ARTICLE IN PRESS
Ocean Engineering 34 (2007) 247–260
www.elsevier.com/locate/oceaneng
Analysis of seabed instability using element free Galerkin method
J.G. Wanga,, M.R. Karima, P.Z. Linb
a
b
Tropical Marine Science Institute
Department of Civil Engineering National University of Singapore 10 Kent Ridge Crescent, Singapore 119260 Singapore
Received 6 September 2005; accepted 11 January 2006
Available online 19 April 2006
Abstract
Wave-induced seabed instability, either momentary liquefaction or shear failure, is an important topic in ocean and coastal
engineering. Many factors, such as seabed properties and wave parameters, affect the seabed instability. A non-dimensional parameter is
proposed in this paper to evaluate the occurrence of momentary liquefaction. This parameter includes the properties of the soil and the
wave. The determination of the wave-induced liquefaction depth is also suggested based on this non-dimensional parameter. As an
example, a two-dimensional seabed with finite thickness is numerically treated with the EFGM meshless method developed early for
wave-induced seabed responses. Parametric study is carried out to investigate the effect of wavelength, compressibility of pore fluid,
permeability and stiffness of porous media, and variable stiffness with depth on the seabed response with three criteria for liquefaction. It
is found that this non-dimensional parameter is a good index for identifying the momentary liquefaction qualitatively, and the criterion
of liquefaction with seepage force can be used to predict the deepest liquefaction depth.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Momentary liquefaction; Shear failure; Wave loading; Pore water pressure; Element-free Galerkin method; Parametric study
1. Introduction
In recent years, wave-induced seabed responses have
engrossed growing interests not only in coastal engineering
but also in geotechnical engineering. This is because seabed
instability may have caused the damage and destruction of
some coastal and offshore installations, such as breakwaters, piers, pipelines and so on (Sumer et al., 2001; Jeng,
2003). Two types of seabed instability may occur in a sandy
seabed: momentary liquefaction and shear failure. Waveinduced momentary liquefaction was reported in laboratory tests (Sekiguchi et al., 1995; Sassa and Sekiguchi, 1999,
2001) and in field observations (Zen and Yamazaki, 1990;
Sakai et al., 1992). Computational results (Madsen, 1978;
Yamamoto et al., 1978; Okusa, 1985; Hsu and Jeng, 1994;
Jeng and Seymour, 1997; Jeng, 2001) also revealed the
potential of momentary liquefaction. If momentary liquefaction occurs routinely in shallow waters, serious stability
Corresponding author. Tel.: +65 6516 6591; fax: +65 779 16 35.
E-mail addresses: [email protected], [email protected]
(J.G. Wang).
0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2006.01.004
problems have to be confronted for the structures laid on a
cohesionless seabed. Because momentary liquefaction and
shear failure are directly related to the excess pore pressure
and effective stresses within seabed sediments, prediction of
seabed responses and evaluation of seabed instability have
become important issues in coastal and ocean engineering.
Liquefaction and shear failure are produced by different
mechanisms (Zen et al., 1998; Jeng, 2001). The liquefaction
is a state that effective stress in any direction becomes zero.
For example, quick sand or boiling is closely related to
vertical seepage flow. When water wave propagates over a
seabed, the fluctuation of water pressure exerts on the
seabed surface, causing pore fluid in the seabed to flow out
or into and producing frictional force on soil skeletons (this
frictional force is called as seepage force). The wave train
generates the fluctuation of pore water pressure, and thus
the transient fluctuation of effective stress in soil masses. If
the effective stress momentarily becomes zero, soil skeleton
loses its structural strength and the seabed becomes
momentarily liquefied. In addition, shear stresses in a
seabed may be big enough to overcome its shear resistance,
resulting in another type of seabed instability, shear failure.
ARTICLE IN PRESS
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J.G. Wang et al. / Ocean Engineering 34 (2007) 247–260
A shear failure refers to a state that stress level reaches to
shear failure envelope which is usually described by
Mohr–Coulomb criterion.
Wave-induced seabed response can be evaluated by Biot
consolidation theory (Biot, 1941) and Verruijt’s storage
equation (Verruijt, 1969). Analytical and numerical methods have been employed to solve the Biot consolidation
equation. Analytical solutions are usually available for
those problems with simple boundary conditions. For
example, Madsen (1978) investigated a hydraulically
anisotropic and partially saturated seabed, whilst Yamamoto et al. (1978) studied an isotropic seabed with infinite
thickness. Okusa (1985) used the compatibility equation
under elastic conditions and reduced the sixth-order
governing equation of Yamamoto et al. (1978) to a
fourth-order differential linear equation. Yamamoto
(1981) developed a semi-analytical solution for a nonhomogeneous layered porous seabed. Later, Hsu and Jeng
(1994) and Jeng and Hsu (1996) further extended the
framework to a finite-thickness seabed as well as a layered
seabed (Hsu et al., 1995).
Several numerical algorithms were proposed to accumulate complex geometry and physical conditions. For
example, Thomas (1989) developed a semi-analytical onedimensional finite element model to simulate the waveinduced stresses and pore water pressure. This method was
later extended to 2D and 3D wave-seabed interaction
problems (Jeng and Lin, 1996; Jeng, 2003). Gatmiri (1990)
developed a simplified finite element model for an isotropic
and saturated permeable seabed. Recently, a meshless
EFGM model was developed for the analysis of transient
wave-induced soil responses (Karim et al., 2002). In order
to improve the computation efficiency, a radial PIM
method (Wang and Liu, 2002a, b; Wang et al., 2002) was
extended for wave-induced seabed responses by introducing repeatability conditions (Wang et al., 2004). This
radial PIM considers not only sinusoidal waves but also
other nonlinear waves such as solitary wave. These
numerical meshless methods provide a useful tool for
further analysis of seabed instability.
