Finite Element Modeling of Shallow Foundations on Nonlinear
Soil Medium
Authors:
Jian Zhang, University of California, Los Angeles, zhangj@ucla.edu
Yuchuan Tang, University of California, Los Angeles, bytang@ucla.edu
ABSTRACT
This paper investigates the dynamic response of shallow foundations on linear and
nonlinear soil medium using finite element method. The study was motivated by the need
to develop macroscopic foundation models that can realistically capture the nonlinear
behavior and energy dissipation mechanism of shallow foundations. An infinitely long
strip foundation resting on soil half-space is analyzed in depth to evaluate the dependence
of its dynamic responses on various parameters, e.g. foundation width, material
properties, input motion amplitude and frequency etc. Special attentions are paid to
choose appropriate domain scale, mesh size and boundary conditions so as to minimize
the often observed numerical oscillations when the outgoing waves are contaminated by
the reflecting waves at boundaries. Such judicious choice results in an excellent
agreement between the finite element analysis and the analytical solution of strip
foundation on linear soil half-space. Closed-form formulas are developed to describe the
frequency-dependent linear dynamic stiffness of strip foundation along both horizontal
and vertical directions. Various nonlinear constitutive models of soil, which exhibit the
yielding and kinematic hardening behavior of soil, are implemented in this study to
evaluate the dynamic stiffness of strip foundation sitting on nonlinear soil medium. The
finite element analyses reveal the strong dependency of response on input motion
amplitude, frequency and yielding of soil. A nonlinearity indicator is developed to
incorporate the combined effects of initial elastic stiffness, yielding stress and excitation
amplitude. The numerical analyses presented here provide improved understanding on the
nonlinear behavior and energy dissipation mechanism of shallow foundations under
dynamic loads.
INTRODUCTION
It is recognized that the dynamic responses of structures with flexible foundations are
affected by the nonlinear dynamic behavior of individual components as well as the
interaction between them, i.e. the soil-structure interaction effects. These interaction
effects are often characterized by changing stiffness and energy dissipation through either
hysteretic or radiation damping [1]. They can be represented by frequency dependent
dynamic stiffness, which subsequently provides information on equivalent spring and
dashpot constants of foundations.
Various analytical models are available to describe the dynamic stiffness of shallow
foundations of different shapes on elastic soil medium, e.g. strip foundation on elastic
half-space [2,3] and on visco-elastic soil layer [4,5], circular foundation on elastic half-
space [6,7] and on visco-elastic half-space [8], rectangular foundation on elastic half-
space or layered medium [9,10], and cylindrical and rectangular embedded foundations
[11]. Gazetas [12] and Mylonakis et al. [13] compiled an extensive set of graphs and
tables for dynamic stiffness of foundations with a variety of geometries and linear soil
conditions. Despite the abundance of analytical solutions for shallow foundations on
linear soil medium, very limited work has been reported on the dynamic stiffness of
shallow foundations on nonlinear soil medium [13]. The nonlinearity of soil has caused
reduced stiffness and modified energy dissipation mechanism. As pointed out by Borja
and his co-workers [14,15], the local yielding in an otherwise homogeneous elastic soil
half-space tends to reduce the radiation damping and create resonance frequencies.
In this study, finite element method is adopted to compute the dynamic response of an
infinitely long strip foundation resting on an elastic and inelastic half-space. Numerical
results from finite element method are compared with the theoretical solution of strip
foundation on elastic half-space so as to provide guidance on choosing appropriate
domain scale, mesh size and boundary condition for correct modeling of the wave
propagation in a half-space. Closed-form formulas are developed to describe the
frequency-dependent linear dynamic stiffness of strip foundation along both horizontal
and vertical directions. Subsequently, the strip foundation on nonlinear soil medium is
analyzed. Nonlinear soil models that exhibit yielding and kinematic hardening are
implemented based on a simple procedure derived from widely available shear modulus
reduction curves. The dynamic stiffness is evaluated and effects of foundation width,
input motion amplitude and frequency, and development of soil nonlinearity are
quantified by a newly developed nonlinearity indicator. The numerical results showed
here revealed that the energy dissipation through radiation damping of nonlinear soil is
significantly reduced due to localized yielding zone in soil.
