Environmental Engineering Reference
In-Depth Information
The finite-element analysis is performed stepwise, following the construction process.
First, the modeling of the in situ stress state is carried out by means of the
K
0
-procedure,
where
K
0
= 1 − sinφ is the lateral earth pressure coefficient at rest for normally consolidated
soils. Next, the sheet pile is installed by activating the corresponding beam and interface ele-
ments. Finally, the excavation is modeled by removing the plane-strain elements correspond-
ing to the trench and applying the necessary loading to establish equilibrium.
5.6.1 Prior probabilistic model
Homogeneous non-Gaussian random fields describe the prior distributions of the uncertain
material properties: Young's modulus
E
, friction angle
φ
,
and unit weight
γ
.
The joint distri-
bution at each pair of locations is modeled by the Nataf distribution (Der Kiureghian and
Liu 1986) with marginal distributions according to
Table 5.5.
The autocorrelation coeffi-
cient function is given by a separable exponential model
ρττ
(,)
=
xp( (
−
τ λ
/
)
−
(
τ λ
/
)),
xz
xz
xx
zz
where
τ
x
, τ
z
are the absolute distances in the x (horizontal) and z (vertical) directions. The
correlation lengths are λ
x
= 20 m and λ
z
= 5 m for all uncertain soil material properties.
Cross-correlation between the different material properties is not included. The random
fields are discretized by the midpoint method (Der Kiureghian and Ke 1988) using a stochas-
tic mesh, consisting of 144 deterministic FE patches. The stochastic discretization resulted
in a total of 3 × 144 = 432 basic random variables gathered in a vector
X
. In
Figure 5.14
,
the
stochastic and deterministic FE meshes are shown.
Figure 5.14c
shows the deformed con-
figuration at the final excavation stage computed with the mean values of the random fields.
5.6.2 updating the soil parameters with
deformation measurements
We assume that a measurement of the horizontal displacement at the top of the trench
u
x,m
= 60 mm is made at full excavation. The measurement is subjected to an additive
error
ε
m
, which is described by a normal PDF
f
∈
with zero mean and standard deviation
σ
ε
m
= 5mm. The likelihood function describing the measurement is
L
()
x
=
f
[
u
−
u
(),
x
]
(5.57)
∈
xm
,
x
m
where
x
describes the material properties at the midpoints of the stochastic elements and
u
x
(
x
) is the displacement evaluated by the FE program. Bayesian updating of the vector
X
is performed with BUS in conjunction with subset simulation. The constant
c
is selected as
c
σ , which satisfies the condition
cL
(
x
) ≤ 1.
=
m
Table 5.5
Prior marginal distributions of the material properties of the soil
Parameter
Distribution
Mean
COV
Unit weight
γ
(kN/m
3
)
Normal
19.0
5%
Young's modulus
E
(MPa)
Lognormal
125.0
25%
-
0.35
-
Poisson's ratio
ν
Beta(0.0, 45.0)
35.0
10%
Friction angle
φ
(°)
Cohesion
c
(MPa)
-
0.0
-
-
5.0
-
Dilatancy angle
ψ
(°)
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