Seabed instability is a complicated topic in marine
geotechnics (Poulos, 1988). Past studies have revealed that
both soil characteristics and wave properties play a
dominant role in the wave-induced seabed responses (Jeng,
2003). Some parameters are proposed to assess the
potential of momentary liquefaction. For example, Sakai
et al. (1992) proposed two parameters to justify the
occurrence of liquefaction based on a boundary layer
theory (Mei and Foda, 1981). They concluded that the
maximum depth of liquefaction is about half the wave
height for surf conditions. However, no single parameter is
available to evaluate the momentary liquefaction. Furthermore, the parameters so far are obtained for a homogeneous porous seabed. Is it suitable for a nonhomogeneous seabed? Seabed is usually non-homogeneous
and its shear modulus of seabed increases with soil depth
such as Gibson soil (Jeng, 2003). This paper will also
explore the effect of depth-variable modulus on seabed
responses.
This paper proposes a non-dimensional parameter to
study the most critical condition governing seabed instability due to momentary liquefaction. This single parameter
includes both wave characteristics and seabed properties.
Numerical examples are studied to verify its effectiveness
through a meshless EFGM model as well as three criteria of
liquefaction. It is noted that this EFGM model has been
critically examined against available analytical and/or semianalytical methods (Karim et al., 2002). A parametric study,
in the perspective of wave-induced soil instability, is carried
out to examine the sensitivity of soil and wave properties on
seabed responses. The shear failure in one-wave period is
briefly discussed, too. The focus is mainly on the potential of
wave-induced momentary liquefaction and liquefaction
depth. This paper is organized as follows: The criteria of
seabed instability, momentary liquefaction and shear failure,
are first discussed in Section 2. Then Biot consolidation
equation and its EFGM meshless method are briefly
introduced in Section 3. A non-dimensional parameter
which includes both soil and wave properties is proposed to
identify the potential of momentary liquefaction in Section
4, and this parameter is validated with a finite-thickness
seabed and three criteria for liquefaction in Section 5.
Parametric study is carried out in Section 6. Parameters
include wavelength, fluid compressibility or degree of
saturation, soil permeability and Young’s modulus, as well
as variable shear modulus along depth. Mohr cycles of
effective stress status within one wave period and shear
failure mechanisms are discussed in Section 7. Finally, the
conclusion and remarks are given.
2. Seabed instability under wave loading
Wave-induced momentary liquefaction and shear failure are
caused by different mechanisms. When a sandy seabed is
subjected to cyclic wave loading, the effective stresses and pore
water pressure fluctuate with the propagation of waves. When
the effective stress attains to some critical value, momentary
liquefaction or shear failure may occur. This section will
discuss the criteria of liquefaction and the shear failure.
2.1. Liquefaction
As discussed in the Introduction, liquefaction is a state
where a soil loses its structural strength and behaves like a
fluid, producing large deformation and the evolution of
seabed such as ripples (Ourmieres and Chaplin, 2004).
Three criteria have been proposed in various reports to
assess momentary liquefaction.
Criterion 1: Soil is liquefied when vertical effective stress
becomes zero (Yamamoto, 1981):
s0zz Xgb z,
s0zz
(1)
stands for wave-induced vertical effective stress,
where
gb ¼ ðgs gw Þ is for the effective unit weight of soil and z
ARTICLE IN PRESS
J.G. Wang et al. / Ocean Engineering 34 (2007) 247–260
denotes the soil depth beneath seabed surface, in which gs
and gw are the unit weights of soil and pore fluid,
respectively. Eq. (1) indicates that liquefaction occurs
when seepage force lifts above soil column and soil
particles are no more in contact.
Criterion 2: Liquefaction occurs when wave-induced
effective volumetric stress in soil becomes identical or
larger than the initial in situ effective volumetric stress
(Okusa, 1985; Tsai, 1995):
s0vol Xs0vol0
(2)
where s0vol is the wave-induced effective volumetric stress,
and s0vol0 the initial in-situ effective volumetric stress. These
stresses are defined as
1
s0vol ¼ ð1 þ nÞðs0zz þ s0xx Þ;
3
s0vol0 ¼
1 ð1 þ nÞ
g z,
3 ð1 nÞ b
(3)
where s0xx stands for wave-induced horizontal effective
stress and n is the Poisson ratio.
Criterion 3: Liquefaction may occur if upward seepage
force is equal or larger than overburden load. This criterion
is mathematically expressed as (Zen and Yamazaki, 1990)
ðPz P0 ÞXgb z,
(4)
where Pz is the pore pressure at the depth z and P0 the pore
pressure amplitude at the seabed surface.
2.2. Shear failure
The ratio of shear stress to normal stress is an important
parameter for investigating the potential of wave-induced
shear failure. Shear failure occurs when stress angle f
becomes equal to or greater than the angle of internal
friction fu (Yamamoto, 1981):
fXfu
For sandy soils, the stress angle f is calculated by
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
ððsxx szz Þ=2Þ2 þ s2xz
1 @
A.
f ¼ sin
ððsxx þ szz Þ=2Þ
(5)
(6)
The effective stress has two components: initial in situ
effective stress and wave-induced effective stress. That is,
szz ¼ s0oz þ s0zz , sxx ¼ s0ox þ s0xx . sxz is the shear stress.
Initial in-situ effective vertical stress is s0oz ¼ gb z, and initial
in situ horizontal effective stress s0ox is related to vertical
effective stress s0oz by
n (7)
s0ox ¼ K 0 s0oz ¼
g z.
1n b
3. Biot consolidation theory and variational formulations
3.1. Governing equations
A seabed soil is assumed to be elastic, isotropic and
homogeneous. Biot consolidation equation (Biot, 1941)
and Verruijt’s storage equation (Verruijt, 1969) have been
249
applied to describe the wave-induced response of a mixture
of compressible pore fluid and compressible porous seabed.
