ff

1. Soil Dynamics and Earthquake Engineering 26 (2006) 648–665
Dynamics of nonlinear porous media with applications
to soil liquefaction
Radu Popescua
, Jean H. Prevostb,, George Deodatisc
, Pradipta Chakraborttya
a
Faculty of Engineering and Applied Science, Memorial University, St. John’s, Nfld, Canada A1B 3X5
b
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
c
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA
Abstract
This paper provides a description of the extension of Biot’s theory for dynamic behavior of saturated porous media into the nonlinear
regime that was introduced by the second author in 1980. It also provides a finite element implementation of this extension and two
numerical applications involving the seismic behavior of saturated soil deposits. In the first numerical application, the dynamic
interaction between liquefying soil and a structure sitting on the ground surface is examined, with emphasis on the interplay between the
seismic loading rate and the (evolving) characteristic frequency of the soil–structure system. The attenuation of seismic energy as the
seismic waves pass through softened soil is also discussed. The second numerical application involves the seismically induced liquefaction
of stochastically spatially variable soils. It is explained why more pore-water pressure is generated in a heterogeneous soil than in a
corresponding uniform soil. Comparisons are also provided with experimental centrifuge data.
r 2006 Elsevier Ltd. All rights reserved.
1. Introduction
Soils consist of an assemblage of particles with different
sizes and shapes which form a skeleton whose voids are
filled with water and air or gas. The word ‘‘soil’’, therefore,
implies a mixture of assorted mineral grains with various
fluids. Hence, soil in general must be looked at as a one-
phase (dry soil), two-phase (saturated soil), or multi-phase
(partially saturated soil) material whose state is to be
described by the stresses and displacements (velocities)
within each phase. There are still uncertainties on how to
deal analytically with partially saturated soils, and this is
currently an area of active research. Attention is restricted
in the following to fully saturated soils. The stresses carried
by the soil skeleton are conventionally called ‘‘effective
stresses’’ in the soil mechanics literature [1], and those in
the fluid phase are called ‘‘pore-fluid pressures’’.
In a saturated soil, when free drainage conditions
prevail, the steady-state pore-fluid pressures depend only
on the hydraulic conditions and are independent of the soil
skeleton response to external loads. Therefore, in that case,
a single-phase continuum description of soil behavior is
certainly adequate. Similarly, a single-phase description of
soil behavior is also adequate when no drainage (i.e., no
flow) conditions prevail. However, in intermediate cases in
which some flow can take place, there is an interaction
between the skeleton strains and the pore-fluid flow. The
solution of these problems requires that soil behavior be
analyzed by incorporating the effects of the transient flow
of the pore-fluid through the voids, and therefore requires
that a two-phase continuum formulation be available for
saturated porous media. Such a theory was first developed
by Biot [2–4] for an elastic porous medium. Applications of
Biot’s theory to finite element analysis of saturated porous
media have been reported by numerous authors (see, e.g.,
Refs. [5,6] for a summary of the wide range of existing
finite element formulations). Mesgouez et al. [7] present a
review of work on applications of Biot’s theory in transient
wave propagation in saturated porous media. In most
formulations, the soil skeleton has either elastic or visco-
elastic behavior.
An extension of Biot’s theory into the nonlinear inelastic
range is presented by Prévost [8], where soil is viewed as a
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doi:10.1016/j.soildyn.2006.01.015
Corresponding author. Tel.: +1 609 258 5424; fax: +1 609 258 2760.
E-mail address: prevost@princeton.edu (J.H. Prevost).

