Environmental Engineering Reference
In-Depth Information
Table 6.9
Settlement of a foundation on a two-layer soil mass—PC expansion features (
p
stands for
the maximal degree of polynomials and
P
is the number of nonzero polynomials in the
sparse expansion)
n
=
100
n
=
200
n
=
500
n
=
1000
8.45
⋅
10
−
5
6.76
⋅
10
−
6
1.25
⋅
10
−
6
1.14
⋅
10
−
7
1.20
⋅
10
−
3
2.39
⋅
10
−
4
1.33
⋅
10
−
5
2.08
⋅
10
−
6
p
4
4
4
5
P
59
126
152
225
It can be observed that the results obtained from the PC expansion compare very well to
the reference as soon as
n
= 200 points are used in the ED. The error is <1% in the general-
ized reliability index, for values as large as β = 4, that is, for probabilities of failure in the
order of 10
−5
. The detailed features of the PC expansions built for each ED of size
n
= 100,
Again, the sparsity of the expansions is clear: a full expansion with all polynomials up to
degree
p
= 4 in
M
= 6 variables has
P
=
(
)
=
64
210 terms, which would typically require
an ED of size 2 × 210 = 440. Using only
n
= 100 points, a sparse PC expansion having 59
terms could be built up. It is also observed that the classical (normalized) empirical error
1 −
R
2
is typically one order of magnitude smaller than the LOO normalized error, the latter
being a closer estimate of the real generalization error.
+
4
6.5.3 Settlement of a foundation on soil mass with
spatially varying Young's modulus
Let us now consider a foundation on an elastic soil layer showing spatial variability in
its material properties (after Blatman, 2009). A structure to be founded on this soil mass
is idealized as a uniform pressure
P
applied over a length 2
B
= 20 m of the free surface
The soil layer thickness is equal to 30 m. The soil mesh width is equal to 120 m. The soil
layer is modeled as an elastic linear isotropic material with Poisson's ratio equal to 0.3. A
plane strain analysis is carried out. The finite-element mesh is made of 448 QUAD4 ele-
ments. Young's modulus is modeled by a 2D homogeneous
lognormal
random field with
mean value μ
E
= 50 MPa and a coefficient of variation of 30%. The underlying Gaussian
random field log
E
(
x
, ω) has a square-exponential autocorrelation function:
2
xx
−
′
ρ
log
E
xx
(, )
′ =
exp
−
,
(6.83)
2
where ℓ = 15 m. The Gaussian random field log
E
(
x
, ω) is discretized using the KL expansion
(Loève, 1978; Sudret and Der Kiureghian, 2000; Sudret and Berveiller, 2007,
Chapter 7
):
∞
∑
log(,)
E
x
ωµ σ
=
+
λ
ξ ωφ
() (),
x
(6.84)
log
E
log
E
i
i
i
i
=
1
where {()}
1
are the
solution of the following eigenvalue problem (Fredholm integral of the second kind):
ξω
i
∞
1
are independent standard normal variables and the pairs {(
λ
i
, }
φ
∞
i
=
i
i
=
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