Environmental Engineering Reference
In-Depth Information
Finally, it should be mentioned that in the case of a log-normal random field,
l
x
and
l
y
in
Equation 15.1
should be replaced by
l
lln x
and
l
lln y
, where
l
ln
x
and
l
ln
y
are, respectively, the
horizontal and vertical autocorrelation lengths of the underlying normal random field. They
are the lengths over which the values of
ln
[
R
(
x
,
y
, θ)] are highly correlated.
15.3 brIeF oVerVIeW oF the SubSet SIMulatIon aPProaCh
SS approach was proposed by Au and Beck (2001) as an alternative to MCS methodology to
compute small failure probabilities. The basic idea of the SS approach is that the small failure
probability can be expressed as a product of larger conditional failure probabilities. In this
section, one presents a brief description of the steps of SS approach in the case of two ran-
dom variables (for more details, the reader may refer to
Chapters 1
and
4
by Phoon 2008). A
quasi-similar procedure will be used later in this chapter for the case of a spatially random
field where a finite number of random variables are involved in the analysis. The steps of the
SS approach in the case of two random variables (
V
1
,
V
2
) can be described as follows:
1. Generate a vector of two random variables (
V
1
,
V
2
) according to a target PDF using
direct MCS.
2. Using the deterministic model, calculate the system response corresponding to (
V
1
,
V
2
).
3. Repeat steps 1 and 2 until obtaining a prescribed number
N
s
of vectors of random
variables and the corresponding values of the system response.
4. Determine the value of the performance function corresponding to each value of
the system response and then arrange the values of the performance function in an
increasing order within a vector
G
0
, where
GG GG
……
. Notice that the
subscripts “0” refer to the first level (level 0) of the SS approach.
1
k
Ns
= {,,
,
,
}
0
0
0
5. Prescribe a constant intermediate conditional failure probability
p
0
for the failure
regions
Fj
12
…
and evaluate the first failure threshold
C
1
, which cor-
is equal to the [(
N
s
×
p
0
) + 1]th value in the increasing list of elements of the vector
G
0
.
j
{
=
,
,
,
m
−
}
F
m
=
F
Level
m
F
j
Level
j
F
2
F
1
Level 2
Level 1
Level 0
C
1
>
C
2
>
0
C
j
-1
>
C
j
>
0
C
m
=
G(
x
)
=
0
C
1
>
0
Figure 15.1
Nested failure domain.
Search WWH ::
Custom Search