2. multi-phase medium and the modern theories of mixtures
developed by Green and Naghdi [9] and Eringen and
Ingram [10] are used. General mixture results can be shown
through formal linearization of the field and constitutive
equations, to reduce Biot linear poroelastic model (see, e.g.,
Ref. [11]). The resulting equations are summarized in
Section 2.
The first part of this paper presents briefly Biot’s basic
theory of dynamics of saturated porous media, its
extension into the nonlinear regime [8], and its implemen-
tation in the finite element code DYNAFLOW [12]. Two
numerical examples are presented next, illustrating this
extension of Biot’s theory into the nonlinear dynamic
behavior of saturated soils. Both examples are presented in
relation to centrifuge experiments of structures on homo-
geneous or non-homogeneous saturated soil deposits
subjected to seismic loads. The first numerical example
provides results of a study aimed at better understanding
the loading and failure mechanisms of structures on
liquefiable soil, while the second example examines the
mechanisms of excess pore-water pressure (EPWP) build-
up and liquefaction of natural soil deposits exhibiting
stochastic spatial variability of their properties. The first
example is an original study extending the results of
previous research on earth dams (e.g., [13]) to structures on
liquefiable soil deposits. The second example is also an
original study that continues and expands on previous
work of the authors on seismically induced liquefaction of
randomly heterogeneous soil [14–17]. Based on detailed
analysis of both experimental and numerical results, it
provides an interpretation for a particularly interesting and
important behavior detected in previous work, namely that
more pore-water pressure is generated by seismic loads in a
heterogeneous soil than in an equivalent uniform soil. The
numerical calculations presented in both examples are
validated based on results of recent centrifuge experiments
[18,19]. Besides demonstrating the capabilities of the
nonlinear extension of Biot’s theory implemented in
DYNAFLOW to handle relatively complex phenomena
(such as dynamic soil–structure interaction and seismic
behavior of randomly variable soils), these studies provide
new insight into the mechanisms of seismic behavior of
soil–structure systems and interesting practical results for
geotechnical design.
2. Dynamics of porous media
2.1. Basic theory
During deformation, the solid particles, which form the
soil skeleton, undergo irreversible motions such as slips at
grain boundaries, creations of voids by particles coming
out of a packed configuration, and combinations of such
irreversible motions. When the particulate nature and the
microscopic origin of the phenomena involved are not
explicitly sought, phenomenological equations then pro-
vide an adequate description of the behavior of the various
phases which form the soil medium. In multi-phase
theories, the conceptual model is, thus, one in which each
phase (or constituent) enters through its averaged proper-
ties obtained as if the particles were smeared out in space.
In other words, the particles nature of the constituents is
described in terms of phenomenological laws as the
particles behave collectively as a continuum. Soil is, thus,
viewed herein as consisting of a solid skeleton interacting
with the pore fluids.
Balance of linear momentum equations for each
constituent then can be written simply as (see, e.g., Ref. [8])
rS
aS
¼ r s0S
ð1 nw
Þ rpw
x ðvS
vw
Þ þ rS
b, (1)
rw dvw
dt
¼ rw
ðvS
vw
Þ rvw
nw
rpw
þ x ðvS
vw
Þ þ rw
b, ð2Þ
where the motion of the solid phase is used as the reference
motion. In Eqs. (1) and (2), s0s
is the solid (effective) stress,
as
the solid acceleration, vs
(vw
) the solid (fluid) velocity, b
the body force (per unit mass), pw
the pore-fluid pressure,
rS
¼ ð1 nw
Þrs and rw
¼ nw
rw with rs the solid mass
density, rw the fluid mass density, and nw
the porosity;
x ¼ nw2
gwk1
where k is the hydraulic conductivity (L/T)
and gw ¼ rwg the fluid unit weight, g ¼ ðjjbjjÞ acceleration
of gravity.
In the case of a compressible pore fluid, the pore-fluid
pressure is determined from the computed velocities
through time integration of the following equation:
dpw
dt
¼ ðlw
=nw
Þ r ð1 nw
ÞvS
þ r ðnw
vw
Þ
, (3)
where lw
is the fluid bulk modulus. In the case of an
incompressible pore fluid, the pore-fluid pressure is
determined from the computed velocities through the
following equation:
pw
¼ ðlw
=nw
Þ r ð1 nw
ÞvS
þ r ðnw
vw
Þ
, (4)
where lw
is a penalty parameter used to enforce the
incompressibility constraint [20,21].
2.2. Weak form/semi-discrete finite element equations
The weak formulation associated with the initial
boundary value problem is obtained by proceeding along
standard lines (see, e.g., Refs. [22,23]). The associated
matrix problem is obtained by discretizing the domain
occupied by the porous medium into non-overlapping finite
elements. Each element is defined by nodal points at which
shape functions are prescribed. In general, two sets of
shape functions are required for the solid displacement and
the fluid velocity field, respectively. However, attention in
the following is restricted to low-order (i.e., four-node
plane and eight-node brick) finite elements which are the
most efficient in nonlinear analysis, and thus the same
shape functions are used for both the solid and the fluid.
The shape functions for the solid displacement and fluid
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3. velocity associated with node A are denoted by NA
in the
following. They satisfy the relation NA
ðxB
Þ ¼ dAB in which
xB
denotes the position vector of node B and dAB ¼ Kro-
necker delta. The Galerkin counterpart of the weak
formulation is expressed in terms of the shape functions
and gives rise to the following system of equation:
Ms
0
0 Mw
#
as
aw
( )
þ
Z Z
ZT
Z
#
vs
vw
( )
þ
ns
nw
( )
¼
fs ext
fw ext
( )
, ð5Þ
where Ma
, aa
, va
, na
, and fa ext
represent the (generalized)
mass matrix, acceleration, velocity, and force vectors,
respectively (a ¼ s; w for the solid and fluid phases,
respectively). The element contributions to node A from
node B for directions i and j to the matrices appearing in
Eq. (5) are defined in the following. The terms are
integrated over the spatial domain occupied by the
element, Oe
. For computational simplicity, a diagonal
‘‘lumped mass’’ matrix is used:
mAB
ij
a
¼ dijdAB
Z
Oe
ra
NA
dO. (6)
The momentum transfer terms give rise to the damping
matrix Z:
ZAB
ij
¼
Z
Oe
NA
xijNB
dO. (7)
The external force ðfa
Þext
(i.e., body force, surface
traction) is computed as follows:
ðfA
i Þa ext
¼
Z
Oe
ra
biNA
dO þ boundary terms: (8)
The internal stress forces na
are computed as follows:
ðnA
i Þs
¼
Z
Oe
ðs0s
ij ns
pwdijÞNA
;j dO (9)
and
ðnA
i Þw
¼
Z
Oe
rw
NA
ðvw
j vs
j Þ dO
Z
Oe
NA
;i nw
pw dO, (10)
where ns
¼ 1 nw
.
2.3. Time integration
Time integration of the semi-discrete finite element
equations is accomplished by a finite difference time-
stepping algorithm. In general, implicit and explicit
integration procedures are available. Explicit procedures
are computationally the simplest since they do not require
(for a diagonal mass matrix) equation solving to advance
the solution. However, stability restricts the size of the
allowable time step. On the other hand, unconditional
stability can usually be achieved in implicit procedures,
which require solution of a system of equations at each
time step. For the problem at hand, a purely explicit
procedure is not usually appropriate because of the
unreasonably stringent time-step restriction resulting from
the presence of the very stiff fluid in the mixture (even for
highly nonlinear solid material models). The methods used
combine the attractive features of explicit and implicit
integration (see, e.g., Refs. [24,25]), and fall under the
category of ‘‘split operator methods’’. Different portions of
the system of equations are treated implicitly and explicitly,
reducing the system of equations to be solved. The specific
split to be made is obviously problem dependent, and the
appropriate implicit/explicit splits for the problem at hand
are detailed in Refs. [20,21].
2.4. Nonlinear iterations
At each step the resulting nonlinear algebraic problem is
solved by Newton–Raphson type iterations. Quasi-Newton
procedures with line searches are typically used.
3. Modeling seismic-induced soil liquefaction
Liquefaction (as defined by Castro and Poulos [26]) is a
phenomenon wherein a saturated sand subjected to
monotonic or cyclic shear loads looses a large percentage
of its shear resistance and flows in a manner resembling a
liquid. Marcuson [27] defines liquefaction as the transfor-
mation of a granular material from a solid to a liquefied
state as a consequence of increased pore-water pressure
and reduced effective stress. This phenomenon occurs most
readily in loose to medium dense granular soils that have a
tendency to compact when sheared. In saturated soils,
pore-water pressure drainage may be prevented due to the
presence of silty or clayey seam inclusions, or may not have
time to occur due to rapid loading—such as in the case of
seismic loads. In this situation, the tendency to compact is
translated into an increase in pore-water pressure. This
leads to a reduction in effective stress, and a corresponding
decrease of the frictional shear strength. If the EPWP
generated at a certain location in a purely frictional soil
(e.g., sand) reaches the initial value of the effective vertical
stress, then, theoretically, all shear strength is lost at that
location and the soil liquefies and behaves like a viscous
fluid. Liquefaction-induced large ground deformations are
a leading cause of disasters during earthquakes. EPWPs are
induced also in moderately dense to dense granular
materials subjected to cyclic loads, but due to their
tendency to dilate during shear, the softening is only
temporary leading to increased cyclic shear strains, but not
to major strength loss and large ground deformations [28].
This phenomenon is known as cyclic mobility.
Two important aspects need to be addressed in a
rigorous seismic analysis of saturated soils involving
EPWP build-up: (1) coupled analysis: solid and fluid
coupled field equations have to be used in a step-by-step
(time domain) dynamic analysis to correctly capture the
inertial and dissipative coupling terms; and (2) soil
constitutive model: accurate simulation of dynamically
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650

