Environmental Engineering Reference
In-Depth Information
standard deviation of the system response were extensively investigated. This was not the
case for the failure probability because MCS methodology requires a large number of calls
of the deterministic model to accurately calculate a small failure probability. This chapter
fills this gap. An extension of the SS to the case of a spatially varying soil (where the soil
property was modeled by a random field) was presented. The random field was discretized
using K-L expansion methodology.
Three example applications were provided. They aim at showing the practical imple-
mentation of (i) the method of generation of random field by K-L expansion, (ii) the
computation of the failure probability against bearing capacity failure (using the SS
approach) of a strip footing resting on a (
c
, φ) soil and subjected to a service vertical load
P
s
where
c
and φ are considered as two random variables, and (iii) the probability of fail-
ure against exceeding a tolerable vertical displacement (using the SS approach) of a strip
footing resting on a spatially varying soil where the soil Young's modulus was considered
as a random field.
It was found that for a prescribed accuracy, the SS approach significantly reduces the
number of realizations as compared to MCS methodology (the reduction was found to be
equal to 93.3% in the present chapter). In other words, for the same computational effort,
the SS approach provides a smaller value of the coefficient of variation of
P
e
than MCS.
Finally, it should be mentioned that the MATLAB
®
codes used for three example applica-
tions are provided in
http:/ / www. univ- nantes. fr/ soubra- ah
for practical use.
aPPenDIX 15a: MoDIFIeD M-h algorIthM
The M-H algorithm is a Markov chain Monte Carlo (MCMC) method. It is used to gener-
ate a sequence of new realizations from existing realizations (that follow a target PDF called
“
P
t
”. Using a proposal PDF “
P
p
”, a next realization
s
k
+1
∈
F
j
that follows the target PDF “
P
t
”
can be simulated from the current realization
s
k
as follows:
a. A candidate realization
ŝ
is generated using the proposal PDF (
P
p
). The candidate real-
ization
ŝ
is centered at the current realization
s
k
.
b. Using the deterministic model, evaluate the value of the performance function
G
(ŝ)
corresponding to the candidate realization
ŝ
. If
G
(
ŝ
) <
C
j
(i.e.,
ŝ
is located in the failure
region
F
j
), set
s
k
+1
=
ŝ
; otherwise, reject
ŝ
and set
s
k
+1
=
s
k
(i.e., the current realization
s
k
is repeated).
c. If
G
(
ŝ
) <
C
j
in the preceding step, calculate the ratio
r
1
=
P
t
(
ŝ
)/
P
t
(
s
k
) and the ratio
r
2
=
P
p
(
s
k
|
ŝ
)/
P
p
(
ŝ
|
s
k
), and then compute the value
r
=
r
1
r
2
.
d. If
r
≥ 1 (i.e.,
ŝ
is distributed according to the
P
t
), one continues to retain the realization
s
k
+1
obtained in step b; otherwise, reject
ŝ
and set
s
k
+1
=
s
k
(i.e., the current realization
s
k
is repeated).
Notice that in step b, if the candidate realization
ŝ
does not satisfy the condition
G
(
ŝ
) <
C
j
,
it is rejected and the current realization
s
k
is repeated. Also in step d, if the candidate realiza-
tion
ŝ
does not satisfy the condition
r
≥ 1 (i.e.,
ŝ
is not distributed according to the
P
t
), it is
rejected and the current realization
s
k
is repeated. The presence of several repeated realiza-
tions is not desired as it leads to high probability that the chain of realizations remains in the
current state. This means that there is high probability that the next failure threshold
C
j
+1
is equal to the current failure threshold
C
j
. This decreases the efficiency of the SS approach.
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