Environmental Engineering Reference
In-Depth Information
Table 6.5
Ultimate bearing capacity of a strip foundation—probability of failure (resp. generalized
reliability index
β
gen
=
Ф
−
1
(
P
f
) between parentheses)—case of dependent (
c
,
ϕ
)
SF
FORM
SORM
MCS (10
7
Runs)
PCE
+
MCS
a
1.5
1.55
⋅
10
−
1
(1.01)
1.52
⋅
10
−
1
(1.03)
1.51
⋅
10
−
1
(1.03)
1.52
⋅
10
−
1
(1.03)
2.0
4.02
⋅
10
−
2
(1.75)
3.85
⋅
10
−
2
(1.77)
3.85
⋅
10
−
2
(1.77)
3.85
⋅
10
−
2
(1.77)
2.5
9.60
⋅
10
−
3
(2.34)
8.98
⋅
10
−
3
(2.37)
8.98
⋅
10
−
3
(2.37)
8.99
⋅
10
−
3
(2.37)
3.0
2.20
⋅
10
−
3
(2.85)
2.00
⋅
10
−
3
(2.88)
2.01
⋅
10
−
3
(2.88)
2.00
⋅
10
−
3
(2.88)
a
n
=
500,
n
MCS
=
10
7
.
The number of runs associated to FORM (resp. SORM) are 31 for
SF
= 1.5 and 2, and
35 for
SF
= 2.5 and 3 (resp. 65 for
SF
= 1.5 and 2, and 69 for
SF
= 2.5 and 3). Note that for
each value of
SF,
a new analysis FORM/SORM analysis shall be run. In contrast to FORM/
SORM, a
single
PC expansion has been used to obtain the reliability associated to all safety
factors. Using 500 points in the ED, that is, 500 evaluations of
q
u
(
X
), the obtained PC
expansion provides a normalized LOO error equal to 1.7 ⋅ 10
−7
(the maximal PC degree is 6
and the number of terms in the sparse expansion is 140).
6.5.1.2 Correlated input variables
For a more realistic modeling of the soil properties, one now considers the statistical depen-
dence between the cohesion
c
and the friction angle ϕ. From the literature (see a review in Al
Bittar and Soubra (2012)), the correlation between these parameters is negative with a value
around −0.5. In this section, we model the dependence between
c
and ϕ by a Gaussian copula
that is parameterized by the
rank correlation coefficient
ρ
R
= −0.5. Owing to the choice of
500 points in the ED, that is, 500 evaluations of
q
u
(
X
), the obtained PC expansion provides
an LOO error equal to 6.8 ⋅ 10
−7
(the maximal PC degree is 6 and the number of terms in
the sparse expansion is 172). The reliability results accounting for correlation are reported
These results show that PCEs may be applied to also solve reliability problems when the
variables in the limit state function are correlated. In terms of accuracy, the PCE results
compare very well with the reference results obtained by MCS, the error on the probability
of failure being, again, <1%. SORM provides accurate results as well, at a cost of 65, 65, 72,
and 83 runs when SF = 1.5, 2, 2.5, and 3 (the associated FORM analysis required are 31, 31,
38, and 49 runs). Moreover, it clearly appears that neglecting the correlation between
c
and
ϕ leads to a conservative estimation of the probability of failure, for example, by a factor 2.5
for
SF
= 3 (10% underestimation of the generalized reliability index).
6.5.2 Settlement of a foundation on an elastic two-layer soil mass
Let us consider an elastic soil mass made of two layers of different isotropic linear elastic
materials lying on a rigid substratum. A foundation on this soil mass is modeled by a uni-
form pressure
P
1
applied over a length 2
B
1
= 10 m of the free surface. An additional load
P
2
Owing to the symmetry, half of the structure is modeled by finite elements
(
Figure 6.7b
).
The mesh comprises 500 QUAD4 isoparametric elements. A plane strain analysis is carried
out. The geometry is considered as deterministic. The elastic material properties of both
layers and the applied loads are modeled by random variables, whose PDF is specified in
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