4. induced EPWP build-up and continuous softening of the
material requires soil models able to reproduce the
experimentally observed nonlinear hysteretic behavior and
shear stress-induced anisotropic effects, and to reflect the
strong dependency of plastic dilatancy on effective stress
ratio. Advanced plasticity models, such as multi-yield or
bounding surface plasticity, in combination with kinematic
hardening rules, can offer a material representation of
considerable power and flexibility (e.g., [29]). As such soil
constitutive models usually require a relatively large number
of parameters, a well-defined methodology for calibrating
those parameters based on results of standard in situ and/or
laboratory soil tests is also desirable.
The first aspect is addressed in the DYNAFLOW code
by the extension of Biot’s theory into the nonlinear regime,
discussed in Section 2. Nonlinear dynamic constitutive
behavior of saturated soil under partially drained condi-
tions is modeled using a kinematic hardening, multi-yield
constitutive model based on a simple plasticity theory [30].
The yield function is described in the principal stress space
by a set of nested rounded Mohr–Coulomb yield surfaces.
A non-associative plastic flow rule is used for the
dilatational component of the plastic deformation. The
model has been tailored (1) to retain the extreme versatility
and accuracy of the simple multi-surface J2 theory in
describing observed shear nonlinear hysteretic behavior
and shear stress-induced anisotropic effects, and (2) to
reflect the strong dependency of the shear-induced dila-
tancy on the effective stress ratio. Accurate simulation of
shear-induced plastic dilation and of hysteretic effects
under cyclic loading, together with full coupling between
solid and fluid equations, allows capturing the build-up
and dissipation of pore-water pressures and modeling the
gradual softening and hardening of soil materials. The
required constitutive model parameters can be derived
from the results of conventional laboratory (e.g., triaxial,
simple shear [31]) or in situ (e.g., standard penetration,
cone penetration, wave velocity [32]) soil tests. The multi-
yield plasticity soil constitutive model, its implementation
algorithm, and the methodology for estimating the
constitutive model parameters have been repeatedly
validated in the past for soil liquefaction computations,
based on both centrifuge experimental results (e.g.,
[31,33,34]) and full-scale measurements (e.g., [35]).
Practical applications of the extension of Biot’s theory
into the nonlinear regime, including its implementation in
DYNAFLOW, are presented in the remaining of this paper
through results of two studies involving simple structures
on homogeneous or non-homogeneous saturated soil
deposits subjected to seismic loads.
4. Example 1: tower structures on liquefiable soil
Any soil–structure system has its own characteristic
frequency, which depends on material properties, geome-
try, degree of saturation of the soil, and contact conditions
at the soil–foundation interface. This characteristic fre-
quency may decrease during dynamic excitation due to
degradation of the soil effective shear modulus as a result
of pore pressure build-up and/or large shear strains. Any
mechanical system is more sensitive to dynamic loading as
its characteristic frequency becomes closer to the frequency
range corresponding to the maximum spectral values of the
excitation. Consequently, both the frequency content of
seismic excitation and the evolution of structural frequency
characteristics can make a significant difference in the
dynamic response of soil–structure systems.
Several centrifugal studies of seismic behavior of tower
structures on soil foundation are reported, among which:
Morris [36] and Weissman and Prevost [37]—tower struc-
tures on dry soil, Madabhushi and Schofield [18]—tower
structures on saturated soil. Several tests with different input
seismic acceleration have been performed to study the effect
of characteristic frequency. Madabhushi and Schofield [18]
concluded that, when the predominant frequency of the
input seismic motion was lower than the characteristic
frequency of the system, the characteristic frequency of the
system decreased during shaking to a value close to the
predominant input seismic frequency due to build up of
excess pore pressure and subsequent degradation of soil
stiffness. Popescu [13] performed nonlinear dynamic finite
element analyses of an embankment dam on liquefiable
foundation and concluded that the interplay between the
frequency content of the seismic motion, the vibration
characteristics of the structure and the possible evolution of
those characteristics during shaking had significant influence
on the predicted dynamic response.
In this example, the behavior of two different tower
structures (with different characteristic frequencies but
with the same mass) resting on a homogeneous liquefiable
soil deposit is investigated for a range of intensities of the
seismic input motion. The saturated soil is modeled as a
two-phase porous medium, and the finite element calcula-
tions are conducted in terms of effective stress, using fully
coupled solid–fluid equations. The soil and structures
analyzed in this study correspond to those used by
Madabhushi and Schofield [18] in a series of centrifuge
experiments. The earthquake motion used here spans a
wide range of seismic intensities and is different from that
used in the centrifuge experiments.
The main purpose of this study is to investigate in detail
the changes in soil strength induced by EPWP build-up,
leading to changes in the vibration characteristics of a
soil–structure system during and after the earthquake. It is
shown how those changes and their interplay with the
frequency content of the seismic excitation affect: (1) the
dynamic excitation of the structure and (2) the amount of
seismic energy transmitted to the system and, eventually,
the structural response.
4.1. Finite element model and analysis method
Two different structures are analyzed. Tower 1 with
a characteristic frequency of 4.8 Hz, corresponding to a
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5. two-story building, and Tower 2 with a characteristic
frequency of 1.4 Hz, corresponding to a seven-story
building. The dynamic characteristics of both structures
are assumed to be dominated by their first mode of
vibration, and the structures are idealized as single degree
of freedom systems. The plane strain assumption is used in
the finite element analyses, and the foundation is idealized
as a 10 m wide continuous strip footing. The factor of
safety for bearing capacity under static conditions and
assuming local bearing capacity failure is about 20 for both
structures. In the numerical analyses, the structure material
is idealized as linear elastic. The soil, assumed saturated
below the foundation (ground water table at 1 m depth), is
idealized as a two-phase porous material and discretized
into four-node quadrilateral continuum two-phase porous
elements with four degrees of freedom per node (two for
the solid phase and two for the fluid phase displacements).
The soil constitutive parameters are listed in Table 1.
Use of a rigid box in an experimental earthquake
simulation induces a series of spurious seismic waves that
are generated by the motion of each end wall of the box
and are subsequently reflected back by the opposite wall.
This phenomenon, which may induce a behavior in the
model different from the real field, is partly attenuated by
placing a certain amount of duxseal (a relatively soft
material) at each end wall of the box to create absorbing
boundaries. It is assumed that those phenomena (wave
reflection due to simulation of rigid end walls and
subsequent attenuation when modeling the presence of
duxseal) are reproduced by the numerical model. Duxseal
has been used in the centrifuge experiments to prevent
seismic wave reflection from the lateral boundaries of the
rigid centrifuge box and thus helping the soil deposit to
behave like a semi-infinite medium [18]. This material is
considered also in the numerical analyses, and idealized as
linear elastic. The material properties assumed for duxseal
and structure are listed in Table 2. The finite element mesh
is shown in Fig. 1.
The seismic acceleration is applied at the base and lateral
boundaries of the analysis domain, same as in the
centrifuge experiment performed in a rigid box. The base
input accelerations are generated using a procedure for
simulation of non-stationary stochastic processes [38]
capable of generating seismic ground motion time histories
that are compatible with prescribed response spectra and
have a prescribed modulating envelope for amplitude
variation. Two hundred acceleration time histories have
been generated to be compatible with the response spectra
type-1 and type-3 recommended by the Uniform Building
Code [39] and corresponding to different soil conditions.
All 100 generated acceleration time histories in each group
are compatible to the corresponding response spectrum,
ARTICLE IN PRESS
Table 1
Parameters of the multi-yield plasticity model, and values used for the saturated soil in Example 1
Constitutive parameter Symbol Value Type
Mass density—solid rs
2710 kg/m3
State parameters
Porosity nw
0.43
Hydraulic conductivity k 0.0002 m/s
Low strain elastic shear modulus G0 20 MPa Low strain elastic parameters
Poisson’s ratio n 0.33
Power exponent n 0.5
Friction angle at failure f 381 Yield and failure parameters
Maximum deviatoric strain (comp/ext) edev
max
0.08/0.06
Coefficient of lateral stress k0 0.426
Stress–strain curve coefficient a 0.21
Dilation angle c 321 Dilation parameters
Dilation parameter Xpp 0.2
Table 2
Constitutive parameters for linear elastic materials used in Examples 1 and 2
Constitutive parameter Duxseal
(Examples 1 and 2)
Tower structures
(Example 1)
Structurea
(Example 2)
ESB boxb
Dural Rubber
Mass density (kg/m3
) 1650 3530 3200 3953 3953
Young’s modulus (MPa) 8 73,000 73,000 73,000 158
Poisson’s ratio 0.46 0.33 0.33 0.33 0.45
a
Mass calculated to obtain a pressure on soil equivalent to the one in the centrifuge experiments.
b
Mass density for both materials and Young’s modulus of rubber have been fitted to obtain equivalent vibration and deformability characteristics with
those of the ESB box used in the centrifuge experiments.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665
652

