to be used in the analysis of SSI . . . . . 142
6.3 Three displacement components: Vertical, Transverse, and Parallel . . . . 143
6.4 Schematic showing the denition of \distance to source" . . . . . . . . . 143
6.5 First model: two-story-single-bay concrete structure (a) 3D view, (b)
Cross-sectional details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.6 Schematic showing the \exterior" nodes 1, 5, and 9 locations relative to
loading source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.7 Vertical displacement component at dierent levels of model (1) vs. Fre-
quency (soft layer / 75 km/h / 15 m away from source) . . . . . . . . . . 147
ix
6.8 Vertical displacement component at dierent levels of model (1) vs. Fre-
quency (soft layer / 75 km/h / 60 m away from source) . . . . . . . . . . 147
6.9 Transverse displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 15 m away from source) . . . . . . . . 148
6.10 Transverse displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 60 m away from source) . . . . . . . . 148
6.11 Parallel displacement component at dierent levels of model (1) vs. Fre-
quency (soft layer / 75 km/h / 15 m away from source) . . . . . . . . . . 149
6.12 Parallel displacement component at dierent levels of model (1) vs. Fre-
quency (soft layer / 75 km/h / 60 m away from source) . . . . . . . . . . 149
6.13 Vertical displacement component at dierent levels of model (1) vs. Fre-
quency (hard layer / 75 km/h / 15 m away from source) . . . . . . . . . 151
6.14 Vertical displacement component at dierent levels of model (1) vs. Fre-
quency (hard layer / 75 km/h / 60 m away from source) . . . . . . . . . 151
6.15 Transverse displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 15 m away from source) . . . . . . . . 152
6.16 Transverse displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 60 m away from source) . . . . . . . . 152
6.17 Parallel displacement component at dierent levels of model (1) vs. Fre-
quency (hard layer / 75 km/h / 15 m away from source) . . . . . . . . . 153
6.18 Parallel displacement component at dierent levels of model (1) vs. Fre-
quency (hard layer / 75 km/h / 60 m away from source) . . . . . . . . . 153
6.19 Second model: three-story concrete structure model (a) 3D view, (b)
Cross-sectional details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.20 Schematic showing the \interior" nodes 11, 27, 43, and 59 locations rel-
ative to loading source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
x
6.21 Vertical displacement component at dierent levels of model (2) vs. Fre-
quency (soft layer / 75 km/h / 20 m away from source) . . . . . . . . . . 157
6.22 Vertical displacement component at dierent levels of model (2) vs. Fre-
quency (soft layer / 75 km/h / 40 m away from source) . . . . . . . . . . 157
6.23 Transverse displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 20 m away from source) . . . . . . . . 158
6.24 Transverse displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 40 m away from source) . . . . . . . . 158
6.25 Parallel displacement component at dierent levels of model (2) vs. Fre-
quency (soft layer / 75 km/h / 20 m away from source) . . . . . . . . . . 159
6.26 Parallel displacement component at dierent levels of model (2) vs. Fre-
quency (soft layer / 75 km/h / 40 m away from source) . . . . . . . . . . 159
6.27 Schematic showing location of nodes 25, 26, 27, and 28 on the second
oor plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.28 Vertical displacement component at second
oor of model (2) vs. Fre-
quency (soft layer / 75 km/h / 20 m away from source) . . . . . . . . . . 160
6.29 Vertical displacement component at second
oor of model (2) vs. Fre-
quency (soft layer / 75 km/h / 40 m away from source) . . . . . . . . . . 161
6.30 Vertical displacement component at dierent levels of model (2) vs. Fre-
quency (hard layer / 75 km/h / 20 m away from source) . . . . . . . . . 163
6.31 Vertical displacement component at dierent levels of model (2) vs. Fre-
quency (hard layer / 75 km/h / 40 m away from source) . . . . . . . . . 163
6.32 Transverse displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 20 m away from source) . . . . . . . . 164
6.33 Transverse displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 40 m away from source) . . . . . . . . 164
xi
6.34 Parallel displacement component at dierent levels of model (2) vs. Fre-
quency (hard layer / 75 km/h / 20 m away from source) . . . . . . . . . 165
6.35 Parallel displacement component at dierent levels of model (2) vs. Fre-
quency (hard layer / 75 km/h / 40 m away from source) . . . . . . . . . 165
6.36 Vertical displacement component at second
oor of model (2) vs. Fre-
quency (hard layer / 75 km/h / 20 m away from source) . . . . . . . . . 166
6.37 Vertical displacement component at second
oor of model (2) vs. Fre-
quency (hard layer / 75 km/h / 40 m away from source) . . . . . . . . . 166
6.38 Third model: three-story concrete structure model (a) 3D view, (b)
Cross-sectional details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.39 Plan view of the ground
oor of model (3) showing nodes 1, 7, and 12
locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.40 Vertical displacement components for three nodes vs. Time at (a) Ground
oor and (b) First
oor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.41 Vertical displacement components for three nodes vs. Time at (a) Second
oor and (b) Third
oor . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.42 Vertical displacement component for the same location on dierent
oor
level vs. Time (a) Nodes 19, 31, and 43 and (b) Nodes 24, 36, and 48 . . 172
6.43 Plan view of the ground
oor of model (3) showing nodes 1, 4, and 12
locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.44 Transverse displacement components for three nodes vs. Time (a) Ground
oor and (b) First
oor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.45 Transverse displacement components for three nodes vs. Time at (a)
Second
oor and (b) Third
oor . . . . . . . . . . . . . . . . . . . . . . . 175
xii
6.46 Transverse displacement component for the same location on dierent
oor level vs. Time (a) Nodes 4, 16, 28, and 40 and (b) Nodes 7, 19, 31,
and 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.1 Schematic of the vibration path caused by trains moving on a viaduct
on nearby structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2 Schematic of the vibration path caused by trains moving underground
nearby structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.3 Schematic showing a steel beam to stien the concrete slab of a vibrating
oor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.4 Schematic showing an open trench to mitigate the vibration caused by
trains nearby structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A.1 (a) Line load on top of a poroelastic layered space, (b) 3D view of axis . 196
B.1 (a) Concentrated harmonic point load on top of a poroelastic layered
space, (b) 3D view of axis . . . . . . . . . . . . . . . . . . . . . . . . . . 201
B.2 Stress element in (a) Spherical coordinates (b) cylindrical coordinates
(Schepers, Savidis, & Kausel, 2010) . . . . . . . . . . . . . . . . . . . . . 203
C.1 Schematic of a moving harmonic point load along the x axis and an
observer at distance D along the y axis. . . . . . . . . . . . . . . . . . . . 206
xiii
LIST OF TABLES
1.1 Symptoms due to whole-body vibration and the frequency range at which
they occur. (after Rasmussen (1983)) . . . . . . . . . . . . . . . . . . . . 15
2.4 Velocities produced given frequencies and wavelengths . . . . . . . . . . . 27
2.5 Summary of the boundary conditions in the case of free plane and two
porous media in contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Material constants and wave velocities for dry frame and water saturated
media with dierent porosities . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1 Dry and saturated medium parameters . . . . . . . . . . . . . . . . . . . 112
5.2 Material constants and wave velocities for dry, poroelastic, and equiva-
lent dry elastic medium with dierent porosities . . . . . . . . . . . . . . 118
xiv
ABSTRACT
The Ministry of Communications, through the Kuwait Overland Transport Union,
raised the need to develop a master plan for the implementation of a rapid transit system
that will reduce the trac congestions in the streets of Kuwait. With such a project,
an understanding of the vibration induced by this mean of transportation on the nearby
structures is needed. The water table in the area proposed to host the lanes
uctuates
depending mainly on rainfall and irrigation process and ranges from 0.50 m in some areas
and as deep as 13.0 m. Due to the saturation of the soil with water, Biots theory of
wave propagation in poroelastic media is applied to calculate the surface horizontal and
vertical displacements for the poroelastic half-space and a poroelastic layered media.
The train load is simulated by a uniformly distributed harmonic line load rst and
then by a concentrated harmonic point load with load speed formulation added to the
latter to simulate the movement of the train and study the eect on horizontal and
vertical displacements. It was found that the displacement values increase with the
increase of the porosity of the medium. Plus, the values of such displacements will
increase even more if the saturating water was trapped due to the impermeability of
the mediums surface. The water table eect was simulated by changing the thickness
of the rst layer below the surface and assuming a permeable interface between this
layer and the half-space underneath it. There existed two critical combinations between
the applied load and the thickness of the rst layer where at the one combination the
horizontal disturbance is at maximum and at the second the vertical is. The change
of porosity contrast between the rst layer and the half-space showed that the higher
the contrast, the higher the displacements values. Plus, the eect of porosity contrast
is highly substantial at the load-layer thickness critical combinations. As for a single
train speed, regardless of the distance from the train track, a signicant increase of
vertical displacements was present as the porosity of the half-space increased. As the
xv
train speed increased, the displacements increased with increasing porosity. With the
introduction of a poroelastic layer, the displacements at closer distances from the track
increased even more than the one obtained for the half-space case. Plus, It was shown
that the rate at which the displacement increase with increasing train speed became
higher as the porosity ratio of the poroelastic layered system increased. Finally, a Soil-
Structure-Interaction (SSI) scheme was developed and showed ,using three structural
models, that SSI plays a critical role in amplied vibration from transit noise and it
was concluded through frequency domain analysis and time history analysis that Soil-
Structure-Interaction is mostly dominated by the lower modes of the superstructure.
xvi
Chapter 1
Introduction
1.1 Motivation:
The Ministry of Communications, through the Kuwait Overland Transport Union,
raised the need to develop a master plan for the implementation of a rapid transit system
that can reduce the trac loads and congestions in the streets of Kuwait. The Kuwait
Rapid Transit System (KRTS) master plan carried by INECO provided a complete
analysis of background information on previous transit studies and dened the main
corridors with its technical characteristics, passenger demands, and also infrastructure,
rail system. It was concluded that a modern and attractive public transport system
comprising of a line haul Light Rail Transit System replacing the heavily used bus
services. Moreover, the transportation network lines should cover the Metropolitan Area
and be supplied gradually with park{and{ride sites. The implementation of a KRTS will
provide a fast, safe & sustainable modern transport, and serve to connect the new towns
and urban developments with Kuwait City, will allow Kuwait to take advantage of the
future connection to G.C.C countries & International regional rail networks, and will
give the opportunity to Kuwait ports to be international hubs for freight, respecting the
1
environmental and other protected areas.
In an eort to reduce construction cost and time, the existing road networks will be
used as the main guides for the proposed transit lanes. The two main options proposed,
depending on the availability of road width, are on grade lanes and elevated lanes as
shown in gure (1.1). Huge portion of the project runs through highly populated areas
where the distance from the proposed tracks and the nearby structures ranges from 20
m to 50 m. In other parts of the project, the distance to the closest structure is about
100 m. Although these options can serve as money and time savers in terms of construc-
tion, the vibration caused by the moving trains being felt in the nearby residential and
governmental buildings is a main concern. How severe are these vibrations? Will they
cause discomfort to the residents or employees?
(b)
Figure 1.1: Schematic of the two main options proposed for train tracks (a) on
grade, (b) elevated
Another aspect of the vibration produced by these trains is the medium carrying
the waves. The water table in the area proposed to host the lanes is as shallow as 0.50
m in some areas and as deep as 13.0 m. The depth of water table
uctuate depending
2
mainly on rainfall and irrigation process. Can the presence of water in the soil aect
the characteristics of the wave propagation to the nearby structures? Can the depth of
this water table play a role in increasing or decreasing the level of vibration in nearby
buildings? To understand the train induced vibrations in saturated soils, the theory of
wave propagation in poroelastic media by Biot is implemented using a two dierent load
sources acting from the train tracks; a distributed harmonic line load and a concentrated
harmonic point load. These loads are to be applied for a poroelastic half-space and for
a poroelastic half-space with a poroelastic layer on top.
1.2 Literature Review:
1.2.1 Wave Propagation and Poroelasticity:
Wave propagation in solids can be dened as the mechanical disturbance that spreads
through a material triggering oscillations of the particles about their equilibrium posi-
tions. Elastic wave propagation is the same disturbance mentioned above, yet it takes
place in an elastic material that deforms proportionally to the applied force and rapidly
return to its original state and dimensions with no permanent damage. Elastic or not,
one can observe that the general theory of wave propagation is being used in many
elds of engineering and science. In the eld of geotechnical engineering, for example,
eld testing techniques utilizing wave propagation theory help identify some underly-
ing soil mechanical properties such as soil moduli and layer thicknesses. In the eld of
structural engineering, some non-destructive techniques which are based on wave propa-
gation theory can be used to improve safety, reliability and operational life of structures.
In geophysics, wave propagation plays a great role in determining earthquake sources,
studying global ray paths, and imaging 3D structure of the Earth's interior.
A major concept in the wave propagation theory is the behavior of the material in
3
which the wave travels through. Homogeneity, isotropy, and elasticity, of the medium
should rst be specied. The simplest media is always assumed homogenous, isotropic,
and elastic which leads to simplest formulations of wave propagation equations due to
the direct application of the well-established theory of elasticity. Unfortunately, these
assumptions are not accurate when describing a medium such as soil. It is well known
that soil is a heterogeneous media due to its lack of uniformity in composition and
constituents. As for isotropy, soil is a perfect example of anisotropic medium since it
does not show similar characteristics in all direction of property measurements.
Elasticity of any material is aected by two factors, the applied load and internal
structure of the material. If the applied load makes the material to behave within its
elastic region, and the inner structure is mainly made of one type of material, then the
material can be classied as elastic. Focusing on the material structure, soil, which can
contain rocks as well, is a porous medium with voids that are lled with either gas (dry
soil) or liquid (saturated soil). The presence of such
uids in the voids, classify the soil
as a poroelastic medium.
The mechanical properties of the soil will be modied due to the presence of the
uids in the pores. Under applied stress, the
uid in the pores can either escape the pore
network or not. If the
uid is allowed to escape, the soil will experience a progressive
deformation. If not, the pore pressure will build up due to the local dilation of the
medium as well as the external applied stress. In both cases, the mechanical properties
of the medium will have a time dependent character. All of the previous aspects are
well described using the theory of poroelasticity.
The theory of poroelasticity is a continuum theory for the analysis of a porous media
including of an elastic matrix containing interconnected
uid-saturated pores. Phys-
ically, the theory suggests that when a porous material is under stress, the resulting
matrix deformation leads to volumetric variations in the pores. Since the pores are
4
uid-lled, the presence of the
uid not only acts as a stiener of the material, but also
results in the
ow of the pore
uid between regions of higher and lower pore pressure
(diusion). The dynamic response of a poroelastic medium is of interest to applica-
tions in geophysics, petroleum engineering, geotechnical engineering, and earthquake
engineering (C. Lin, Lee, & Trifunac, 2001). Moreover, the theory of poroelasticity has
many applications in biological tissue mechanics as well as applications in industrial
ltration systems (Lam, Zhang, & Zong, 2004).
According to De Boer (1996), the origin of the theory of poroelasticity can be traced
back to the late eighteenth century when it was called porous media theory. Early
contributions to that theory were made by Woltman (1794) when he developed earth
pressure theory and introduced the concept of volume fractions. In the nineteenth cen-
tury, further important contributions were published on the concept of surface fractions,
the diusion problem, ground-water
ow, and the mixture theory. These publications
were by Delesse (1866), Fick (1855), Darcy (1856), and Stefan in (1871),(1872), and
(1875).
In the twentieth century, Fillunger (1913) started the discussion of porous media
theories in a paper about the uplift in saturated rigid porous solids. In subsequent arti-
cles (1914), (1929), (1930), and (1935), Fillunger investigated the phenomena of friction
and capillarity where he discovered the eect of eective stresses and in 1936. Plus,
Fillunger founded the concept of the mechanical theory of liquid-saturated deformable
porous solids (Fillunger, 1936). In the frame work of the calculation of the permeabil-
ity coecient of clay, the founder of modern soil mechanics, Karl Von Terzaghi (1923),
started his studies on saturated deformable porous solids. The research of Von Terza-
ghi and Fillunger was continued by Biot (1935), (1941a), and (1941b), Heinrich (1938),
Heinrich and Desoyer (1955) and (1956) then Frenkel (1944) in the next decades.
The theory of propagation of elastic waves in a
uid-saturated porous medium was
5
rst established by Biot in a series of papers (1956a), (1956b), and (1962) to deal with soil
consolidation and wave propagation problems in geomechanics. Unlike the two waves,
one dilatational (P) and one rotational (S) wave, that exist in an elastic medium, Biots
theory predicted the existence of an additional P-wave in a poroelastic medium. This
additional wave is a result of the relative motion of the
uid component of the porous
material with respect to the solid. The original P-wave was referred to as the fast P-
wave, where the new predicted P-wave was denoted as the slow P wave, which is slower
and is more attenuated and dispersed than the original P wave. More than twenty years
later, Berrymann (1980) conrmed the existence of the slow P-wave experimentally.
In 1956, Biot presented the theory of wave propagation in poroelastic medium appli-
cable to lower frequencies based on the assumption that the
ow of the
uid in the pores
is Poiseuille (M. A. Biot, 1956a). In the same year (M. A. Biot, 1956b), he expanded
his theory beyond the critical frequency for which the Poiseuille assumption becomes
invalid. Moreover, Biot (1962) extended his theory to media with solid dissipation and
anisotropic media.
Deresiewicz (1960) studied the eects of boundaries on wave propagation in fully
saturated poroelastic media by considering incident plane body and surface waves onto
a traction free poroelastic half-space. Deresiewicz (1960) investigated Love waves in
a half-space saturated with a viscous liquid. The existence of Rayleigh-type surface
wave in poroelastic medium was investigated by Jones (1961). Deresiewicz and Rice
(1962) studied incident plane P and SV waves onto a half-space saturated by a viscous
uid. Based on the principles of conservation of mass and continuity of momentum,
Deresiewicz and Skalak (1963) discussed the boundary conditions at free plane as well as
for two dierent saturated porous media in contact. It was concluded that the boundary
conditions needed to solve the wave propagation equation proposed by Biot (1956a), in
general, depend on the average displacements of the solid component of the porous
6
medium, the normal solid particle displacement relative to the liquid component of the
medium, the stresses experienced by the bulk material, and the pore pressure of the
liquid in the medium.
Hajra and Mukhopadhyay (1982), based on work of Biot, Deresiewicz and Skalak,
studied the problem of re
ection and refraction of a wave propagating obliquely at the
boundary between elastic and poroelastic media by applying the continuity of normal and
tangential displacements and stresses at that interface. Yang and Sato (1998) considered
the hydraulic boundary condition for the interface between the elastic and poroelastic
media to be either permeable or impermeable. Later, Yang (2001) applied permeable
boundary conditions only to study the eects of dierent degrees of saturations in a
multi-layered soil-bedrock system.
Some investigators studied the wave propagation through multi-layered poroelastic
mediums. Chattopadhyay et al. (1986) investigated the propagation of Love waves in an
isotropic porous layer overlying an inhomogeneous half-space generated by a point source
at the interface of the layer and the half-space. Sharma and Gogna (1991) examined the
propagation of Love waves in an initially stressed medium consisting of a slow elastic
layer lying over a liquid-saturated porous solid half-space. Degrande et al. (1998)
examined the harmonic and transient wave propagation in multi-layered dry, saturated
and unsaturated isotropic poroelastic media. Wang and Zhang (1998) discussed the
propagation of Love waves in a transversely isotropic
uid-saturated porous layered
half-space using an iterative method. Vashishth and Khurana (2004) studied the wave
propagation in a multi-layered anisotropic poroelastic medium. Ke et al. (2005) and
(2006) studied the propagation of Love waves in a porous, layered half-space saturated
with an inhomogeneous
uid with the elastic constants varying with the depth.
Some investigation was based on the theory of wave propagation in porous media
to study problems in oil production, transportation, and sea
oor imaging. Edelman
7
(2006) studied the low-magnitude earthquakes produced resulting from the injection of
borehole
uid into reservoirs during oil production process. He rigorously derived an
analytical method to monitor the value of pore pressure and assess values of hydro-
mechanical parameters of porous rocks. Theodorakopoulos and Beskos (2006) reviewed
their exact and approximate solutions in applying the theory of poroelasticity in prob-
lems encountered in transportation engineering. The problems involved the analysis of
retaining wall resting on poroelastic soil and subjected to moving loads with harmonic
excitation. Moreover, they examined the in
uence of dynamical and hydro-mechanical
properties of the soil as well as the seismic frequency of the retaining structure in the
dynamic response of the medium. Realizing the importance of the variation of acoustic
waves incident angle re
ection in the eld of acoustic sea
oor imaging and seismology,
Bouzidi and Schmitt (2012) conrmed the boundary conditions proposed by Deresiewicz
and Skalak (1963) experimentally using an ultrasonic re
ectometer.
Other recent work on wave propagation in
uid saturated poroelastic media includes
that of: Cai et al. (2007) studied the dynamic steady state responses of a poroelastic
half-space soil medium subjected to a moving rectangular load by solving the governing
equations of motion semi-analytically. It was concluded that the load velocity has an
apparent eect on the dynamic response of the medium and pore water pressures. Xu
et al. (2008) investigated the dynamic response of a layered water-saturated half space
to a moving load using the transmission and re
ection matrix method (TRM). They
concluded that the presence of a soft layer will enhance the vertical displacement and
the pore pressure of the layered half space to a certain degree. Hu et al. (2010), who
presented an approximate analytical for the dynamic response of a rigid cylindrical
foundation embedded in a poroelastic soil layer under the excitation of a time-harmonic
rocking moment; Lo et al. (2010), who presented theoretical analysis for the dynamic
response of a semi-innite
uid-bearing porous medium to external harmonic loading
8
based on the decoupled poroelasticity equations of Biot; Cai et al. (2010), based on
Biots poroelastic dynamic theory, used an analytical approach to study the torsional
vibrations of a rigid circular foundation resting on saturated soil to obliquely incident
SH waves; Chattaraj et al. (2011), who investigated the possibility of propagation of
torsional surface wave in
uid saturated poroelastic layer lying over non-homogeneous
elastic half space; Son and Kang (2012a) and (2012b), studied how the thickness ratio,
layer boundary condition, porosity, and anisotropy of poroelastic layered medium can
aect the phase and group velocities of shear waves. Liu et al. (2013) extended the
generalized transfer matrix method to the dynamic analysis of multilayered poroelastic
media. That resulted in an efcient recursive scheme for determination of the equivalent
interface sources resulting from excitation by a source at an arbitrary depth for multi-
layered poroelastic media. An analytical wave form and dispersion solutions of Rayleigh
surface wave for a poroelastic model at low frequencies was obtain by Zhang et al. (2014)
using viscoelastic representation. It was shown that the proposed model is sensitive to
the critical frequency of the medium yet it requires less computational power which can
become an issue for large models.
1.2.2 Train Induced Vibrations:
Beside natural occurring sources of vibrations, like earthquakes, surface and under-
ground trains can produce a common source of vibration. The irregularities in tracks
where train move cause vibrations which are classied as environmental impacts that
can aect nearby structures and its occupants. These vibrations can be characterized
and studied using the wave propagation theory. Plus, if the tracks are resting on a
poroelastic medium, the ground disturbance can be described using Biots theory of
wave propagation in poroelastic media.
It is safe to say that the problems related to vibrations induced by trains are not
9
recent since it can be traced back to the time where trains came to serve the public. For
example, in (1897) and (1900) a central London newspaper, The Lancet, had two articles
regarding the vibrations of the railway vibrations. In the rst, the newspaper compared
the produced vibration between dierent railway routes and wondered why people feel
the vibration along one route more than the other. In the second, The Lancet was urging
people to take actions if they suered from any discomfort from the vibrations cause by
electric trains while living close to the train routes.
Researchers were trying to understand how to predict these kinds of vibrations on
nearby structures and then how to control it or even prevent it. According to Gutowski
and Dym (1976), Sir Horace Lamb (1903), Professor of Mathematics at the Univer-
sity of Adelaide in South Australia, investigated the response of isotropic, homogeneous
elastic half-spaces to various harmonic and impulsive loads which formed the basis of
almost all of the analytical work on sources and transmission paths in soils. Verhas
(1979) presented three models for the prediction of the propagation of train-induced
ground vibration. The models where: the line source, the point source, and the super-
posed models. He concluded that each model is acceptable given a certain number of
assumptions. Ford (1987) showed the relation between the vibration frequencies and
the wagon length and discussed how vehicle vibration (inertia eects) can contribute to
ground vibration. Alabi (1989) introduced a three-dimensional model for the evaluation
of ground displacements due to forces transmitted to the ground from the wheels of a
moving surface train through the track. The model showed that the vertical component
of displacement was dominant and displacements showed great sensitivity to the num-
ber of wagons. Sheng et al. (1999) modeled the train track as an innite layered beam
structure resting on a ground made up of innite parallel homogeneous elastic layers.
Jones et al. (2000) tested that model and found that for loads travelling at high speeds,
comparable to the speeds of vibration propagation in the track/ground structure high
10
levels of vibration occur in the ground under, or very near to, the track. Bahrekazemi
(2004) developed an empirical model to be used in an early design phase to decide on
possible routes for a new railway line, identifying areas where excessive ground motion
can be expected producing results relatively quickly with a limited amount of site in-
formation. With et al. (2006) used some eld data to validate the proposed empirical
method by Bahrekazemi and conclude that it was fairly accurate and easy to use. Chen
et al. (2007) presented a hybrid method alongside with a numerical method to evaluate
the vibration on nearby buildings resulting from moving trains on bridges. They found
that the hybrid method, which consists of eld measurements, numerical modeling, and
design handbook equations, can be used for preliminary design stages only. On the other
hand, the numerical method provides a more detailed analysis that can count for some
factors overlooked by the hybrid method such as building stiness, attenuation within
the building, and resonance eects.
Other recent work on the prediction of vibrations induced by trains on nearby struc-
tures includes that of: Cai et al. (2008) studied three dimensional dynamic responses
of track{ground system subjected to a moving train over fully saturated soil medium.
They showed that dynamic responses of the track{ground system are highly aected by
the load velocity and rail rigidity. Ju et al. (2009) investigated the dominant frequencies
of tain{induced vibrations on bridges, embankments, and in tunnels. They concluded
that the dominant frequencies are only dependent on the ratio of the train speed to
the carriage length and any frequencies beyond twenty ve times this ratio (usually
generated from contact of wheels and rails) are considered as sound noise. Salvador et
al. (2011) proposed a procedure to estimate the vibration path caused by the passing
of a train where the equations which describe the physical phenomenon are set in the
frequency and wavenumber domain and solved by using the Fourier Transform and then
the analytical solution is transformed back into the time and space domain by means of
11
the Fourier series. Verbraken et al. (2011) suggested a procedure that allows predicting
ground surface vibrations and re-radiated noise in buildings where ground vibrations are
calculated based on force densities. The procedure, however, can only be used when an
appropriate estimation of the force density is available.
Jones et al. (2012) pointed the increased level of uncertainty when using homogenous
soil properties in numerical prediction models and how stochastic variability of the soils
elastic modulus, if introduced properly, can minimize such uncertainty eect. Cao et
al. (2013) investigated the eects of acceleration and deceleration of trains on a fully
saturated poroelastic half-space. It was found that the deceleration can cause an increase
in ground vibration as well as pore water pressure for train speeds of 200 km/h. Yang
et al. (2014) investigated, semi-analytically, the dynamic response of railway track
resting on a poroelastic half-space supported by sleepers. They found that the vertical
displacement under the tracks increases for a train velocity above the Rayleigh-wave
velocity of the soil medium. These displacements decrease rapidly if the space between
the sleepers is shortened. The transient responses in poroelastic media due to moving
surface pulses was investigated by Zhang et al. (2014) for permeable, impermeable, and
partially permeable boundaries. It was concluded that the Rayleigh surface wave carries
most of the energy where it is more sensitive to the surface stiness condition that the
body waves.
