Environmental Engineering Reference
In-Depth Information
6.4.3 reliability analysis
Reliability analysis aims at computing the probability of failure associated to a performance
criterion related to the QoI
Y
= M(
X
). In general, the failure criterion under consideration
is represented by a
limit state function g
(
X
) defined in the space of parameters as follows
(Ditlevsen and Madsen, 1996):
• D
s
= {
x
∈ D
X
:
g
(
x
) > 0} is the
safe domain
of the structure;
• D
f
= {
x
∈ D
X
:
g
(
x
) < 0} is the
failure domain;
• The set of realizations {
x
∈ D
X
:
g
(
x
) = 0} is the so-called
limit state surface.
Typical performance criteria are defined by the fact that the QoI shall be smaller than an
admissible threshold
y
adm
. According to the above definition, the limit state function then
reads
g
(
x
) =
y
adm
− M(
x
).
(6.57)
Then the probability of failure of the system is defined as the probability that
X
belongs
to the failure domain:
∫
P
=
f
()
xx
d
=
E 1
(),
X
(6.58)
f
X
{:
x
y
−
M
() }
x
≤
0
adm
{:
x
y
−
M
() }
x
≤
0
adm
y
adm
− M
is the indicator function of the failure
domain. In all but academic cases, this integral cannot be computed analytically, since the
failure domain is defined from a QoI
Y
= M(
X
) (e.g., displacements, strains, stresses, etc.),
which is obtained by means of a computer code (e.g., finite-element code) in industrial
applications.
Once a PC expansion of the QoI is available though, the probability of failure may be
where
f
X
is the joint PDF of
X
and 1
{:
x
() }
x
0
∫
P
PC
=
f
()
xx
d
=
E
1
().
X
(6.59)
f
X
PC
{:
x
M
()
x
≥
y
}
PC
adm
{:
x
y
−
M
() }
x
≤
0
adm
The latter can be estimated by crude MCS. Using the sample set in
Equation 6.52
,
one
computes the number
n
f
of samples such that M
PC
(
x
i
) ≥
y
adm
. Then the estimate of the prob-
ability of failure reads:
n
n
·
=
f
P
PC
.
(6.60)
f
MCS
This crude Monte Carlo approach will typically work efficiently if
P
f
≤ 10
−4
, that is, if at
most 10
6
runs of PC expansion are required. Note that any standard reliability method such
as importance sampling (IS) or subset simulation could also be used.
6.4.4 Sensitivity analysis
6.4.4.1 Sobol decomposition
Global sensitivity analysis (GSA) aims at quantifying which input parameters {
X
i
,
i
= 1,
…,
M
} or combinations thereof best explain the variability of the QoI
Y
= M(
X
) (Saltelli
Search WWH ::
Custom Search