6. and consequently have similar frequency content. They
also have similar duration of the strong motion. Aside
from the above common characteristics, each individual
acceleration time history is different from the others in the
same group (e.g., the peak values occur at different time
instants in each sample), as each time history constitutes a
different sample of the target non-stationary stochastic
process. The accelerations in each group have been scaled
to a range of Arias intensities [40], IA ¼ 0:1922:1 m=s,
corresponding to a range of peak ground accelerations,
PGA ¼ 0:120:4 g. The duration of seismic motion con-
sidered here is about 15 s and the total analysis time is 40 s.
4.2. Analysis results
Contours of EPWP ratio with respect to the initial
effective vertical stress calculated for one case (Tower 1
subjected to type-3 acceleration sample #51, with IA ¼
1:12 m=s and PGA ¼ 0:25 g) are shown in Color Plate 1 (at
the end of the paper). The soil in the free field is predicted
to liquefy, but, due to increased overburden pressure, the
soil below the structure does not liquefy. There is, however,
significant pore-water pressure build-up and subsequent
softening of the soil in this area, leading to large
settlements and rotations of the structure. The numerical
model also reproduces gradual dissipation of pore-water
pressure after the end of the earthquake (Color Plate 1c).
Color Plate 2 illustrates the predicted failure mechanism
for the case presented in Color Plate 1. At time T ¼ 5 s, the
average EPWP ratio in the area below the structure is
about 75% (Color Plate 2b), and the corresponding
reduction in effective stress leads to a factor of safety for
local bearing capacity failure equal to 1. The failure
mechanism illustrated by the maximum shear strain
contours in Color Plate 2a is characteristic for a local
bearing capacity failure, consisting of an active wedge
below the structure, clearly shown by the shear strain
contours, and fading transition and passive wedges. The
failure wedges are not symmetric, due to the presence of
lateral inertial forces.
Due to softening of soil below the structure, the seismic
motion is strongly attenuated, especially for large Arias
intensities (IA), and the acceleration at the base of the
structure has significantly less specific energy compared to
the input. In Fig. 2, it is shown how at low seismic intensity
(sample #1 with IA ¼ 0:19 m=s) the seismic energy is
entirely transmitted to the structure (Figs. 2a and b), while
for high seismic intensities (sample #90 with IA ¼ 1:84 m=s,
where the soil liquefies), the input seismic energy is largely
attenuated by the soil (Fig. 2c), especially for the high
frequency components of the base motion (Fig. 2d). For
the second case, it can be noted from Fig. 2c how the
acceleration amplitudes at the base of the structure are
similar to those of the input motion for the first 2–3 s of
shaking, when the pore-water pressures in the soil are still
small. After that, as the soil starts to soften due to build-up
ARTICLE IN PRESS
Fig. 1. Seismic analysis of a tower structure: finite element mesh.
-1
0
1
m/s2
0 5 10 15 20
-3
-2
-1
0
1
2
3 input
base of struct.
m/s2
Time (sec)
(c)
(a)
(b)
(d)
10-1
100
101
10-2
100
Sample # 1
Sample # 1
10-1
100
101
10-2
100
Period (sec)
Response
Spectrum(m/s
2
)
input
base of struct.
Sample # 90
Sample # 90
Fig. 2. Comparison between acceleration time histories (a, c) and response spectra (b, d) used as input at the base of the mesh and computed at the base of
the structure for two samples of type-1 acceleration used for Tower 1 (corresponding to two different Arias intensity values).
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665 653

7. of EPWP, the maximum accelerations transmitted to the
structure base become significantly lower than the ones
input at the base of the analysis domain. It can be noted
from the computed acceleration response spectrum shown
in Fig. 2d that the high frequency components of the
seismic acceleration are predominantly attenuated, while
the energy of the low frequency components that are closer
to the natural frequency of the liquefied soil deposit is not
very much affected by the seismic wave propagation
through soil. This is also evident from the low frequency
sinusoidal shape exhibited by the computed acceleration
time history at the base of the structure (red line in the
electronic version and darker line in the paper version in
Fig. 2c). The shape of this computed acceleration time
history is similar to accelerations recorded at shallow depth
in a landfill that liquefied during the Hyogoken-Nambu
Earthquake (e.g., [41]).
It is mentioned that the computed reduction in accel-
eration amplitude at the base of the structure for large
Arias intensities of the input seismic motion should not be
misinterpreted as a safer situation for the case in Fig. 2c
(sample #90 with IA ¼ 1:84 m=s) than for the case in
Fig. 2a (sample #1 with IA ¼ 0:19 m=s). Even if structure
accelerations are in general smaller for sample #90 than for
sample #1, the structure shaken by sample #90 is predicted
to fail due to bearing capacity failure of the liquefied soil,
while the structure shaken by sample #1 is predicted to
be safe.
The effects of earthquake intensity and loading rate on
the settlement and rotation of the rigid foundation for the
two types of structures are presented in Fig. 3 in terms of
fragility curves, which are an illustrative and practical way
of expressing the probability of exceeding a certain degree
of structural response as a function of load intensity.
Fragility curves have been used extensively in earthquake
engineering to describe the seismic vulnerability of
structures as a function of the severity of the seismic event
(e.g., [42,43]). The procedure used here for establishing the
fragility curves for foundation settlements and rotation
follows that proposed by Shinozuka et al. [42]. After
selecting a specific response threshold, all earthquake
intensities IA for which the response exceeds this threshold
are assigned to the unity probability level, while all the
other intensities are assigned to the zero probability level.
Then, a shifted lognormal distribution function is fitted to
those points using the maximum likelihood method.
Fragility curves can include effects of multiple sources of
uncertainty related to material resistance or load char-
acteristics. The curves used in this example reflect only
uncertainty in the seismic ground motion.
The curves presented here express the probability of
exceeding certain thresholds in the response, namely
foundation settlement of 20 cm in Fig. 3a and base
rotations of 1/300 in Fig. 3b. Those results represent
earthquake effects in terms of rigid motion of the structure.
From both figures it can be concluded that, for the same
level of seismic intensity, type-3 acceleration has stronger
effect on both settlements and rotations than type-1
acceleration. This is due to a change in vibration
characteristics of the soil to fundamental frequencies close
to the dominant frequency values of type-3 acceleration.
The same behavior has also been observed by Madabhushi
and Schofield [18] in the centrifuge experiments. The initial
characteristic periods of the soil–structure system (before
the earthquake) are T1 ¼ 0:6 s, for Tower 1, and
T2 ¼ 0:8 s, for Tower 2. During the earthquake, due to
significant increase in pore-water pressure and subsequent
soil softening, this characteristic period goes up. For most
cases analyzed here, when the soil liquefies in the free field
and the EPWP increases significantly below structure, the
characteristic period of the soil–structure system for both
Towers 1 and 2 goes up to about 1.5 s. Consequently, the
seismic energy input at the base of the soil layer is
delivered, for most of the earthquake duration, to a system
that has a characteristic period of about 1.5 s. As shown in
Fig. 4, this value is closer to the range of maximum spectral
amplitudes of type-3 acceleration than to those of type-1,
and this results in significantly more energy delivered at the
resonance frequency to the soil–structure system when
acted by the type-3 input than when acted by type-1 input.
ARTICLE IN PRESS
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Arias Intensity (m/s)
Prob.
that
rotation
exceeds
1/300
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
Arias Intensity (m/s)
Prob.
that
settlement
exceeds
20cm
Tower1 - Type 1
Tower2 - Type 1
Tower1 - Type 3
Tower2 - Type 3
Tower1 - Type 1
Tower2 - Type 1
Tower1 - Type 3
Tower2 - Type 3
(a) (b)
Fig. 3. Comparison between analysis results in terms of fragility curves: (a) settlements and (b) rotations (type-1 and type-3 refer to the corresponding
UBC response spectra, and Towers 1 and 2 refer to the two structures considered).
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665
654