DYNAMIC STIFFNESS OF STRIP FOUNDATION ON ELASTIC SOIL
Consider an infinitely-long rigid strip foundation sitting on elastic soil half-space subject
to harmonic excitations, as shown in Figure 1. Its dynamic stiffness can be obtained
analytically [2,3]. Under a harmonic motion, the reacting forces are related to
displacements by the general form shown below:
⎭
⎬
⎫
⎩
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
+
+
=
⎭
⎬
⎫
⎩
⎨
⎧
)(
)(
i0
0i
)(
)(
2222
1111
tU
tU
dc
dc
G
tP
tP
h
v
h
v
π
(1)
FIGURE 1
FOUNDATION GEOMETRY AND EXCITATION CONDITIONS
U
v0
sinωt
U
h0
sinωt
soil half-plane
finite domain
absorbin
g
boundar
y
2H
D
2b
FIGURE 2
FINITE DOMAIN AND ABSORBING BOUNDARY
where G is the shear modulus of soil, πG(c
11
+id
11
) and πG(c
22
+id
22
) are the dynamic
stiffnesses in vertical and horizontal directions respectively. The force-displacement
relationship in (1) is analogous to that of a spring-dashpot system with spring constant
πGc
11
(or πGc
22
) and dashpot coefficient πGd
11
/ω (or πGd
22
/ω). The dynamic stiffness
parameters c
11
, d
11
, c
22
, d
22
depend on both the frequency of excitation and soil properties
and are often plotted against dimensionless frequency a
0
= ωb/v
s
for a given Poisson’s
ratio with b as the half-width of strip foundation and v
s
as the shear wave velocity of soil.
Finite element method is used in this study to conduct the dynamic analyses of strip
foundation under harmonic displacement excitation in vertical and horizontal directions
respectively. The soil half-space is represented by a finite domain where an absorbing
boundary condition needs to be present to correctly model the outgoing waves of an
infinite medium (Figure 2). Maximum element size, boundary conditions and scale of the
finite domain dominate the accuracy of finite element analysis. Previous research [16,17]
suggested that the maximum element size l
max
should satisfy
max
11
~
85
lL
⎛⎞
≤
⎜⎟
⎝⎠
(2)
where L is shear wave length. For a given finite element mesh, (2) equivalently puts an
upper limit on the applicable dimensionless excitation frequency a
0
:
max
0
2
5
1
~
8
1
l
b
a
π
⎟
⎠
⎞
⎜
⎝
⎛
≤
(3)
An absorbing boundary is also required to simulate the waves transmitting outward in
a half-space. Either viscous damping boundary [18] or infinite element boundary [19] can
be used for this purpose. However, their implementation in ABAQUS results in
unexpected numerical oscillations as observed in Figure 3, where the dynamic stiffness
parameters c
11
and c
22
are plotted for a finite domain of D=H=10b with absorbing
boundaries. Similar oscillation has been observed for a even larger domain of D=H=20b.
To eliminate the oscillation, a large finite domain is needed so that the steady state
response can be achieved before the wave reflection at boundary contaminates the
response [14]. For this purpose, the scale of the finite domain needs to satisfy
a
0
c
11
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
viscous boundary
infinite element
Hryniewicz1981
(a)
a
0
c
22
0
0.1
0.2
0.3
0.4
0.5
0.6
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Hryniewicz1981
viscous boundary
infinite element
(b)
FIGURE 3
DYNAMIC STIFFNESS OF STRIP FOUNDATION COMPUTED BY FEM (WITH NUMERICAL OSCILLATION) AND
ANALYTICAL SOLUTION FOR HORIZONTAL (a) AND VERTICAL (b) DIRECTIONS
rp
LnTv
≤
(4)
where
L
r
is the length of the shortest wave reflection path within the finite domain, v
p
is
the longitudinal wave velocity,
T is the period of harmonic excitation, n is the number of
periods from beginning of excitation which includes one full cycle of steady state
response. Substituting dimensionless frequency
a
0
into (4) for period T yields the lower
bound on the applicable dimensionless excitation frequency
0
21
22
2 a
L
b
n
r
≤
−
−
⋅
ν
ν
π
(5)
A finite mesh of H=250m, D=250m, l
max
=1.25m (refer to Figure 2) was set up for a
strip foundation of half-width b=1m on elastic soil medium of v
s
=201.5m/s, ν=0.25,
ρ=1600kg/m
3
. Equation (5) gives lower bound of excitation frequency as a
0
≥0.04 while
(3) gives upper bound of excitation frequency as a
0
≤1.0. For input frequency within this
range, the numerical oscillation is well eliminated. Figure 4 compares the dynamic
stiffness parameters computed by finite element and the theoretical solution given by
Hryniewicz [3], where an excellent agreement is achieved.
The dynamic stiffness of strip foundation depends on foundation width, Young’s
modulus and Poissson’s ratio of soil. The first two parameters can be incorporated by
using dimensionless frequency a
0
. Simplified formulas are developed to characterize the
effects of Poisson’s ratio as shown in (6) to (9). These simplified formulas showed
excellent agreement with finite element results as shown in Figure 5, where the family of
(b)
a
0
d
11
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.
0
Hryniewicz1981
ABAQUS
(a)
c
11
a
0
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.2 0.4 0.6 0.8 1.0
Hryniewicz1981
ABAQUS
(c)
c
22
a
0
0.0
0.1
0.2
0.3
0.4
0.5
0.00.20.40.60.81.0
Hryniewicz1981
ABAQUS
(d)
d
22
a
0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Hryniewicz1981
ABAQUS
FIGURE 4
D
YNAMIC STIFFNESS OF STRIP FOUNDATION FOR VERTICAL (a,b) AND HORIZONTAL DIRECTIONS (c,d)