These two equations are expressed as follows:
Dq2 u þ qs p þ b ¼ 0,
qs
qu
qp k
nb q2s p ¼ 0,
qt
qt gw
(8)
(9)
where u is the displacement, p the pore water pressure, b the
body force vector, n the porosity of soil skeleton, k the soil
permeability, and gw the unit weight of pore fluid, and t as
real time. The constitutive law of soils is given by
s0 ¼ Dqu
(10)
For a plane strain problem, the material matrix D, and
operators qs and q are given by
2
3
1n
n
0
6
7
E
6 n
1n
0 7
D¼
6
7,
ð1 þ nÞð1 2nÞ 4
1 2n 5
0
0
2
3
2
q
2 3
0
7
6 qx
q
7
6
7
6
7
6
q 7
6 qx 7
6
ð11Þ
q¼6 0
7; qs ¼ 6 7.
qz 7
4 q 5
6
7
6
4 q
qz
q 5
qz qx
The compressibility of pore fluid b is a function of the
degree of saturation Sr, the bulk modulus of fluid Kf, and
the absolute fluid pressure Pa, such as (Okusa, 1985)
b¼
1
1 Sr
þ
.
Kf
Pa
(12)
The repeatability conditions are used to implement
periodic temporal and spatial conditions (Karim et al.,
2002; Wang et al., 2004). Incorporating these virtual
boundaries together with physical boundary conditions, a
modified variational formulation was proposed for the
EFGM model whose final discrete system equation is as
½R½S tþ1 ¼ ½F þ ½Q½St .
(13)
A Crank–Nicholson scheme is adopted to discretize the
time domain in Eq. (13). Appendix A gives the definitions
of various coefficient matrices in Eq. (13).
3.2. Moving least-square approximation
Excess pore water pressure and displacements in the Biot
consolidation theory are approximated by moving least
square (MLS) approximants (Lancaster and Salkauskas,
1981). The MLS approximant uh ðxÞ for a function uðxÞ has
the following form:
uh ðxÞ ¼
n
X
I¼1
fI ðxÞuI ,
(14)
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J.G. Wang et al. / Ocean Engineering 34 (2007) 247–260
250
where n is the number of nodes I in the neighbourhood
of x for which the weight function wðxÞa0, and uI is the
nodal index of u at x ¼ xI . The shape function fI ðxÞ is
obtained as
one-dimensional and two-dimensional problems (Karim et
al., 2002). In situ soil condition is presented by a and avol as
follows:
fI ðxÞ ¼ sT A1 BI ,
a¼
(15)
where A ¼ sT wðxÞs and B ¼ sT wðxÞ. For a linear basis,
sj ðxÞ is as
sT ðxÞ ¼ ½ 1
x z in 2D space
(16)
Then the shape functions can be expressed as
fI ðxÞ ¼
m
X
sj ðxÞðA1 ðxÞBðxÞÞjI .
(17)
Weight function usually takes radial function as
wI ðxÞ wðx xI Þ ¼ wI ðd I Þ, where d I ¼ kx xI k is the
distance between two points xI and x. The size of the
influence domain of xl is defined as d mI ¼ d max d I , where
dmax is known as support size factor (usually taken as 2.5).
In this study, cubic spline is used to express the weight
function for different ranges within an influence domain:
for
for
for
dI
1
p ;
2
d mI
1
dI
o
p1;
2
d mI
dI
41:
d mI
ð18Þ
Because the shape function in Eq. (17) has no property
of Kronecker delta functions, Lagrange multiplier method
is employed to implement essential boundary conditions
(Belytschko et al., 1994).
(21)
A 2-D wave-induced transient problem is defined in
Fig. 1(a). The meshless model for soil domain is shown in
Fig. 1(b). This domain is discretized with regular distributed nodes (441 nodes) for function approximation and
regular background cells for integration. Domain integrals
use 4 4 Gauss points in each background cell and 4
Gauss points in each boundary integral cell. The bottom of
the soil domain is assumed to be rigid and impermeable:
qp
¼ 0.
qn
(22)
Ignoring the relative acceleration between water and soil
skeleton, the boundary conditions at the seabed surface are:
s0z ¼ 0;
sxz ¼ 0
and
p ¼ P0 cosðWx otÞ,
(23)
where W ¼ 2p=L is the wave number, o ¼ 2p=T the wave
frequency, x the horizontal coordinate, and t is the time. L is
the wavelength, and T the wave period. The amplitude P0 is
obtained from the linear theory of a monochromatic wave
(Madsen, 1976):
P0 ¼
gw H
,
2 cosh Wd w
o2 ¼ Wg tanh Wd w .
Wave-induced effective stresses at any point reach their
extremes when wave crest or trough goes directly over.
Momentary liquefaction is likely to occur under a wave
trough due to uplift seepage force on the soil skeleton. We
propose a non-dimensional parameter to assess the liquefaction potential for given soil and wave properties:
P0 nba 1 e2ah
,
gb ðmv þ nbÞ 1 þ e2ah
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
iogw ðmv þ nbÞ
ð1 þ nÞð1 2nÞ
; mv ¼
,
a¼ k
Eð1 nÞ
ð1 þ nÞ gb h
.
3ð1 nÞ P0
(24)
where H is the wave height and dw water depth. The dispersion
equation determines the relationship among o, W and dw:
4. Non-dimensional parameter for the evaluation of
liquefaction potential
k¼
avol ¼
5. Validation of parameter j for identifying liquefaction
potential
u ¼ 0;
wðx xI Þ
and
5.1. Meshless model and computation parameters
j¼0
8
>
2
dI 2
dI 3
>
>
þ4
> 4
>
>
3
d mI
d mI
>
>
>
>
<
4
dI
dI 2 4 dI 3
¼
4
þ4
>
3
3 d mI
d mI
d mI
>
>
>
>
>
>
>
>
>
:0
gb h
P0
(19)
(20)
where h is the thickness of seabed. o is the wave frequency.