8. For example, for the case illustrated in Fig. 4, more than
two times more energy is delivered at the resonance
frequency by the type-3 input acceleration than by the
type-1 input. It is mentioned that when computing the
characteristic period before the earthquake and at various
time instants during the dynamic analysis, the coupling
effects in Eqs. (1) and (2) have been neglected.
It can be noticed from Fig. 3 that, for type-1 input
motion, the fragility curves calculated for Tower 1 are
located to the left of the ones calculated for Tower 2. This
means that a lower value of seismic intensity corresponds
to a given probability of seismic damage for Tower 1 than
for Tower 2. Apparently Tower 2, more flexible, can
attenuate some part of the seismic energy by structural
deflections. It is also interesting that, when subjected
to seismic motion type-3, where the degree of soil
liquefaction is higher than for type-1 input, the computed
responses of both structures are similar, as both structures
behave as rigid blocks compared to the deformability of
liquefied soil.
5. Example 2: effects of stochastic soil variability on
liquefaction
5.1. Results from previous studies
Natural soil properties vary from one point to another,
even within so-called ‘‘uniform’’ soil layers. In addition to
inducing uncertainty in the computed response, natural
spatial variability of soil properties within geologically
distinct layers affects the mechanical behavior of geotech-
nical structures. For example, for phenomena involving the
presence of a failure surface (such as encountered in
landslides or in bearing capacity failures) the actual failure
surface can deviate from its theoretical position to pass
selectively through weaker soil zones and thus the average
mobilized strength is reduced when compared to that of a
corresponding uniform soil (e.g., [44,45]). For the case of
seismically induced soil liquefaction, it was shown (e.g.,
[14]) that a larger amount of EPWP is generated in a
heterogeneous soil than in the corresponding uniform soil
having geomechanical properties equal to the average
properties of the variable soil.
Experimental evidence of the effects of soil heterogeneity
on liquefaction resistance is presented in Refs. [46,47].
Budiman et al. [46] performed a series of undrained cyclic
triaxial tests on sand specimens containing up to 25%
gravel inclusions. They found that the liquefaction
resistance of sand with inclusions was lower than that of
uniform sand. Moreover, the reduction in liquefaction
resistance was more pronounced for samples with a higher
content of gravel. Konrad and Dubeau [47] conducted
undrained cyclic triaxial tests on fine Ottawa sand and on
silica silt. The study consisted of three series of tests: the
first two series were conducted to characterize the cyclic
strength of the sand and of the silt, respectively. The third
series was performed on layered samples, consisting of one
horizontal silt layer sandwiched between two sand layers.
The study results revealed that layering induced a much
lower cyclic resistance than that developed in either of the
materials in uniform samples.
Pioneering work in numerical analysis of the effects of
soil heterogeneity on liquefaction resistance was presented
by Ohtomo and Shinozuka [48]. Later on, Popescu [32] and
Popescu et al. [14,15] conducted a systematic study of those
effects using the multi-yield surface constitutive model
implemented in DYNAFLOW. A Monte-Carlo simulation
(MCS) method, combining digital generation of non-
Gaussian stochastic vector fields representing the spatial
distribution of various soil properties over the analysis
domain with dynamic effective stress finite element
analyses, was used to quantify the effects of soil hetero-
geneity on the amount and pattern of EPWP build-up in a
saturated soil deposit subjected to seismic loads. It was
shown that, for the same average values of soil parameters,
more EPWP build-up was predicted in stochastic analyses,
accounting for soil heterogeneity, than in so-called
‘‘deterministic analyses’’, assuming uniform soil properties.
Characteristic percentiles of soil strength were proposed for
use in deterministic analyses, resulting in a response similar
to that predicted by more expensive stochastic analyses.
Effects of the degree of soil variability and of the seismic
loading rate were also investigated. Fenton and Vanmarcke
[49] addressed the three-dimensional (3D) aspect of
liquefaction of randomly variable soils through an analysis
of liquefaction potential at the Wildlife Site, Imperial
Valley, California. The soil properties were modeled as a
3D random field; however, because of limited computa-
tional resources, the liquefaction analysis was carried out
in 1D vertical columns, without any coupling in the
horizontal plane. Consequently, the study could only
address the pointwise initiation of liquefaction, and did
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10-1
100
101
10-1
100
101
Period - T (sec)
Spectral
acc.
-Sa
(m/s
2
)
UBC-Type 1
UBC-Type 3
2.1m/s2
1.0m/s2
T1
=1.5s
Fig. 4. Comparison of the spectral acceleration values of the two types of
spectra used for the input motion at the characteristic period of the
soil–structure system during shaking. This characteristic period increases
to the value of about T1 ¼ 1:5 s from the initial values of 0.6 and 0.8 s.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665 655

9. not include the effects of 3D pore-water pressure redis-
tribution before and after initial liquefaction. Another
analysis of the same site was presented by Elkateb et al.
[50]. A 3D stochastic analysis was performed, but the
liquefaction potential was assessed separately at each point
in space using an empirical method (e.g., [51,52]). Although
this approach could not capture any interaction between
zones of soil with different liquefaction strengths and
different amounts of pore-water pressure build-up, it
provided some very interesting conclusions on the effect
of loose soil zones on the probability of liquefaction
failure. It was found that liquefaction assessments using
mean values of soil properties (e.g., from in situ CPT
results) are not able to capture the effect of loose soil
pockets within the soil mass and consequently are on the
unsafe side. Popescu et al. [17] performed MCSs including
fully coupled 3D dynamic finite element analyses to
investigate the effects of true 3D soil variability on
liquefaction of soil deposits. They concluded that 2D plane
strain stochastic analyses can provide sufficiently accurate
information for liquefaction strength assessment of hor-
izontally layered soil deposits, while liquefaction-induced
differential settlements can only be captured accurately by
a 3D analysis, with correct simulation of pore-water
migration. Koutsourelakis et al. [53] and Popescu et al.
[16] analyzed the effects of spatial variability on soil
liquefaction for a wide range of earthquake intensities
using again MCSs, and presented fragility curves for simple
structures on liquefiable soil.
To illustrate the effects of soil heterogeneity on
seismically induced EPWP build-up, some of the results
obtained in Ref. [14] for a loose to medium dense soil
deposit, with geomechanical properties and spatial
variability characteristics estimated based on a series of
piezocone test results from a hydraulic fill deposit
are reproduced in Fig. 5. The results in Fig. 5b show
ARTICLE IN PRESS
20 one - phase finite
480 two - phase finite elements
for porous media (4 dof/node)
with dimensions: 3m x 0.5m
Finite elements
analysis set-up
free field boundary
conditions
elements (2 dof/node)
12.0m
1.6m
Water table
Vertical scale is two
times larger than
horizontal scale
60.0m
(a)
(b)
(c)
Excess pore pressure ratio predited using six samples functions of a stochastic vector field with
cross-correlation structure and probability distribution functions estimated from piezocone test results
Grey scale range
for the excess pore
pressure ratio (u/σv0)
0.90...1.00
0.70...0.90
0.50...0.70
0.00...0.50
Sample function # 1 Sample function # 2 Sample function # 3
Sample function # 6
Sample function # 5
Sample function # 4
Excess pore pressure ratio predicted using deterministic input soil parameters
Deterministic - average 50 - percentile 60 - percentile
90 - percentile
80 - percentile
70 - percentile
(u/σv0
)max = 0.44
´
´
Fig. 5. Monte-Carlo simulations of seismically induced liquefaction in a saturated soil deposit, accounting for natural variability of the soil properties
(after [14]): (a) finite element analysis setup; (b) contours of EPWP ratio for six sample functions used in the Monte-Carlo simulations; (c) contours of
EPWP ratio using deterministic (i.e., uniform) soil parameters and various percentiles of soil strength.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665
656