The research on considering the eects of the soil-structure interaction (SSI) in train
induced vibration is not extensive compared to the SSI research done in the earthquake
engineering eld. Yet, it can be seen that it focuses on two areas; soil{track interaction
(STI) and soil-structure interaction. For the STI, Bode et al. (2002) demonstrated the
in
uence of the soiltrack interaction when analyzing the dynamic response of a railway
track due to a moving wheel. Considering STI, Auersch in (2005) developed a prediction
model for xed and moving point or track loads to generate the transfer function for
12
testing site. A three dimensional nite element model was developed in the time domain
by Galv n et al. (2010) to predict vibrations due to train passage at the vehicle, the track
and the free eld considering the traintracksoil interaction. Xia et al. (2010) presented,
on the basis of vehicle dynamics, track dynamics and the Green's Functions of subsoil,
an integrated train-track-subsoil dynamic interaction model of moving-train induced
ground vibration. In an eort to reduce computational cost, Triepaischajonsak et al.
(2015) proposed a hybrid model that where the vehicle/track interaction is obtained in
the time domain then extracting track forces to used them a second model to predict the
ground response at arbitrary locations. This method took into account the train-track-
subsoil dynamic interaction and could be compound with other train-track interaction
models.
1.2.3 Soil-Structure-Interaction and Trac Induced Vibrations:
One of the early studies done relating SSI and trac as a source of vibration was
done by L. Pyl (2004) when he proposed a model to predict the response of a single
family residence using a coupled BEM-FEM formulation. When investigating the role
of SSI eects on building due to trac induced vibration, Fran cois et al. (2007) showed
that the vibration levels in buildings depend on the relative stiness between the soil
and foundation. Plus, increased attenuation of the
oor response is observed due to SSI
since the energy in the building is dissipated via geometric damping in the soil. Based
on Pyl's work, Auersch in (2010), concluded that if SSI is considered, resonance eects
are mild and decrease at high frequencies. Romero et al. (2013) accounted for the
non-linear soil-structure contact when analyzing the soil-foundation interaction. Using
SSIFiBo, a MATLAB toolbox they developed, Galv n and Romero (2014) investigated a
soil-structure interaction problem having a structure exposed to dierent incident wave
elds considering the vertical component of disturbance only.
13
1.2.4 Vibrations and Human Comfort:
The impact of vibration on humans cannot be neglected as well. In terms of occupants
comfort, Connolly et al. (2014) identied this impact on occupants of a given structure
to be a combination of whole body response, low frequency noise from disturbed
oors
and walls, and noise/vibration from non-structural objects within a structure such as
windows, doors, and furniture. In fact, Lee et al. (2013) concluded that overall annoy-
ance caused collectively by noise and vibration is signicantly more than the annoyance
caused by vibration only.
Physiologically, vibrations can have an impact on respiratory, cardiovascular, metabolic,
and skeletal systems. According to Harris et al. (2002), increased blood pressure, respi-
ration and heart rates (which are usually associated with moderate exercise) can occur
as a response to body vertical vibrations. Thinking of the human body organs and parts
as masses that can vibrate, leads to the possibility that these parts can vibrate at their
own resonance causing discomfort or even damage to the human body. Moreover, Ran-
dall et al. (1997) showed that maximum displacement between the skeletal structure
and the organ takes place at the resonant frequency which causes strain on the body
tissue surrounding that organ. According to Al Suhairy (2000), the most critical fre-
quency of vibration resides between 4 Hz and 8 Hz which according to Auvinen (2011)
is in the range of the natural frequecy of the human body (4 Hz12 Hz). Vibrations
between 2.5 Hz and 5 Hz can amplify the vertebration of the neck and lumbar resonance
up to 240%. The resonance of the trunk of a human body can be amplied by 200%
between the frequencies of 4 Hz and 6 Hz. And nally, between 20 Hz and 30 Hz, the
resonance between the head and shoulders amplied by up to 350%. As shown in table
(1.1), Rasmussen (1983) provided a list of some predominant symptoms experienced by
humans when exposed to frequencies ranging from 4 Hz to 20 Hz.
14
Table 1.1: Symptoms due to whole-body vibration and the frequency
range at which they occur. (after Rasmussen (1983))
Symptoms Frequency (Hz)
General feeling of discomfort 4-9
Head symptoms 13-20
Lower jaw Symptoms 6-8
In
uence on speech 13-20
\Lump in the throat" 12-16
Chest pains 5-7
Abdominal pains 4-10
Urge to urinate 10-18
Increased muscle tome 13-20
In
uence on breathing 4-8
Muscle contractions 4-9
15
Chapter 2
Biot's Theory of Poroelasticity
2.1 Theory Summary:
The theory of poroelasticity established by Biot assumes a cube of unit size that
represents a solid-
uid system in which the stress tensors are divided into solid stresses
and
uid stresses as seen in gure (2.1). When developing the theory, Biot (1956a)
assumed that rst; the wavelength of the motion is much larger than the size of the
unit size cube of the solid-
uid system and therefore, larger than the size of the pores.
Second, the
ow of the
uid in the pores is laminar. Lastly, the elastic and isotropic
matrix is completely saturated with a single
uid phase. These assumptions will be
discussed in more details later in the chapter.
16
Figure 2.1: Unit size cube with solid stress tensors (
ij
) and
uid stress tensor ()
The unit cube system has the following stress-strain relations:
On the solid part:
0
B
B
B
B
@
xx
xy
xz
yx
yy
yz
zx
zy
zz
1
C
C
C
C
A
(2.1)
On the
uid part:
0
B
B
B
B
@
0 0
0 0
0 0
1
C
C
C
C
A
(2.2)
If is the porosity, then is proportional to the
uid pressure p according to:
=p (2.3)
Let:
xx
;
yy
;
zz
;
xy
;
yz
;
zx
: The stresses on the solid part of the cube.
xx
;
yy
;
zz
;
xy
;
yz
;
zx
: The strains of the solid part of the cube.
17
" =
@Ux
@x
+
@Uy
@y
+
@Uz
@z
: Dilatational strain of the pore
uid where U
x
, U
y
, and
U
z
are the components of the average
uid displacement
vector U.
;;P;Q;R : Elastic moduli for the solid
uid system whereP =+2
where and are Lam es constants.
: The stress of the pore
uid.
Assuming a conservative physical system which is at equilibrium, the seven com-
ponent of stress (
xx
;
yy
;
zz
;
xy
;
yz
;
zx
;) will be linear functions of the seven strain
components (
xx
;
yy
;
zz
;
xy
;
yz
;
zx
;" ) and can be represented in matrix form as:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
xx
yy
zz
xy
yz
zx
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
P Q 0 0 0
P Q 0 0 0
P Q 0 0 0
Q Q Q R 0 0 0
0 0 0 0 2 0 0
0 0 0 0 0 2 0
0 0 0 0 0 0 2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
xx
yy
zz
"
xy
yz
xz
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
(2.4)
From this point on, Biot (1956a) derived the wave propagation equations for cases
when dissipation is neglected rst and then when considered. For the case with no
dissipation, the wave equations derived are:
r
2
u + grad [( +) e + Q"] =
@
2
@t
2
(
11
u +
22
U) (2.5a)
grad [Qe + R"] =
@
2
@t
2
(
12
u +
22
U) (2.5b)
18
For the case with dissipation, the wave equations are:
r
2
u + grad [( +)e +Q"] =
@
2
@t
2
(
11
u +
22
U) +
b
b
@
@t
(u + U) (2.6a)
grad [Qe +R"] =
@
2
@t
2
(
12
u +
22
U)
b
b
@
@t
(u + U) (2.6b)
Where:
" =
@U
x
@x
+
@U
y
@y
+
@U
z
@z
: Dilatational strain of the pore
uid.
e =
@u
x
@x
+
@u
y
@y
+
@u
z
@z
: Dilatational strain of the solid.
11
;
22
;
12
: Dynamic mass coecients.
b
b : Dissipative coecient (related to permeabilityk, porosity
, and
uid viscosity through
b
b =
2
/k)
u : Solid displacement vector.
U : Fluid displacement vector.
Introducing the operation div (divergence) and curl, both pairs of equations (2.5)
and (2.6) can be uncoupled. Let div (u) =e, div (U) =", curl (u) =!, and curl (U) =
the operators that will be applied to equations (2.5) and (2.6). For simplicity, we will
assume no dissipation (
b
b = 0) and we will let P = + 2. When applying divergence
operator to equations (2.5), we obtain a pair of equations that govern the propagation
of dilatational waves as follows:
r
2
(Pe +Q") =
@
2
@t
2
(
11
e +
12
") (2.7a)
r
2
(Qe +R") =
@
2
@t
2
(
12
e +
22
") (2.7b)
When applying curl operator to equations (2.6), we obtain a pair of equations that
govern the propagation of rotational waves as follows:
r
2
! =
@
2
@t
2
(
11
! +
12
) (2.8a)
@
2
@t
2
(
12
! +
22
) = 0 (2.8b)
19
For equations (2.7), let
11
=
P
H
;
22
=
R
H
;
12
=
Q
H
: Non-dimensional parameters dening the elastic
properties of the material.
11
=
11
;
22
=
22
;
12
=
12
: Non-dimensional parameters dening the dynamic
properties of the material.
V
c
=
H
: A reference velocity with H =P +R + 2Q and is
the mass per unit volume of the
uid-soil aggregate.
Therefore, equations (2.7) can be represented as:
r
2
(
11
e +
12
") =
1
V
c
2
@
2
@t
2
(
11
e +
12
") (2.9a)
r
2
(
12
e +
22
") =
1
V
c
2
@
2
@t
2
(
12
e +
22
") (2.9b)
The solution of equations (2.9) has the form of:
e =Ae
i(lx+!t)
(2.10a)
" =Be
i(lx+!t)
(2.10b)
Substituting (2.10) into (2.9) and letting z =
V
c
2
V
2
where V =
!
l
we get:
z (
11
A +
12
B) =
11
A +
12
B (2.11a)
z (
12
A +
22
B) =
12
A +
22
B (2.11b)
Eliminating A and B, an equation of z is obtained in the form:
(
11
22
12
2
)z
2
(
11
22
+
22
11
2
12
12
)z +
11
22
12
2
= 0 (2.12)
Equation (2.12) has two roots corresponding to two propagation velocitiesV
1
andV
2
as:
V
1
2
=
V
c
2
z
1
(2.13a)
V
2
2
=
V
c
2
z
2
(2.13b)
20
The velocities V
1
and V
2
correspond to two dilatational waves. Now, starting with
equations (2.6) and eliminating
in these equations to get:
r
2
! =
11
1
12
2
11
22
@
2
!
@t
2
(2.14)
Equation (2.14) indicates one type of rotational wave with a velocity V
s
which can
be represented as:
V
s
=
11
(1
12
2
/
11
22
)
1
2
(2.15)
For non-viscous
uids, Biots wave equations for porous medium become non-dissipative
(
b
b = 0) and can be presented as:
For dilatational waves:
r
2
2
6
4
sat
+ 2 Q
Q R
3
7
5
8
>
<
>
:
9
>
=
>
;
=
@
2
@t
2
2
6
4
11
12
12
22
3
7
5
8
>
<
>
:
9
>
=
>
;
(2.16)
For rotational waves:
r
2
2
6
4
0
0 0
3
7
5
8
>
<
>
:
9
>
=
>
;
=
@
2
@t
2
2
6
4
11
12
12
22
3
7
5
8
>
<
>
:
9
>
=
>
;
(2.17)
Where ;;Q, and R are elastic moduli. Following the harmonic wave presentation
as in Deresiewicz (1960):
= (x;y;z)e
i!t
; = (x;y;z)e
i!t
(2.18)
= (x;y;z)e
i!t
; = (x;y;z)e
i!t
(2.19)
When substituting equation (2.18) into equation (2.16), the general solution for di-
latational waves become:
=
1
+
2
(2.20)
21
Which leads to the fast and slow dilatational wave velocities:
V
fast
=
s
2A
B (B
2
4AC)
1/2
(2.21a)
V
slow
=
s
2A
B + (B
2
4AC)
1/2
(2.21b)
Where
A =PRQ
2
, B =
11
R +
22
P 2
12
Q , C =
11
22
2
12
The P-wave potential for the
uid component can be determined from:
=
1
+
2
=f
1
1
+f
2
2
(2.22)
Where:
f
1
=
11
R
12
QA
V
2
fast
(
22
Q
12
R) (2.23a)
f
2
=
11
R
12
QA
V
2
slow
(
22
Q
12
R) (2.23b)
When substituting equation (2.19) is substituted into equation (2.16), the general solu-
tion can be expressed as:
=
1
=f
3
1
(2.24)
Where:
f
3
= (
12
/
22
) (2.25)
and the velocity of the rotational waves is obtained as:
V
shear
=
r
22
C
(2.26)
From the derivation presented, it is concluded that in a poroelastic medium three
body waves exist; two dilatational waves with velocities given by equation (2.21) and
plus one rotational with a velocity given by equation (2.26). The two dilatational waves
have dierent velocities as shown above. Moreover, according to Biot (1956a), the high-
velocity is designated as the wave of the rst kind and the low velocity is designated
22
as the wave of the second kind. The rst kind of dilatational wave and the shear wave
experience slight dispersion during propagation, whereas the second kind of dilatational
wave is highly attenuated in the nature of a diusion process.
2.1.1 Evaluation of Material Constants for Poroelastic Media:
Elastic moduli (;;P;Q; and R) and dynamic mass coecients (
11
;
22
; and
12
)
are required to nalize the solution of the proposed model by Biot. Experimental mea-
surements proposed by Biot and Wills (1957) suggest that for an isotropic case, the
elastic moduli (;;P;Q; and R) can be determined from the shear modulus, the jack-
eted and unjacketed compressibility of the porous solid, and the compressibility of pore
uid respectively.
Lin et al. (2005) expressed elastic moduli in terms of bulk modulus of solid skeleton
K
dry
, bulk modulus of solid grainK
g
, and bulk modulus of the pore
uidK
f
as follows:
=
3 (1 2)
2 (1 +)
K
dry
sat
=
dry
+
Q
2
R
=
3
1 +
K
dry
+
(1K
dry
/K
g
)
2
(1K
dry
/K
g
) +K
g
/K
f
K
g
P =
sat
+ 2
Q =
(1K
dry
/K
g
)
(1K
dry
/K
g
) +K
g
/K
f
R =
2
(1K
dry
/K
g
) +K
g
/K
f
where
: Poissons ratio for the solid skeleton : Porosity
dry
: Lam e constant for the dry system : Shear modulus for solid skeleton
sat
: Lam e constant for the saturated system K
s
: Bulk modulus of solid skeleton
23
and
K
dry
=K
cr
+
1
cr
(K
g
K
cr
)
WhereK
cr
is the critical bulk modulus of the solid frame at
cr
which is the critical
porosity of the material. The solids in the porous material is supported by the frame
of the solid under critical porosity and becomes suspended in
uid (when saturated)
or loose (if dry) beyond this critical porosity. Yang and Sato (1998) expressed Biots
parameters M and in terms of bulk modulus of solid skeleton K
b
, bulk modulus of
solid grain K
s
, and bulk modulus of the pore
uid K
f
as follows:
M =
K
2
s
K
d
K
b
= 1
K
b
K
s
where
K
d
=K
s
1 +
K
s
K
f
1
Combining the expressions from Lin et al. (2005) and Yang and Sato (1998), Biots
parametersM and, the elastic moduli;;P;Q; andR can be compactly represented
as follows:
= 1
K
b
K
s
sat
=
dry
+
Q
2
R
=
3
1 +
K
b
+M()
2
M =K
s
+
K
s
K
f
1
1
P =
sat
+ 2 =
3 (1)
1 +
K
b
+M()
2
=
3 (1 2)
2 (1 +)
K
b
Q =M ()
dry
=
3
1 +
K
b
R =M
2
Where M and are Biots parameters, is the shear modulus of the skeleton, is
the Poissons ratio of the skeleton,
sat
is the Lam e constant of the saturated skeleton,
24
dry
is the Lam e constant of the dry skeleton,K
b
is the bulk modulus of the skeleton,K
s
is the bulk modulus of the solid grain. The dynamic mass coecients (
11
;
22
; and
12
)
introduced by Biot (1956a) are related to (;
1
;
2
;
s
; and
f
) through the following
relations:
1
=
11
+
12
= (1)
s
(2.27a)
2
=
12
+
22
=
f
(2.27b)
=
1
+
2
= (1)
s
+
f
(2.27c)
11
: The total eective mass of the solid
1
: Unit mass of solid
22
: The total eective mass of the
uid
2
: Unit mass of
uid
12
: The additional apparent density
s
: Density of solid
: Unit mass of solid-
uid cube
f
: Density of
uid
Berryman (1980) suggested the following relations to determine the dynamic mass
coecients proposed by Biot:
11
= (1) (
s
+r
o
f
) (2.28a)
22
=
f
(2.28b)
Where r
o
f
is the induced mass due to the oscillation of solid particles in the
uid
(Lamb, 1945) and
is the dynamic toruosity parameter. The dynamic toruosity is a
dimensionless parameter, where on a microscopic scale, characterizes the deviation of
uid
ow from straight paths in a porous medium. Plus, the values of ranges from 1 in
the case of pure
uid ( = 1) to1 in the case of pure solid ( = 0) (Al Rjoub, 2007).
Berryman (1980) also suggested that the value of
to be calculated from a microscopic
model of the frame moving in the
uid. Moreover, he suggested the value of r
o
to be
equal to 0.5 for spherical solid particles. The dynamic toruosity parameter
can be
25
related to porosity through
= 1r
o
1
1
(2.29)
Assuming that r
o
= 0:5 as suggested by Berryman (1980), equation (2.29) will sim-
plify to:
=
(1 +)
2
(2.30)
Substituting (2.30) in (2.28) and then in (2.27) will lead to:
11
=
1
2
(1) (
f
+ 2
s
) (2.31a)
12
=
1
2
(1)
f
(2.31b)
22
=
1
2
(1 +)
f
(2.31c)
2.1.2 Validity of Biot's Theory:
As mentioned earlier, when he rst developed the theory, Biot (1956a) assumed that
rst; the wavelength of the motion is much larger than the size of the unit size cube of
the solid-
uid system and therefore, larger than the size of the pores. Second, the
ow
of the
uid in the pores is laminar. Lastly; the elastic and isotropic matrix is completely
saturated with a single
uid phase.
The basic idea behind the assumption of the wavelength being larger than both
the unit cube and the pore is to avoid diraction of the incident waves in case of the
wavelength being short enough to feel the presence of the pores. As for the pore size of
the medium, the soil is made of dierent solid particles with dierent sizes; the eective
pore diameter for the soil needs to be determined to represent the entire soil sample.
According to Cedergren (1989), the eective pore diameter for soil can be assumed to
be one-fth of eective grain size D
10
in permeability studies, where D
10
is the grain
26
diameter at 10% passing. Lin et al. (2005) and Al Rjoub (2013) showed that soils made
up of sands and gravel with grain diameter D
10
= 0:05 to 1 mm are eligible to apply
Biots theory.
The wavelength of incident wave can be calculated from the frequency and the veloc-
ity of the wave. To achieve a small wavelength that is close to the pore size, the incident
wave should have a very low velocity and a high frequency. Considering the average
pore diameters in porous rocks to be 10
3
mm to 1 mm (C.-H. Lin et al., 2005), table
(2.4) summarizes the wave velocities that must be produced in order for diraction to
take place within the pores of the medium.
Table 2.4: Velocities produced given frequencies and wavelengths
Frequency Range Wavelength Velocity
(Hz) (mm) (m/s)
1 1.00E-03 1 1.00E-06 1.00E-03
100 1.00E-03 1 1.00E-04 1.00E-01
From the results in Table 2.4, we conclude that the propagating wave must be ex-
tremely slow for the wavelength to be close to the average diameter of the pores which
contradicts the fact that waves traveling in ground, regardless of their source, are much
faster. Therefore, the rst assumption of the wavelength of the propagating wave being
much larger than the average pore diameter is fullled.
The second assumption made by Biot is about the
ow of the
uid being laminar.
When the relative
ow of
uid is laminar, it takes a streamline form. According to
Biot (1956a), this
ow pattern can be uniquely determined by six generalized veloci-
ties _ u
x
; _ u
y
; _ u
z
;
_
U
x
;
_
U
y
;
_
U
z
. But, this laminar
ow breaks down if the frequency of the
propagating wave exceeds the value of f
t
which can be dened as:
f
t
=
4d
2
(2.32)
Where is the kinematic viscosity of
uid (for underground water = 1:310
6m
2
s
)
27
and d is the pore diameter.
The saturation degree of the cube is a key assumption in Biots theory. If the cube is
fully saturated, that guarantees the existence of a two phase medium (
uid and solid).
Biots derivation was based on the presence of one
uid relative motion. In case of
partial saturation of soils, the voids are lled partially with
uid, and partially with gas
and these possess dierent partial pressures. Such materials that represent three-phase
medium are referred to as partially saturated media. One can still use Biots theory using
some approximation techniques to transfer three-phase medium properties to two-phase
medium properties as the case of reduced bulk modulus of the
uid in (J. Yang, 2001).
The main challenge of with the method explained by (J. Yang, 2001) and (Al Rjoub,
2007), is how sensitive the change in the value of the
uid bulk modulus with the change
of the saturation degree. The bulk modulus of the
uid, K
f
can be represented as in
Santamarina et al. (2001):
K
f
=
S
K
w
+
1S
K
a
1
(2.33)
where S is the medium degree of saturation, K
w
= 2:18 GPa is the bulk modulus
of water and K
a
= 142 kPa is the bulk modulus of air at 1.0 atmosphere. Figure (2.2)
shows how sensitive is the
uid modulus K
f
to air content. It can be concluded that a
small decrease in saturation of the medium leads to a large drop in the value of the
uid
modulus. Therefore, in this study, the saturation degree is considered higher than 90%
with the assumption that the uniformly distributed bubbles through the
uid represent
the embedded air in that
uid.
28
22
where S is the medium degree of saturation,
2.18
w
K
GPa is the bulk modulus of water
and
142
a
K
kPa is the bulk modulus of air at 1.0 atmosphere. Figure 2-2 shows how
sensitive is the fluid modulus
f
K
to air content. It can be concluded that a small
decrease in saturation of the medium leads to a large drop in the value of the fluid
modulus. Therefore, in this study, the saturation degree is considered higher than 90%
with the assumption that the uniformly distributed bubbles through the fluid represent the
embedded air in that fluid.
Figure 2-2: The variation in the fluid modulus Kf with air content (after Santamarina et al. (2001))
2.1.3 Boundary Conditions for Porous Media:
Understanding and defining the boundary conditions at the interface between
porous medium and open air or between two porous media will ultimately define the
solution to the problem of wave propagation in both cases. Based on the principles of
conservation of mass and continuity of momentum, Deresiewicz et al. (1963) discussed
the boundary conditions at free plane as well as for two different saturated porous media.
Deresiewicz et al. (1963) defined the boundary condition as permeable, partially
permeable, and impermeable as in the figure below.
Figure 2.2: The variation in the
uid modulus K
f
with air content (after Santamarina et
al. (2001))
2.1.3 Boundary Conditions for Porous Media:
Understanding and dening the boundary conditions at the interface between porous
medium and open air or between two porous media will ultimately dene the solution
to the problem of wave propagation in both cases. Based on the principles of conser-
vation of mass and continuity of momentum, Deresiewicz et al. (1963) discussed the
boundary conditions at free plane as well as for two dierent saturated porous media
where the boundary conditions were dened as as permeable, partially permeable, and
impermeable as in gure below.
29
23
Medium (1)
Interface
Medium (2)
Permeable Partially Permeable Impermeable
Solid Phase Liquid Phase
Case (1) Case (2) Case (3)
Figure 2-3: Boundary conditions between two porous media (Deresiewicz et al. (1963))
It was concluded that the boundary conditions, in general, depend on the average
normal displacement of the solid component of the medium, the average tangential
displacement of the solid component of the medium, the normal solid particle
displacement relative to the liquid component of the medium, the normal stress
experienced by the bulk material, the tangential stress experienced by the bulk material,
and the pore pressure of the liquid in the medium.
In the case of permeable interface for a free plane, the following conditions apply
at the interface:
a- Vanishing of the normal stress (
0
yy
).
b- Vanishing of the tangential stress (
0
xy
).
c- Vanishing of the pore pressure (
0
f
p
).
In the case of impermeable interface, the following conditions apply at the
interface:
a- Vanishing of the total normal stress (
0
yyf
p
).
b- Vanishing of the tangential stress (
0
xy
).
c- Vanishing of the normal solid particle displacement relative to the liquid
component of the medium (
0
r
V ).
In the case of two porous media (1 & 2) in contact, the following conditions apply
across the permeable interface:
a- Continuity of the normal stress (
12 yy yy
).
b- Continuity of the tangential stress (
12 xy xy
).
Figure 2.3: Boundary conditions between two porous media (Deresiewicz & Skalak,
1963)
It was concluded that the boundary conditions, in general, depend on the average
normal displacement of the solid component of the medium, the average tangential dis-
placement of the solid component of the medium, the normal solid particle displacement
relative to the liquid component of the medium, the normal stress experienced by the
bulk material, the tangential stress experienced by the bulk material, and the pore
pressure of the liquid in the medium.
In the case of permeable surface for a free plane, the following conditions apply:
Vanishing of the normal stress (
yy
= 0)
Vanishing of the tangential stress (
xy
= 0)
Vanishing of the pore pressure (p
f
= 0)
In the case of impermeable surface, the following conditions apply:
Vanishing of total normal stress (
yy
+p
f
= 0)
Vanishing of the tangential stress (
xy
= 0)
Vanishing of the normal solid particle displacement relative to the liquid compo-
nent of the medium (V
r
= 0)
30
In the case of two porous media (1 & 2) in contact, the following conditions apply
across the permeable interface:
Continuity of the normal stress (
yy1
=
yy2
)
Continuity of the tangential stress (
xy1
=
xy2
)
Continuity of pore water pressure (p
f1
=p
f2
)
Continuity of the normal solid particle displacement relative to the liquid compo-
nent of the medium (V
r1
=V
r2
)
Continuity of solid particle normal displacement (v
1
=v
2
)
Continuity of solid particle tangential displacement (u
1
=u
2
)
Finally, in the case of two porous media (1 & 2) in contact, the following conditions
apply across the impermeable interface:
Continuity of the total normal stress (
yy1
+p
f1
=
yy2
+p
f2
).
Continuity of the tangential stress (
xy1
=
xy2
).
Vanishing of the normal solid particle displacement relative to the liquid compo-
nent of rst medium (V
r1
= 0).
Vanishing of the normal solid particle displacement relative to the liquid compo-
nent of second medium (V
r2
= 0).
Continuity of solid particle normal displacement (v
1
=v
2
)
Continuity of solid particle tangential displacement (u
1
=u
2
)
Figure (2.4) summarizes the four boundary situations listed above.
31
24
c- Continuity pore pressure (
12 ff
pp
).
d- Continuity of the normal solid particle displacement relative to the liquid
component of the medium (
12 rr
VV ).
e- Continuity of solid particle normal displacement (
12
vv ).
f- Continuity of solid particle tangential displacement (
12
uu ).
Finally, in the case of two porous media (1 & 2) in contact, the following
conditions apply across the impermeable interface:
a- Continuity of the total normal stress (
1 1 22 yy f yy f
pp ).
b- Continuity of the tangential stress (
12 xy xy
).
c- Vanishing of the normal solid particle displacement relative to the liquid
component of first medium (
1
0
r
V ).
d- Vanishing of the normal solid particle displacement relative to the liquid
component of second medium (
2
0
r
V
).
e- Continuity of solid particle normal displacement (
12
vv ).
f- Continuity of solid particle tangential displacement (
12
uu ).
Figure 2-4 summarizes the four boundary situations listed above.