This non-dimensional parameter k is the ratio of the slopes
for one-dimensional depth-wise effective stress profile to
the initial in-situ effective stress. It can be derived from
the analytical solutions of wave-induced soil responses for
(25)
Unless otherwise specified, parameters used in the computation are given in this section. In the fluid domain, the wave
conditions for a 5 s wave with height H ¼ 0.5 m in water
depth d w ¼ 4:86 m; such given P0 ¼ 1:656 kN=m2 and
wavelength L ¼ 30 m. In the soil domain, the parameters of
seabed soil are for soil thickness h ¼ 20 m, Young’s modulus
E ¼ 2:5 105 N=m2 , Poisson ratio n ¼ 0:3, porosity n ¼ 0:4,
and isotropic permeability k ¼ 2:5 102 m=s. The density of
the pore fluid and sea water is gw ¼ 10 kN=m3 and the fluid
compressibility of b ¼ 3 103 m2 =kN ðS r ¼ 0:97Þ. Time
step size taken is 0.0625 s for the whole wave period of 5 s in
the computation. In order to better express the results, a linear
normalization procedure as shown in Fig. 2 is carried out. The
effective stresses and excess pore water pressure are normalized
by P0 while the depth is normalized by the seabed thickness h.
It is noted that the seabed condition is ‘finite’ when hoL
(Hsu and Jeng, 1994).
ARTICLE IN PRESS
J.G. Wang et al. / Ocean Engineering 34 (2007) 247–260
251
(x, t) = H cos(a x − t)
2
H/2
X
Z
dw
Fluid domain
Γt
Γvl
h
Γvr
Soil domain
Γb
(a)
L
(b)
Fig. 1. Wave-induced seabed response problem and its meshless model.
′z z / P0
′z z
Liquefied zone
Liquefied zone
Z
b
Z
b
P0
=
z
h
Z
Original expression
Normalized expression
Fig. 2. Normalized expression of effective stress along depth.
5.2. Non-dimensional parameter k
When the line representing effective self-weight (called as
a-line) intercrosses with the effective stress lines, the soil
above the intercrossing point is regarded as liquefied and
the depth of crossing point is the liquefaction depth
because the liquefaction criterion of Eq. (1), (2) or (4) is
satisfied in this zone. At this time, the kX1. The parameter
k is obtained through changing fluid compressibility
(degree of saturation), Young’s modulus, and permeability
of seabed soil, respectively, based on those parameters
in Section 5.1. Fig. 3 shows that all liquefaction criteria
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J.G. Wang et al. / Ocean Engineering 34 (2007) 247–260
252
0.00
0.00
Non-dimensional κ
6.85
0.05
3.95
0.04
1.99
0.10
1.53
1.08
0.08
α
vol
0.15
0.41
0.20
Non-dimensional κ
z/h
z/h
0.88
0.07
0.12
2e-4
0.25
0.30
0.16
α=
0.20
0.0
(a)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.35
5
0.9
0.40
1.0
σzz′ / P0
0.0
0.1
0.2
0.3
0.4
= 3.1
6.85
3.95
1.99
1.53
1.08
0.88
0.41
0.07
2e-4
0.5
0.6
′ /P
σvol
0
(b)
0.00
0.04
0.08
z/h
3.95
1.99
0.16
1.53
0.20
1.08
0.88
0.24
0.41
0.07
Non-dimensional κ
6.85
0.12
α=
5
2e-4
0.28
0.0
0.1
0.2
0.3
0.4
(c)
0.5
0.6
0.7
0.8
0.9
1.0
(Pz-P0)/P0
Fig. 3. Implication of non-dimensional parameter k.
(Eqs. (1)–(4)) are satisfied if the value of k is equal to or
greater than unity. In other words, the seabed soil is
liquefied if kX1. Ideally, when ko1, the liquefaction may
not occur and precaution for protection may not be
necessary for the seabed or offshore structure against
possible damage induced by liquefaction. Fig. 4(a) and (b)
show the variations of soil responses with k at the depth of
1.5 m. The soil response enhances exponentially with k.
Therefore, the condition kX1 can be used as a criterion to
predict the vulnerability of momentary liquefaction without the details of the wave-induced soil response. Detail
analysis reveals that the three criteria mentioned in Eq.
(1)–(4) perform slightly different in the assessment of
liquefaction potential. Criterion 2 always predicts the
occurrence of liquefaction at the seabed surface because
it is expressed by volumetric effective stress including
horizontal effective stress. The parameter k is obtained
from a one-dimensional formulation, thus it does not take
the effect of horizontal effective stress into account,
consequently, the condition that kX1 might overestimate
the liquefaction zone at seabed surface.
5.3. Verification of numerical algorithm with centrifuge test
data
Experimental results of two centrifuge tests1 (Wang and
Lin, 2004) are used to verify the numerical algorithm in this
1
Centrifuge test was carried out by Cheng Chen for his Master thesis.
His work is appreciated.
paper. Table 1 lists the computational parameters obtained
from experiments. These parameters are used for meshless
method, and the analytical solutions. Fig. 5 is the
comparison of excess pore water pressure predicted by
meshless method, Madsen’s solution (1978), and Hsu and
Jeng’s solution (1994). The experimental data obtained by
Centrifuge tests are also plotted for comparison. They
generally agree well in the whole seabed whether the seabed
soil is fine sand or coarse sand. It is noted that the pore
water pressure predicted by meshless method is between
those of Madsen’s solution and Hsu and Jeng’s solution.
6. Parametric study for identifying liquefaction potential
6.1. Effect of wavelength
This section reports the effect of wavelength on soil
response. Wavelength varies from site to site. For example,
Yamamoto et al. (1978) took the design wavelength for
North Sea as 324 m, while Jeng (2003) used the value of
200 m in his analysis. In this study, the wavelength is
assumed to vary from 10 to 180 m which corresponds to a
reasonable range of wave periods (Demirbilek and Vincent,
2002) according to the dispersion equation of Eq. (25) at
the water depth of 4.86 m. Typical seabed response for
wavelengths of 40–100 m is shown in Fig. 6(a) for vertical
effective stress, Fig. 6(b) for volumetric effective stress, and
Fig. 6(c) for vertical seepage force. These curves have
almost the same slopes before maximum values, and the
slopes are not affected by the variation in wavelength L.