10. computed contours of EPWP ratio with respect to the
initial effective vertical stress for six sample functions
of a stochastic field representing six possible realizations of
soil properties over the analysis domain (see Ref. [14] for
more details on the soil properties). Soil liquefaction
(EPWP ratio larger than approximately 0.9) was predicted
for most sample functions shown in Fig. 5b. Analysis of an
assumed uniform soil deposit, with strength characteristics
corresponding to the average strength of the soil samples
used in MCSs, resulted in no soil liquefaction (the
maximum predicted EPWP ratio was 0.44). It can be
concluded from these results that both the pattern and
the amount of dynamically induced EPWP build-up
are strongly affected by the spatial variability of soil
properties. For the same average values of the soil
parameters, more EPWP build-up was predicted in the
stochastic analysis (MCS), accounting for spatial varia-
bility, than in the deterministic analysis, assuming uniform
soil properties.
Popescu et al. [14] suggested that in order to predict
more accurate values of EPWP build-up in a deterministic
analysis assuming uniform soil properties, one has to use a
modified soil strength which is lower than the average
strength of the variable soil. The results presented in
Fig. 5c illustrate the idea of a ‘‘characteristic percentile’’
of soil strength for dynamic liquefaction analysis of
randomly heterogeneous soils. This is a percentile of
the field-recorded soil strength values which is lower
than the average value. The characteristic percentile is
selected in such a way that, when used in a deterministic
dynamic analysis assuming uniform soil strength, it
results in a computed EPWP ratio equivalent to that
obtained from the stochastic analysis accounting for
soil spatial variability. For example, for the soil properties
and earthquake intensity considered in Ref. [14],
the resulting characteristic percentile was somewhere
between 70% and 80% (Fig. 5c). A comparison between
the results presented in Figs. 5b and c also shows clearly
the differences in the predicted liquefaction pattern of
natural heterogeneous soil vs. that of assumed uniform
soil.
The effects of soil spatial variability on structures
founded on liquefiable soil deposits are illustrated in
Fig. 6 (from Ref. [16]). The structure is the same as Tower
2 presented in the previous example, founded on
either heterogeneous soil, or on corresponding uniform
soil, with the same strength as the average strength of
the variable soil. It is mentioned that the soil properties
for uniform soil, as well as the lateral boundary conditions,
are different in the study presented in Fig. 6 from those
used in Example 1. Fragility curves expressing the
exceedance probability of two thresholds in the response,
namely foundation settlements larger than 20 cm and
base rotations larger than 1/150, are plotted in Figs. 6a
and b, respectively. As both the spatial distribution of
soil properties and the input seismic motion vary
randomly from one sample to another, the curves shown
in Fig. 6 reflect uncertainties in both spatial distribution
of soil strength and in the seismic ground motion.
The fragility curves are plotted for two types of seismic
input (Types 1 and 3 discussed in Example 1), and for
two assumptions related to soil properties: (1) variable
soil, with properties described by a random field with
correlation distances yH ¼ 8 m in horizontal direction
and yV ¼ 2 m in vertical direction, and a coefficient
of variation (CV) ¼ 0.5 (see Ref. [16] for more details)
and (2) corresponding uniform soil. For all cases
analyzed, the fragility curves calculated for variable
soil are shifted to the left as compared to the ones
for uniform soil, indicating that, for a given exceedance
probability, a significantly lower seismic intensity is
needed when soil variability is accounted for than for an
assumed uniform soil. For example, from Fig. 6a, for
type-1 input acceleration, a 50% exceedance probability
of 20 cm foundation settlement is predicted at Arias
intensity of the seismic motion IA ¼ 1:7 m=s for uniform
soil, and at IA ¼ 1:0 m=s when the soil variability is
accounted for.
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Fig. 6. Comparison between analysis results for variable soil and
equivalent uniform soil, for Tower 2 in Example 1 (after [16]): (a)
settlements and (b) rotations. The foundation conditions considered in this
analysis are different from those discussed in Example 1.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665 657

11. 5.2. Mechanisms of liquefaction of heterogeneous soils
It was postulated by Popescu et al. [14] that presence of
loose pockets would lead to earlier initiation of EPWP and
local liquefaction than in a corresponding uniform soil.
After that, the pressure gradient between loose and dense
zones would lead to water migration, softening, and
liquefaction of the dense sand. This assumption that
presence of loose pockets is responsible for lower liquefac-
tion resistance was theoretically verified by two types of
numerical results. First, for a given average value of the soil
strength and a given earthquake intensity, the resulting
amount of EPWP build-up in a variable soil was increasing
with the assumed degree of variability of soil properties,
expressed by the CV, which also controls the amount
of loose pockets in the soil mass (e.g., [15]). Second, for
the same CV of spatial variation, more EPWP was
predicted when the soil strength fluctuations followed
a symmetrical probability distribution function (PDF)—
e.g., truncated Gaussian—than for a positively skewed
PDF—e.g., lognormal [54]. For the same mean value,
a symmetrical PDF has a fatter left tail (indicating a
larger amount of loose pockets) than a positively
skewed PDF.
Several practical guidelines for assessment of liquefac-
tion potential and for design of structures on liquefiable
soil with spatially varying properties resulted from
the numerical studies discussed before. DYNAFLOW
and the multi-yield plasticity constitutive model have
been repeatedly validated for liquefaction assessment
based on centrifuge experimental results and field observa-
tions. All those validation studies were based on determi-
nistic analyses, assuming uniform soil properties in
horizontal direction. To validate the geotechnical design
recommendations for liquefaction of heterogeneous soil
mentioned above, the capability of the numerical model
to capture the real behavior of spatially variable soil
also needs to be verified based on experimental data. Such
a verification is performed as part of this numerical
example.
There are very few experimental studies dealing with
liquefaction of spatially variable soil. The two studies
mentioned in Section 5.1 were based on undrained cyclic
triaxial tests for samples made of different soils (Budiman
et al. [46] used sand and gravel, and Konrad and Dubeau
[47] used sand and silt). A detailed analysis of the behavior
of non-homogeneous soils was performed by Chakrabortty
et al. [55] using DYNAFLOW. They reproduced the cyclic
undrained triaxial tests on sandwiched samples made of
dense sand and silt layers described in Ref. [47], and
analyzed the mechanism by which a sample made of two
different soils liquefies faster than each of the soils tested
separately in uniform samples. The explanation, resulting
from a detailed analysis of the numerical results, was that
water was squeezed from the more deformable silt layer
and injected into the neighboring sand, leading to
liquefaction of the dense sand.
Regarding liquefaction mechanisms of soil deposits
involving the same material, but with spatially variable
strength, Ghosh and Madabhushi [19] performed a series
of centrifuge experiments to analyze the effects of localized
loose patches in a dense sand deposit subjected to seismic
loads. They observed that EPWP is first generated in the
loose sand patches, and then the water migrates into the
neighboring dense sand, reduces the effective stress and
softens the dense soil that can liquefy. As discussed before,
it is believed that a similar phenomenon is responsible for
lower liquefaction resistance of heterogeneous soil than
that of the corresponding uniform soil (e.g., [14]). There-
fore, this phenomenon is analyzed in details in the
following numerical example, based on the backanalysis
of Ghosh and Madabhushi’s centrifuge experiments,
and on a study for a hypothetical chess-board like
heterogeneous soil. This is an original study aimed
at: (1) improving understanding of the mechanisms
of liquefaction of spatially variable sandy soil deposits
and (2) verifying if the nonlinear Biot model implemented
in DYNAFLOW is able to capture experimentally
observed phenomena including water migration from loose
to dense sand zones, with subsequent softening and
liquefaction of the entire soil mass.
A detailed description of the centrifuge tests analyzed in
this example is presented in Ref. [19]. Most of the
experimental results are posted on the WWW [56]. The
general layout of the tests discussed here—structure on
dense sand deposit with included horizontal loose sand
layer (test BG4) and included loose sand patch (test
BG5)—is sketched in Fig. 7a, along with transducer
locations and the finite element mesh used in the back-
analyses. The location of the loose sand layer in test BG4 is
indicated in Fig. 7a by two thick horizontal lines, and the
loose sand patch in test BG5 is shown by the orange
rectangular area below the structure in the electronic
version (darker area in the paper version). The centrifuge
container used in the experiments was an equivalent shear
beam (ESB) box, ideally matching the stiffness of a soil
column. The vibration and deformability characteristics of
the Cambridge centrifuge ESB box are presented in details
in Ref. [57]. Those characteristics have been reproduced for
plane strain analysis in the finite element model used here.
A duxseal padding placed between soil and the lateral walls
of the box in the centrifuge model was also included in the
numerical model, as shown in Fig. 7a. Duxseal properties,
as well as the properties of the ESB box materials (dural
and rubber), all modeled as linear-elastic materials, are
listed in Table 2. The base input accelerations used in the
two tests [56] are presented in Figs. 7b and c.
The soil properties at two relative densities used in the
centrifuge experiments (Dr ¼ 85% for the dense sand and
Dr ¼ 45% for the loose sand) have been estimated based
on information provided in Ref. [19] and on the back-
analysis of test BG4. Those soil properties have been
subsequently checked in a backanalysis of test BG5. A
comparison of numerical results obtained in this study with
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R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665
658