Medium (1)
Medium (2)
yy2
yy1
xy2
xy1
p
f2
p
f1
V
r2
V
r1
v
2
v
1
u
2
u
1
Permeable Interface
Medium (1)
Medium (2)
xy2
xy1
V
r2
= 0
V
r1
= 0
v
2
v
1
u
2
u
1
Impermeable Interface
Medium (1)
Air
yy2
= 0
yy1
xy2
= 0
xy1
p
f2
= 0
p
f1
Permeable Interface
Medium (1)
Air
yy2
+ p
f2
=0 x y2
= 0
xy1
V
r2
= 0
V
r1
Impermeable Interface
yy1
+ p
f1
Free Plane Two Porous Media
x
z
y
Positive Cartesian
Coordinate System
yy2
+ p
f2
yy1
+ p
f1
Figure 2-4: Illustration of the boundary conditions in the case of free plane and two porous media in contact
Figure 2.4: Illustration of the boundary conditions in the case of free plane and two
porous media in contact
In short, the four boundary situations can be put in a compact form in Table (2.5)
Table 2.5: Summary of the boundary conditions in the case of free plane and two
porous media in contact
Free plane Two-media in contact
Permeable Impermeable Permeable Impermeable
yy
= 0
yy
+p
f
= 0
yy1
=
yy2
yy1
+p
f1
=
yy2
+p
f2
xy
= 0
xy
= 0
xy1
=
xy2
xy1
=
xy2
p
f
= 0 V
r
= 0 p
f1
=p
f2
V
r1
= 0
xy
= 0 V
r1
=V
r2
V
r2
= 0
xy
= 0 v
1
=v
2
v
1
=v
2
xy
= 0 u
1
=u
2
u
1
=u
2
2.2 Comparison of Results:
As mentioned earlier, the expressions for the material constants from Lin et al. (2005)
and Yang and Sato (1998) where combined in a more compact form eliminating the need
32
to the critical bulk modulusK
cr
and the critical porosity of the material
cr
. The plane
incident wave method procedure used by Lin et al. (2005) with the compactly expressed
form of the material constants was implemented to compare results. Figure (2.5) shows
a comparison of re
ected waves amplitudes obtained using material constants from this
study with the ones obtained by Lin et al. (2005). From gure (2.5), it can be seen that
the more compact expressions produce a solution in a perfect agreement with the one
produced by Lin et al. (2005).
33
26
Figure 2-5: Comparison of reflected wave amplitudes using material constants from between results from
Lin et al. (2005) and the ones developed in this study
Figure 2.5: Comparison of re
ected wave amplitudes using material constants
resulting from Lin et al. (2005) and the ones developed in this study
34
2.3 Medium Constants and Velocities:
The data presented in table (2.6) describe the material constants and wave veloci-
ties for water saturated sands with three dierent porosities. In an eort to make the
data easily generated, it was assumed that the rst step is to obtain the in-situ shear
wave velocity of the medium through the standard penetration test (SPT) or the cone
penetration test (CPT). Using the shear wave velocity equation
V
s
=
r
=
v
u
u
t
3
2
(1+)
(12)
K
b
s
(1)
(2.34)
where is the shear modulus of the medium, is the mass density of the soil, K
b
is the bulk modulus of the solid frame,
s
is the mass density of the solid grains, is
Poissons ratio, and is the porosity. Solving for K
b
leads to:
K
b
=
2
3
(1 2)
(1 +)
s
(1) (V
s
)
2
(2.35)
With the value ofK
b
obtained, the other material constants presented in table (2.6) are
calculated based on the equations presented in section 2.1.1.
35
Table 2.6: Material constants and wave velocities for dry frame and water saturated media with
dierent porosities
Porosity () 0.1 0.2 0.3
Dymanic
mass
coecients
11
kg/m
3
2835 2835 2835 2520 2520 2520 2205 2205 2205
12
kg/m
3
-450 -450 -450 -400 -400 -400 -350 -350 -350
22
kg/m
3
550 550 550 600 600 600 650 650 650
- 0.991 0.964 0.857 0.992 0.968 0.872 0.993 0.972 0.888
Dry frame
properties
dry
kg/m
3
2385 2385 2385 2120 2120 2120 1855 1855 1855
K
dry
MPa 323 1292 5168 287 1148 4593 251 1005 4019
- 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333
dry
MPa 149 596 2385 133 530 2120 116 464 1855
dry
MPa 298 1193 4770 265 1060 4240 232 928 3710
V
p
km/s 0.500 1.000 2.000 0.500 1.000 2.00 0.500 1.000 2.00
V
s
km/s 0.25 0.50 1.00 0.25 0.50 1.00 0.25 0.50 1.00
Saturated
media
properties
sat
kg/m
3
2485 2485 2485 2320 2320 2320 2155 2155 2155
K
sat
MPa 11018 11678 14428 5495 6276 9463 3147 3915 7028
sat
MPa 149 596 2385 133 530 2120 116 464 1855
sat
MPa 10919 11280 12838 5407 5922 8050 3070 3606 5791
V
pfast
km/s 2.364 2.450 2.783 1.963 2.074 2.502 1.766 1.882 2.351
V
pslow
km/s 0.101 0.199 0.360 0.183 0.347 0.583 0.246 0.462 0.746
V
s
km/s 0.246 0.492 0.983 0.242 0.485 0.970 0.239 0.479 0.959
36
Chapter 3
Line Load on top of a Poroelastic Half-space
/ Layered Media
3.1 Introduction:
The study of plane waves can give a useful insight about the physics of the problem.
Yet, these plane waves solutions are limited in terms of obtaining vibration charac-
teristics a function of distance from a surface vibration source. After the comparison
presented in chapter (2), a dierent loading source is introduced here to account for
the change of vibration as a function of distance. In this chapter, the train loading is
represented as a uniformly distributed line load along the x-axis.
37
3.2 Mathematical Derivation:
3.2.1 Problem setup:
In this section, the vertical and horizontal displacement expressions will be derived
for a line load acting on top of a poroelastic half-space as shown in gure (3.1). Let the
loading function have the following form:
F (x)j
z=0
=F
0
e
i(kx!t)
=F
0
e
ikx
e
i!t
By omitting the e
i!t
term for simplicity, the nal loading function will be:
F (x)j
y=0
=F
0
e
ikx
(3.1)
In the following derivation, the subscript m represent the medium number, g = 1; 2
represent fast and slow P-waves respectively and g = 3 represents shear wave. Plus,
d = 1; 2 represent transmitted waves and re
ected waves respectively. Noting that for
the case of a half-space d = 1.
28
CHAPTER (3): Line Load on top of Poroelastic Half-space
3.1 Introduction:
The study of plane waves can give a useful insight about the physics of the
problem. Yet, these plane waves’ solutions are limited in terms of obtaining vibration
characteristics a function of distance from a surface vibration source. After the
comparison presented in chapter 2, a different loading source is introduced here to
account for the change of vibration as a function of distance. In this chapter, the train
loading is represented as a uniformly distributed line load along the x axis.
3.2 Mathematical Derivation:
Medium (m): Poroelastic
x
y
y = 0
m
, m
, M
m
, m
, m
F=F
0
e
ikx
Medium (m+1): Poroelastic
m+1
, m+1
, M
m+1
, m+1
, m+1
y = -h
1
m31
m21
m11
m22
m32
m12
(m+1)31
(m+ 1)21
(m+1)11
(a)
x
y
F=F 0 e
ikx
z
(b)
Figure 3-1: (a) Line load on top of a poroelastic layered space, (b) 3D view of axis
3.2.1 Problem setup:
In this section, the vertical and horizontal displacement expressions will be
derived for a line load acting on top of a poroelastic half-space as shown in Figure 3-1.
Let the loading function have the following form:
00
0
ikxt ikx it
z
F x Fe Fee
By omitting the
it
e
term for simplicity, the final loading function will be:
0
0
ikx
y
F x Fe
(3-1)
Figure 3.1: (a) Line load on top of a poroelastic layered space, (b) 3D view of axis
38
The propagating waves potentials can be expressed as follows:
mgd
=A
mgd
exp (ikx
mgd
y); g = 1; 2 (3.2)
mgd
=A
mgd
exp (ikx
mgd
y); g = 3 (3.3)
where:
2
mgd
=k
2
k
2
mgd
and k
mgd
=
!
V
mgd
where:
! : The frequency
k : The apparent wave number.
k
mgd
: The wave number associated with wave in medium m with type g.
V
mgd
: The wave velocity associated with wave in medium m with type g.
The total wave potentials for a poroelastic medium can be expressed as:
In solid component In liquid Component
m
=
2
P
g=1
2
P
d=1
mgd
m
=
2
P
g=1
2
P
d=1
f
mgd
mgd
m
=
3
P
g=3
2
P
d=1
mgd
m
=
3
P
g=3
2
P
d=1
f
mgd
mgd
where:
f
mgd
=
11m
R
m
12m
Q
m
Am
V
2
mgd
(
22m
Q
m
12m
R
m
)
, g = 1; 2
f
mgd
=
12m
22m
, g = 3
And (
11m
;
12m
; and
22m
) are the dynamic mass coecients (
11
;
12
; and
22
)
associated with medium (m). Moreover,(A
m
; R
m
; andQ
m
) are the (A; R; andQ) dened
for the fast and slow dilatational wave for medium (m).
The absolute displacements of solid and liquid components of the poroelastic medium
(m) can be written as:
39
solid component in the x-axis solid component in the y-axis
u
m
=
@
m
@x
+
@
m
@y
v
m
=
@
m
@y
@
m
@x
liquid component in the x-axis liquid component in the y-axis
U
m
=
@
m
@x
+
@
m
@y
V
m
=
@
m
@y
+
@
m
@x
The relative displacements of solid component with respect to the liquid components
of the poroelastic medium can be written as:
x-direction y-direction
U
rm
=
@
@x
(
m
m
) +
@
@y
(
m
m
) V
rm
=
@
@y
(
m
m
)
@
@x
(
m
m
)
If the liquid is trapped inside the pores when stress is applied, pore water pressure
(p
f
) for layer (m) is developed and can be calculated from:
p
fm
=M
m
r
2
(
m
(
m
m
) +
m
m
)
Where for Cartesian coordinate system, the Laplacian operatorr
2
=
@
2
@x
2
+
@
2
@y
2
+
@
2
@z
2
.
The stresses developed in the poroelastic medium are:
Normal stress in the x-direction:
xxm
=
m
r
2
m
+ 2
m
@
2
m
@x
2
+
@
2
m
@y@x
+
m
p
fm
(3.4)
Normal stress in the y-direction:
yym
=
m
r
2
m
+ 2
m
@
2
m
@y
2
@
2
m
@x@y
+
m
p
fm
(3.5)
Tangential stress:
xym
=
m
2
@
2
m
@y@x
+
@
2
m
@y
2
@
2
m
@x
2
(3.6)
For equations (3.4) and (3.5), if the water in the pores is not trapped, the term p
fm
40
is set to zero since no pore water pressure will develop which will reduce equations (3.4)
and (3.5) to the stress equations for the elastic medium.
3.2.2 Boundary Conditions:
The boundary conditions used to solve the wave propagation equation depend on the
nature of the free surface in case if the medium has an interface with air or depends on
the interface between two consecutive layers. The free surface or the interface between
the layers can be permeable or impermeable.
3.2.2.1 Permeable Surface:
At the free surface, three wave potentials exist which means three boundary condi-
tions are needed to solve for the wave amplitudes. Assuming that the free surface to be
permeable, the boundary conditions equations can be presented as:
1. The normal stress in the medium is equal to the applied stress at the surface
m
r
2
m
+ 2
m
@
2
m
@y
2
@
2
m
@x@y
=F (x) (3.7)
2. Vanishing of total shear stress across the surface
m
2
@
2
m
@y@x
+
@
2
m
@y
2
@
2
m
@x
2
= 0 (3.8)
3. Vanishing of pore pressure across the surface
M
m
r
2
(
m
(
m
m
) +
m
m
) = 0 (3.9)
3.2.2.2 Impermeable Surface:
Assuming that the free surface to be impermeable, the boundary conditions equations
can be presented as:
41
1. The total normal stress in the medium is equal to the applied stress at the surface
m
r
2
m
+ 2
m
@
2
m
@y
2
@
2
m
@x@y
+
m
p
fm
=F (x) (3.10)
2. Vanishing of total shear stress across the surface
m
2
@
2
m
@y@x
+
@
2
m
@y
2
@
2
m
@x
2
= 0 (3.11)
3. Vanishing of relative vertical displacement at the surface
@
@y
(
m
m
)
@
@x
(
m
m
) = 0 (3.12)
3.2.2.3 Permeable Intermediate Interface:
At an intermediate interface between two consecutive poroelastic layers, (m and
m+1), six wave potentials exist which means six boundary conditions are needed to solve
for the wave amplitudes. Assuming that the interface to be permeable, the boundary
conditions equations can be presented as:
1. Continuity of normal stress across the interface
m
r
2
m
+2
m
@
2
m
@y
2
@
2
m
@x@y
=
m+1
r
2
m+1
+ 2
m+1
@
2
m+1
@y
2
@
2
m+1
@x@y
(3.13)
2. Continuity of shear stress across the interface
m
2
@
2
m
@y@x
+
@
2
m
@y
2
@
2
m
@x
2
=
m+1
2
@
2
m+1
@y@x
+
@
2
m+1
@y
2
@
2
m+1
@x
2
(3.14)
3. Continuity of pore pressure across the interface
M
m
r
2
(
m
(
m
m
) +
m
m
) =
M
m+1
r
2
(
m+1
(
m+1
m+1
) +
m+1
m+1
) (3.15)
42
4. Continuity of relative vertical displacement across the interface
@
@y
(
m
m
)
@
@x
(
m
m
) =
@
@y
(
m+1
m+1
)
@
@x
(
m+1
m+1
) (3.16)
5. Continuity of normal displacement across the interface
@
m
@y
@
m
@x
=
@
m+1
@y
@
m+1
@x
(3.17)
6. Continuity of tangential displacement across the interface
@
m
@x
+
@
m
@y
=
@
m+1
@x
+
@
m+1
@y
(3.18)
3.2.2.4 Permeable Intermediate Interface:
Assuming that the interface to be permeable, the boundary conditions equations can
be presented as:
1. Continuity of normal stress across the interface
m
r
2
m
+2
m
@
2
m
@y
2
@
2
m
@x@y
+
m
p
fm
=
m+1
r
2
m+1
+ 2
m+1
@
2
m+1
@y
2
@
2
m+1
@x@y
+
m+1
p
fm+1
(3.19)
2. Continuity of shear stress across the interface
m
2
@
2
m
@y@x
+
@
2
m
@y
2
@
2
m
@x
2
=
m+1
2
@
2
m+1
@y@x
+
@
2
m+1
@y
2
@
2
m+1
@x
2
(3.20)
3. Vanishing of relative vertical displacement in medium (m) at the interface
@
@y
(
m
m
)
@
@x
(
m
m
) = 0 (3.21)
4. d. Vanishing of relative vertical displacement in medium (m + 1) at the interface
@
@y
(
m+1
m+1
)
@
@x
(
m+1
m+1
) = 0 (3.22)
43
5. Continuity of normal displacement across the interface
@
m
@y
@
m
@x
=
@
m+1
@y
@
m+1
@x
(3.23)
6. Continuity of tangential displacement across the interface
@
m
@x
+
@
m
@y
=
@
m+1
@x
+
@
m+1
@y
(3.24)
3.2.3 Solution of Wave Amplitudes:
Let the solution of the boundary condition equations, for the half-space or the layered
case, satisfy the following relation:
[a] [Z] = [c] (3.25)
Where [a] is a 6m 3 by 6m 3 matrix with elementsa
i;j
wherei = 1; 2:::; 6m 3
and j = 1; 2:::; 6m 3. The matrix [Z] = [A
mgd
]
T
is a 6m 3 by 1 column matrix
containing the amplitudes of the transmitted fast compressional, slow compressional,
and shear waves respectively for each interface. Solution to equation (3.25) will results
in the amplitudes of the waves [A
mgd
(k)]
T
.
3.2.4 Surface Displacements:
The surface tangential and vertical displacements can be obtained from the Equations
3.23 and 3.24:
For the half-space case, (m = 1)
u =ie
ikx
(kA
111
(k) +kA
121
(k) +i
131
A
131
(k)) (3.26)
v =ie
ikx
(i
111
A
111
(k) +i
121
A
121
(k)kA
131
(k)) (3.27)
44
For a single poroelastic layer on top of the half-space (m> 1)
u = e
ikx
(ikA
111
(k) + ikA
121
(k)
131
A
131
(k) + ikA
112
(k) + ikA
122
(k) +
132
A
132
(k)) (3.28)
v =e
ikx
(
111
A
111
(k)
121
A
121
(k)ikB
131
(k) +
112
A
112
(k) +
122
A
122
(k)ikB
132
(k)) (3.29)
For the load function F (x)j
y=0
= F
0
e
i(kx!t)
, and since the solution must be satis-
ed at all values of x, the solution must also have a factor e
ikx
. Consider the Fourier
Transform pair:
F (k) =
Z
1
1
f ()e
ik
d (3.30)
and
f (x) =
1
2
Z
1
1
F (k)e
ikx
dk (3.31)
If f () is a delta function, f () = (), then
F (k) = 1 =
Z
1
1
()e
ik
d (3.32)
therefore,
(x) =
1
2
Z
1
1
e
ikx
dk (3.33)
or
F
0
(x) =
1
2
Z
1
1
F
0
e
ikx
dk (3.34)
If
mgd
(k) = A
mgd
(k)e
ikx
mgd
y
is a solution to the loading function F
0
e
ikx
, then a
solution to F
0
(x) can be expressed as
1
2
R
1
1
A
mgd
(k)e
ikx
mgd
y
dk.
45
3.3 Numerical Results:
For the analysis of a uniformly distributed harmonic load, the data presented in (2.6)
are used with the addition of the train load F
0
which was taken to be 20 kN/m.
3.3.1 Poroelastic Half-space:
For the poroelastic half-space case, three medium porosity values were studied 0:10
(elastic behavior), 0.20, and 0.30. It can be seen from gure (3.2) that as the porosity
of the medium increases, the displacements in the horizontal and vertical direction for
both permeable and impermeable surfaces increases. However, the eect of this porosity
change diminish at a short distance away from the source compared to the impermeable
case which required longer distance to reduce. Examining the eect of the surface
boundary condition on the amplitudes of both horizontal and vertical displacements, it
can be seen that the displacements amplitudes for the case of impermeable surface are
higher than those for the permeable case. That is due to the fact that the pore pressure
developed by the trapped
uid weakens the structure of the soil matrix by reducing the
friction between the soil particles. Figure (3.3) shows that for dierent distances away
from the source, the amplitude of the horizontal and vertical displacements decreases
with the increase of frequency. It is noted that for close distances, displacements values
decrease slowly with frequency where they decrease rapidly for points away from the
loading source.
The eect of the shear velocity of the medium, which is an indication of the soil
softness, can be seen in gure (3.4) and gure (3.5). For a poroelastic half-space with
permeable surface and porosity of 0.30, the lower the shear wave velocity (softer soil),
the higher the displacement for all distances away from the source. However, for an
impermeable surface, gure (3.6) and gure (3.7), the pore pressure developed in the
46
medium has an eect in the case of soft and medium soils (V
s
= 250 m/s and V
s
= 500
m/s). The pore pressure weakens the soil matrix for these two types of soils, where it
has less noticeable eect on harder soil since the soil matrix is much stronger (i.e. larger
K
b
). To explain the dierence in magnitude for the matrix bulk modulus, let us examine
equation (2.35):
K
b
=
2
3
(1 2)
(1 +)
s
(1) (Vs)
2
which can be written as:
K
b
=K(Vs)
2
(3.35)
where K is a constant. As a result, we see that K
b
is directly proportional to V
2
s
and
that explains why at V
s
= 250 m/s and V
s
= 500 m/s the soil was aected by the pore
pressure compared to the case with V
s
= 1000 m/s where it had K
b
4 times larger than
the one for V
s
= 500 m/s and 16 times larger than the one for V
s
= 500 m/s. Changing
the porosity of the medium changes its strength where the higher the porosity, the weaker
the soil since the solid matrix occupies less space of the unit cube. Less porosity means
more solid matrix to resist the stresses. These eects are shown in gure (3.8), gure
(3.9), gure (3.10), and gure (3.11) for all types of surfaces, permeable or impermeable,
as a function of frequency. It is concluded that horizontal and vertical displacement
amplitudes increase with increasing porosity at any given distance from the source.
47
36
Figure 3-2: Horizontal and vertical displacement values versus distance from the source for different porosity ratios (V
s
= 500 m/s)
Figure 3.2: Horizontal and vertical displacement values versus distance from the source for
dierent porosity ratios (V
s
= 500 m/s)
48
37
Figure 3-3: Horizontal and vertical displacement values versus frequency for different distances from the source for = 0.30 and V
s
= 500 m/s
Figure 3.3: Horizontal and vertical displacement values versus frequency for dierent
distances from the source for = 0:30 and V
s
= 500 m/s
49
38
Figure 3-4: Horizontal displacement versus frequency at different distances from the source with different
medium shear velocities, = 0.30 (permeable surface)
Figure 3.4: Horizontal displacement versus frequency at dierent distances from the source
with dierent medium shear velocities, = 0:30 (permeable surface)
50
39
Figure 3-5: Vertical displacement versus frequency at different distances from the source with different
medium shear velocities, = 0.30 (permeable surface)
Figure 3.5: Vertical displacement versus frequency at dierent distances from the source with
dierent medium shear velocities, = 0:30 (permeable surface)
51
40
Figure 3-6: Horizontal displacement versus frequency at different distances from the source with different
medium shear velocities, = 0.30 (impermeable surface)
Figure 3.6: Horizontal displacement versus frequency at dierent distances from the source
with dierent medium shear velocities, = 0:30 (impermeable surface)
52
41
Figure 3-7: Vertical displacement versus frequency at different distances from the source with different
medium shear velocities, = 0.30 (impermeable surface)
Figure 3.7: Vertical displacement versus frequency at dierent distances from the source with
dierent medium shear velocities, = 0:30 (impermeable surface)
53
42
Figure 3-8: Horizontal displacement versus frequency at different distances from the source with different
medium porosities Vs = 500 m/s (permeable surface)
Figure 3.8: Horizontal displacement versus frequency at dierent distances from the source
with dierent medium porosities V
s
= 500 m/s (permeable surface)
54
43
Figure 3-9: Vertical displacement versus frequency at different distances from the source with different
medium porosities Vs = 500 m/s (permeable surface)
Figure 3.9: Vertical displacement versus frequency at dierent distances from the source with
dierent medium porosities V
s
= 500 m/s (permeable surface)
55
44
Figure 3-10: Horizontal displacement versus frequency at different distances from the source with different
medium porosities Vs = 500 m/s (impermeable surface)
Figure 3.10: Horizontal displacement versus frequency at dierent distances from the source
with dierent medium porosities V
s
= 500 m/s (impermeable surface)
56
45
Figure 3-11: Vertical displacement versus frequency at different distances from the source with different
medium porosities Vs = 500 m/s (impermeable surface)
Figure 3.11: Vertical displacement versus frequency at dierent distances from the source
with dierent medium porosities V
s
= 500 m/s (impermeable surface)
57
3.3.2 Poroelastic Layer on Top of a Poroelastic Half-space:
In the case of a poroelastic layer on top of a poroelastic half-space, for all cases, it was
assumed that the
uid can move between the half-space and the top layer (permeable
interface). Yet, the permeable and impermeable top surface conditions are still valid.
For the following results, if
L
is the porosity of the top layer, and
H
is the porosity of
the half-space underlying this layer, then the porosity ratio,
can be calculated from:
=
L
H
(3.36)
And, ifV
S
L
shear wave velocity of the top layer, andV
S
H
is the shear wave velocity of the
half-space underlying this layer, then the shear wave velocity ratio,V
s
can be calculated
from:
V
s
=
V
S
L
V
S
H
(3.37)
From gure (3.12), it can be noted that the change in porosity ratio for a shallow
layer (1 m) deep does not aect the horizontal and vertical displacement values for both
permeable and impermeable surfaces. However, some changes in these displacements will
be clear if the layer thickness was increased to (10 m) as in gure (3.13). Moreover, for
the deeper layer, as the porosity ratio increases the vertical and horizontal displacement
increase.
For a porosity ratio of 1.20 as shown in gure (3.14), the deeper the rst layer,
the higher the displacement values. However, this relation between the depth and the
displacement is not true for all depths and all loading values. Two factors play a role
here: load magnitude and frequency. If the load applied is small in magnitude, then a
layer of 10 m can act as a half-space for this loading. And if the load value increased
substantially, this 10 m can be considered as shallow depth and has no eect on the
displacement values. As for frequency, if the frequency of the applied load coincides with
58
the natural frequency of the layer (which depends on the layer thickness), resonance will
take eect which amplies the response immensely. In short, to maximize a displacement,
we need a combination of load magnitude and rst layer thickness.
As a function of distance and frequency, the eect of dierent layer thicknesses, with
a single porosity was studied. From gure (3.15), it can be seen that shallow layer
of (1 m), with our loading value, shows a behavior of a half-space. When the depth
is increased to (5 m) as in gure (3.16), a peak in the displacement is introduced at
frequency of 54 Hz for all distances. This peak exists because frequency of the loading
force equals the natural of the soil layer (resonance). As the layer depth is doubled from
(5 m) to (10 m) as in gure (3.17), the peak is shifted toward 27 Hz and we notice a
new peaks appear at 54 Hz and 79 Hz that could have been captures in gure (3.16) if
the range of frequency was extended beyond 100 Hz.
Figure (3.18) shows that the porosity ratio of the top layer has no noticeable eect on
the vertical displacement value with changing frequency in the case of a shallow layer of
(1 m). If the layer thickness is increased to (10 m), as stated earlier, at a frequency f 27
Hz a peak should be expected to appear. Yet, it can be shown, according to gure (3.19),
three peaks appear at a frequency of 27 Hz. The amplitude of these three peaks show
again that at higher porosity ratios, higher displacements will be expected at resonance.
We examine gure (3.20) to understand the eect of layer thickness as a function of
porosity for a single porosity ratio. It can be seen that for a small layer thickness (1
m), no resonance aect takes place within the frequency range presented in the gure.
When the layer thickness is set to (5 m), the resonance eect takes place at frequency
of 54 Hz and this frequency will be reduced to half (27 Hz) this value when the layer
thickness is doubled to 10 m. The higher modes of vibrations for a layer thickness of
(10 m) can be captured in the frequency range of presented in gure (3.20).
59
48
Figure 3-12: Horizontal and vertical displacement values versus distance from the source for different porosity ratios and a layer thickness of 1 m with a
stiffness ratio Vs* = 0.50
Figure 3.12: Horizontal and vertical displacement values versus distance from the source for
dierent porosity ratios and a layer thickness of 1 m with a stiness ratio V
s
= 0:50
60
49
Figure 3-13: Horizontal and vertical displacement values versus distance from the source for different porosity ratios and a layer thickness of 10 m with
a stiffness ratio Vs* = 0.50
Figure 3.13: Horizontal and vertical displacement values versus distance from the source for
dierent porosity ratios and a layer thickness of 10 m with a stiness ratio V
s
= 0:50
61
50
Figure 3-14: Horizontal and vertical displacement values versus distance from the source for different top layer thicknesses with * = 1.20 and
Vs*=0.50
Figure 3.14: Horizontal and vertical displacement values versus distance from the source for
dierent top layer thicknesses with
= 1.20 and V
s
= 0:50
62
51
Figure 3-15: Horizontal and vertical displacements values versus frequency for different distances from the source with a layer thickness of 1 m,
*=1.20, and Vs* = 0.50
Figure 3.15: Horizontal and vertical displacements values versus frequency for dierent
distances from the source with a layer thickness of 1 m,
= 1.20, and V
s
= 0:50
63
52
Figure 3-16: Horizontal and vertical displacements values versus frequency for different distances from the source with a layer thickness of 5 m,
*=1.20, and Vs* = 0.50
Figure 3.16: Horizontal and vertical displacements values versus frequency for dierent
distances from the source with a layer thickness of 5 m,
=1.20, and V
s
= 0:50
64
53
Figure 3-17: Horizontal and vertical displacements values versus frequency for different distances from the source with a layer thickness of 10 m,
*=1.20, and Vs* = 0.50
Figure 3.17: Horizontal and vertical displacements values versus frequency for dierent
distances from the source with a layer thickness of 10 m,
=1.20, and V
s
= 0:50
65
54
Figure 3-18: Vertical displacement values versus frequency for different porosity ratios for a layer
thickness of 1 m and Vs* = 0.50 (permeable surface)
Figure 3.18: Vertical displacement values versus frequency for dierent porosity ratios for a
layer thickness of 1 m and V
s
= 0:50 (permeable surface)
66
55
Figure 3-19: Vertical displacement values versus frequency for different porosity ratios for a layer
thickness of 10 m and Vs* = 0.50 (permeable surface)
Figure 3.19: Vertical displacement values versus frequency for dierent porosity ratios for a
layer thickness of 10 m and V
s
= 0:50 (permeable surface)
67
56
Figure 3-20: Vertical displacement values versus frequency for different top layer thicknesses for *=1.50
and Vs* = 0.50 (permeable surface)
Figure 3.20: Vertical displacement values versus frequency for dierent top layer thicknesses
for
=1.50 and V
s
= 0:50 (permeable surface)
68
Chapter 4
Concentrated Harmonic Point Load on top
of a Poroelastic Half-space / Layered Media
4.1 Introduction:
In chapter (3), the uniformly distributed line load was used as a loading force to
simulate the dynamic eects of a train. Although it established the basis toward un-
derstanding the behavior of the half-space and layered media, one more aspect should
not be neglected in the analysis, the speed of the train. The dynamic load of a train is
a moving load acting through its wheels. Hence, a better presentation of such a load
is a concentrated harmonic point load instead of a uniformly distributed line load as
previously discussed. With this new formulation, the eect of the speed of the point
load (speed of the train) can be studied.