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J.G. Wang et al. / Ocean Engineering 34 (2007) 247–260
253
concluded that the most unstable bed thickness varied
between 0.20L (Yamamoto et al., 1978) and 0.25L
(Yamamoto, 1981). For a finite seabed, our results show
that maximum soil responses are more likely one-dimensional within some depth near surface. The depth increases
with wavelength until some value. When wavelength
exceeds this value, for example two times of seabed
thickness in our study, the maximum response is independent of wavelength. As indicated in Fig. 2, the maximum
liquefaction depth is determined by taking the intersection
Fig. 7 further compares the vertical effective stress for onedimensional and two-dimensional conditions with three
wavelengths (20, 40, and 160 m). Take vertical effective
stress as an example for detailed analysis. The waveinduced maximum vertical effective stress occurs at 0.08 h
for L ¼ 10 m, 0.143 h for L ¼ 20 m and 0.23 h for LX40 m.
Thomas (1995) obtained the maximum soil response at
0.15L depth if the seabed is deep enough, i.e., h L.
Yamamoto compared the North Sea data with his
analytical solution for a seabed in infinite thickness. He
0
0.8
At z = 1.5 m
-2
0.7
-4
σzz′ /P0
0.5
Depth (m)
0.6
Actual soil response
A2 + (A1 -A2)/(1 + exp((x-x0)/Δx))
0.4
Meshless method
Madsen's solution
Hsu & Jeng's solution
Centrifuge data
-6
-8
A1 = -0.09898; A2 = 0.76154
x0 = 2.1199; Δx = 1.1657
0.3
-10
0.2
Liquefied Zone
-12
0. 1
0.1
0
1
2
3
4
κ
(a)
5
6
7
8
0. 2
0. 3
0. 4
0. 5
9
0. 6
0. 7
0. 8
0. 9
1
P/ Po
(a)
0
0.5
At z = 1.5 m
-2
0.4
-4
Depth (m)
Actual soil response
0.3
σvol
′ /P
0
A2 + (A1 -A2)/(1 + exp((x-x0)/Δx))
0.2
A1 = -0.09505; A2 = 0.44201
-8
x0 = 1.87563; Δx = 0.96981
0.1
Meshless method
Madsen's solution
Hsu & Jeng's solution
Centrifuge data
-6
-10
0.0
Liquefied Zone
-12
0. 1
-0.1
0
1
2
3
4
(b)
κ
5
6
7
8
0. 2
0. 3
0. 4
(b)
9
0. 5
0. 6
0. 7
0. 8
0. 9
P/ Po
Fig. 5. Comparison of pore water pressure for different sandy seabeds. (a)
Fine sand, (b) coarse sand.
Fig. 4. Variation of effective stresses versus non-dimensional parameter k.
Table 1
Computation parameters for sand and wave
Seabed type
Parameters of sandy bed
n
Fine sand
Coarse sand
0.41
0.48
n
0.35
0.33
Wave parameters
E (Pa)
k (m/s)
7
3.4 10
2.7 106
4
3 10
2.5 102
b
8
4 10
0
h (m)
T (s)
dw(m)
12.5
12.5
5
5
5
5
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254
0.0
0.0
0.1
0.1
L = 40 m
L = 60 m
L = 80 m
L = 100 m
0.2
α=4
α=6
0.2
α
vol = 4
0.3
z/h
0.3
z/h
αvol = 6
L = 40 m
L = 60 m
L = 80 m
L = 100 m
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.0
0.1
0.2
(a)
0.3
0.4
0.5
0.6
0.7
0.8
0.7
-0.075
0.9
′ /P
σzz
0
0.000
0.075
0.150
0.225
0.300
0.375
0.450
′ /P
σvol
0
(b)
0.0
α=6
0.1
z/h
α=4
L = 40 m
L = 60 m
L = 80 m
L = 100 m
0.2
0.3
0.4
0.5
0.6
0.7
0.0
(c)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
(Pz-P0)/ P0
Fig. 6. Effect of wavelength on wave-induced soil response.
6.2. Effect of fluid compressibility
0.0
0.2
0.4
z/h
of the stress profile (s0zz , s0vol or (Pz–P0)) with the initial
effective stress line (a-line or avol-line). Therefore, onedimensional analysis may suffice to identify seabed
liquefaction, especially when Criteria 1 and 3 are considered. Because volumetric effective stress s0vol includes the
horizontal effective stress and is difficult to obtain
accurately close to seabed surface, Criterion 2 always
predicts the liquefied status at the surface. Within the zone
close to seabed surface, both momentary liquefaction and
shear failure may occur, and the later mechanism may turn
out to be more important.
1-D (Independent of L)
0.6
2-D (L = 20 m)
2-D (L = 60 m)
2-D (L = 160 m)
0.8
1.0
0.0
The degree of saturation has been recognized as a
dominant factor for the wave-induced seabed response.
Pore water in seabed soils is compressible due to gas bubbles
(Okusa, 1985; Thomas, 1989; Jeng and Lin, 1996). The
structure of an unsaturated marine soil can vary significantly
depending on the relative size of gas bubbles to soil particles.
The in-site degree of saturation of unsaturated marine
sediments normally lies on the range of 85–100% (Esrig and
Kirby, 1977; Pietruszczak and Pande, 1996). The compressibility b in Eq. (12) is assumed to vary from 0 to
1 102 m2/kN which corresponds to S r ¼ 1:020:9.
The maxima of s0zz , s0vol and (Pz–P0) increase as the
degree of saturation (Sr) decreases. Fig. 8 indicates the
effect of the degree of saturation on vertical seepage force.