12. ARTICLE IN PRESS
Fig. 8. Recorded (after [19]) and computed excess pore-water pressure time histories at three locations: below the structure (P2 and P5) and in the free field
(P6) (transducer locations are shown in Fig. 7). The calculations are done using the acceleration time histories shown in Figs. 7b and c for tests BG4 and
BG5, respectively. The figure is plotted at model scale.
Fig. 7. Finite element simulation of tests BG4 and BG5: (a) finite element mesh, (b) input acceleration for test BG4, and (c) input acceleration for
test BG5.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665 659

13. centrifuge experimental records in terms of EPWP at three
locations presented in Fig. 8 shows acceptable match. The
soil constitutive parameters are listed in Table 3.
To analyze in detail the mechanism of water migra-
tion between loose and dense sand observed in the
experiments, and to have a better basis for comparison,
both centrifuge tests have been numerically simulated
with DYNAFLOW using the same base input acceleration,
presented in Fig. 7c. Some results in terms of EPWP
ratio with respect to the initial effective vertical stress
are presented in Fig. 9. Evolution of EPWP ratio at
neighboring locations in test BG4 below the structure
and in the free field are presented in Figs. 9a and b,
respectively. In those figures, P4 and P7 are located in
the loose sand layer, while P5 and P6 are located in dense
sand. While some pore-water pressure dissipation is
predicted to occur after the earthquake in the loose
sand, increase in pore-water pressure after the end of
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Table 3
Parameters of the multi-yield plasticity model used for the saturated soil in Example 2
Constitutive parameter Symbol Dr ¼ 45% Dr ¼ 65% Dr ¼ 85%
Mass density—solid (kg/m3
) rs
2650 2650 2650
Porosity nw
0.45 0.425 0.40
Hydraulic conductivity (m/s) k 1.35 104
1.1 104
8 105
Low strain elastic shear modulus (MPa) G0 12 16 20
Poisson’s ratio n 0.25 0.25 0.25
Power exponent n 0.5 0.5 0.5
Friction angle at failure (deg.) f 39 42.5 45
Maximum deviatoric strain (comp/ext) edev
max
0.07/0.04 0.07/0.04 0.07/0.04
Coefficient of lateral stress k0 0.7 0.7 0.7
Stress–strain curve coefficient a 0.182 0.182 0.182
Dilation angle (deg.) c 32 32 32
Dilation parameter Xpp 0.05 0.02 0.006
(c)
(a)
(b) (d)
P7
P6
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
Test BG4
Time (sec)
EPWP
Ratio
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
Test BG5
Time (sec)
EPWP
Ratio
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
Test BG4
Time (sec)
EPWP
Ratio
P4
P5
P4
P5
Fig. 9. Computed excess pore-water pressure (EPWP) ratios at pairs of transducers, located one in dense sand and one in loose sand: (a) test BG4, below
structure; (b) test BG4, free field; (c) test BG5, below structure; (d) transducer locations. The calculations are done using the acceleration time history in
Fig. 7c as input motion for both tests.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665
660

14. shaking is simulated at the locations in dense sand, due
to migration of water from loose sand. The same trend
is observed in test BG5—Fig. 9c shows computed
EPWP ratios at two neighboring locations below
the structure (P4 in the loose sand patch and P5 in dense
sand).
Fig. 10 compares computed EPWP ratios at the same
location (transducer P6 in dense sand in the free field)
for tests BG4 and BG5. An increase in residual EPWP
ratio from about 0.2 in test BG5 to about 0.4 in test
BG4 is attributed to water migration from the loose
sand, present in the free field close to location P6 in
test BG4, but not present in the free field in test BG5.
The effects of water migration with gradual increase in
EPWP in test BG4 as compared to test BG5 can be
observed starting at time T ¼ 20 s. Simulation of water
migration from loose to dense sand by the numerical model
is also illustrated in Fig. 11 by computed time histories
of water displacements (relative to the solid phase) at
two locations at the border between loose and dense sand
in test BG5.
To quantify the effects of alternating loose and dense soil
patches, as encountered in natural soil deposits, a
hypothetical soil–structure system (Fig. 12) is analyzed
next. The soil is made of a chess-board like pattern of
loose (Dr ¼ 45%) and dense (Dr ¼ 85%) sand zones.
It is mentioned that natural soils exhibit a smooth
(gradual) variation in relative density between loose
and dense zones, while the example analyzed here
assumes sudden variation in soil properties. The reason
for selecting this chess-board like layout is that sudden
changes in soil properties may amplify the effects of
spatial variability, leading to easier result analysis and
understanding of the underlying mechanisms. To avoid
any boundary effects, free field boundary conditions
are imposed in the following calculations (corresponding
solid and fluid nodal displacements for all pairs of
nodes located at the same elevations on the lateral
boundaries of the mesh are identical). The input motion
used in this example corresponds to the acceleration
time history in Fig. 7c divided by a factor of 2 to simulate
a small to moderate intensity earthquake. For compa-
rison, three other analyses are performed for a structure
ARTICLE IN PRESS
0 10 20 30 40 50
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time (sec)
EPWP
Ratio
Test BG4-P6
Test BG5-P6
Fig. 10. Comparison of computed excess pore-water pressure (EPWP)
ratios in the free field (dense sand) for tests BG4 and BG5. The
calculations are done using the acceleration time history in Fig. 7c as input
motion for both tests.
0 10 20 30 40 50
-5
0
5
10
15
Time (sec)
Relative
water
movement
[water-solid]
(mm)
Vertical displacements at Node A
Horizontal displacements at Node B
Fig. 11. Computed relative movement of water with respect to the solid
phase at the interface between loose and dense sand in test BG5. Node
locations are shown in Fig. 9d. Positive displacements are upward
(vertical) and left to right (horizontal).
Fig. 12. Finite element mesh and distribution of loose and dense soil patches for a hypothetical soil–structure system analyzed in Example 2.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665 661