69
4.2 Mathematical Derivation:
4.2.1 Problem setup:
In this section, the vertical and horizontal displacement expressions will be derived
for a concentrated harmonic point load acting on top of a poroelastic half-space as shown
in gure (4.1). We note that axial symmetry about z axis is assumed. Let the loading
function have the following form:
Z (r) =ZJ
0
(kr) (4.1)
In the following derivation, the subscript m represent the medium number, g = 1; 2
represent fast and slow P-waves respectively and g = 3 represents shear wave. Plus,
d = 1; 2 represent transmitted waves and re
ected waves respectively. Noting that for
the case of a half-space d = 1.
57
CHAPTER (4): Concentrated Harmonic Point Load on top of a
Poroelastic Half-space
4.1 Introduction:
In chapter three, the uniformly distributed line load was used as a loading force to
simulate the dynamic effects of a train. Although it established the basis toward
understanding the behavior of the half-space and layered media, one more aspect should
not be neglected in the analysis, the speed of the train. The dynamic load of a train is a
moving load acting through its wheels. Hence, a better presentation of such a load is a
concentrated harmonic point load instead of a uniformly distributed line load as
previously discussed. With this new formulation, the effect of the speed of the point load
(speed of the train) can be studied.
4.2 Mathematical Derivation:
Medium (m): Poroelastic
r
z
z = 0
m
, m
, M
m
, m
, m
Medium (m+1): Poroelastic
m+1
, m+1
, M
m+1
, m+1
, m+1
z = -h
1
m31
m21
m11
m22
m32
m12
(m+1)31
(m+ 1)21
(m+1)11
(a)
F=Z
J
0
(k r)
r
z
(b)
F=Z
J
0
(k r)
Figure 4-1: (a) Concentrated harmonic point load on top of a poroelastic layered space, (b) 3D view of axis
4.2.1 Problem setup:
In this section, the vertical and horizontal displacement expressions will be
derived for a concentrated harmonic point load acting on top of a poroelastic half-space
as shown in Figure 4-1. We note that axial symmetry about z axis is assumed. We note
Figure 4.1: (a) Concentrated harmonic point load on top of a poroelastic layered space, (b)
3D view of axis
70
The propagating waves potentials can be expressed as follows:
mgd
=A
mgd
J
0
(kr)e
mgd
z
g = 1; 2 (4.2)
mgd
=A
mgd
J
0
(kr)e
mgd
z
; g = 3 (4.3)
where:
2
mgd
=k
2
k
2
mgd
and k
mgd
=
!
V
mgd
where:
! : The frequency
k : The apparent wave number.
k
mgd
: The wave number associated with wave in medium m with type g.
V
mgd
: The wave velocity associated with wave in medium m with type g.
The total wave potentials for a poroelastic medium can be expressed as:
In solid component In liquid Component
m
=
2
P
g=1
2
P
d=1
mgd
m
=
2
P
g=1
2
P
d=1
f
mgd
mgd
m
=
3
P
g=3
2
P
d=1
mgd
m
=
3
P
g=3
2
P
d=1
f
mgd
mgd
where:
f
mgd
=
11m
R
m
12m
Q
m
Am
V
2
mgd
(
22m
Q
m
12m
R
m
)
, g = 1; 2
f
mgd
=
12m
22m
, g = 3
And (
11m
;
12m
; and
22m
) are the dynamic mass coecients (
11
;
12
; and
22
)
associated with medium (m). Moreover,(A
m
; R
m
; andQ
m
) are the (A; R; andQ) dened
for the fast and slow dilatational wave for medium (m).
The absolute displacements of solid and liquid components of the poroelastic medium
(m) can be written as:
71
solid component in the r-axis solid component in the z-axis
u
rm
=
@
m
@r
+
@
2
m
@r@z
u
zm
=
@
m
@z
1
r
@
m
@r
@
2
m
@r
2
liquid component in the r-axis liquid component in the z-axis
U
rm
=
@
m
@r
+
@
2
m
@r@z
U
zm
=
@
m
@z
1
r
@
m
@r
@
2
m
@r
2
The relative displacements of solid component with respect to the liquid components
of the poroelastic medium can be written as:
In the r-direction: U
rrm
=
@
@r
(
m
m
) +
@
2
@r@z
(
m
m
)
In the z-direction: U
rzm
=
@
@z
(
m
m
)
1
r
@
@r
(
m
m
)
@
2
@r
2
(
m
m
)
If the liquid is trapped inside the pores when stress is applied, pore water pressure
(p
f
) for layer (m) is developed and can be calculated from:
p
fm
=M
m
r
2
(
m
(
m
m
) +
m
m
)
Where for cylindrical coordinate system with the axial symmetry assumed previously,
the Laplacian operatorr
2
=
@
2
@r
2
+
1
r
@
2
@r
+
@
2
@z
2
.
The stresses developed in the poroelastic medium are:
Normal stress in the z-direction:
zzm
=
m
r
2
m
+ 2
m
@u
zm
@z
m
p
fm
(4.4)
Tangential stress:
rzm
=
m
@u
rm
@z
+
@u
zm
@r
(4.5)
For equations (4.4), if the water in the pores is not trapped, the term p
fm
is set to
zero since no pore water pressure will develop which will reduce equations (4.4) to the
stress equations for the elastic medium.
72
4.2.2 Boundary Conditions:
The boundary conditions used to solve the wave propagation equation depend on the
nature of the free surface in case if the medium has an interface with air or depends on
the interface between two consecutive layers. The free surface or the interface between
the layers can be permeable or impermeable.
4.2.2.1 Permeable Surface:
At the free surface, three wave potentials exist which means three boundary condi-
tions are needed to solve for the wave amplitudes. Assuming that the free surface to be
permeable, the boundary conditions equations can be presented as:
1. The normal stress in the medium is equal to the applied stress at the surface
m
r
2
m
+ 2
m
@u
zm
@z
=Z (4.6)
2. Vanishing of total shear stress across the surface
m
@u
rm
@z
+
@u
zm
@r
= 0 (4.7)
3. Vanishing of pore pressure across the surface
M
m
r
2
(
m
(
m
m
) +
m
m
) = 0 (4.8)
4.2.2.2 Impermeable Surface:
Assuming that the free surface to be impermeable, the boundary conditions equations
can be presented as:
1. The total normal stress in the medium is equal to the applied stress at the surface
m
r
2
m
+ 2
m
@u
zm
@z
m
p
fm
=Z (4.9)
73
2. Vanishing of total shear stress across the surface
m
@u
rm
@z
+
@u
zm
@r
= 0 (4.10)
3. Vanishing of relative vertical displacement at the surface
@
@z
(
m
m
)
1
r
@
@r
(
m
m
)
@
2
@r
2
(
m
m
) = 0 (4.11)
4.2.2.3 Permeable Intermediate Interface:
At an intermediate interface between two consecutive poroelastic layers, (m and
m+1), six wave potentials exist which means six boundary conditions are needed to solve
for the wave amplitudes. Assuming that the interface to be permeable, the boundary
conditions equations can be presented as:
1. Continuity of normal stress across the interface
m
r
2
m
+ 2
m
@u
zm
@z
=
m+1
r
2
m+1
+ 2
m+1
@u
zm+1
@z
(4.12)
2. Continuity of shear stress across the interface
m
@u
rm
@z
+
@u
zm
@r
=
m+1
@u
rm+1
@z
+
@u
zm+1
@r
(4.13)
3. Continuity of pore pressure across the interface
M
m
r
2
(
m
(
m
m
) +
m
m
) =
M
m+1
r
2
(
m+1
(
m+1
m+1
) +
m+1
m+1
) (4.14)
4. Continuity of relative vertical displacement across the interface
@
@z
(
m
m
)
1
r
@
@r
(
m
m
)
@
2
@r
2
(
m
m
) =
@
@z
(
m+1
m+1
)
1
r
@
@r
(
m+1
m+1
)
@
2
@r
2
(
m+1
m+1
)
(4.15)
5. Continuity of normal displacement across the interface
@
m
@z
1
r
@
m
@r
@
2
m
@r
2
=
@
m+1
@z
1
r
@
m+1
@r
@
2
m+1
@r
2
(4.16)
74
6. Continuity of tangential displacement across the interface
@
m
@r
+
@
2
m
@r@z
=
@
m+1
@r
+
@
2
m+1
@r@z
(4.17)
4.2.2.4 Permeable Intermediate Interface:
Assuming that the interface to be permeable, the boundary conditions equations can
be presented as:
1. Continuity of normal stress across the interface
m
r
2
m
+ 2
m
@uzm
@z
+
m
p
fm
=
m+1
r
2
m+1
+ 2
m+1
@u
zm+1
@z
+
m+1
p
fm+1
(4.18)
2. Continuity of shear stress across the interface
m
@u
rm
@z
+
@u
zm
@r
=
m+1
@u
rm+1
@z
+
@u
zm+1
@r
(4.19)
3. Vanishing of relative vertical displacement in medium (m) at the interface
@
@z
(
m
m
)
1
r
@
@r
(
m
m
)
@
2
@r
2
(
m
m
) = 0 (4.20)
4. d. Vanishing of relative vertical displacement in medium (m + 1) at the interface
@
@z
(
m+1
m+1
)
1
r
@
@r
(
m+1
m+1
)
@
2
@r
2
(
m+1
m+1
) = 0 (4.21)
5. Continuity of normal displacement across the interface
@
m
@z
1
r
@
m
@r
@
2
m
@r
2
=
@
m+1
@z
1
r
@
m+1
@r
@
2
m+1
@r
2
(4.22)
6. Continuity of tangential displacement across the interface
@
m
@r
+
@
2
m
@r@z
=
@
m+1
@r
+
@
2
m+1
@r@z
(4.23)
75
4.2.3 Solution of Wave Amplitudes:
Let the solution of the boundary condition equations, for the half-space or the layered
case, satisfy the following relation:
[a] [Z] = [c] (4.24)
Where [a] is a 6m 3 by 6m 3 matrix with elementsa
i;j
wherei = 1; 2:::; 6m 3
and j = 1; 2:::; 6m 3. The Matrix [Z] = [A
mgd
]
T
is a 6m 3 by 1 column matrix
containing the amplitudes of the transmitted fast compressional, slow compressional,
and shear waves respectively for each interface. Solution to equation (4.24) will results
in the amplitudes of the waves [A
mgd
(k)]
T
.
4.2.4 Surface Displacements:
The surface tangential and vertical displacements can be obtained from equations
(4.22) and (4.23):
For the half-space case, (m = 1)
u
r
=kJ
1
(kr) (A
111
(k) +A
121
(k) +
131
A
131
(k)) (4.25)
u
z
=J
0
(kr)
111
A
111
(k) +
121
A
121
(k) +k
2
A
131
(k)
(4.26)
For a single poroelastic layer on top of the half-space (m> 1)
u
r
=kJ
1
(kr) (A
111
(k) +A
121
(k) +
131
A
131
(k) +A
112
(k) +A
122
(k)
132
A
132
(k)) (4.27)
u
z
=J
0
(kr) (
111
A
111
(k) +
121
A
121
(k) +k
2
A
131
(k)
112
A
112
(k)
122
A
122
(k) +k
2
A
132
(k)) (4.28)
Assuming a concentrated force Ze
i!t
that acts at the origin, the loading function
(stress) ZJ
0
(kr) at that point can be represented using Fourier-Bessel integral as:
Z (r)j
z=0
=ZJ
0
(kr) =
Z
1
0
J
0
(kr)kdk
Z
1
0
f (s)J
0
(ks)sds (4.29)
76
If f (s) vanishes for all but minute values of s, where it becomes innite in such a way
that:
Z
1
0
f (s) 2sds =Z (4.30)
is nite. Then equation (4.29) is simplied as
Z (r)j
z=0
=
Z
2
Z
1
0
J
0
(kr)kdk (4.31)
SettingZ in the loading function of equation (4.1) toZ =
Zkdk
2
, the surface displace-
ments for a point source can be obtained by:
~ u
r
=
Z
2
Z
1
0
u
r
J
1
(kr)kdk (4.32)
~ u
z
=
Z
2
Z
1
0
u
z
J
0
(kr)kdk (4.33)
4.3 Numerical Results:
For the analysis of a concentrated harmonic load, the data presented in (2.6) are
used with the addition of the train load Z which was taken to be 20 kN.
4.3.1 Poroelastic Half-space
For poroelastic half-space with a harmonic point load, as the porosity of the medium
increases, the magnitude of the horizontal and vertical displacements increase as shown in
gure (4.2). Moreover, having an impermeable surface for the poroelastic half-space will
increase displacement magnitudes since the pore pressure produced by the trapped water
will weaken the soil structure by reducing the friction between the soil grains. From gure
(4.3), for a single porosity, as the shear wave velocity of the medium increases (i.e. harder
soil) the displacement magnitudes decrease for both cases of permeable and impermeable
77
surface. For a permeable surface, as the shear wave velocity of the medium increases, for
all distances from the source, the horizontal and vertical displacement values decrease as
the frequency of the loading harmonic load increase as shown in gure (4.4) and gure
(4.5).
The higher the porosity values, in the case of permeable surface, the higher the
displacement values for all distances away from the source as the frequency of the applied
load increase as seen in gures (4.6) and (4.7). For the case of impermeable interface,
as in gure (4.8), for all porosity values, the displacement decreases with increasing
frequency for all distances from the source. However, as the porosity value of the medium
increase, the range of frequency for the displacement to vanish increase. Moreover, for a
porosity of 0.30 (weakest soil of the three tested), it can be seen at all distances from the
source the value of the initial displacement is much greater than the other displacements
with other porosities which indicate that the pore water pressure plays a huge role in
weakening the soil with high porosity.
78
65
Figure 4-2: Horizontal and vertical displacement values versus distance from the source for different medium porosities for both permeable and
impermeable surface, Vs = 500 m/s
Figure 4.2: Horizontal and vertical displacement values versus distance from the source for
dierent medium porosities for both permeable and impermeable surface (V
s
= 500 m/s)
79
66
Figure 4-3: Horizontal and vertical displacement values versus distance from the source for different medium shear wave velocities for both permeable
and impermeable surface, = 0.20
Figure 4.3: Horizontal and vertical displacement values versus distance from the source for
dierent medium shear wave velocities for both permeable and impermeable surface, = 0:20
80
67
Figure 4-4: Horizontal displacement values versus frequency at different medium shear wave velocities at
different distances from the source and a porosity of 0.20 (permeable surface)
Figure 4.4: Horizontal displacement values versus frequency at dierent medium shear wave
velocities at dierent distances from the source and a porosity of = 0:20 (permeable surface)
81
68
Figure 4-5: Vertical displacement values versus frequency at different medium shear wave velocities at
different distances from the source and a porosity of 0.20 (permeable surface)
Figure 4.5: Vertical displacement values versus frequency at dierent medium shear wave
velocities at dierent distances from the source and a porosity of = 0:20 (permeable surface)
82
69
Figure 4-6: Horizontal displacement values versus frequency for different medium porosity values at
different distances from the source and a shear wave velocity of 500 m/s (permeable surface)
Figure 4.6: Horizontal displacement values versus frequency for dierent medium porosity
values at dierent distances from the source for V
s
= 500 m/s (permeable surface)
83
1.00
0.80
~
1
~ 0.60
"
"
E
"
g 040
].
i:5
0.20
0.00
0.15
Vertical displacement vs_ Frequency at different porosity values
for x = 50 m, Vs = 500 mis (Open B.C)
/'\
I \
- -- -- -·- . . .\
_.·· \
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
-- 13 = 0.10
......... 13 = 0.20
- - 13 = 0.30
Vertical displacement vs. Frequency at different porosity values
for x = 250 m, Vs = 500 mis (Open B C)
~
0.13 'l
I
.I
/1
f 0.10
<::
E 0.08
"
<.)
"'
]. 0 05
i:5
003
0 00
0.08
'? 0 06
I
"
~ 0.04
;;;
<.)
"'
].
i:5 0.02
0.00
I
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
-- 13 = 0.10
......... 13 = 0.20
- - 13 = 0.30
Vertical displacement vs. Frequency at different porosity values
for x = 500 m, Vs = 500 mis (Open B C)
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)
-- 13=0.10
......... 13 = 0.20
- - 13 = 0.30
Figure 4.7: Vertical displacement values versus frequency for dierent medium porosity
values at dierent distances from the source for V
s
= 500 m/s (permeable surface)
84
71
Figure 4-8: Vertical displacement values versus frequency for different medium porosity values at different
distances from the source and a shear wave velocity of 500 m/s (impermeable surface)
Figure 4.8: Vertical displacement values versus frequency for dierent medium porosity
values at dierent distances from the source for V
s
= 500 m/s (impermeable surface)
85
4.3.2 Poroelastic Layer on Top of a Poroelastic Half-space:
As in the case of line load, for a layer thickness of (1 m), the change of porosity ratios
had no noticeable impact of the values of vertical and horizontal displacements as shown
in gure (4.9). As the layer thickness increases to (5 m) as in gure (4.10), a noticeable
dierence in the displacement values appear in the case of impermeable surface only.
The higher the porosity ratio, the higher the displacement. However, comparing the
impermeable surface horizontal and vertical displacements we notice that the highest
displacements occurs when the porosity ratio is 1.35 not 1.50 which suggests that the
load and layer thickness combination reaches its maximum eect on displacements at
a porosity ratio of 1.35. A dierent load-layer thickness combination will indicate a
dierent critical porosity ratio.
When analyzing the eect of the layer thickness for a single porosity ratio as in
gure (4.11), for the horizontal displacements, for both permeable and impermeable
surfaces, the maximum value occurs at a layer thickness of (5 m). However, for the
vertical displacements, the maximum value occur at a layer of thickness (10 m). This
leads to the conclusion that the load-layer thickness eect is dierent for displacements
in the horizontal and vertical direction. In other words, if we start with a thin layer,
the presence of the loading force is felt the same way for both horizontal and vertical
direction in the same manner as a half-space. As the layer thickness increase to (5 m), the
displacements in both direction increase. As the layer thickness increase to (10 m), the
horizontal displacement will not be aected by the new layer thickness. On the contrary,
the medium will act as a half-space and the value for the horizontal displacement will
drop to that of a thin layer. On the other hand, the value of the vertical displacement
will continue to increase. There will be a thickness where the vertical displacement will
reach its maximum value. Beyond this thickness, the load will not feel the presence of
this deep layer and the response of the surface displacements will be as the response
86
captured in the half-space case.
In gure (4.12), the eect of dierent porosity ratios for a layer thickness of (5 m)
is studied at 3 dierent distances from the source, (50 m), (250 m), and (500 m). This
eect is examined in relation to the horizontal displacements in the case of permeable
surface. There exist two peaks at frequencies of 21 Hz and 41 Hz which will shift to 11 Hz
and 21 Hz when the layer thickness is doubled to (10 m) as in gure (4.13). In the case
for vertical displacement for a permeable surface, the same peaks are present at the same
frequencies as in the horizontal displacement case as seen in gures (4.14) and (4.15).
From both cases, horizontal and vertical displacements, we conclude that the higher
the porosity ratio, the higher the displacement magnitude. For the eect of the layer
thickness at a xed porosity, we see in gure (4.16) that a thin layer of (1 m) does not
produce any peaks in horizontal displacements within the frequency range shown. The
peaks within the current range of frequency appear for layers of thickness (5 m) and (10
m). The peaks for a layer thickness of (5 m) are larger for all distances compared to the
peaks for a layer of (10 m) .That shows again that the critical load-thickness combination
is present for horizontal displacements at smaller layers thicknesses compared to the case
of vertical displacement in gure (4.17) where the peaks for the a layer of (10 m) are
greater in magnitudes.
87
74
Figure 4-9: Horizontal and vertical displacements values versus distance from the source for different porosity ratios with a layer thickness of 1 m and
Vs* = 0.50
Figure 4.9: Horizontal and vertical displacements values versus distance from the source for
dierent porosity ratios with a layer thickness of 1 m and V
s
= 0.50
88
75
Figure 4-10: Horizontal and vertical displacements values versus distance from the source for different porosity ratios with a layer thickness of 5 m and
Vs* = 0.50
Figure 4.10: Horizontal and vertical displacements values versus distance from the source for
dierent porosity ratios with a layer thickness of 5 m and V
s
= 0.50
89
76
Figure 4-11: Horizontal and vertical displacement values versus distance from the source for different top layer thicknesses with * = 1.50
Figure 4.11: Horizontal and vertical displacement values versus distance from the source for
dierent top layer thicknesses with
= 1.50
90
77
Figure 4-12: Horizontal displacement values versus frequency at different distances from the source for
different porosity ratios and a layer thickness h = 5 m (permeable surface)
Figure 4.12: Horizontal displacement values versus frequency at dierent distances from the
source for dierent porosity ratios and a layer thickness h = 5 m (permeable surface)
91
78
Figure 4-13: Horizontal displacement values versus frequency at different distances from the source for
different porosity ratios and a layer thickness h = 10 m (permeable surface)
Figure 4.13: Horizontal displacement values versus frequency at dierent distances from the
source for dierent porosity ratios and a layer thickness h = 10 m (permeable surface)
92
79
Figure 4-14: Vertical displacement values versus frequency at different distances from the source for
different porosity ratios and a layer thickness h = 5 m (permeable surface)
Figure 4.14: Vertical displacement values versus frequency at dierent distances from the
source for dierent porosity ratios and a layer thickness h = 5 m (permeable surface)
93
80
Figure 4-15: Vertical displacement values versus frequency at different distances from the source for
different porosity ratios and a layer thickness h = 10 m (permeable surface)
Figure 4.15: Vertical displacement values versus frequency at dierent distances from the
source for dierent porosity ratios and a layer thickness h = 10 m (permeable surface)
94
81
Figure 4-16: Horizontal displacement values versus frequency at different distances from the source for
different layer thicknesses, = 1.20 (permeable surface)
Figure 4.16: Horizontal displacement values versus frequency at dierent distances from the
source for dierent layer thicknesses,
=1.20 (permeable surface)
95
82
Figure 4-17: Vertical displacement values versus frequency at different distances from the source for
different layer thicknesses, = 1.20 (permeable surface)
Figure 4.17: Vertical displacement values versus frequency at dierent distances from the
source for dierent layer thicknesses,
=1.20 (permeable surface)
96
4.4 Moving Load Formulation:
As stated in the introduction, the concentrated harmonic point load formulation in-
troduced in this chapter can be used to study the eect of the train speed on the resulting
displacements at a distance perpendicular to the track. That can be accomplished by
changing the location of the concentrated point load as a function of time t. Referring
to gure (4.18), a harmonic loadZ moving with a speedc along thex-axis with a point
of interest (observer) is located at a distance D along the y-axis.
83
4.4 Moving load formulation:
As stated in the introduction, the concentrated harmonic point load formulation
introduced in this chapter can be used to study the effect of the train speed on the
resulting displacements at a distance perpendicular to the track. That can be
accomplished by changing the location of the concentrated point load as a function of
time t. Referring to Figure 4-18, A harmonic load Z moving with a speed c along the x
axis with a point of interest, observer, is located at a distance D along the y axis.
z
x
y
c
D
Z
Figure 4-18: Schematic of a moving harmonic point load along the x axis and an observer at distance D
along the y axis.
Let vD be the vertical displacement at point D. it can be calculated from
22 i t xc
D
v f x De
(4-34)
where
i t xc
e
is a time shift factor for the load moving in the positive x direction and
the quotient xc has the unit of time. Noticing that vD is an accumulation of all loads at
different times along the x axis, the vertical displacement at point D can be obtained by
integrating equation (4-34) over x as
22
x
i
it
c
D
v f x D e dxe
(4-35)
4.4.1 Numerical results:
Equation (4-35) was applied to the case of poroelastic half-space with three
different porosities, 0.10, 0.20, and 0.30. The vertical displacement was calculated, as
indicated by Figure 4-19, at distances ranging from 20 m to 100 m away from the train
Figure 4.18: Schematic of a moving harmonic point load along the x-axis and an observer at
distance D along the y-axis.
Letv
D
be the vertical displacement at the observation pointD. It can be calculated
from:
v
D
=f
p
x
2
+D
2
e
i!(tx/c)
(4.34)
wheree
i!(tx/c)
is a time shift factor for the load moving in the positivex-direction and
the quotient x/c has the unit of time. Noticing that v
D
is an accumulation of all loads
at dierent times along thex-axis, the vertical displacement at pointD can be obtained
by integrating equation (4.34) over x as:
97
v
D
=
Z
1
1
f
p
x
2
+D
2
e
i
!x
c
dx
e
i!t
(4.35)
A more appropriate representation of the displacement is to include the eects of
n number of axial moving loads simulating the loads provided by dierent axles of
the moving trains. Referring to gure (4.19), and using the superposition of multiple
displacements calculated by equation (4.34) , the total displacement caused byn number
of moving axial loads can be expressed as:
v
D
=
n1
X
k=0
f
q
x
2
k
+D
2
e
i!
t
k
x
k
c
!!
(4.36)
z
x
y
D (0, D, 0)
1
l
,0,0
oo
xx
11
,0,0 xx
,0,0
kk
xx
2
l
Z
2
Z
n
Z
1
Z
o
k
l
22
,0,0 xx
22
2
xD 2 2
1
x D 22
n
xD 2 2
o
x D c
Figure 4.19: Schematic showing the moving loads at x
1
, x
2
, x
3
, and x
n
If we assume that the rst load passes through the origin x (0; 0; 0) at time t
o
, then
the n
th
load will pass through the origin at:
t
k
=t
o
+
l
k
c
(4.37)
Moreover, the coordinate x
k
appearing in gure (4.19) can be expressed in terms of the
98
coordinate x
o
. From gure (4.19), one can see that
x
k
=x
o
l
k
(4.38)
Substituting equations (4.37) and (4.38) in (4.36) leads to:
v
D
=e
i!to
n1
X
k=0
f
q
(x
o
l
k
)
2
+D
2
e
i!
0
@
x
o
2l
k
c
1
A
(4.39)
To accumulate the eects ofn number of loads moving along thex-axis, equation (4.39)
is integrated over the entire x-axis and presented as:
v
D
=e
i!to
n1
X
k=0
Z
1
1
0
B
@
f
p
(x
o
l
k
)
2
+D
2
e
i!
0
@
x
o
2l
k
c
1
A
1
C
A
dx (4.40)
4.4.1 Numerical Results:
4.4.1.1 Poroelastic Half-space
Equation (4.40) was applied to the case of poroelastic half-space with three dierent
porosities, 0.10, 0.20, and 0.30. The vertical displacement was calculated, as indicated
by gure (4.20), at distances ranging from (20 m) to (100 m) away from the train tracks
for three dierent train speeds of 20, 60, and 120 km/hr. From gure(4.20), it can be
seen that at any given speed, the vertical displacement increases as the porosity of the
medium increases. From gure (4.20a), doubling the porosity from = 0:1 to = 0:2
will increase the the vertical displacement by about 170% at a distance of 20 m. Plus,
when increases to 0.30, the vertical displacement is increased by 270%.
As for the eect of the train speed on displacements, it is noted from (4.21) increasing
the speed of the train will increase the vertical displacement regardless of the medium
porosity. Moreover, at any given distance from the train tracks, the vertical displacement
increases as the porosity of the medium increases as the speed of the train increases. For
a speed of 20 km/hr, increasing the porosity from 0.10 to 0.20 will increase the vertical
99
displacement by 23% and when increasing from 0.10 to 0.30, the vertical displacement
will increase by 28%. As the speed of the train increases, increasing the porosity of the
medium will have a bigger eect on the vertical displacements. For the largest value of
train speed in this study (120 km/hr), increasing the porosity from 0.10 to 0.20 will
increase the vertical displacement by 62% and when increasing from 0.10 to 0.30, the
vertical displacement will increase by 73%.