It reveals that unsaturated soil is more vulnerable to
liquefaction. The mechanism for this fragility to liquefaction is complicated. Fluid compressibility increases the
0.2
0.4
σzz′ / P0
0.6
0.8
Fig. 7. Comparison of soil responses for 1-D and 2-D problems.
absorbing rate of wave energy in this surface zone. This
prevents the pore fluid pressure from infiltrating easily into
subsurface layers and produces a phase lag in pore pressure
response near the surface zone (see Fig. 9). This phase lag
increases when degree of saturation decreases. Due to this
phase difference, the soil response at any given time could
be greater than the initial load, thus enhancing the
possibility towards liquefaction. Therefore, wave-induced
soil response is sensitive to the fluid compressibility or
degree of saturation. Numerical results again reveal that
Criterion 2 overestimates liquefaction potential, and that
Criterion 3 or ðPz P0 ÞXgb z predicts deepest liquefaction
zone. Therefore, Criterion 3 is the most critical one.
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0.0
255
0.0
0.1
0.1
α=6
Sr = 0.90
α=
Sr = 0.97
0.2
4
0.2
k = 7.5e-3 m/s
k = 1e-2 m/s
k = 2.5e-2 m/s
k = 5e-2 m/s
z/h
z/h
Sr = 0.975
Sr = 0.98
0.3
0.3
Sr = 0.985
Sr = 0.99
Sr = 0.995
0.4
0.4
Sr = 1
0.0
0.1
k = 2.5e-1 m/s
k = 9.5e-1 m/s
0.5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
α=4
k = 7.5e-2 m/s
k = 1e-1 m/s
k = 1.5e-1 m/s
Sr = 0.99965
0.5
α=6
k = 2.5e-3 m/s
k = 5e-3 m/s
0.0
0.1
0.2
0.3
0.4
(Pz-P0)/ P0
Fig. 8. Effect of fluid compressibility on normalized seepage force.
1.0
Sr = 0.97
0.6
Sr = 0.975
0.0
0.8
0.9
1.0
0.1
α=6
0.2
E = 2.5e8 N/m2
r
Applied surface pressure
2
E = 1e7 N/m
E = 2.5e6 N/m2
0.4
-0.2
α=4
E = 2.5e7 N/m2
0.3
z/h
P/ P0
0.2
0.7
0.0
Sr = 0.98
Sr = 0.985
Sr = 0.99
Sr = 0.995
0.4
0.6
Fig. 10. Effect of soil permeability on normalized seepage force.
Sr = 0.90
0.8
0.5
(Pz- P0)/ P0
E = 1.5e6 N/m2
E = 1e6 N/m2
-0.4
0.5
At z = 0.5 m
-0.6
E = 7.5e5 N/m2
E = 5e5 N/m2
0.6
-0.8
E = 2.5e5 N/m2
E = 2.5e4 N/m2
-1.0
0.0
0.1
0.2
0.3
0.4
0.5
t/T
0.6
0.7
0.8
0.9
1.0
Fig. 9. Phase shift due to variation in the degree of saturation.
0.7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
(Pz-P0)/ P0
Fig. 11. Effect of Young’s modulus on normalized seepage force.
6.3. Effect of soil permeability
Soil permeability is assumed to vary between 2:5 101 m=s (gravel) and 2:5 107 m=s (clay). The effect of
permeability on seepage force, (PzP0), is shown in
Fig. 10. It indicates that the seabed response is sensitive
to the permeability k. The seepage force decreases with
permeability. When the permeability is low, the seabed is
more vulnerable to liquefaction. The maximum response of
pore water pressure occurs at deeper position when
soil permeability is higher. Again, Criterion 3 predicts the
deepest liquefaction zone.
shallow zone, Young’s modulus E has almost no effect on
vertical effective stress s0zz and vertical seepage force
(PzP0). The maximum effective stress increases with
Young’s modulus E. This implies that the seepage force
becomes higher and the soil mass may become more
susceptible to liquefaction when E increases. When
Young’s modulus E is very large (45 107 N=m2 such as
gravels), the maximum response (s0zz , s0vol or (PzP0)) is
not affected. At this stage, a seabed can be regarded as
rigid porous medium for the analysis of liquefaction.
Again, Criterion 3 is the most critical one because it
predicts the deepest liquefaction zone.
6.4. Effect of Young’s modulus
6.5. Effect of variable shear modulus
Young’s modulus is assumed to vary between 2.5 104
and 2.5 108 N/m2 but keeps constant along depth. This
range is suitable for a wide range of soil masses (Jeng and
Lin, 1996). The seepage force is shown in Fig. 11. In the
Variable shear modulus along depth is a feature of
seabed soil in ocean engineering and has been studied by
many researchers (Thomas, 1989; Lin and Jeng, 2000). A
typical distribution of shear modulus along depth is shown
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256
0
Depth, z (m)
5
Variable G
Constant G
Approx. variable G
10
15
20
0
1000
2000
3000
4000
5000
2
(a)
Shear Modulus, G (kN/m )
0.0
0.1
0.2
0.3
z/h
0.4
Variable G
Constant G
0.5
0.6
0.7
0.8
0.9
1.0
0.0
(b)
0.1
0.2
0.3
0.4
0.5
0.6
(Pz- P0 )/ P0
0.7
0.8
0.9
1.0
Fig. 12. Effect of variable shear modulus on normalized seepage force.
in Fig. 12(a). Here the effect of variation of shear modulus
on seabed response is studied. For comparison, an
equivalent constant modulus (called constant modulus),
which has the same area over the entire thickness, is also
used. The meshless method approximates this variable
shear modulus with stepwise constants over background
cells. Typical response is shown in Fig. 12(b) for excess
pore water pressure. The contours of effective stresses are
compared in Fig. 13(a)–(c), where solid lines are for the
variable modulus and dashed lines are for constant
modulus. Vertical effective stress is larger for variable
modulus than for constant modulus, and horizontal
effective stress is more sensitive than vertical effective
stress. The maximum response occurs at deeper zone for
variable modulus, and the liquefaction depth is larger for
variable modulus.
7. Stress angle under wave loading
Shear failure may occur in the seabed. The stress angle f
is used to describe the mobilization of soil shear strength.