15. founded on an assumed uniform soil deposit, for three
different relative densities: dense (Dr ¼ 85%) and loose
(Dr ¼ 45%), corresponding to the patches in the chess-
board example, and medium dense (Dr ¼ 65%), correspon-
ding to the average relative density of the heterogeneous
soil.
Some results in terms of EPWP ratio time histories
computed in the free field are presented in Fig. 13. Fig. 13a
shows the evolution of the EPWP ratio in two adjacent
elements of the chess-board variable soil, one located
in a loose soil patch (element 1—shown in Fig. 12) and
the other in a dense one (element 2). The effects of
water migration are evident here: the dense sand in the
chess-board deposit reaches about 95% EPWP ratio,
while at the same location in a uniform dense sand deposit
the maximum EPWP ratio is about 25% (compare
Figs. 13a and c). Figs. 13b and c show computed EPWP
ratios in the free field for the chess-board soil deposit and
for the other three cases with assumed uniform soil. It is
apparent from those results that the excess pressures
calculated in the heterogeneous soil (at both loose and
dense sand locations) are closer to the ones for uniform
loose sand than to those calculated in the uniform medium
dense sand. Computed structure settlements shown in
Fig. 13d also indicate a significantly softer behavior of the
heterogeneous soil than that of the corresponding uniform
soil (i.e., Dr ¼ 65%). Color Plates 3 and 4 present contours
of EPWP ratio at time T ¼ 20 s and of maximum shear
strains at the end of analysis, respectively. From both color
plates it is obvious that the heterogeneous soil response
(Color Plates 3a and 4a) is much closer to the response of
the uniform loose sand (Color Plates 3c and 4c) than to
that of the uniform medium dense sand (Color Plates 3b
and 4b).
6. Summary and conclusions
The theory of dynamics of saturated porous media, its
extension into the nonlinear regime, and its implementa-
tion in the DYNAFLOW code have been presented, along
with numerical applications illustrating the capabilities of
the numerical model to deal with complex phenomena,
such as seismic behavior of structures on liquefiable soil
deposits. The numerical results are validated through
comparisons with centrifuge experimental data.
The frequency content of seismic motion has a major
role in predicted structural response. In the first numerical
example, it is shown how the characteristic frequency of the
soil–structure system is reduced to values close to the
dominant frequency range of type-3 inputs, due to soil
softening after build-up of pore pressures. Consequently,
considerably more damage is predicted for this type of
seismic motion. The results of this study also point out to
important attenuation of the earthquake motion when
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0 10 20 30 40 50
0.25
0.2
0.15
0.1
0.05
0
Time (sec)
Settlement
(m)
Dense sand
Medium dense sand
Chess-board
Loose sand
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
0.8
1
Element 1
Time (sec)
EPWP
Ratio
Loose sand
Chess-board
Medium dense sand
Dense sand
(b)
(a)
(c) (d)
Settlement at base of structure
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
0.8
1
Time (sec)
EPWP
Ratio
Element 2
(Dense sand)
Element 1
(Loose sand)
0 10 20 30 40 50
-0.2
0
0.2
0.4
0.6
0.8
1
Element 2
Time (sec)
EPWP
Ratio
Medium dense sand
Chess-board
Loose sand
Dense sand
Fig. 13. (a) EPWP ratio time histories for the chess-board type heterogeneous soil computed at two adjacent locations in the free field (one in the loose
soil—element 1, and the other in the dense soil—element 2); (b) EPWP ratio in the free field in element 1; (c) EPWP ratio in the free field in element 2; (d)
computed settlements of the structure. The four curves in (b)–(d) correspond to results from four different analyses defined in the text. Element locations
are shown in Fig. 12.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665
662

16. seismic waves pass through soft soil after considerable
increase in pore-water pressure. The reduction in specific
seismic energy was found to range between zero, for
small earthquake intensities, to about 75%, for large
earthquake intensities (IA about 2 m/s and PGA about
0.4 g). As expected, the numerical model predicts higher
attenuation for the high frequency components of the base
motion.
In the second numerical example, it was mentioned that
soil heterogeneity leads to more EPWP build-up than
that predicted for corresponding uniform soil and, for
structures founded on liquefiable soils, to reduction in
average mobilized soil strength due to deviation of
the failure surface to pass through looser soil zones.
Those effects result in larger predicted damage when
accounting for soil spatial variability, as compared to
assuming a corresponding uniform soil. The mechanism
of liquefaction of heterogeneous soils was also studied in
this example through backanalysis of centrifuge tests
including loose soil zones in a dense sand deposit. From
the experimental results, it was concluded that the
pore pressure gradient created from the initial liquef-
action of loose zones leads to migration of water in the
adjacent dense zones and makes the dense soil liable to
liquefaction. Several numerical examples were presented
that confirmed the capabilities of the numerical model to
reproduce those experimentally observed phenomena
and validated previous research results and design recom-
mendations for liquefaction of stochastically spatially
variable soils.
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Plate 4. (Example 2) Computed maximum shear strain contours and deformed meshes at the end of analysis (time T ¼ 50 s) for four different soil
structure systems, defined as: (a) chess-board distributed loose and dense patches; (b) uniform medium dense soil (Dr ¼ 65%); (c) uniform loose soil
(Dr ¼ 45%); (d) uniform dense soil (Dr ¼ 85%).
Plate 3. (Example 2) Computed EPWP ratio contours at time T ¼ 20 s for four different soil structure systems, defined as: (a) chess-board distributed
loose and dense patches; (b) uniform medium dense soil (Dr ¼ 65%); (c) uniform loose soil (Dr ¼ 45%); (d) uniform dense soil (Dr ¼ 85%).
Plate 1. (Example 1) EPWP ratio contours for Tower 1, seismic motion
type-3, sample #51 (IA ¼ 1:12 m=s). Deformation magnification
factor ¼ 5.
Plate 2. (Example 1) Tower 1, seismic motion type-3, sample #51 at time
T ¼ 5 s: (a) contours of maximum shear strains and deformed mesh and
(b) contours of EPWP ratio.
R. Popescu et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 648–665 663

17. Acknowledgments
The financial support provided by NSERC under
Research Grant No. RG203795-02 and by NSF under
Grant No. 0075998 is gratefully acknowledged. The
authors are also indebted to Dr. B. Ghosh for providing
some of the results of centrifuge experiments analyzed in
Example 2.
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