100
0.00
1.00
2.00
3.00
4.00
5.00
20 30 40 50 60 70 80 90 100
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different medium
porosities (speed = 120 km/hr, Open B.C)
0.00
1.00
2.00
3.00
4.00
5.00
20 30 40 50 60 70 80 90 100
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different medium
porosities (speed = 60 km/hr, Open B.C)
0.00
1.00
2.00
3.00
4.00
5.00
20 30 40 50 60 70 80 90 100
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different medium
porosities (speed = 20 km/hr, Open B.C)
Figure 4.20: Vertical displacement for dierent values of half-space porosities as a function of
distance from the source for train speeds of (a) 120 km/hr, (b) 60 km/hr, and (c) 20 km/hr
101
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs Distance from source for different medium
porosities (D = 20 m, Open B.C)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs Distance from source for different medium
porosities (D = 50 m, Open B.C)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs Distance from source for different medium
porosities (D = 100 m, Open B.C)
(a)
(b)
(c)
Figure 4.21: Vertical displacement for dierent values of half-space porosities as a function of
train speed for away distances from the tracks of (a) 20 m, (b) 50 m, and (c) 100 m
102
4.4.1.2 Poroelastic Layer on Top of a Poroelastic Half-space:
In the case of a single poroelastic layer on top a poroelastic half-space, the goal is to
see how the properties, mainly thickness and porosity, of the top layer aect the resulting
displacements on the surface. Referring to gure (4.22), one can conclude that for a given
layer thickness the larger the porosity ratio, the larger the vertical displacements. For
example, for a layer thickness of (10 m) and a distance (20 m) away from the train tracks,
increasing porosity ratio
from 1.20 to 1.35 will increase the vertical displacement by
200% and will increase by 400% if the porosity ratio
increased from 1.20 to 1.50.
For a given porosity ratio
, the thicker the top layer, the higher the displacement.
This is true under the assumption that the load/layer combination make the medium
behave as layered. If the rst layer is very thick compared to the following layer, the
surface behavior will depend on the mechanical properties of the layer regardless of the
thickness. Therefore, this top layer will behave as a half-space. Plus, in the case of a
extremely thin layer, the applied load will not feel the presence of the layer and the
behavior will depend on the properties of the supporting half-space beneath this thin
layer.
The surface displacement is highly dependent on the frequency of the applied load.
Figures (4.23) and (4.24), show the results from analyzing the vertical displacement as a
function of train speeds for dierent distances from the train track. The two gures are
input data for the two gures (4.23) and (4.24) are identical except for the frequency. A
low frequency was used for the data presented in gure (4.23) which was in close to the
resonance frequency of that layer (h = 10 m). It is noted that even at large distances
from the tracks, the vertical displacement for that layer is signicantly higher than the
other thicknesses. When a higher frequency was used for the data in gure (4.23), the
thinner layer (h = 5 m) was excited more than the rest of the layers.
The same conclusions can be reached by studying gures (4.25) and (4.26) which
103
relate the vertical displacement to the distance from the train tracks for three dierent
layer thicknesses (h = 1 m, h = 5 m, h = 10 m). This indicates that a complete
understanding of the layered media behavior cannot be reached just by knowing the
layer mechanical properties but added to that is the frequency at which the external
load is applied which plays a huge role in the nal values of displacements.
104
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs. Distance from the train track for different
medium porosities (speed = 120 km/hr, h = 1 m, Open B.C)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs. Distance from the train track for different
medium porosities (speed = 120 km/hr, h = 5 m, Open B.C)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs. Distance from the train track for different
medium porosities (speed = 120 km/hr, h = 10 m, Open B.C)
(a)
(c)
(b)
Figure 4.22: Vertical displacement for dierent values of layer porosity ratios as a function of
distance from train track speed for layer thickness of (a) 1 m, (b) 5 m, and (c) 10 m
105
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 20 m, = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 50 m, = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 100 m, * = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 150 m, * = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
( a) (c)
(b) (d)
Figure 4.23: Vertical displacement for dierent values of layer thicknesses as a function of
train speeds for distance from train track of (a) 20 m, (b) 50 m, (c) 100 m, and (d) 150 m
(lower frequency)
106
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 20 m, = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 50 m, = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 150 m, * = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
20 40 60 80 100 120
Displacement (mm)
Speed (km/hr)
Vertical displacement vs. Train speed for different top layer
thickness (D = 100 m, * = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
( a)
(c)
(d) (b)
Figure 4.24: Vertical displacement for dierent values of layer thicknesses as a function of
train speeds for distance from train track of (a) 20 m, (b) 50 m, (c) 100 m, and (d) 150 m
(higher frequency)
107
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different top
layer thickness (speed = 60 km/hr, = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different top
layer thickness (speed = 20 km/hr, = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different top
layer thickness (speed = 120 km/hr, * = 1.50, Open B.C)
h = 1 m
h = 5 m
h = 10 m
( a)
(b)
(c)
Figure 4.25: Vertical displacement for dierent values of layer thickness as a function of
distance from train tracks for speeds of (a) 120 km/hr, (b) 60 km/hr, and (c) 20 km/hr
(lower frequency)
108
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different top
layer thickness (speed = 60 km/hr, = 1.20, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different top
layer thickness (speed = 120 km/hr, * = 1.20, Open B.C)
h = 1 m
h = 5 m
h = 10 m
0.00
1.00
2.00
3.00
4.00
5.00
6.00
20 30 40 50 60 70 80 90 100 110 120 130 140 150
Displacement (mm)
Distance (m)
Vertical displacement vs Distance from source for different top
layer thickness (speed = 20 km/hr, = 1.20, Open B.C)
h = 1 m
h = 5 m
h = 10 m
( a)
(c)
(b)
Figure 4.26: Vertical displacement for dierent values of layer thickness as a function of
distance from train tracks for speeds of (a) 120 km/hr, (b) 120 km/hr, and (c) 120 km/hr
(higher frequency)
109
Chapter 5
The Equivalent Dry Elastic Medium
5.1 Introduction:
After introducing the theory of poroelasticity in chapter (2), the uniformly harmonic
distributed line load was used in chapter (3) as a loading force to simulate the dynamic
eects exerted by train on underlying medium. In chapter (4), the loading force of a
harmonic line load was replaced by a concentrated harmonic point load simulating the
load from the train through its wheel. This new formulation was essential to introduce
the eect of the speed of the train by moving the harmonic point load along the track
axes. In chapters (3) and (4), the half-space and layered medium were analyzed as being
poroelastic using the theory of poroelasticity proposed by Biot (1956a) which introduced
the existence of two pressure body waves (fast and slow) and a single shear wave as stated
earlier in chapter (2). One of the assumptions ensuring a correct application of Biot's
theory of poroelasticity, as mentioned in section (2.1.2) is a complete saturation of the
medium resulting in what is known as two-phase medium. If a relative movement takes
place between the
uid and the solid part of the poroelastic cube matrix, a second
P-wave is generated (slow P-wave). In case of partial saturation, three-phase medium
110
exists where Biots theory can be used with some approximation techniques to transfer
three-phase medium properties to two-phase medium properties as the case of reduced
bulk modulus of the
uid in (J. Yang, 2001). Using similar analogy, depending on the
frequency range under investigation, the two-phase medium can be further reduced to an
equivalent dry medium (single phase) with modied mechanical properties allowing the
use of the well established classical wave propagation techniques to study the response
of the dynamic moving point load introduced in chapter (4).
5.2 Mathematical Background:
Biot (1956a) introduced a characteristic frequency!
o
which can be seen as the \cut-
o" frequency that governs the transition between low and high frequency behavior of
the poroelastic medium and was presented as:
!
o
=
2
f
(5.1)
Schevenels et al. (2004) presented that frequency as:
!
o
=
f
f
(
f
)
(5.2)
where
: The kinematic viscosity of pore
uid
: Porosity
k : Permeability of the medium
f
: The specic weight of the pore
uid (
f
= 9:80 kN/m
3
for water)
f
: Density of the pore
uid
: The mixture density dened in equation (2.27c) as: =
f
+
s
(1)
: Dynamic toruosity parameter
The dispersion relation for the P and S-waves can be represented as in Schevenels et
111
al. (2004) as:
k
2
p
V
2
fast
!
2
k
2
p
V
2
slow
!
2
+
i!
2
k
2
p
V
2
low
!
2
= 0 (5.3)
k
2
s
V
2
s
!
2
i
k
2
s
V
2
s low
!
2
= 0 (5.4)
where V
low
and V
s low
are the P and S-wave velocity in the low- frequency limit re-
spectively. V
fast
, V
slow
, and V
s
are the fast, slow P-wave, and S-wave velocities in the
poroelastic medium respectively. The ratio of the excitation frequency ! to the charac-
teristic frequency!
o
denes, a dimensionless frequency. The parametersk
p
andk
s
are
the complex wave numbers for the P and S-waves.
Solving the bi-quadratic equation (5.3) shows the existence of two P-waves (as pre-
dicted by Biot (1956a)) with velocities V
j
=
!
Re(k
j
)
, where j = 1; 2 for fast and slow
P-wave respectively. On the other hand, solving the quadratic equation (5.4) shows the
existence of a single S-wave with a velocity V
s
=
!
Re(ks)
. To demonstrate, let us consider
a medium with the parameters shown in table (5.1):
Table 5.1: Dry and saturated medium parameters
Dry poroelastic medium properties Saturated poroelastic medium properties
Parameter Unit Value Parameter Unit Value
s
kg/m
3
2650
f
kg/m
3
1000
E
s
MPa 95.4 K
f
MPa 2200
- 0.40 m/s 110
4
- 0.333 V
pfast
km/s 1.651
V
pdry
km/s 0.500 V
pslow
km/s 0.296
V
sdry
km/s 0.250 V
s
km/s 0.238
Based on the values shown in table (5.1), the characteristic frequency of the saturated
poroelastic medium, using equation (5.2), was found to be equal to!
o
= 25.3310
3
rad/s
( 4000 Hz). Fiala et. al (2007) stated that for frequency range between 1 Hz and 80
Hz, building vibration causes a form of a mechanical vibration of the human body,
112
whereas between 16 Hz and 250 Hz, these vibrations can cause structure-borne noise by
exciting
oors and walls. Sanayei et al. (2013) and (2014) identied the frequency range
of interest to the vibrations of building to lie between 10 Hz and 200 Hz with a peak
around 50 Hz. This frequency band (1 Hz to 200 Hz) includes the critical frequencies
that will not only produce the mechanical vibration suggested by Fiala et. al (2007),
but also will have a direct impact on the physiology and comfort of the occupants that
was discussed in section (1.2.2). Therefore, the frequency range of interest for structural
mechanical vibrations associated with this study is limited at most to 200 Hz which can
be seen as a low frequency value compared to the characteristic frequency of 4000 Hz.
In other words, using the data presented in table (5.1), the dimensionless frequency
will have a maximum value of 510
2
which is clearly a low frequency value.
Figure (5.1) shows the relation between the dimensionless frequency () of the poroe-
lastic saturated medium and the velocities of the waves generated. Since the second P-
wave, as noted by Biot (1956a), is highly attenuated, and by comparing plots (a) and (b)
in gure (5.1), the wave propagation process in the low frequency range is dominated by
a single equivalent P-wave and S-wave. Moreover, It can be seen that at low frequency
(regardless of the type of soil), the wave velocities approach the velocity indicated by the
subscript \low" in equations (5.3) and (5.4). Therefore, for low frequency range (closer
to the frequency range of this study), the two P-wave velocities can be replaced by a
single \equivalent" P-wave velocity.
This was also noted by Gupta et al. (2009) while investigating the in
uence of tunnel
and soil parameters on vibrations from underground railways. They indicated that the
relative motion between the
uid and the solid particles is prevented in the low frequency
range which lead the medium to behave as a \frozen" mixture rather than a two-phase
medium. They added that the medium can be modeled as a dry elastic medium with
\equivalent" mechanical parameters.
113
The values of the fast P-wave velocities at low frequency (equivalent velocity) shown
in gure (5.1a) are obviously higher than the ones for dry medium. According to Lom-
baert et al. (2015), the velocity of the equivalent P-wave is higher due to the presence
of the pore
uid that lowers the compressibility of the medium. They added that due to
the change in density of the medium, a small change will occur to the S-wave velocity.
This can be seen in the data shown in table (5.2).
No doubt that transforming the poroelastic medium to an equivalent dry elastic one,
given that the transformation is justied, will eliminate the need to develop new com-
puter algorithms to calculate responses based on the theory of poroelasticity. Instead,
having an equivalent dry elastic medium will utilize the well established classical solu-
tions found in many publications as in Ewing et al. (1957), Cagniard (1962), and Bullen
(1985).
114
0.00
1.00
2.00
3.00
4.00
5.00
6.00
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
Fast P-wave velocity (km/s)
Dimensionless frequency ()
Dimensionless frequency ( ) vs. Fast P-wave velocity for different
in-situ V
s
values ( = 0.40)
Vs = 250 m/s
Vs = 500 m/s
Vs = 1000 m/s
0.00
1.00
2.00
3.00
4.00
5.00
6.00
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
Slow P-wave velocity (km/s)
Dimensionless frequency ( )
Dimensionless frequency ( ) vs. Slow P-wave velocity for different
in-situ V
s
values ( = 0.40)
Vs = 250 m/s
Vs = 500 m/s
Vs = 1000 m/s
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
S-wave velocity (km/s)
Dimensionless frequency (χ)
Dimensionless frequency ( ) vs. S-wave velocity for different
in-situ V
s
values
Vs = 250 m/s
Vs = 500 m/s
Vs = 1000 m/s
(a)
( b)
( c)
Figure 5.1: Dimensionless frequency () vs. Wave velocities of the medium for
dierent in-situ S-wave velocities (a) Fast P-wave (b) Slow P-wave (c) S-wave
115
5.3 Equivalent Dry Medium Parameters:
Due to the modication of the medium mechanical properties, the equivalent veloc-
ities of generated waves will depend on \modied" mechanical properties of the equiv-
alent dry medium. The equivalent P and S-wave velocities can be calculated as in Biot
(1956a), Gupta et al. (2009), and Wang et al. (2011):
V
pe
=
s
eq
+
eq
eq
(5.5)
V
se
=
r
eq
eq
(5.6)
where
eq
=
s
,
eq
=
s
+
K
f
are the modied (or equivalent) Lam e constants for the
equivalent elastic dry medium and
eq
=
f
+
s
(1) is the equivalent (mixture)
density of the medium (which was introduced earlier in equation (2.27c). Using the newly
dened P and S-wave velocities, the equivalent Poisson's ratio
eq
, can be calculated
from:
eq
=
V
2
pe
2V
2
se
V
2
pe
+V
2
se
(5.7)
When substituting equations (5.5) and (5.6) into equation (5.7), a relation among the
equivalent Poisson's ratio
eq
, the original Lam e constants (
s
and
s
), porosity (), and
bulk modulus of the
uid (K
f
) is established as:
eq
=
1
2
3
s
2 ( (
s
+ 2
s
) +K
f
)
(5.8)
In terms of equivalent material constants (
eq
,
eq
), the equivalent Poisson's ratio
eq
can be expressed as:
eq
=
1
4
3
eq
(
eq
+ 2
eq
)
1
(5.9)
Calculating all equivalent mechanical properties of the equivalent dry medium de-
116
pends on few principle properties that can be obtained either by eld or laboratory tests.
These principle properties are: modulus of elasticity of soil medium (E
s
), Poisson's ratio
(), the density of the medium (
s
), the density of the pore
uid (
f
), the porosity of
the medium (), and the bulk modulus of the
uid (K
f
). When these properties are
obtained, the next steps are followed to create a list of all mechanical properties of the
equivalent dry medium as shown in table (5.2) using the saturated media data from
table (2.6):
1. The rst Lam e constant (
s
) of the medium is obtained from:
s
=
s
V
2
s
=
E
s
2 (1 +)
2. The second Lam e constant (
s
) of the medium is obtained from:
s
=
2
s
1 2
=
E
s
(1 +) (1 2)
3. The equivalent (mixed) density of the medium (
eq
) is calculated from:
eq
=
f
+
s
(1)
4. The equivalent Lam e constants (
eq
and
eq
) of the medium are obtained from:
eq
=
s
and
eq
=
s
+
K
f
5. The equivalent P-wave and S-wave velocities (V
pe
and V
se
) are obtained from:
V
pe
=
s
eq
+
eq
eq
and V
se
=
r
eq
eq
6. The equivalent Poisson's ratio (
eq
) is calculated as:
eq
=
1
2
3
s
2 ( (
s
+ 2
s
) +K
f
)
=
1
4
3
eq
(
eq
+ 2
eq
)
1
Before proceeding with the transformation, a look at the data obtained from table
(5.2) is a must. A visual representation of some data observations is presented in the
next section.
117
Table 5.2: Material constants and wave velocities for dry, poroelastic, and equivalent dry elastic
medium with dierent porosities
Porosity () 0.1 0.2 0.3 0.4
Dry
medium
properties
dry
kg/m
3
2385 2385 2385 2120 2120 2120 1855 1855 1855 1590 1590 1590
dry
MPa 149 596 2385 133 530 2120 116 464 1855 99 398 1590
dry
MPa 298 1193 4770 265 1060 4240 232 928 3710 199 795 3180
V
p
km/s 0.500 1.000 2.000 0.500 1.000 2.000 0.500 1.000 2.000 0.500 1.000 2.000
V
s
km/s 0.250 0.500 1.000 0.250 0.500 1.000 0.250 0.500 1.000 0.250 0.500 1.000
- 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333
Poroelastic
medium
properties
sat
kg/m
3
2485 2485 2485 2320 2320 2320 2155 2155 2155 1990 1990 1990
sat
MPa 149 596 2385 133 530 2120 116 464 1855 99 398 1590
sat
MPa 11564 11772 12753 5798 6225 8041 3305 3789 5792 1970 2442 4371
V
pfast
km/s 2.364 2.450 2.783 1.963 2.074 2.502 1.766 1.882 2.351 1.651 1.761 2.248
V
pslow
km/s 0.102 0.199 0.360 0.183 0.347 0.583 0.246 0.462 0.746 0.296 0.555 0.875
V
s
km/s 0.246 0.492 0.983 0.243 0.485 0.970 0.240 0.480 0.959 0.238 0.475 0.950
- 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333 0.333
eq
kg/m
3
2485 2485 2485 2320 2320 2320 2155 2155 2155 1990 1990 1990
eq
MPa 149 596 2385 133 530 2120 166 464 1855 99 398 1590
Equivalent
dry
medium
properties
eq
MPa 22298 23193 26770 11265 12060 15240 7565 8261 11043 5699 6295 8680
V
pe
km/s 3.006 3.094 3.425 2.216 2.330 2.735 1.888 2.012 2.446 1.707 1.834 2.272
V
se
km/s 0.245 0.490 0.980 0.239 0.478 0.956 0.232 0.464 0.928 0.223 0.447 0.894
eq
- 0.489 0.463 0.387 0.478 0.427 0.337 0.476 0.424 0.311 0.475 0.416 0.299
118
5.4 Numerical Results:
We start by looking at the characteristic frequency presented by equation (5.2). From
gure (5.2), it can be seen that for any given medium porosity (), as the permeability
of the medium () increases, the characteristic frequency (!
o
) of the medium decreases.
That can be seen from equation (5.2) where the characteristic frequency (!
o
) is inversely,
linearly proportional to the permeability of the medium. As for the eect of porosity,
in general, for any given medium permeability (), as the porosity of the medium ()
increases, the characteristic frequency (!
o
) increases. However, this relation is not as
straight forward as the case with permeability ().
Figure (5.3) shows the non-linear relation between medium porosity () and charac-
teristic frequency (!
o
). Simple regression analysis shows that the line follows the form
of a \power" trend-line in the form of !
o
= A
B
. When substituting the values from
table (5.1), the power \B" is found to be equal to (1.9). The term \A" changes values
depending on the permeability (), and since () is inversely linearly proportional to
characteristic frequency, (!
o
) can be expressed as !
o
= C
1:9
. When substituting the
values from table (5.1), the value of \C" is found to be equal to (14.4). Therefor, an
approximate equation is produced to calculate the characteristic frequency (!
o
) can be
presented as:
!
o
= 14:4
1:9
(5.10)
Equation (5.10) is only used to show that the characteristic frequency (!
o
) experiences
an approximate \quadratic growth" as the porosity varies linearly while preserving the
inverse proportionality with the permeability of the medium (). Equation (5.2) must
be used to calculate the characteristic frequency (!
o
).
Figure (5.4a) shows the values for the P and S-wave velocities in relation with the
porosity of the medium for dierent values of in-situ S-wave velocities. It is observed
119
that at any given porosity, the P-wave velocity is higher for soils with high in-situ S-wave
velocity (i.e. harder soil). That is expected from classical theory of wave propagation
since the P-wave velocity is higher in denser soils. At any given in-situ S-wave velocity,
as the porosity of the medium () increases the P-wave velocity decreases. When the
porosity of the medium increases, the density of the medium decreases therefore the
P-wave velocity decreases. As for gure (5.4b), it can be seen that the porosity has a
minor aect on the shear wave velocity due to the minor change the porosity has on the
equivalent density (
eq
) supporting what was observer by Lombaert et al. (2015).
Figure (5.5) represents the relation between the calculated and the equivalent P and
S-wave velocities and the porosity of the medium for various in-situ S-wave velocities. It
can be seen that the calculated and equivalent S-wave velocities are almost identical for
all porosity range. However, a dierence is shown between the values of the calculated
and equivalent P-wave velocities specially at low porosity values and the value of equiva-
lent P-wave velocity is always higher for all values in the porosity range. As the porosity
of the medium increases, the dierence between the values of calculated and equivalent
P-wave velocities decreases. The percentage dierence between these values are shown
in gure (5.6). From gure (5.6), the dierence decreases below 15% at porosities higher
than = 0:20 where below = 0:20 the material behavior is closer to being elastic
rather than poroelastic. Moreover, the dierence between the calculated and equivalent
P-wave velocities is lower for harder soils (higher in-situ V
s
).
As for the dierence between calculated and equivalent S-wave velocity values, gure
(5.7) shows that the dierence increases as the porosity of the medium increases and
reaches a dierence of about 6% at porosity = 0:40. Since both calculated and
equivalent are obtained based on the same shear modulus of the medium (
s
=
eq
), we
observe that the dierence in values between calculated and equivalent S-wave velocities
is the same across the porosity range.
120
The calculated, equivalent, and dry P-wave velocities are all presented in gure (5.8).
This gure shows the dierence between the velocity values in the dry medium and how
the presence of the pore
uid lowers the compressibility of the medium resulting in the
increases of the P-wave velocity value. It is shown in gure (5.8) that in the case of dry
medium, the P-wave velocity is not aected by the porosity of the medium since it is a
constant value across the porosity range.
The calculated equivalent Poisson's ratios for dierent medium porosities are pre-
sented in gure (5.9). For all values of medium porosity, the value of the equivalent
Poisson's ratio is higher for lower in-situ S-wave velocity. The reason being that for
lower in-situ S-wave mediums, the soil is weaker and more deformable. For all values of
in-situ S-wave velocity, the higher the porosity of the medium, the lower the equivalent
Poisson's ratio (harder soil). In general, the presence of the pore
uid will weaken the
soil and make it more deformable (higher
eq
) except for the case when in-situ S-wave
velocity is high (V
s
= 1000 m/s). For the present case, the presence of pore
uid changes
the characteristics of the medium after a porosity value of = 0:214 where it acts as a
stiener for the soil-
uid cube system resulting in a low (
eq
).
The shear modulus () for a medium is not aected by the presence or absence
of the pore
uid. That is the reason why the calculated S-wave velocity is slightly
aected only by the equivalent density of the medium which is a function of porosity.
Therefore, shear modulus () is aected by the porosity as well as the in-situ S-wave of
the medium. Figures (5.10) and (5.11) show, (for dry, saturated, and equivalent), the
variation of the value of te shear modulus () along the porosity range and for dierent
in-situ S-wave values of the medium. From gure (5.10), it can be seen that as the
porosity () increases, the shear modulus of the medium () decreases for all values of
in-situ S-wave velocities. With higher porosity, more voids are present to host the pore
uid which has, in general, low shear modulus leaving less solid material to resist the
121
shear stress and that leads to a lower shear modulus. In gure (5.11), it is shown that, in
general, for any medium porosity the higher the in-situ S-wave velocity of the medium,
the higher is the shear modulus. Again, it is obvious that the higher the porosity (),
regardless of the in-situ S-wave velocity value, the higher is the shear modulus ().
The dry and equivalent density of the medium (
dry
and
eq
) are plotted in gure
(5.12) for dierent values of medium porosity. The contribution of the pore
uid is
evident in increasing the value of (
eq
) compared with (
dry
) for any porosity of the
medium. Moreover, it can be seen both densities (
dry
and
eq
) decrease as the porosity
of the medium increase. The percentage dierence between the dry and equivalent
density, as shown in gure (5.10), increases from 4% at = 0:10 to 25% at = 0:40
suggesting that with more pores to host the
uid, more contribution the presence of the
uid will have on the (mixture) density making the value of the equivalent density (
eq
)
higher than that of dry medium.
122
1.0E+00
1.0E+02
1.0E+04
1.0E+06
1.0E+08
1.0E+10
0.10 0.20 0.30 0.40
Characteristic frequency (
o
) (rad/s)
Porosity ( )
Porosity ( ) vs. Characteristic frequency (
o
) for different values of
permeability ( )
κ = 1.E-03 m/s
κ = 1.E-04 m/s
κ = 1.E-05 m/s
κ = 1.E-06 m/s
κ = 1.E-07 m/s
Figure 5.2: Porosity () vs. Characteristic frequency (!
o
) for dierent values of
medium permeability ()
0.0E+00
5.0E+03
1.0E+04
1.5E+04
2.0E+04
2.5E+04
3.0E+04
0.10 0.20 0.30 0.40
Characteristic frequency (
o
) (rad/s)
Porosity ( )
Porosity ( ) vs. Characteristic frequency (
o
) for permeability
= 1E-04 m/s
κ = 1.E-04
Figure 5.3: Porosity () vs. Characteristic frequency (!
o
) for permeability
= 110
04
m/s
123
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Equivalent P-wave velocity (km/s)
Porosity
Equivalent P-wave velocities vs. Porosity of medium at different
field measured S-wave velocities
Vs = 250 m/s
Vs = 500 m/s
Vs = 1000 m/s
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Equivalent S-wave velocity (km/s)
Porosity
Equivalent S-wave velocities vs. Porosity of medium at different
field measured S-wave velocities
Vs = 250 m/s
Vs = 500 m/s
Vs = 1000 m/s
(b)
( a )
Figure 5.4: Equivalent wave velocity vs. porosity of the medium for dierent in-situ
S-wave velocities (a) P-wave (b) S-wave
124
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40 P-wave and S-wave Velocities (km/s)
Porosity
Calculated / Equivalent P-wave and S-wave velocities vs.
Porosity of medium at in-situ Vs = 250 m/s
Vps
Vpe
Vss
Vse
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40 P-wave and S-wave Velocities (km/s)
Porosity
Calculated / Equivalent P-wave and S-wave velocities vs.
Porosity of medium at in-situ Vs = 500 m/s
Vps
Vpe
Vss
Vse
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40 P-wave and S-wave Velocities (km/s)
Porosity
Calculated / Equivalent P-wave and S-wave velocities vs.