Seabed is only stable when fofu . It is noted that the angle
of internal friction fu is between 201 and 301 for sandy
seabed (Poulos, 1988). The stress angle at each node is
computed with Eq. (5). Typical Mohr circles within a wave
period are shown in Fig. 14(a). The line AOB (fu line)
passes through the crown point of the Mohr circle, and
fu ofu . When the crown envelope crosses the line AOB,
part of the seabed may be subject to shear failure.
However, a soil is liquefied when the instantaneous stress
at the horizontal plane reaches the point O or the stress
crown is on the line COD, i.e. s03 ¼ 0 for Criterion 1 and
0:5ðs01 þ s03 Þ ¼ 0 for Criterion 2. Because a liquefied soil
behaves like fluid, the stress status cannot be obtained by
Biot’s consolidation equation. Theoretically, the crown
envelope cannot go beyond the line COD as indicated in
the current elastic analysis of Fig. 14(a). The wave-induced
seabed instability may be induced by a complex coupled
process combining shear failure with momentary liquefaction. Once shear failure occurs, seabed soil becomes highly
nonlinear and the current theory is inappropriate to deal
with the situation. Mohr circles are also drawn at two
particular depths for different degrees of saturation as
shown in Fig. 14(b)–(c). The wave and seabed parameters
are the same as those in Section 6.3 with a ¼ 4. These
Mohr circles correspond to the maximum vertical effective
stress at that point. If Criterion 1 is used, a soil is liquefied
because the minor principal stress is zero or negative.
Fig. 15 shows a typical relationship of shear failure depth
and liquefaction depth when a ¼ 4. It can be seen that
shear failure occurs at the surface and is shallower than
that for the liquefaction (Zen et al., 1998). Shear failure
occurs before liquefaction if internal frictional angle is
fu ¼ 301. According to Criterion 2, a seabed soil is always
liquefied near seabed surface, and thus protection work for
seabed surface, such as covering the seabed by a layer of
concrete blocks or rubble, is necessary (Jeng, 2001).
8. Concluding remarks
Wave-induced seabed instability, both momentary liquefaction and shear failure, is studied under various soil and
wave properties. A non-dimensional parameter is proposed
to evaluate liquefaction potential. The response of a seabed
with finite thickness is numerically studied when a twodimensional progressive wave is applied on the surface of
seabed. Parametric study on soil and wave properties is
carried out and their effects on the seabed responses and
liquefaction potential are analyzed. From these studies,
following conclusions can be made.
Momentary liquefaction may occur within the shallow
zone of a seabed and the non-dimensional parameter k can
be used to identify the momentary liquefaction. The seabed
is likely liquefied if kX1 for any one of the three criteria of
liquefaction. Three criteria of liquefaction, which are based
on vertical effective stress, effective volumetric stress and
dynamic excess pore pressure or seepage force, respectively,
are discussed for the identification of soil liquefaction. For
the same soil and wave properties, Criterion 3 (for seepage
force) predicts the deepest liquefaction zone and Criterion
ARTICLE IN PRESS
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257
1.9
Vertical distance (m)
0.0
0.13
0.27
0.53
0
-1.9
4
0.53
8
0.27
0.13 0
0 -0.13
-0.27
12
0.40
0.40 0.53
-0.67
-0.53
-0.53
-0.40
-0.40
0.40
0.40
-0.27-0.13
16
0
0
0
4
8
12
16
20
24
28
0.27
0.27
0.13
0.13
20
(a)
P (k N/m2)
Vertical effective stress
32
36
40
Horizontal distance (m)
1.9
Vertical distance (m)
0.0
0
-1.9
0.050
0
-0.15 -0.050
-0.10
-0.15
-0.10
-0.20
-0.050
0.050
4
8
12
-0.10
0.10
0.20
0.15
0
-0.20
0.10
0.15
16
-0.050 -0.15
-0.10
-0.050 -0.10
0
0
0
20
0
4
P (k N/m2)
Horizontal effective stress
8
12
16
20
24
28
32
36
40
Horizontal distance (m)
(b)
1.9
Vertical distance (m)
0.0
0
-0.037
4
-0.29
8
12
-0.16
-0.29-0.22
-0.10 0.025
0.025
0.088 0.15
0.28
0.28
0.21
0.21
-0.037
16
0.15
-0.10
0.088
20
0
(c)
-1.9
-0.16
-0.22
4
P (k N/m2)
Shear stress
8
12
16
20
24
28
32
36
40
Horizontal distance (m)
Fig. 13. Comparison of stress contours for variable and constant shear modulus.
2 is the least critical one. Criterion 2 always predicts soil
liquefaction at the seabed surface. Therefore, Criterion 3
becomes the most critical condition for liquefaction.
The sensitivity of wave and seabed properties is different
in the evaluation of liquefaction potential. Within the
shallow zone, wavelength has almost no effect on the
maximum seabed response. Seabed response is similar to
that in one-dimensional case within the shallow depth near
seabed surface. As an approximation, one-dimensional
analysis suffices for the identification of soil liquefaction.
However, seabed characteristics have dominant effects on
wave-induced seabed response. Among all the soil parameters described, compressibility of pore fluid (degree of
saturation) is the most critical one. The higher the fluid
compressibility is, the more vulnerable condition for the
occurrence of soil liquefaction. The coefficient of permeability also plays an important role. The lower the
permeability is, the more vulnerable to soil liquefaction.
In the shallow zone near the seabed surface, Young’s
modulus of soil skeleton has almost no effect on vertical
effective stress and excess pore pressure, but has some
effect on effective volumetric stress. If Young’s modulus is
very high (45 107 N=m2 ), the effect on soil response may
be ignored and the seabed can be regarded as a rigid one.
Such simplification can predict vertical effective stress and
excess pore pressure with reasonable accuracy. However,
the predicted effective volumetric stress is slightly larger.
Variable shear modulus predicts bigger maximum response
and deeper liquefaction zone. Therefore, variable shear
modulus along depth has to be considered.
