Environmental Engineering Reference
In-Depth Information
This means that the value of the conditional failure probability of the first level
P
(
F
1
)
will be equal to the prescribed
p
0
value.
6. Among the
N
s
vectors of random variables, there are [
N
s
×
p
0
] ones whose values of
the performance function are less than
C
1
(i.e., they are located in the failure region
F
1
). These vectors are used as “mother vectors” to generate
N
s
new vectors of random
variables (according to a proposal PDF
P
p
) using the Markov chain method based on
the modified Metropolis-Hastings (M-H) algorithm by Santoso et al. (2011). This
algorithm is presented in Appendix 15A.
7. Using the deterministic model, calculate the values of the system response corre-
sponding to the new vectors of random variables (which are located in level 1). Then,
calculate the corresponding values of the performance function. Finally, gather the
values of the performance function in an increasing order within a vector
G
1
, where
GG GG
1
……
k
Ns
= {,,
,
,
}.
1
1
1
8. Evaluate the second failure threshold
C
2
as the [(
N
s
×
p
0
) + 1]th value in the increasing
list of elements of the vector
G
1
.
9. Repeat steps 6-8 to evaluate the failure thresholds
C
3
,
C
4
, …,
C
m
corresponding to
the failure regions
F
3
,
F
4
, …,
F
m
. Notice that contrary to all other thresholds, the last
failure threshold
C
m
is negative. Thus,
C
m
is set to zero and the conditional failure
probability of the last level
PF
mm
(
1
) is calculated as
−
Ns
1
∑
PF F
(
)
=
Is
(
)
(15.14)
mm
−
1
N
Fk
m
s
k
=
1
where
I
F
m
= 1 if the performance function
G
(
s
k
) is negative and
I
F
m
= 0 otherwise.
Finally, the failure probability
P
(
F
) is evaluated as follows:
m
∏
P
(
F
)
=
(
F
)
(
FF
j
)
(15.15)
1
j
−
1
j
=
2
It should be mentioned that a normal PDF was used herein as a target PDF
P
t
. However,
a uniform PDF was used as a proposal PDF
P
p
(for more details, refer to Appendix 15A).
15.4 MethoD oF CoMPutatIon oF the FaIlure
ProbabIlItY bY the SS aPProaCh In the CaSe
oF a SPatIallY VarYIng SoIl ProPertY
This section aims at employing the SS methodology for the computation of the failure prob-
ability in the case of a spatially varying soil property modeled by a random field. The ran-
dom field was discretized in this chapter using the K-L expansion. In order to calculate the
failure probability, a link between the SS approach and the K-L expansion was performed.
In fact, the K-L expansion (cf.
Equation 15.2
)
includes two types of parameters (deter-
ministic and stochastic). The deterministic parameters are the eigenvalues λ
i
and eigenfunc-
tions ϕ
i
of the covariance function. The role of these parameters is to ensure the correlation
between the values of the random field at different points in the space. On the other hand,
the stochastic parameters are represented by the vector of the standard normal random vari-
ables {ξ
i
}
i
=1, …,
M
. The role of these parameters is to ensure the random nature of the uncertain
property. The link between the SS approach and the K-L expansion was performed through
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