Porosity of medium at in-situ Vs = 1000 m/s
Vps
Vpe
Vss
Vse
( a )
( b)
( c)
Figure 5.5: Calculated and equivalent P and S-wave velocities vs. Porosity of the
medium for (a) V
s
= 250 m/s (b) V
s
= 500 m/s (c) V
s
= 1000 m/s
125
0
5
10
15
20
25
30
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Percentage difference (%)
Porosity
Percentage difference between calculated P-wave velocities
and equivalent P-wave vel ocities for different medium
porosities
Vp @ Vs = 250 m/s
Vp @ Vs = 500 m/s
Vp @ Vs = 1000 m/s
Figure 5.6: Percentage dierence between calculated P-wave velocities and
equivalent P-wave velocities for dierent medium porosities
0
2
4
6
8
10
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Percentage difference (%)
Porosity
Percentage difference between calculated S-wave velocities
and equivalent S-wave vel ocities for different medium
porosities
Vs @ Vs_in-situ = 250 m/s
Vs @ Vs_in-situ = 500 m/s
Vs @ Vs_in-situ = 1000 m/s
Figure 5.7: Percentage dierence between calculated S-wave velocities and
equivalent S-wave velocities for dierent medium porosities
126
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40
V
pc
/ V
pe
/ V
p
velocities (km/s)
Porosity
Calculated / Equivalent / Dry P-wave velocities vs. Porosity
of medium at in-situ Vs = 250 m/s
Vpc
Vpe
Vp
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40
V
pc
/ V
pe
/ V
p
velocities (km/s)
Porosity
Calculated / Equivalent / Dry P-wave velocities vs. Porosity
of medium at in-situ Vs = 500 m/s
Vpc
Vpe
Vp
0.00
1.00
2.00
3.00
4.00
5.00
0.10 0.15 0.20 0.25 0.30 0.35 0.40
V
pc
/ V
pe
/ V
p
velocities (km/s)
Porosity
Calculated / Equivalent / Dry P-wave velocities vs. Porosity
of medium at in-situ Vs = 1000 m/s
Vpc
Vpe
Vp
( a )
( b)
( c)
Figure 5.8: Calculated / Equivalent / Dry P-wave velocities vs. Porosity of medium
at (a) V
s
= 250 m/s (b) V
s
= 500 m/s (c) V
s
= 1000 m/s
127
0.20
0.30
0.40
0.50
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Equivalent Poisson's ratio (
eq
)
Porosity
Equivalent Poisson's ratio vs. Porosity of medium at
different in-situ S-wave velocity of equivalent dry medium
Vs = 250 m/s
Vs = 500 m/s
Vs = 1000 m/s
Original
0.214
0.33
Figure 5.9: Equivalent Poisson's ratio vs. Porosity of medium at dierent in-situ
S-wave velocity of equivalent dry medium
0
500
1000
1500
2000
2500
3000
0.10 0.15 0.20 0.25 0.30 0.35 0.40
dry
/
sat
/
eq
of the medium (MPa)
Porosity
Dry / Saturated / Equivalent shear modulus of the medium vs.
Porosity of the medium for different values of in-situ Vs
Vs = 250 m/s
Vs = 500 m/s
Vs = 1000 m/s
Figure 5.10: Dry / Saturated / Equivalent shear modulus of the medium vs.
Porosity of the medium for dierent values of in-situ V
s
128
0
500
1000
1500
2000
2500
3000
250 500 750 1000
dry
/
sat
/
eq
of the medium (MPa)
In-situ S-wave velocity of the medium (m/s)
Dry / Saturated / Equivalent shear modulus of the medium vs.
In-situ S-wave velocity for different values of porosities
Figure 5.11: Equivalent wave velocity vs. porosity of the medium for dierent
in-situ S-wave velocities (a) P-wave (b) S-wave
0
500
1000
1500
2000
2500
3000
0.10 0.15 0.20 0.25 0.30 0.35 0.40
dry
/
eq
(
sat
) density (MPa)
Porosity
Dry / Equivalent (saturated) density of the medium vs. Porosity of
the medium
Dry
Equivalent
0
5
10
15
20
25
30
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Percentage difference between
dry
and
eq
(
sat
) of the medium (%)
Porosity
Percentage difference between dry and equivalent (saturated)
density of the medium vs. Porosity of the medium
Difference
Figure 5.12: Dry / Equivalent (saturated) density of the medium vs. Porosity of the
medium
129
0
500
1000
1500
2000
2500
3000
0.10 0.15 0.20 0.25 0.30 0.35 0.40
dry
/
eq
(
sat
) density (MPa)
Porosity
Dry / Equivalent (saturated) density of the medium vs. Porosity of
the medium
Dry
Equivalent
0
5
10
15
20
25
30
0.10 0.15 0.20 0.25 0.30 0.35 0.40
Percentage difference between
dry
and
eq
(
sat
) of the medium (%)
Porosity
Percentage difference between dry and equivalent (saturated)
density of the medium vs. Porosity of the medium
Difference
Figure 5.13: Percentage dierence between dry and equivalent (saturated) density
of the medium vs. Porosity of the medium
130
Chapter 6
Soil-Structure Interaction of Buildings
Nearby Train Tracks
6.1 Introduction:
Environmental issues related to vibrations induced by trains raised considerable con-
cerns over the past years. The development of several studies and models for the pre-
diction of the vibrations induced by this mean of transportation has been achieved due
to the signicant eorts apportioned by technical and scientic communities. These
studies explored how trains movements constitute a source of vibration, examined the
way energy propagates in the medium surrounding the train tracks, and how close struc-
tures are aected by such radiation of waves. However, the majority of these studies
concentrated on high-speed trains due to many reasons. The two major ones are: rst,
high{speed trains are becoming increasingly popular and reliable source of transporta-
tion in many major parts of the word. Second, the energy produced by these trains
are noticeable specially for trains with speeds reaching the Rayleigh wave speed of the
mediums they pass through. For the latter, this behavior is accompanied with large
131
amplitudes of vibrations that can be felt not only close tracks but at far distances as
well. In some extreme cases, specially for high{speed trains on when the speed of the
train is larger than the shear wave velocity of the medium, a form of a \shock wave" is
produced which carries a considerable amount of energy that can cause damage to near
by structures. A literature review of train induced vibrations was introduced earlier in
section (1.2.2). The waves generated by trains are several orders of magnitude lower
than that caused by an earthquake. Therefore, the main focus of this chapter is not
related to the structural damage that an incident wave can cause, but rather, related to
the vibration level of the structures near the train track and the eects these vibrations
have on the occupants as discussed in section (1.2.2). Due to the fact that the frequency
of excitation is higher for transit systems incident waves than those of earthquakes,
ex-
ible foundation analysis must be used. The assumption of large rigid foundation will
lter out the incoming high frequency waves.
An amplied vibration can take place regardless of the order of source vibration
magnitude. One reason is due to the fact that the structures are in a close proximity to
the train tracks as shown in the schematic gure (6.1). Another reasonable reason for
larger than usual vibration level is that caused by resonance phenomena of the structure.
Lastly, vibration levels are in
uenced by the soil stratication since wave propagation
in layered media depend on frequency (dispersive wave propagation).
In this chapter, the soil{structure interaction (SSI) between the foundation of nearby
structures and surrounding soil will be studied due to a vibration source caused by a
moving conventional train with a maximum speed of 120 km/hr over stratied media.
From observations of occupants of many residential housing and buildings, the vibrations
were reported higher on upper levels, consistent with the suggestion that (SSI) can play
a role in transit vibration analysis. Three concrete frames with dierent heights will be
examined to identify which building conguration is more vulnerable to train induced
132
ground vibration.
Figure 6.1: Schematic of how the buildings are in a close proximity of the train
tracks
6.2 Superstructure Formulation
It is proposed to analyze a structure with multiple footings using a substructures
approach. That will allow the two major components, (a) the structure and (b) the
system of footings to be analyzed independently, and later, combined for an interaction
analysis. This method is feasible for all linear problems and for this particular case it is
a good application because of the complexity of the problem.
Rearranging the node numbers of the superstructure model such that the degrees of
freedom for the building and those at the foundation are separated so that the equation
of motion for the superstructure can be written in a partition matrix equation of the
form:
2
6
4
[M
bb
] [M
bf
]
[M
fb
] [M
ff
]
3
7
5
8
>
<
>
:
f x
b
g
f x
f
g
9
>
=
>
;
+
2
6
4
[C
bb
] [C
bf
]
[C
fb
] [C
ff
]
3
7
5
8
>
<
>
:
f _ x
b
g
f _ x
f
g
9
>
=
>
;
+
2
6
4
[K
bb
] [K
bf
]
[K
fb
] [K
ff
]
3
7
5
8
>
<
>
:
fx
b
g
fx
f
g
9
>
=
>
;
=
8
>
<
>
:
f0g
ff
f
g
9
>
=
>
;
(6.1)
in which the subscribesb andf refer to degrees of freedom assigned to the building and
133
to the foundation, respectively. Noting that, if the base nodes are xed in the nite
element model, all the matrix elements with a subscript off would be eliminated in the
initial phase to retain the foundation degree of freedom. It is proposed that ctitious
springs be placed at a the base nodes to stabilize the structure, and then these ctitious
springs can be removed to obtain the matrices required for this procedure.
Using a Fourier Transformation, the equation of motion can be written in the fre-
quency domain as:
2
6
4
[Z
bb
(!)] [Z
bf
(!)]
[Z
fb
(!)] [Z
ff
(!)]
3
7
5
8
>
<
>
:
fX
b
(!)g
fX
f
(!)g
9
>
=
>
;
=
8
>
<
>
:
f0g
fF
f
(!)g
9
>
=
>
;
(6.2)
in which X
b
(!), X
f
(!) and F
f
(!) are Fourier Transforms of x
b
(t), x
f
(t) and f
f
(t) of
equation (6.1), respectively, and the sub-matrices,
Z
ii
(!)
, are dened as:
Z
ii
(!)
=!
2
M
ii
+i!
C
ii
+
K
ii
(6.3)
A separation procedure can now be implemented by dividing equation (6.2) into two
smaller matrix equations as:
Z
bb
~
X
b
=
Z
bf
~
X
f
(6.4)
Z
ff
~
X
f
=
~
F
f
Z
fb
~
X
b
(6.5)
in which the vector notation
~
X is used interchangeably with an earlier notation of
X
.
Proceeding now to solve equation (6.4) by inverting the matrix,
Z
bb
, and the har-
monic displacement
~
X
b
can be expressed in terms of
~
X
f
as:
~
X
b
=
Z
bb
1
Z
bf
~
X
f
(6.6)
Substituting
~
X
b
from equation (6.6) into equation (6.5) yields a matrix equation with
134
the unknown
~
X
f
as:
Z
ff
~
X
f
=
~
F
f
+
Z
fb
Z
bb
1
Z
bf
~
X
f
(6.7)
Gathering the terms involving
~
X
f
on the left side as:
Z
ff
Z
fb
Z
bb
1
Z
bf
~
X
f
=
~
F
f
(6.8)
and the solution for unknown
~
X
f
can be expressed as:
~
X
f
=
Z
ff
Z
fb
Z
bb
1
Z
bf
1
~
F
f
(6.9)
After the motion at the foundation level,
~
X
f
, is determined, the motion on the
building can be computed using equation (6.6).
The solution procedure above has to be performed for each frequency ! and the
inversion for the large superstructural matrix
Z
bb
can require a large number of
oating
point operations. A more ecient method is to use modal superposition of the xed-base
modes of the building since [Z
bb
] represents the impedance matrix of a building with its
base xed.
First, dene the matrix
with the mass normalized eigenfunctions of
K
bb
as
column vectors, i.e.,
having the properties,
T
M
bb
=
I
(6.10)
and
T
K
bb
=
diag(!
2
i
)
(6.11)
in which !
i
are the natural frequencies of the building.
Using the matrix
, the dynamic matrix
Z
bb
(!)
can be diagonalized as:
T
Z
bb
(!)
=
D(!)
(6.12)
135
with the diagonal elements of
D(!)
are dened as:
D
ii
(!) =
!
2
i
!
2
+ 2i!!
i
i
(6.13)
The parameter,
i
, in equation (6.13) is the modal damping factor assigned to the
i-th mode. For reinforced concrete structures,
i
= 0:02 is a typical value.
To implement the modal analysis, express the motion of the superstructure,
~
X
b
, as
a linear combination of the modal ordinate ~ as:
~
X
b
=
~ (6.14)
and its substitution into equation (6.4) and the subsequent pre-multiplication of the
equation by
T
yields:
T
Z
bb
~ =
T
Z
bf
~
X
f
(6.15)
D(!)
~ =
T
Z
bf
~
X
f
Since the matrix on the left side of of equation (6.15) is now diagonal, the solution of
the modal ordinate can then be obtained by easily inverting the diagonal matrix
D(!)
as:
~ =
D(!)
1
T
Z
bf
~
X
f
(6.16)
Using equation (6.16), the motion in the superstructure can be expressed in terms
of the foundation motion as:
~
X
b
=
~ =
D(!)
1
T
Z
bf
~
X
f
(6.17)
and the subsequent substitution into equation (6.5) yields:
Z
ff
~
X
f
=
~
F
f
+
Z
bf
D(!)
1
T
Z
bf
~
X
f
(6.18)
136
Introduce the modal participation factor
as:
=
T
Z
bf
(6.19)
Equation (6.18) can now be simplied to:
Z
ff
T
D(!)
1
~
X
f
=
~
F
f
(6.20)
and the solution for
~
X
f
follows easily. The computation eciency of equation (6.20)
should be superior to equation (6.8) since the modal matrices
and
are constant
throughout the spectrum of frequencies and the diagonal matrix
D(!)
can be inverted
easily. Also, the number of modes used in the analysis can be signicantly reduced from
the total number of degrees of freedom of the superstructure.
6.3 A Simplied Model
The eects of the foundation stiness, to couple with that of the superstructure as
described in section (6.2), is known in soil-structure interaction circle as the impedance
matrix. Analytically, using the continuum mechanics approach, the interaction of the
foundation mats can be related using the Green's Functions of the layered soil medium,
where a load, or a disturbance, at one location can in
uence the motion of another
location. This relationship is established as:
~ u(~ r
o
) =
G(~ r
o
j~ r
p
)
~
P (~ r
p
) (6.21)
in which ~ u is a column vector which contains the x, y and z components of harmonic
displacement at observation point~ r
o
,
~
P contains thex,y andz components of harmonic
point loads at source point~ r
p
, and
G(~ r
o
j~ r
p
)
is the 33 Green's Functions matrix. The
values of
G
are complex in the frequency domain.
Using the principle of superposition, the displacement vector~ u can be that generated
137
by a distributed load over a
at surface area S as:
~ u(~ r
o
) =
Z
S
G(~ r
o
j~ r
p
)
~ p(~ r
p
)dS (6.22)
in which ~ p is a distributed load. Equation (6.22) is correct for a load on the surface of
the soil medium, it would be more complicated if loads are also beneath the surface. For
the development targeted in this chapter, surface foundations can provide many answers
without dealing with the issues of foundation embedment.
The solution to the integral equation (6.22) was rst attempted by Kobori (1966)
and later in an embedded version by Ohsaki (1973). Wong (1975) and subsequent work,
Wong and Luco (1976) were able to eliminate singularities of the Green's Functions and
transform the integral equation into a complex matrix equation.
Once the distributed load,~ p, is determined, it represents the tractions under the rigid
footing. The summation of the tractions, multiplied the subregions' areas and proper
moment arms, will yield the forces and moments required to drive the rigid footing
into six rigid body motions. The relationship between the normalized forces and the
normalized displacements can be expressed in the form of a matrix product,
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
F
x
F
y
F
z
M
x
=b
M
y
=b
M
z
=b
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
= [K
ss
(!)]
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
x
y
z
b
x
b
y
b
z
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
(6.23)
In equation (6.23), K
ss
is known as the complex impedance matrix for the footing,
the real part of the matrix elements represents the stiness and the inertia properties of
the soil medium and the imaginary part is equivalent to the damping of the soil medium,
including material damping and radiation damping. The latter damping is an energy
138
loss generated by the spreading of waves away from the footing into the innite soil
medium. The displacement vector is normalized by a length, b, a selected dimension of
the footing.
x
,
y
and
z
are the harmonic displacements in thex,y andz directions,
respectively, whileM
x
andM
y
are the rocking moments andM
z
is the torsional moment.
The impedance matrix concept was used in many linear earthquake engineering anal-
ysis. For the present application to transit problems, there are many footings supporting
the columns of a structure. This type of foundation is
exible and each footing could
respond to the ground motion independently; unlike the massive rigid foundation de-
signed for nuclear reactor containment buildings. Since there are many footings, the
impedance matrix is of the size 6N 6N, in which N is the number of footings.
The main dierence in the present formulation away from the traditional earthquake
engineering formulation is how the driving forces and moments are obtained. From a
traveling train, the waves travel outward into the soil medium and the motion directly
under a footing cause forces and moments on the footing and they cause the superstruc-
ture to move. For the simplicity of formulation, the normalized driving force vector (
~
F
)
is obtained by assuming the footing is held motionless.
One method for obtaining the driving force vector is solving the large matrix equa-
tion at the same time as when the rigid motion is prescribed for the calculation of the
impedance functions. But the above method requires advance planning for all incoming
wave patterns and is impractical for the present application because there are many
parameters which deals with train speed, train conguration and source-to-footing dis-
tances. Using the Betti-Rayleigh reciprocal theorem, the driving forces can be obtained
using the tractions obtained for the impedance functions, i.e., rigid body input motion.
The scalar product of the traction vectors and the ground motion directly under the
footing would yield the desired normalized driving force vector.
139
Mathematically, the driving force vector can be calculated as
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
F
x
F
y
F
z
M
x
=b
M
y
=b
M
z
=b
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
=
m
X
i=1
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
T
x
(~ r
i
;
x
) T
y
(~ r
i
;
x
) T
z
(~ r
i
;
x
)
T
x
(~ r
i
;
y
) T
y
(~ r
i
;
y
) T
z
(~ r
i
;
y
)
T
x
(~ r
i
;
z
) T
y
(~ r
i
;
z
) T
z
(~ r
i
;
z
)
T
x
(~ r
i
;b
x
) T
y
(~ r
i
;b
x
) T
z
(~ r
i
;b
x
)
T
x
(~ r
i
;b
y
) T
y
(~ r
i
;b
y
) T
z
(~ r
i
;b
y
)
T
x
(~ r
i
;b
z
) T
y
(~ r
i
;b
z
) T
z
(~ r
i
;b
z
)
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
8
>
>
>
>
<
>
>
>
>
:
u
x
(~ r
i
)
u
y
(~ r
i
)
u
z
(~ r
i
)
9
>
>
>
>
=
>
>
>
>
;
(6.24)
in whichi is the index of them subregions for the footing model and~ r
i
is position vector
of the centroid of that subregion. This formulation is practical because the traction
vectors for each footing and for each frequency could be stored in a large binary le and
then the driving force vector could be prepared readily by changing the ground motion
to the case of interest.
6.4 Dynamic Soil-Structure Interaction
Consider a
exible superstructure support on a
exible foundation system subjected
to ground excitation and to forces acting on the structure. The equation of motion for
the structure-foundation system can be written, in the frequency domain, in the form:
2
6
4
[Z
bb
(!)] [Z
bf
(!)]
[Z
fb
(!)] [Z
ff
(!)]
3
7
5
8
>
<
>
:
fX
b
(!)g
fX
f
(!)g
9
>
=
>
;
=
8
>
<
>
:
fF
e
b
(!)g
F
e
f
(!)
9
>
=
>
;
8
>
<
>
:
f0g
F
s
f
(!)
9
>
=
>
;
(6.25)
in which the sub-matrices [Z (!)] were dened in equation (6.3). The force vectors
on the right hand side of the equation (6.25) arefF
e
(!)g, the external forces on the
structure andfF
s
(!)g, the generalize force (forces and moments) that the soil exerts
on the foundation. The subscripts b and f represent the nodes on the building and
on the foundation, respectively. For the present application, the external force vector,
140
fF
e
(!)g, is zero as the forcing functions are all provided by train vibrations.
The force vector,fF
s
(!)g, can be written in the form:
F
s
f
(!)
= [K
ss
]
fU
f
g
U
f
(6.26)
where [K
ss
] is the impedance matrix for the soil-foundation system and
U
f
is the
foundation input motion. Physically,
U
f
is the response of the set of massless footings
to the free eld motion. If the size of the footings are small compared to the incident
wavelength,
U
f
is basically the same as the free eld motion. For higher frequency
waves, there is kinematic interaction between the rigid footings and the incident wave
and the input motion is typically less than the free eld motion. But the reduced
translational response leads to increased rotational response.
To be consistent with section (6.3), the driving force vector,
F
f
(!)
, can be derived
from equation (6.26) as:
F
f
(!)
= [K
ss
]
U
f
(6.27)
or it could be calculated using rigid body tractions as shown in equation (6.24).
Now, by setting the external forces,fF
e
b
(!)g and
F
e
f
(!)
, to zero, the interaction
equation of motion can be written as:
2
6
4
[Z
bb
(!)] [Z
bf
(!)]
[Z
fb
(!)] [Z
ff
(!)] + [K
ss
(!)]
3
7
5
8
>
<
>
:
fX
b
(!)g
fX
f
(!)g
9
>
=
>
;
=
8
>
<
>
:
f0g
F
f
(!)
9
>
=
>
;
(6.28)
With the addition of [K
ss
(!)] to the free-free structure represented by the [Z (!)]
matrix, the matrix on the left hand side is no longer singular because of the additional
stiness of the foundation. The solution of equation (6.28) must be solved for each
frequency in the spectrum, a Fourier Transform then performed numerically to obtain
the nodal responses in time domain.
141
6.5 Analysis Setup:
6.5.1 Soil Proles:
To demonstrate the SSI eects role in our analysis, two soil proles are introduced
as in gure (6.2). These proles are to show the response when a structure is placed on
a soft soil layer on top of a half-space and on a hard hard layer on top of the half-space.
For simplicity, we will refer to these proles as soft and hard cases respectively.
Soft
Hard
Figure 6.2: Soft and hard cases soil proles to be used in the analysis of SSI
6.5.2 Displacement Components:
For a full understanding of the behavior of the structures under considerations, three
displacements components are selected for analysis, vertical, transverse, and parallel.
The vertical component is perpendicular to the train track acting \vertically". The
transverse component is perpendicular to the train track along the horizontal distance
to the point of interest. Finally the parallel is component is, as the name indicate, is
parallel to the train track at the point of interest. A better representation can be found
in gure(6.3).
142
Vertical
Parallel
Transverse
Point of interest
Figure 6.3: Three displacement components: Vertical, Transverse, and Parallel
6.5.3 Distance from Structures:
The train track (or the loading source), is placed at dierent distances from the
structure under analysis. This distance is dened as the \Transverse" distance from the
loading source to the center of the building under analysis as shown in gure(6.4).
Distance to
source
2
nd
floor plan
25
26
27
28
Distance to
source
Building
Train
1
5
9
Distance to
source
11
Distance to source
27
43
59
Figure 6.4: Schematic showing the denition of \distance to source"
6.5.4 SAP2000 and Material Denitions:
The nite element analysis software (SAP2000 v.16) by Computers & Structures.
Inc, is used to produce the stiness and mass matrices of the free-free models to be
analyzed. The material used in all models is normal weight concrete (unit weight = 24
kN/m
3
) and its behavior is assumed to be isotropic. Using ACI-318M (2011) section
8.5.1, the modulus of elasticity of concrete (E
c
) is calculated byE
c
= 4700
p
f
c
0
wheref
c
0
is the compressive strength of concrete in MPa. It is determined that E
c
= 24900 MPa
143
when f
c
0
= 28 MPa (McCormac & Brown, 2013). Following ACI-318M (2011) section
10.10.4.1, the eective moment of inertia of the beams is calculated as I
beam
= 0:35I
g
,
for columns asI
column
= 0:70I
g
, and for solid slabs asI
slab
= 0:25I
g
whereI
g
is the gross
moment of inertia of the reinforced concrete section. Damping was chosen to be 2%.
But SSI adds more damping eect to the building from radiation damping, vibration
energy that dissipates through the innite soil medium due to the spreading of waves
outward and never return.
6.6 Analysis Models:
6.6.1 First Model:
The rst model to be analyzed, as shown in gure (6.5), is a two-story-single-bay
concrete structure that is 15 m by 15 m in plan dimensions with a typical story height
taken to be 3 m. All
oor slabs are solid with a thickness of 0.15 m where the beams
have a depth of 0.70 m and a width of 0.30 m. All
oors are supported by 4 square
columns with cross section of 0.30 m by 0.30 m.
The number of footings is 4 for Model (1). Each footing is allowed to move indepen-
dently in six degrees of freedom: Two horizontal, one vertical translation, two rocking
and one torsional rotation. Each footing of size 2 m by 2 m is assumed to be rigid but
the entire foundation system, as a whole, is
exible as each footing is allowed to move
independently. The superstructure adds the stiness to the foundation system to keep
the footings together.
144
Figure 6.5: First model: two-story-single-bay concrete structure (a) 3D view, (b)
Cross-sectional details
6.6.1.1 Results for the First Model:
Soft Case:
For the rst model, the loading source (the train) had a speed of 75 km/h and was
placed at 15 and 60 meters away from the building. The analysis is based on results of
the three displacement components of \exterior" nodes 1, 5, and 9 as shown in gure
(6.6).
Distance to
source
Building
Train
Distance to
source
2
nd
floor plan
25
26
27
28
Train Direction
11
Distance to source
27
43
59
1
5
9
Distance to
source
Figure 6.6: Schematic showing the \exterior" nodes 1, 5, and 9 locations relative to
loading source
145
From gure(6.7), at a distance 15 m from the source, it can be observed that at
low frequency (up to 9 Hz), all three nodes displace vertically with the same amplitude
where as at frequency of 12.5 Hz node 9 (the second
oor) reaches a maximum amplitude
followed by the rst and the ground
oor nodes. The same behavior exists at a distance
of 60 m away from the source, as shown in gure (6.8), however, with smaller amplitudes
due to the loss of energy in the medium since the loading source is at a higher distance
from the points of interest. As for the transverse displacement component, it can be
seen from gures (6.9) and (6.10) regardless of the frequency range, the ground
oor's
response is negligible compared to the other
oors. Plus, the maximum amplitude for
the transverse displacement takes place around 3 Hz at the rst
oor (node 5) followed
by the the second
oor (node 9). The parallel displacement components shown in gures
(6.11) and (6.12) show that at a frequency of 1.5 Hz the second
oor (node 9) has the
largest parallel displacement component. Moreover, this displacement amplitude is not
aected by the distance from the loading source which suggest minimal loss of energy
from the source.
146
0.0E+00
1.0E-02
2.0E-02
3.0E-02
4.0E-02
5.0E-02
6.0E-02
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component at different levels of the structure
vs. Frequency
(Soft layer / 75 km/hr / 15 m away from source)
Ground floor
First floor
Second floor
Figure 6.7: Vertical displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 15 m away from source)
0.0E+00
1.0E-02
2.0E-02
3.0E-02
4.0E-02
5.0E-02
6.0E-02
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component at different levels of the structure
vs. Frequency
(Soft layer / 75 km/hr / 60 m away from source)
Ground floor
First floor
Second floor
Figure 6.8: Vertical displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 60 m away from source)
147
0.0E+00
1.0E-04
2.0E-04
3.0E-04
4.0E-04
5.0E-04
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse to the track displacement component at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 15 m away from source)
Ground floor
First floor
Second floor
Figure 6.9: Transverse displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 15 m away from source)
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse to the track displacement component at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 60 m away from source)
Ground floor
First floor
Second floor
Figure 6.10: Transverse displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 60 m away from source)
148
0.0E+00
5.0E-03
1.0E-02
1.5E-02
2.0E-02
2.5E-02
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel to the track displacement component at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 15 m away from source)
Ground floor
First floor
Second floor
Figure 6.11: Parallel displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 15 m away from source)
0.0E+00
5.0E-03
1.0E-02
1.5E-02
2.0E-02
2.5E-02
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel to the track displacement component at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 60 m away from source)
Ground floor
First floor
Second floor
Figure 6.12: Parallel displacement component at dierent levels of model (1) vs.
Frequency (soft layer / 75 km/h / 60 m away from source)
149
Hard Case:
In general, the same observations mentioned in the case of soft layer can be obtained
by analyzing the response in the case of hard soil layer with some exceptions. From
gures (6.13) and (6.14), it can be seen that the peaks for the vertical displacement
component take place at a frequency of 15 Hz. As for amplitudes, it is observed that the
vertical displacement amplitudes are higher in the case of soft layer. Although the values
of the displacements are small compared to the vertical ones, for a close distance, the
transverse displacement component peaks at 1 Hz for the second
oor (node 9) where
for a far distance the rst
oor (node 5) transverse displacement shows a peak at 3.75
Hz along with peak for the second
oor at 1 Hz as shown in gures (6.15) and (6.16).
Unlike the soft layer case, the parallel displacement component peaks for the rst
oor
(node 5) at a frequency of 3.75 Hz as shown in gures (6.17) and (6.18).