ARTICLE IN PRESS
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258
D
0.6
B
z = 1 m, α = 8,
δ = 0.4135, κ = 2.4185
*
φu = 30°
0.4 Envelop of crown
of stresses as
time passes
0.0
O
t = T/2
-0.2
-0.4
σ xz′/ P0
σ xz′/ P0
0.2
t=0
0.2
S = 0.90
r
S = 0.97
0.1
Sr = 0.975
S = 0.98
Sr = 0.985
S = 0.99
r
S = 0.995
r
z = 1 m, α = 4
r
φu = 30°
r
0.0
-0.1
A
In-situ stress conditions
-0.2
-0.6
C
-0.6 -0.4 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
σ′/ P0
(a)
-0.6
-0.5
-0.4
-0.3
-0.1
0.0
0.1
0.2
φ = 30°
z = 2 m, α = 4
0.2
-0.2
σ′/ P0
(b)
u
0.1
σ xz′/ P0
0.0
-0.1
Sr = 0.98
-0.2
S = 0.90
S = 0.985
Sr = 0.97
S = 0.99
S = 0.975
Sr = 0.995
r
-0.3
r
r
-0.4
-0.6
-0.5 -0.4 -0.3 -0.2 -0.1
0.0
r
0.1
0.2
0.3
0.4
σ′/ P0
(c)
Fig. 14. Shear failure status by Mohr–Coulomb failure criterion.
Stress angle is another important parameter leading to
seabed instability due o shear failure. Shear failure may
occur near and at the surface. The stress angle of soil has
nothing to do with the liquefaction except for causing shear
failure. Elastic analysis indicates that shear failure takes
place before momentary liquefaction. Once a soil failed in
shear, soil deformation becomes highly nonlinear. Therefore, the present linear theory would not be appropriate for
the prediction of further failure. It is then necessary to
employ the transition mechanism from shear failure to
liquefaction as a progressive process.
0.0
L = 40 m
φu = 20°
0.1
L = 60 m
L = 80 m
L = 100 m
0.5
φu = 30°
0.4
φu = 40°
0.3
Maximum liquefaction
depth for α = 4
z/h
0.2
0.6
0
10
20
30
40
50
φ (degrees)
60
70
80
90
Acknowledgement
This work is financially supported by the US Office of
Navy Research under grant number N00014-01-1-0457.
Fig. 15. Vertical distribution of stress angle.
Appendix A
2
K
LT
6
6
6
6
GT
6
½R ¼ 6
0
6
6 vl T
6 ðG GvrT Þ
4
0
L
G
ðM yDtHÞ 0
T
0
yDtG 0
0
0
0
G 0T
0
0
0
0
0
0
0
T
ðG 0vl G 0vr Þ
3
ðG vl G vr Þ
0
7
0
ðG 0vl G 0vr ÞDty 7
7
7
0
0
7
7,
0
0
7
7
7
0
0
5
0
0
(A.1)
ARTICLE IN PRESS
J.G. Wang et al. / Ocean Engineering 34 (2007) 247–260
2
0
6 LT
6
6
6 0
½Q ¼ 6
6 0
6
6
4 0
0
0
ðM þ Dtð1 yÞHÞ 0
0
h
½Stþ1 T ¼ utþ1
ptþ1
½St T ¼ ut
pt
lt1
h
½F T ¼ f u
Dtf p
0
Dtð1 yÞG0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ltþ1
1
ltþ1
2
lt3
lt4 ,
lt2
f l1
f l2
DtðG 0vl
3
0
G 0vr Þð1 yÞ 7
7
7
7
0
7,
7
0
7
7
5
0
(A.3)
(A.4)
i
0 .
0
(A.2)
0
i
ltþ1
,
4
ltþ1
3
259
(A.5)
The superscript ðt þ 1Þ denotes the current time ðt þ DtÞ. The repeatability conditions create two virtual boundaries at
both ends, as denoted by Gnl and Gnr (unl ¼ unr and pnl ¼ pnr). Other notations in Eqs. (A.1)–(A.5) are given by
Z
Z
Z
Z
k
BTI DBJ dO; LIJ ¼
fI AJ dO; M IJ ¼ nb fI :fJ dO; H IJ ¼
AT :AJ dO,
K IJ ¼
gw O I
O
O
O
Z
G IK ¼
G0IK ¼
N K fI dG;
Gu
G 0vl
IK
f uI
Z
0
N K fI
0
Gp
Z
G 0vl
IK
G vlIK ¼
N K fI dG;
Z
N K fI dG;
Gvl
G vr
IK ¼
Z
N K fI dG,
Gvr
0
dG;
¼
N K fI dG,
Gvl
Gvr
Z
Z
t:fI dG þ
b:fI dO; f pI ¼
j:fI dG,
¼
Z
¼
Gs
Gj
O
Z
f l1 I ¼
Z
Z
N K u dG;
Gu
2
fI;x
6
BI ¼ qðfI Þ ¼ 4 0
fI;z
f l2 I ¼
0
0
Gp
N K p dG,
3
fI;z 7
5;
fI;x
"
AI ¼
fI;x
fI;z
#
"
;
NK ¼
#
Nk
0
0
Nk
;
0
N K ¼ ½N k .
Following boundary conditions are also used during the variational formulation:
uðx; tÞ ¼ uðx; tÞ on Gu and pðx; tÞ ¼ pðx; tÞ on Gp ,
k qp
ðx; tÞ ¼ jðx; tÞ on Gj ,
gw qn^
j and t indicate pore water flux and traction, respectively. n^ is the unit normal to boundary Gs, Gu, Gp, Gs and Gj are the
boundaries where displacement, pore water pressure, total stress and flux of pore water are prescribed. Obviously, they
satisfy the following relations: Gu [ Gs ¼ G and Gu \ Gs ¼ +; Gp [ Gj ¼ G and Gp \ Gj ¼ +.
^ tÞ ¼ tðx; tÞ on Gs and
s:nðx;
References
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