150
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
3.0E-03
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component at different levels of the structure
vs. Frequency
(Hard layer / 75 km/hr / 15 m away from source)
Ground floor
First floor
Second floor
Figure 6.13: Vertical displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 15 m away from source)
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
3.0E-03
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component at different levels of the structure
vs. Frequency
(Hard layer / 75 km/hr / 60 m away from source)
Ground floor
First floor
Second floor
Figure 6.14: Vertical displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 60 m away from source)
151
0.0E+00
1.0E-05
2.0E-05
3.0E-05
4.0E-05
5.0E-05
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse to the track displacement component at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 15 m away from source)
Ground floor
First floor
Second floor
Figure 6.15: Transverse displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 15 m away from source)
0.0E+00
1.0E-05
2.0E-05
3.0E-05
4.0E-05
5.0E-05
6.0E-05
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse to the track displacement component at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 60 m away from source)
Ground floor
First floor
Second floor
Figure 6.16: Transverse displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 60 m away from source)
152
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
1.4E-03
1.6E-03
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel to the track displacement component at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 15 m away from source)
Ground floor
First floor
Second floor
Figure 6.17: Parallel displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 15 m away from source)
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
1.4E-03
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel to the track displacement component at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 60 m away from source)
Ground floor
First floor
Second floor
Figure 6.18: Parallel displacement component at dierent levels of model (1) vs.
Frequency (hard layer / 75 km/h / 60 m away from source)
153
Generally, looking at both soft and hard layers cases for model (1), one can conclude
that occupant comfort is aected mainly by vertical displacement component followed
by parallel displacement component then by the transverse displacement components.
What is to note though, at high frequency all components show minimal response to the
loading source regardless of the source distance to the point of interest.
6.6.2 Second Model:
The second model to be analyzed, as shown in gure (6.19), is a three-story concrete
structure that is 18 m by 18 m in plan dimensions with a typical story height taken to
be 3 m. All
oor slabs are solid with a thickness of 0.10 m where the beams have a
depth of 0.40 m and a width of 0.20 m. All
oors are supported by 16 square columns
per
oor with cross section of 0.30 m by 0.30 m.
Model (2) has 16 footings. Each footing is allowed to move independently in six de-
grees of freedom: Two horizontal, one vertical translation, two rocking and one torsional
rotation. Each footing of size 2 m by 2 m is assumed to be rigid but the entire foundation
system, as a whole, is
exible as each footing is allowed to move independently. The
superstructure adds the stiness to the foundation system to keep the footings together.
154
Figure 6.19: Second model: three-story concrete structure model (a) 3D view, (b)
Cross-sectional details
6.6.2.1 Results for the Second Model:
Soft Case:
For the second model, the loading source (the train) had a speed of 75 km/h and
was placed at 20 and 40 meters away from the building. The analysis is based on results
of the three displacement components of interior nodes 11, 27, 43, and 59 as shown in
gure (6.20).
Distance to
source
Building
Train
Distance to
source
2
nd
floor plan
25
26
27
28
Train Direction
11
Distance to source
27
43
59
1
5
9
Distance to
source
Figure 6.20: Schematic showing the \interior" nodes 11, 27, 43, and 59 locations relative
to loading source
155
From gures(6.21) and (6.22), at a distance 20 m from the source, it can be observed
that at low frequency (up to 9 Hz), the third
oor (node 59) displace slight higher
than the other
oors however, this displacement becomes greater at a frequency of 15
Hz followed by nodes 43, 27, then 11 respectively. Smaller amplitudes, yet similar
behavior, are observed in gure (6.22) due to the loss of energy in the medium since the
loading source is at a higher distance from the points of interest. As for the transverse
displacement component, it can be seen from gures (6.23) and (6.24) regardless of the
frequency range, the ground
oor's response is negligible compared to the other
oors.
Plus, the maximum amplitude for the transverse displacement takes place around 3 Hz at
the rst
oor (node 27) followed by the the second
oor (node 43). What is noticeable
that the transverse displacement component is very minor in close proximity to the
load and become prominent at further distances. The parallel displacement components
shown in gures (6.25) and (6.26) show that at a frequency of 3.25 Hz the second
oor
(node 27) has the largest parallel displacement component. Moreover, this displacement
amplitude is not aected by the distance from the loading source which suggest minimal
loss of energy from the source.
Figures (6.28) and (6.28) shows the vertical displacement components amplitudes for
four nodes on the second level of model (2) as shown in the schematic gure (6.27). It
is observed that in the low frequency range (1 Hz to 9 Hz) all nodes displace vertically
with the same amplitude. At a frequency of 15 Hz, nodes 26 pairs with node 27 and
node 25 pairs with node 28 move together to local maximum displacement. All nodes
displace dierently at frequency 20.5 Hz which represents a higher vibrational mode.
156
0.0E+00
5.0E-03
1.0E-02
1.5E-02
2.0E-02
2.5E-02
3.0E-02
3.5E-02
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component (interior node) at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 20 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.21: Vertical displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 20 m away from source)
0.0E+00
1.0E-02
2.0E-02
3.0E-02
4.0E-02
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component (interior node) at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 40 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.22: Vertical displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 40 m away from source)
157
0.0E+00
4.0E-05
8.0E-05
1.2E-04
1.6E-04
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse displacement component (interior node) at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 20 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.23: Transverse displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 20 m away from source)
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
3.0E-04
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse displacement component (interior node) at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 40 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.24: Transverse displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 40 m away from source)
158
0.0E+00
2.0E-02
4.0E-02
6.0E-02
8.0E-02
1.0E-01
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel displacement component (interior node) at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 20 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.25: Parallel displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 20 m away from source)
0.0E+00
2.0E-02
4.0E-02
6.0E-02
8.0E-02
1.0E-01
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel displacement component (interior node) at different levels of the
structure vs. Frequency
(Soft layer / 75 km/hr / 40 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.26: Parallel displacement component at dierent levels of model (2) vs.
Frequency (soft layer / 75 km/h / 40 m away from source)
159
Distance to
source
Building
Train
1
5
9
Distance to
source
11
Distance to source
27
43
59
Distance to
source
2
nd
floor plan
25
26
27
28
Train Direction
Figure 6.27: Schematic showing location of nodes 25, 26, 27, and 28 on the second
oor
plan
0.0E+00
4.0E-03
8.0E-03
1.2E-02
1.6E-02
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vertical displacement component at 2nd floor vs. Frequency
(Soft layer / 75 km/hr / 20 m away from source)
Node 25
Node 26
Node 27
Node 28
Figure 6.28: Vertical displacement component at second
oor of model (2) vs. Frequency
(soft layer / 75 km/h / 20 m away from source)
160
0.0E+00
4.0E-03
8.0E-03
1.2E-02
1.6E-02
2.0E-02
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vertical displacement component at 2nd floor vs. Frequency
(Soft layer / 75 km/hr / 40 m away from source)
Node 25
Node 26
Node 27
Node 28
Figure 6.29: Vertical displacement component at second
oor of model (2) vs. Frequency
(soft layer / 75 km/h / 40 m away from source)
Hard Case:
Again, the same observations mentioned in the case of soft layer can be obtained by
analyzing the response in the case of hard soil layer with some exceptions. From gures
(6.30) and (6.31), it can be seen that the peaks for the vertical displacement component
take place at a frequency of 5 Hz. As for amplitudes, it is observed that the vertical
displacement amplitudes are higher in the case of soft layer. Although the values of
the displacements are small compared to the vertical ones, for a close and far distances,
the transverse displacement component peaks at 6.5 Hz for the second
oor (node 27)
with smaller amplitude in the case of further distance from the loading source as shown
in gures (6.32) and (6.33). Similar to the soft layer case, the parallel displacement
component peaks for the rst
oor (node 27) at a frequency of 3.75 Hz with smaller
amplitudes compared to the soft layer case as shown in gures (6.34) and (6.35).
161
Figures (6.36) and (6.36) shows the vertical displacement components amplitudes for
four nodes on the second level of model (2) as shown in the schematic gure (6.27). It
is observed that in the low frequency range (1 Hz to 9 Hz) all nodes displace vertically
with the same amplitude. At a frequency of 12.5 Hz, nodes 26 pairs with node 27 and
node 25 pairs with node 28 move together to local maximum displacement. All nodes
displace dierently at frequency 23 Hz which represents a higher vibrational mode.
162
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component (interior node) at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 20 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.30: Vertical displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 20 m away from source)
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vetrical displacement component (interior node) at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 40 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.31: Vertical displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 40 m away from source)
163
0.0E+00
2.0E-06
4.0E-06
6.0E-06
8.0E-06
1.0E-05
1.2E-05
1.4E-05
1.6E-05
1.8E-05
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse displacement component (interior node) at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 20 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.32: Transverse displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 20 m away from source)
0.0E+00
5.0E-06
1.0E-05
1.5E-05
2.0E-05
2.5E-05
3.0E-05
3.5E-05
0 5 10 15 20 25 30 35 40 45 50
Transverse Displacement Component (m)
Frequency (Hz)
Transverse displacement component (interior node) at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 40 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.33: Transverse displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 40 m away from source)
164
0.0E+00
1.0E-03
2.0E-03
3.0E-03
4.0E-03
5.0E-03
6.0E-03
7.0E-03
8.0E-03
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel displacement component (interior node) at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 20 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.34: Parallel displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 20 m away from source)
0.0E+00
1.0E-03
2.0E-03
3.0E-03
4.0E-03
5.0E-03
6.0E-03
7.0E-03
0 5 10 15 20 25 30 35 40 45 50
Parallel Displacement Component (m)
Frequency (Hz)
Parallel displacement component (interior node) at different levels of the
structure vs. Frequency
(Hard layer / 75 km/hr / 40 m away from source)
Ground floor
First floor
Second floor
Third floor
Figure 6.35: Parallel displacement component at dierent levels of model (2) vs.
Frequency (hard layer / 75 km/h / 40 m away from source)
165
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vertical displacement component at 2nd floor vs. Frequency
(Hard layer / 75 km/hr / 20 m away from source)
Node 25
Node 26
Node 27
Node 28
Figure 6.36: Vertical displacement component at second
oor of model (2) vs. Frequency
(hard layer / 75 km/h / 20 m away from source)
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
0 5 10 15 20 25 30 35 40 45 50
Vertical Displacement Component (m)
Frequency (Hz)
Vertical displacement component at 2nd floor vs. Frequency
(Hard layer / 75 km/hr / 40 m away from source)
Node 25
Node 26
Node 27
Node 28
Figure 6.37: Vertical displacement component at second
oor of model (2) vs. Frequency
(hard layer / 75 km/h / 40 m away from source)
166
6.6.3 Third Model:
The third model to be analyzed, as shown in gure (6.38), is a three-story \L-Shape"
concrete structure having the same column spacing as the second and third model with
a typical story height taken to be 3 m. All
oor slabs are solid with a thickness of 0.10 m
where the beams have a depth of 0.40 m and a width of 0.20 m. All
oors are supported
by 12 square columns per
oor with cross section of 0.30 m by 0.30
Model (3) has 12 footings. Each footing is allowed to move independently in six de-
grees of freedom: Two horizontal, one vertical translation, two rocking and one torsional
rotation. Each footing of size 2 m by 2 m is assumed to be rigid but the entire foundation
system, as a whole, is
exible as each footing is allowed to move independently. The
superstructure adds the stiness to the foundation system to keep the footings together.
Figure 6.38: Third model: three-story concrete structure model (a) 3D view, (b)
Cross-sectional details
167
6.6.3.1 Results for the Third Model:
Displacement Components Time History:
Distance to
source
Plan View
Ground Floor
7
1
12
Train Direction
Distance to
source
Plan View
Ground Floor
7
1
12
Train Direction
Distance to
source
Plan View
Third Floor
43
37
48
Train Direction
Distance to
source
Plan View
Third Floor
40
37
48
Train Direction
Figure 6.39: Plan view of the ground
oor of model (3) showing nodes 1, 7, and 12 locations
To examine the response of a structure over a period of time during and after the
application of excitation, a full time history is presented. Time history is a great tool
to visualize the behavior of the component under consideration, displacements in our
case. It can provide some information regarding the response in a more intuitive way
compared to the analysis in the frequency domain. For model (3), time history analysis
is presented here replacing the frequency domain analysis provided in sections (6.6.1.1)
and (6.6.2.1).
The train in this analysis will be placed at a distance of 80 m away from the center of
the structure as shown in gure (6.39) with a speed of 75 km/h. From sections (6.6.1.1)
and (6.6.2.1), it was shown that the soft soil prole will provide more vibrations in the
structure due to the Soil-Structure-Interaction phenomena. Therefore, the time history
analysis presented in this section will consider the soft soil prole shown in gure (6.2).
168
The vertical displacement of the nodes labeled 1, 7, and 12 in gure (6.39) and
the nodes directly above them in the upper
oors will be examined. It can be seen
that in gure (6.40a) the large amplitudes of the vertical displacement take place in
the negative region. The reason being that the excitation of the train in modeled as a
moving load acting in the negative vertical direction according to our sign convention.
Plus, from gure (6.40a) one can see the dierent arrival times of each vibration due to
the dierent locations of the nodes under observation with respect to the moving load.
The time history for the upper
oors node in gures (6.40b) and (6.41) have a common
feature. The two \edge" nodes pair together in all upper
oor motion. The vertical
mode of vibration of the structure requires these \edge" nodes to move together.
It is observed that from gures (6.42) the higher the level of the structure, the higher
the vibration it will experience. This fact was attained previously in model (1) and model
(2) using frequency domain analysis. One more common feature among all gures is the
fact that vibration level is negligible after 7 seconds and that is due to the damping of
the structure and the fact that load has moved away from the source and has minimal
eect.
169
-7.50E-04
-5.00E-04
-2.50E-04
0.00E+00
2.50E-04
5.00E-04
7.50E-04
0 1 2 3 4 5 6 7 8 9 10
Vertical Displacement
Component (m)
Time (seconds)
Vertical displacement components for three nodes on the ground floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 1
Node 7
Node 12
-7.50E-04
-5.00E-04
-2.50E-04
0.00E+00
2.50E-04
5.00E-04
7.50E-04
0 1 2 3 4 5 6 7 8 9 10
Vertical Displacement
Component (m)
Time (seconds)
Vertical displacement components for three nodes on the first floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 13
Node 19
Node 24
(a)
(b)
Figure 6.40: Vertical displacement components for three nodes vs. Time at (a) Ground
oor and (b) First
oor
170
-7.50E-04
-5.00E-04
-2.50E-04
0.00E+00
2.50E-04
5.00E-04
7.50E-04
0 1 2 3 4 5 6 7 8 9 10
Vertical Displacement
Component (m)
Time (seconds)
Vertical displacement components for three nodes on the second floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 25
Node 31
Node 36
-7.50E-04
-5.00E-04
-2.50E-04
0.00E+00
2.50E-04
5.00E-04
7.50E-04
0 1 2 3 4 5 6 7 8 9 10
Vertical Displacement
Component (m)
Time (seconds)
Vertical displacement components for three nodes on the third floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 37
Node 43
Node 48
( a )
(b)
Figure 6.41: Vertical displacement components for three nodes vs. Time at (a) Second
oor and (b) Third
oor
171
-7.50E-04
-5.00E-04
-2.50E-04
0.00E+00
2.50E-04
5.00E-04
7.50E-04
0 1 2 3 4 5 6 7 8 9 10
Vertical Displacement Component
(m)
Time (seconds)
Vertical displacement components for same node location on different floors vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 19
Node 31
Node 43
-7.50E-04
-5.00E-04
-2.50E-04
0.00E+00
2.50E-04
5.00E-04
7.50E-04
0 1 2 3 4 5 6 7 8 9 10
Vertical Displacement Component
(m)
Time (seconds)
Vertical displacement components for same node location on different floors vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 24
Node 36
Node 48
( a )
(b)
Figure 6.42: Vertical displacement component for the same location on dierent
oor level vs. Time (a) Nodes 19, 31, and 43 and
(b) Nodes 24, 36, and 48
172
Distance to
source
Plan View
Ground Floor
7
1
12
Train Direction
Distance to
source
Plan View
Ground Floor
4
1
12
Train Direction
Distance to
source
Plan View
Third Floor
43
37
48
Train Direction
Distance to
source
Plan View
Third Floor
40
37
48
Train Direction
Figure 6.43: Plan view of the ground
oor of model (3) showing nodes 1, 4, and 12 locations
The transverse displacement of the nodes labeled 1, 4, and 12 in gure (6.43) and
the nodes directly above them in the upper
oors will be examined. It can be seen that
in gure (6.44a) the amplitudes of the transverse displacement is negligible compared to
the one in gures (6.44b) and (6.45). Moreover, in gure (6.44), we notice high frequency
of vibration between the second and the sixth seconds. That indicates higher modes of
vibrations are taking place for ground and rst
oor levels. On the other hand, and in
the same time interval for upper levels, we see that only fundamental frequencies being
excited as shown in gure (6.45).
Nodes 25 and 38 from gure (6.45a) move in the same direction where node 36 moves
in the other direction (180
out of phase) causing the structure to experience torsional
eect. This eect could have been concluded earlier with some level of diculty using
the domain frequency. Here we see one of the many advantages of using the time history
analysis. From gure (6.46), it can be seen that the torsional eect increases as we move
up to the upper levels as in gure (6.45b).
173
-1.00E-05
-5.00E-06
0.00E+00
5.00E-06
1.00E-05
0 2 4 6 8 10 12 14 16 18 20
Transverse Displacement
Component (m)
Time (seconds)
Transverse displacement components for three nodes on the ground floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 1
Node 4
Node 12
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
1.00E-03
0 2 4 6 8 10 12 14 16 18 20
Transverse Displacement
Component (m)
Time (seconds)
Transverse displacement components for three nodes on the first floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 13
Node 16
Node 24
(a)
(b)
Figure 6.44: Transverse displacement components for three nodes vs. Time (a) Ground
oor and (b) First
oor
174
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
1.00E-03
0 2 4 6 8 10 12 14 16 18 20
Transverse Displacement
Component (m)
Time (seconds)
Transverse displacement components for three nodes on the second floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 25
Node 28
Node 36
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
1.00E-03
0 2 4 6 8 10 12 14 16 18 20
Transverse Displacement
Component (m)
Time (seconds)
Transverse displacement components for three nodes on the third floor vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 37
Node 40
Node 48
(a)
(b)
Figure 6.45: Transverse displacement components for three nodes vs. Time at (a) Second
oor and (b) Third
oor
175
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
1.00E-03
0 1 2 3 4 5 6 7 8 9 10
Transverse Displacement
Component (m)
Time (seconds)
Transverse displacement components for same node location on different floors vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 4
Node 16
Node 28
Node 40
-1.00E-03
-5.00E-04
0.00E+00
5.00E-04
1.00E-03
0 1 2 3 4 5 6 7 8 9 10
Transverse Displacement
Component (m)
Time (seconds)
Transverse displacement components for same node location on different floors vs. Time
(Soft layer / 75 km/hr / 80 m away from source)
Node 7
Node 19
Node 31
Node 43
(a )
(b)
Figure 6.46: Transverse displacement component for the same location on dierent
oor level vs. Time (a) Nodes 4, 16, 28, and
40 and (b) Nodes 7, 19, 31, and 43
176
6.7 Discussion:
Looking at the results presented in the previous analysis for model (1) and (2), it
is seen that the vertical displacement component is the major one due to the nature
of the applied vertical load at the loading source. It is most critical when the source
is close to the receiver and decreases in amplitudes as we move away from the source.
The transverse displacement component shows very minor eect near the loading source,
however became more prominent at further distances from the point of interest. It shows
that at low distances, body waves dominate the response where the surface waves (like
Rayleigh waves) need more time to develop. The parallel component would have been
zero if the load is stationary because of the symmetry of the model. The fact the train
is in dierent location at dierent time causes phase shifts. We can still expect the
amplitude of the ground motion to be the same even though the phases are dierent,
but the added structure changed the situation.
When the same model is placed on dierent soil proles, in the soft soil case cause
larger amplication because the structure plays a larger role in the interaction. The
natural frequency of the building becomes lower because an additional softer spring
(soil) is added beneath the structure. For that reason, the shift in building resonant
frequency is larger for the soft soil than for the hard soil.
The frequency range of analysis was from 0 to 50 Hz. There are higher frequencies
for the train analysis but the SSI analysis is mostly dominated by the lower modes of
the superstructure. Most graphs generated showed a signicant drop-o after 20 Hz or
so. Per human observation, most of the higher frequency disturbance is from the sound
and the vibration of the structure is of the lower frequency range.
The results from time history analysis of model (3) not only conrmed the ndings
obtained from using the frequency domain analysis on models (1) and (2), but made
177
it easier to visualize the vibration patterns the nodes under consideration experience.
That lead to the clear demonstration of the torsional eects the structure experiences.
There are way too much data to decipher, component by component. But the end
result is that the superstructure plays a critical role in amplied vibration from transit
noise. The data from this chapter demonstrated, using simplied models, how waves
could enter the structure through the ground. It is to be expected that a massive high-
rise would be able to lter out all the high frequencies and that the low level of vibration
could be dampened. But most of the smaller residential building could be aected more
seriously.
Lastly, there is the interaction between nearby buildings which could be explored but
that is beyond the scope of this thesis. It could be assumed those eects are secondary.
178
Chapter 7
Summary and Future work
7.1 Summary:
In an eort to reduce the trac congestions in the streets of Kuwait, the Ministry
of Communications, through the Kuwait Overland Transport Union, raised the need
to develop a master plan for the implementation of a rapid transit system. To reduce
construction cost and time, the existing road networks will be used as the main guides
for the future transit lanes. This raised the need to study the eects of the vibration
caused by this mean of transportation on nearby structures and occupants. Moreover,
the water table in the area proposed to host the lanes is as shallow as 0.50 m in some
areas and as deep as 13.0 m. The depth of water table
uctuate depending mainly on
rainfall and irrigation process. Consequently, a thorough understanding of the eects of
the depth of this water table on the vibration characteristics is a must.
Due to the saturation of the soil with water, Biots theory of poroelasticity was
introduced in chapter (2) as the appropriate theory to be applied in the analysis of the
vibration caused by the transit system. All material constants and limitations of the
theory were discussed and new compact representation of the materials constants was
179
provided. Moreover, the concept of permeable and impermeable surfaces and interfaces
was presented. The analysis started with simulating the train load by a uniformly
distributed harmonic line load acting on a poroelastic half-space and on a poroelastic
soil layer laying on top a poroelastic half-space. For the poroelastic half-space, it was
shown that as the porosity of the medium increases, the displacements in the horizontal
and vertical direction for both permeable and impermeable surface boundary conditions
increase. Due to the fact that the pore pressure developed by the trapped
uid weakens
the structure of the soil matrix by reducing the friction between the soil particles, the
amplitudes of both horizontal and vertical displacements for the case of impermeable
surface are higher than those for the permeable case. At a xed porosity, the lower the
shear wave velocity (softer soil), the higher the displacement for all distances away from
the source. Plus, the pore pressure aects the softer soils more than the harder ones
because the harder soils have larger solid frame modulus (K
b
).
In the case of a poroelastic soil layer laying on top a poroelastic half-space, it was
assumed that the
uid can move between the half-space and the top layer (permeable
interface) to study the eect of the water depth on the response and the concept of
porosity ratio (
) was introduced. It was noted that for shallow layers, changing the
porosity ratio will not aect the magnitudes of horizontal and vertical displacements
regardless of the surface permeability condition. Moreover, depending of the load-layer
thickness combination, the layered media can act as a poroelastic half-space. If the
load-layer thickness can create a behavior of a layered media, horizontal and vertical
displacement peaks will appear when the frequency of the harmonic load resonates with
the natural frequency of the soil layer (resonance eect). This resonance eect is related
to the mechanical properties of the layer plus its thickness where it was shown that when
the layer is doubled in thickness, the resonance frequency dropped to half the value and
with dierent layer properties, the magnitude of the peaks are dierent.
180
In chapter (4), the loading force of a line load was replaced by a concentrated har-
monic point load simulating the load from the train through its wheel. This new for-
mulation was the key to introduce the eect of the speed of the train by moving the
harmonic point load along the track axes. Like the case of line load, the concentrated
point load was assumed to be acting on a poroelastic half-space then on a poroelastic soil
layer laying on top a poroelastic half-space. All of the observations mentioned regarding
the line load applies to the case with harmonic point load for both poroelastic half-space
and the poroelastic soil layer laying on top a poroelastic half-space. The exception is
with the concentrated point load, the eect of the train speed was studied.
For the poroelastic half-space, it was shown that doubling the porosity from = 0:1
to = 0:2 increased the the vertical displacement by about 170% at a distance of 20 m.
Plus, when increases to 0.30, the vertical displacement is increased by 270%. As for
the eect of the train speed on displacements, it was found that increasing the speed of
the train increased the vertical displacement regardless of the medium porosity. More
over, at any given distance from the train tracks, the vertical displacement increased
as the porosity of the medium and the speed of the train increased. It was shown that
for the closest distance studied (20 m) with the fastest possible train speed for this
study (120 km/hr), the vertical displacement values increased as the porosity of the
medium increased. As we move away from the source at larger distances, this increase
in the values of vertical displacements is not signicant. As for the speed eect, it was
concluded that the vertical displacement increased as we increased both the porosity of
the medium and the speed of the train, again, very visible for the short distance (20 m)
away from the track.
For a single poroelastic layer on top a poroelastic half-space, it was concluded that
for any given layer thickness the larger the porosity ratio, the larger the vertical displace-
ments. For example, for a layer thickness of (10 m) and a distance (20 m) away from
181
the train tracks, increasing porosity ratio (
) from 1.20 to 1.35 increased the vertical
displacement by 200% and was increased by 400% when the porosity ratio (
) increased
from 1.20 to 1.50. For any given porosity ratio (
), the thicker the top layer, the higher
the displacement. This was shown to be true under the assumption that the load/layer
combination make the medium behave as layered media not a half-space. Plus, it was
also shown that the surface displacement is highly dependent on the frequency of the ap-
plied load. When the frequency of the applied load coincides with the natural frequency
of the layer, high displacements amplitudes were observed regardless of the distance of
the point of interest from the source.
In chapter (5), the concept of the equivalent dry elastic medium was introduced.
This transformation was introduced based on the fact that at low frequency ranges,
less than the characteristic frequency of the medium introduced by Biot (1956a), no
relative motion takes place between the solid and liquid component of the medium and
the medium behaves as a \frozen" mixture rather than a two-phase medium. Plus, it
was proven that the frequency range related to the vibration caused by moving trains is
signicantly small to cause this relative motion. Using these facts, the saturated poroe-
lastic medium was transformed to a classic dry elastic one with some modication to the
mechanical properties of the medium to count for the presence of the pore
uid. with
the \equivalent" properties of the medium obtained, new P and S-waves are calculated
and a new set of parameters for the medium was obtained.
In chapter (6) a Soil-Structure-Interaction (SSI) scheme was developed to examine
how the SSI aect the response of the structure. Two models were examined with dif-
ferent dimensions and overall structural properties. It was concluded that the vertical
displacement component is the dominates the response due to amplitude and the trans-
verse displacement component became more prominent at further distances from the
point of interest. It was shown that softer soil causes larger amplication plus causing
182
the natural frequency of the structure to be lower. It was found that the SSI analysis
is mostly dominated by the lower modes of the superstructure where analysis showed
signicant drop-o after 20 Hz or so. The end result is that the superstructure plays
a critical role in amplied vibration from transit noise. The results from time history
analysis of model (3) showed clearly the torsional behavior of the structure. This behav-
ior could have been reached by studying the data in the frequency domain. However,
time history analysis made it easier to visualize the vibration patterns the nodes under
consideration.
The data from chapter (6) demonstrated, using simplied models, how waves could
enter the structure through the ground. It is to be expected that a massive high-rise
would be able to lter out all the high frequencies and that the low level of vibration
could be dampened. But most of the smaller residential building could be aected more
seriously.
7.2 Future work:
7.2.1 Part of a Bigger Model:
The theory and modeling presented in this work assumed that the source of vibration
(moving trains) acted on the ground surface. Based on that modeling, an understanding
of the response of the medium surrounding the loads and the structures nearby the track
was established. An extension to broaden the current study is to make it part of a larger
model to study the vibration caused by moving trains on viaduct and their eects on
nearby structures as shown in gure (7.1).
183
Figure 7.1: Schematic of the vibration path caused by trains moving on a viaduct
on nearby structures
The source of vibration in this model is moving. However, the source on the ground
is stationary represented by the foundation of the viaduct. This analysis should open
the door for discussing the role of the viaduct geometric and mechanical properties
and their eects on the resulting nal load (reaction on the foundation) and a better
understanding of the produced load can be achieved. This can also open the possibility
to study the soil{structure{interaction between the foundation of the viaduct and the
soil supporting it.
7.2.2 Below Ground Source:
Trains can run on ground level, viaduct, and in many cases underground as shown
in gure (7.2). The mathematical model of the wave propagation problem is dierent
from the one presented in this study. However, the same concepts of structural response
presented in chapter (6) can be applied once the vibration prole from this new source
of vibration is dened.
184
Figure 7.2: Schematic of the vibration path caused by trains moving underground
nearby structures
7.2.3 Vibration Mitigation:
When the two sources of vibration are studied (on ground or viaduct), a complete
assessment of the vibration prole of the site can be accomplished. That leads to the
idea of vibration mitigation which is the next logical step. Vibration mitigation can
be applied in two dierent approaches depending on \state" of the structures within
proximity of vibration source. For structures in the design stage, a complete vibration
prole of the site can be presented to the structural engineer to apply any necessary
modication to the design to minimize or eliminate undesirable vibrations. For existing
buildings, some modication methods can be used to minimize the vibration in these
structures such as stiening the slabs of the
oors with vibration problems (gure 7.3)
or installing open/closed trenches along the path of vibration propagation as shown in
the schematic in gure (7.4)
185
Figure 7.3: Schematic showing a steel beam to stien the concrete slab of a
vibrating
oor
Figure 7.4: Schematic showing an open trench to mitigate the vibration caused by
trains nearby structures
With proper understanding of the vibration prole of a site, the nearby structures
will suer less cosmetic damage and will attract more tenants reaching their optimum
life expectancy structurally and economically.
186
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bedrock system due to inclined sv waves. Soil Dynamics and Earthquake Engineering,
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195
Appendix: Mathematical Formulas
A.1 Cartesian Coordinate System:
28
CHAPTER (3): Line Load on top of Poroelastic Half-space
3.1 Introduction:
The study of plane waves can give a useful insight about the physics of the
problem. Yet, these plane waves’ solutions are limited in terms of obtaining vibration
characteristics a function of distance from a surface vibration source. After the
comparison presented in chapter 2, a different loading source is introduced here to
account for the change of vibration as a function of distance. In this chapter, the train
loading is represented as a uniformly distributed line load along the x axis.
3.2 Mathematical Derivation:
Medium (m): Poroelastic
x
y
y = 0
m
, m
, M
m
, m
, m
F=F
0
e
ikx
Medium (m+1): Poroelastic
m+1
, m+1
, M
m+1
, m+1
, m+1
y = -h
1
m31
m21
m11
m22
m32
m12
(m+1)31
(m+ 1)21
(m+1)11
(a)
x
y
F=F 0 e
ikx
z
(b)
Figure 3-1: (a) Line load on top of a poroelastic layered space, (b) 3D view of axis
3.2.1 Problem setup:
In this section, the vertical and horizontal displacement expressions will be
derived for a line load acting on top of a poroelastic half-space as shown in Figure 3-1.
Let the loading function have the following form:
00
0
ikxt ikx it
z
F x Fe Fee
By omitting the
it
e
term for simplicity, the final loading function will be:
0
0
ikx
y
F x Fe
(3-1)
Figure A.1: (a) Line load on top of a poroelastic layered space, (b) 3D view of axis
The propagating waves potentials can be expressed as follows:
mgd
=A
mgd
exp (ikx
mgd
y); g = 1; 2 (A-1)
mgd
=A
mgd
exp (ikx
mgd
y); g = 3 (A-2)
where:
2
mgd
=k
2
k
2
mgd
and k
mgd
=
!
V
mgd
196
where:
! : The frequency
k : The apparent wave number.
k
mgd
: The wave number associated with wave in medium m with type g.
V
mgd
: The wave velocity associated with wave in medium m with type g.
m: Layer number
g: Wave type whereg = 1, 2 represent fast and slow p-waves respectively
and g = 3 represents shear wave.
d: d = 1, 2 represent transmitted waves and re
ected waves respectively.
The total wave potentials for a poroelastic medium can be expressed as:
In solid component In liquid Component
m
=
2
P
g=1
2
P
d=1
mgd
m
=
2
P
g=1
2
P
d=1
f
mgd
mgd
m
=
3
P
g=3
2
P
d=1
mgd
m
=
3
P
g=3
2
P
d=1
f
mgd
mgd
where:
f
mgd
=
11m
R
m
12m
Q
m
Am
V
2
mgd
(
22m
Q
m
12m
R
m
)
, g = 1; 2
f
mgd
=
12m
22m
, g = 3
And (
11m
;
12m
; and
22m
) are the dynamic mass coecients (
11
;
12
; and
22
)
associated with medium (m). Moreover,(A
m
; R
m
; andQ
m
) are the (A; R; andQ) dened
for the fast and slow dilatational wave for medium (m).
When using the potential notations for wave equation, Helmholtz decomposition is
applied to the displacement vectors as follows:
u = grad () + curl ( ) (A-3)
U = grad () + curl ( ) (A-4)
Resulting in the following expressions for absolute displacements:
197
in the x-axis in the y-axis
u
m
=
@
m
@x
+
@
m
@y
v
m
=
@
m
@y
@
m
@x
Elastic Half-Space:
Dilatation of solid component:
e
m
= div (u
m
) =
@u
m
@x
+
@v
m
@y
=
@
@x
@
m
@x
+
@
m
@y
+
@
@y
@
m
@y
@
m
@x
=r
2
m
(A-5)
Normal stress in the x-direction:
xxm
=e + 2e
xx
=
m
@u
m
@x
+
@v
m
@y
+ 2
m
@u
m
@x
=
m
r
2
m
+ 2
m
@
2
m
@x
2
+
@
2
m
@y@x
(A-6)
Normal stress in the y-direction:
yym
=
m
e + 2
m
e
yy
=
m
@u
m
@x
+
@v
m
@y
+ 2
m
@v
m
@y
=
m
r
2
m
+ 2
m
@
2
m
@y
2
@
2
m
@x@y
(A-7)
Tangential stress:
xym
= 2
m
e
xym
= 2
m
1
2
@u
m
@y
+
@v
m
@x
=
m
2
@
2
m
@y@x
+
@
2
m
@y
2
@
2
m
@x
2
(A-8)
198
Poroelastic Half-Space:
Relative displacement in the x-direction:
U
rm
=U
m
u
m
=
@
m
@x
+
@
m
@y
@
m
@x
+
@
m
@y
=
@
@x
(
m
m
) +
@
@y
(
m
m
) (A-9)
Relative displacement in the y-direction:
V
rm
=V
m
v
m
=
@
m
@y
+
@
m
@x
@
m
@y
@
m
@x
=
@
@y
(
m
m
)
@
@x
(
m
m
) (A-10)
The variable
m
:
m
=div [
m
(U u)] =
m
@U
rm
@x
+
@V
rm
@y
=
m
r
2
(
m
m
) (A-11)
Dilatation of solid component:
e
m
= div (u
m
) =
@u
m
@x
+
@v
m
@y
=
@
@x
@
m
@x
+
@
m
@y
+
@
@y
@
m
@y
@
m
@x
=r
2
m
(A-12)
Pore water pressure:
p
fm
=M
m
(
m
m
e
m
)
=M
m
m
r
2
(
m
m
)
m
r
2
m
=M
m
r
2
(
m
(
m
m
) +
m
m
) (A-13)
199
Normal stress in the x-direction:
xxm
=
m
e + 2
m
e
xx
m
p
fm
=
m
r
2
m
+ 2
m
@
2
m
@x
2
+
@
2
m
@y@x
+
m
M
m
r
2
(
m
(
m
m
) +
m
m
) (A-14)
Normal stress in the y-direction:
yym
=
m
e + 2
m
e
xx
m
p
fm
=
m
r
2
m
+ 2
m
@
2
m
@y
2
@
2
m
@x@y
+
m
M
m
r
2
(
m
(
m
m
) +
m
m
) (A-15)
Tangential stress:
xym
= 2
m
e
xym
= 2
m
1
2
@u
m
@y
+
@v
m
@x
=
m
2
@
2
m
@y@x
+
@
2
m
@y
2
@
2
m
@x
2
(A-16)
200
B.1 Cylindrical Coordinate System:
57
CHAPTER (4): Concentrated Harmonic Point Load on top of a
Poroelastic Half-space
4.1 Introduction:
In chapter three, the uniformly distributed line load was used as a loading force to
simulate the dynamic effects of a train. Although it established the basis toward
understanding the behavior of the half-space and layered media, one more aspect should
not be neglected in the analysis, the speed of the train. The dynamic load of a train is a
moving load acting through its wheels. Hence, a better presentation of such a load is a
concentrated harmonic point load instead of a uniformly distributed line load as
previously discussed. With this new formulation, the effect of the speed of the point load
(speed of the train) can be studied.
4.2 Mathematical Derivation:
Medium (m): Poroelastic
r
z
z = 0
m
, m
, M
m
, m
, m
Medium (m+1): Poroelastic
m+1
, m+1
, M
m+1
, m+1
, m+1
z = -h
1
m31
m21
m11
m22
m32
m12
(m+1)31
(m+ 1)21
(m+1)11
(a)
F=Z
J
0
(k r)
r
z
(b)
F=Z
J
0
(k r)
Figure 4-1: (a) Concentrated harmonic point load on top of a poroelastic layered space, (b) 3D view of axis
4.2.1 Problem setup:
In this section, the vertical and horizontal displacement expressions will be
derived for a concentrated harmonic point load acting on top of a poroelastic half-space
as shown in Figure 4-1. We note that axial symmetry about z axis is assumed. We note
Figure B.1: (a) Concentrated harmonic point load on top of a poroelastic layered
space, (b) 3D view of axis
The propagating waves potentials can be expressed as follows:
mgd
=A
mgd
J
0
(kr)e
mgd
z
; g = 1; 2 (B-1)
mgd
=A
mgd
J
0
(kr)e
mgd
z
; g = 3 (B-2)
where:
2
mgd
=k
2
k
2
mgd
and k
mgd
=
!
V
mgd
where:
! : The frequency
k : The apparent wave number.
k
mgd
: The wave number associated with wave in medium m with type g.
V
mgd
: The wave velocity associated with wave in medium m with type g.
m: Layer number
g: Wave type whereg = 1, 2 represent fast and slow p-waves respectively
and g = 3 represents shear wave.
d: d = 1, 2 represent transmitted waves and re
ected waves respectively.
201
The total wave potentials for a poroelastic medium can be expressed as:
In solid component In liquid Component
m
=
2
P
g=1
2
P
d=1
mgd
m
=
2
P
g=1
2
P
d=1
f
mgd
mgd
m
=
3
P
g=3
2
P
d=1
mgd
m
=
3
P
g=3
2
P
d=1
f
mgd
mgd
where:
f
mgd
=
11m
R
m
12m
Q
m
Am
V
2
mgd
(
22m
Q
m
12m
R
m
)
, g = 1; 2
f
mgd
=
12m
22m
, g = 3
And (
11m
;
12m
; and
22m
) are the dynamic mass coecients (
11
;
12
; and
22
)
associated with medium (m). Moreover,(A
m
; R
m
; andQ
m
) are the (A; R; andQ) dened
for the fast and slow dilatational wave for medium (m).
When using the potential notations for wave equation, Helmholtz decomposition is
applied to the displacement vectors as follows:
u = grad () + curl ( ) (B-3)
U = grad () + curl ( ) (B-4)
Resulting in the following expressions for absolute displacements:
in the r-direction in the z-direction
u
rm
=
@
m
@r
+
@
2
m
@r@z
u
zm
=
@
m
@z
1
r
@
m
@r
@
2
m
@r
2
The governing equations are now listed with reference to gure (B.2) based on the
axial symmetry of the vertical load at the origin.
202
r
R
z
y
x
F
z
F
x
z
x
y
r
rr
r
rz
zr
zz
z
z
r
(a) (b)
Figure B.2: Stress element in (a) Spherical coordinates (b) cylindrical coordinates
(Schepers et al., 2010)
Elastic Half-Space:
Dilatation of solid component:
e
m
= div (u
m
) =
1
r
@
@r
(ru
rm
) +
@
@z
(u
zm
)
=
@
2
m
@r
2
+
1
r
@
m
@r
+
@
2
m
@z
2
=r
2
m
(B-5)
Normal stress in the z-direction:
zzm
=
m
e
m
+ 2
m
e
zzm
=
m
r
2
m
+ 2
m
@u
zm
@z
(B-6)
Normal stress in the r-direction:
rrm
=
m
e
m
+ 2
m
e
rrm
=
m
r
2
m
+ 2
m
@u
rm
@r
(B-7)
Normal stress in the -direction:
m
=
m
e
m
+ 2
m
e
m
=
m
r
2
m
+ 2
m
@u
m
@
+
@u
rm
r
(B-8)
203
Tangential stress:
rzm
=
m
e
rzm
=
m
@u
rm
@z
+
@u
zm
@r
(B-9)
Poroelastic Half-Space:
Relative displacement in the r-direction:
U
rrm
=U
rm
u
rm
=
@
@r
(
m
m
) +
@
2
@r@z
(
m
m
) (B-10)
Relative displacement in the z-direction:
U
rzm
=U
zm
u
zm
=
@
@z
(
m
m
)
1
r
@
@r
(
m
m
)
@
2
@r
2
(
m
m
) (B-11)
The variable
m
:
m
= - div [
m
(Uu)] =
m
1
r
@
@r
(rU
rrm
) +
@
@z
(U
rzm
)
=
m
r
2
(
m
m
) (B-12)
Dilatation of solid component:
e
m
= div (u
m
) =
1
r
@
@r
(ru
rm
) +
@
@z
(u
zm
) =r
2
m
(B-13)
Pore water pressure:
p
fm
=M
m
(
m
m
e
m
)
=M
m
m
r
2
(
m
m
)
m
r
2
m
=M
m
r
2
(
m
(
m
m
) +
m
m
) (B-14)
204
Normal stress in the z-direction:
zzm
=
m
e
m
+ 2
m
e
zzm
m
p
fm
(B-15)
=
m
r
2
m
+ 2
m
@u
zm
@z
+
m
M
m
r
2
([
m
(
m
m
) +
m
m
]) (B-16)
Tangential stress:
rzm
=
m
e
rzm
=
m
@u
rm
@z
+
@u
zm
@r
(B-17)
205
C.1 Moving Load Formulation:
z
x
y
D (0, D, 0)
1
l
,0,0
oo
xx
11
,0,0 xx
,0,0
kk
xx
2
l
Z
2
Z
n
Z
1
Z
o
k
l
22
,0,0 xx
22
2
xD 2 2
1
x D 22
n
xD 2 2
o
x D c
Figure C.1: Schematic of a moving harmonic point load along the x axis and an
observer at distance D along the y axis.
Formulation for three moving loads:
Let the total displacement caused by the rst three moving loads be expressed as:
v
D
=f
p
x
2
o
+D
2
e
i!
to
x
o
c
!!
+f
q
x
2
1
+D
2
e
i!
t
1
x
1
c
!!
+
f
q
x
2
2
+D
2
e
i!
t
2
x
2
c
!!
(C-1)
Assuming that the rst load passes through the origin x (0; 0; 0) at timet
o
, then the
second and the third loads will pass at times:
t
1
=t
o
+
l
1
c
(C-2a)
t
2
=t
o
+
l
2
c
(C-2b)
206
Substituting (C-2) in (C-1) leads to:
v
D
=f
p
x
2
o
+D
2
e
i!
to
x
o
c
!!
+f
q
x
2
1
+D
2
e
i!
0
@
0
@
to+
l
1
c
1
A
x
1
c
!
1
A
+
f
q
x
2
2
+D
2
e
i!
0
@
0
@
to+
l
2
c
1
A
x
2
c
!
1
A
(C-3)
Let f
p
x
2
o
+D
2
=f
o
, f
p
x
2
1
+D
2
=f
1
, and f
p
x
2
2
+D
2
=f
2
then, with some manipulation, (C-3) can be simplied to
v
D
=
0
B
@
f
o
e
i!
x
o
c
!
+f
1
e
i!
x
1
c
!
e
i!
0
@
l
1
c
1
A
+f
2
e
i!
x
2
c
!
e
i!
0
@
l
2
c
1
A
1
C
A
e
i!to
(C-4)
Integrating equation (C-4) along the entire x-axis leads to:
v
D
=
0
B
@
Z
1
1
0
B
@
f
o
e
i!
x
o
c
!
+f
1
e
i!
x
1
c
!
e
i!
0
@
l
1
c
1
A
+f
2
e
i!
x
2
c
!
e
i!
0
@
l
2
c
1
A
1
C
A
dx
1
C
A
e
i!to
v
D
=
0
@
Z
1
1
f
o
e
i!
x
o
c
!
dx
1
A
e
i!to
+
0
B
@
Z
1
1
f
1
e
i!
x
1
c
!
e
i!
0
@
l
1
c
1
A
dx
1
C
A
e
i!to
+
0
B
@
Z
1
1
f
2
e
i!
x
2
c
!
e
i!
0
@
l
2
c
1
A
dx
1
C
A
e
i!to
Leading to
v
D
=
0
@
Z
1
1
f
o
e
i!
x
o
c
!
dx
1
A
e
i!to
+
0
B
@
Z
1
1
f
1
e
i!
0
@
x
1
l
1
c
1
A
dx
1
C
A
e
i!to
+
0
B
@
Z
1
1
f
2
e
i!
0
@
x
2
l
2
c
1
A
dx
1
C
A
e
i!to
(C-5)
207
or
v
D
=
0
@
Z
1
1
f
p
x
2
o
+D
2
e
i!
x
o
c
!
dx
1
A
e
i!to
+
0
B
@
Z
1
1
f
q
x
2
1
+D
2
e
i!
0
@
x
1
l
1
c
1
A
dx
1
C
A
e
i!to
+
0
B
@
Z
1
1
f
q
x
2
2
+D
2
e
i!
0
@
x
2
l
2
c
1
A
dx
1
C
A
e
i!to
(C-6)
Equation(C-6) represents the superposition of the three moving loads along the x-axis.
For n number of loads moving along the x-axis, equation (C-6) can be generalized to
v
D
=e
i!to
n1
X
k=0
0
B
@
Z
1
1
f
q
x
2
k
+D
2
e
i!
0
@
x
k
l
k
c
1
A
dx
1
C
A
(C-7)
If coordinatesx
1
andx
2
are expressed in terms ofx
o
, and from gure (C.1), one can
see that
x
1
=x
o
l
1
(C-8a)
x
2
=x
o
l
2
(C-8b)
Substituting (C-8) in (C-5) will lead to
v
D
=
0
@
Z
1
1
f
o
e
i!
x
o
c
!
dx
1
A
e
i!to
+
0
B
B
@
Z
1
1
f
1
e
i!
0
@
(x
o
l
1
)l
1
c
1
A
dx
1
C
C
A
e
i!to
+
0
B
B
@
Z
1
1
f
2
e
i!
0
@
(x
o
l
2
)l
2
c
1
A
dx
1
C
C
A
e
i!to
(C-9)
Simplifying (C-9) will lead to
v
D
=
0
B
@
Z
1
1
0
B
@
f
o
+f
1
e
i!
0
@
2l
1
c
1
A
+f
2
e
i!
0
@
2l
2
c
1
A
1
C
A
e
i!
x
o
c
!
dx
1
C
A
e
i!to
(C-10)
208
For n number of loads moving along the x-axis, equation (C-10) can be generalized as:
v
D
=e
i!to
n1
X
k=0
Z
1
1
0
B
@
f
p
(x
o
l
k
)
2
+D
2
e
i!
0
@
x
o
2l
k
c
1
A
1
C
A
dx (C-11)
209
List of Symbols
Symbol Denition
b
b : Dissipation coecient
c : Speed of moving load (train)
d : Pore diameter
d : (Subscript)d = 1; 2 represent transmitted waves and re
ected waves respectively
e : Dilatational strain of solid
f
c
0
: The compressive strength of concrete
f
t
: Upper limit of propagation frequency to maintain laminar
ow
ffg : Force vector
g : (Subsript) g = 1; 2; 3 represent fast P-waves, slow P-waves, and shear wave
k : Apparent wave number
k
mgd
: Wave number associated with wave in medium (m) with wave type (g)
m : (Subscript) m represent the medium number
p
fm
: Pore water pressure for layer (m)
r : Radial distance from the point load source to a point of interest on the surface
~ r
ik
: The centroid of the i-th subregion of the k-th foundation
r
o
: Radius of solid particle in the poroelastic solid-
uid element
u
m
: Tangential displacement of solid particles in medium (m)
~ u : Harmonic displacement vector
u : Solid displacement vector
v
m
: Normal displacement of solid particles
v
D
: Vertical displacement at point D away from the load source
f xg : Acceleration vector
f _ xg : Velocity vector
fxg : Displacement vector
[C] : Damping matrix of the structure
D
n
: The grain diameter at n% passing
210
[D] : Diagonalized dynamic matrix [Z(!)]
E
c
: Modulus of elasticity of reinforced concrete
E
s
: Modulus of elasticity of soil layer
~
F
f
: Force vector at building level
F
: Driving force vector
F
e
(!)
: The external forces vector on the structure
F
s
(!)
: The generalized forces and moments exerted by the soil on the foundation
[G] : Green's Functions matrix
I
beam
: Moment of inertia for reinforced concrete beam
I
column
: Moment of inertia for reinforced concrete column
I
g
: Gross moment of inertia of the reinforced concrete section
I
slab
: Moment of inertia for reinforced concrete solid slab
K
a
: Bulk modulus of air at 1.0 atmosphere
K
b
: Bulk modulus of the skeleton
K
cr
: Critical bulk modulus of the solid frame at critical porosity
K
dry
: Bulk modulus of dry solid skeleton
K
f
: Bulk modulus of pore
uid
K
g
: Bulk modulus of solid grain
K
s
: Bulk modulus of solid skeleton
K
w
: Bulk modulus of water
[K] : Stiness matrix of the structure
[K
ss
] : The impedance matrix for the soil-foundation system
[M] : Mass matrix of the structure
~
P : Harmonic point loads at source
P;Q;R : Elastic moduli for the solid
uid system
S : Saturation degree of the medium
U : Fluid displacement vector
U
f
: The foundation input motion
U
m
: Tangential solid particle displacement in medium (m)
211
U
rm
: Tangential solid particle displacement relative to the liquid component of the
medium (m)
V
s
: Velocity ratio between te surface layer and the underlying half-space
V
c
: Reference velocity
V
fast
: Fast dilatational wave velocity
V
m
: Normal solid particle displacement in medium (m)
V
mgd
: The wave velocity associated with wave in medium (m) with type (g)
V
pe
: Equivalent P-wave velocity in the dry elastic medium
V
rm
: Normal solid particle displacement relative to the liquid component
of the medium (m)
V
se
: Equivalent S-wave velocity in the dry elastic medium
V
S
L
: Shear wave velocity of the top layer
V
S
H
: Shear wave velocity of the half-space underlying surface layer
V
slow
: Slow dilatational wave velocity
V
shear
: Shear wave velocity
~
X
b
: Harmonic displacement vector at building level
~
X
f
: Harmonic displacement vector at foundation level
: Porosity of the medium
, M : Biot's parameters
L
: Porosity of the layer on top of the half-space
H
: The porosity of the half-space underlying surface layer
: Porosity ratio between te surface layer and the underlying half-space
cr
: Critical porosity of the material
[] : Modal participation factor
ii
: Normal strain
ij
: Shear strain
11
;
22
;
12
: Non-dimensional parameters dening the elastic properties of the material
: Kinematic viscosity of
uid
~ : Modal ordinate
" : Dilatational strain of the pore
uid
212
k : Permeability of the medium
i
: Modal damping factor
dry
: Lam e constant for the dry system
eq
: Modied Lam e constant for the equivalent dry system
sat
: Lam e constant for the saturated system
: Poisson's ratio for the solid skeleton
eq
: Modied Poisson's ratio for the equivalent dry elastic medium
: Shear modulus for solid skeleton
eq
: Equivalent shear modulus for solid skeleton
ii
;
ij
: Dynamic mass coecients
eq
: Modied (mixture) density of the equivalent elastic medium
: Mass per unit volume of the
uid-soil aggregate
f
: Density of pore
uid
s
: Density of solid skeleton
1
: Unit mass of solid skeleton
2
: Unit mass of pore
uid
: The stress of the pore
uid
ii
: Normal stress
ij
: Shear stress
: Dynamic toruosity parameter
m
: Total compressional wave potential in solid component for medium (m) with
type (g) in direction (d)
m
: Total compressional wave potential in liquid component for medium (m) with
type (g) in direction (d)
mgd
: Compressional wave potential in medium (m) with type (g) in direction (d)
: The ratio of the excitation frequency ! to the characteristic frequency !
o
11
;
22
;
12
: Non-dimensional parameters dening the dynamic properties of the material
m
: Total shear wave potential in solid component for medium (m) with
type (g) in direction (d)
m
: Total shear wave potential in solid component for medium (m) with
213
type (g) in direction (d)
mgd
: Shear wave potential in medium (m) with type (g) in direction (d)
! : Angular frequency
!
i
: Natural frequency of the structure
!
o
: Characteristic frequency governing the transition between low and high frequency
behavior of a poroelastic medium
214
Abstract (if available)
Abstract
The Ministry of Communications, through the Kuwait Overland Transport Union, raised the need to develop a master plan for the implementation of a rapid transit system that will reduce the traffic congestion in the streets of Kuwait. With such a project, an understanding of the vibration induced by this mean of transportation on the nearby structures is needed. The water table in the area proposed to host the lanes fluctuates depending mainly on rainfall and irrigation process and ranges from 0.50 m in some areas and as deep as 13.0 m. Due to the saturation of the soil with water, Biot’s theory of wave propagation in poroelastic media is applied to calculate the surface horizontal and vertical displacements for the poroelastic half-space and a poroelastic layered media. The train load is simulated by a uniformly distributed harmonic line load first and then by a concentrated harmonic point load with load speed formulation added to the latter to simulate the movement of the train and study the effect on horizontal and vertical displacements. It was found that the displacement values increase with the increase of the porosity of the medium. Plus, the values of such displacements will increase even more if the saturating water was trapped due to the impermeability of the mediums surface. The water table effect was simulated by changing the thickness of the first layer below the surface and assuming a permeable interface between this layer and the half-space underneath it. There existed two critical combinations between the applied load and the thickness of the first layer where at the one combination the horizontal disturbance is at maximum and at the second the vertical is. The change of porosity contrast between the first layer and the half-space showed that the higher the contrast, the higher the displacements values. Plus, the effect of porosity contrast is highly substantial at the load-layer thickness critical combinations. As for a single train speed, regardless of the distance from the train track, a significant increase of vertical displacements was present as the porosity of the half-space increased. As the train speed increased, the displacements increased with increasing porosity. With the introduction of a poroelastic layer, the displacements at closer distances from the track increased even more than the one obtained for the half-space case. Plus, It was shown that the rate at which the displacement increase with increasing train speed became higher as the porosity ratio of the poroelastic layered system increased. Finally, a Soil-Structure-Interaction (SSI) scheme was developed and showed, using three structural models, that SSI plays a critical role in amplified vibration from transit noise and it was concluded through frequency domain analysis and time history analysis that Soil-Structure-Interaction is mostly dominated by the lower modes of the superstructure.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Alshaiji, Mohammad S.
(author)
Core Title
Train induced vibrations in poroelastic media
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Civil Engineering (Structural Engineering)
Publication Date
08/19/2017
Defense Date
07/20/2015
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Green's functions,layer,mode shapes,OAI-PMH Harvest,poroelasticity,SAP2000,soil-structure-interaction,Train,vibration,wave,wave propagation
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Wong, Hung Leung (
committee chair
), Lee, Vincent W. (
committee member
), Moore, James E., II (
committee member
)
Creator Email
alshaiji@usc.edu,malshaiji@yahoo.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-637049
Unique identifier
UC11305646
Identifier
etd-AlshaijiMo-3833.pdf (filename),usctheses-c3-637049 (legacy record id)
Legacy Identifier
etd-AlshaijiMo-3833.pdf
Dmrecord
637049
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Alshaiji, Mohammad S.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
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Repository Location
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Tags
Green's functions
layer
mode shapes
poroelasticity
SAP2000
soil-structure-interaction
vibration
wave propagation