Environmental Engineering Reference
In-Depth Information
5.3.5 updating reliability and risk estimates
Bayesian analysis can be applied to compute characteristic values of soil parameters. These
are defined as lower or upper quantiles of the posterior distribution, depending on whether
the parameters act as a load or resistance variable in the relevant limit state. These charac-
teristic values are then used as an input to a geotechnical assessment based on codes and
standards, typically following the partial safety factor format (LRFD). This approach is
straightforward for individual parameters, but may be less trivial if the parameters are sta-
tistically dependent after the measurements
Z.
The fact that the parameters are already available in a probabilistic format facilitates the
application of a reliability-based design or assessment. This approach should be selected
when the relevant code does not cover the specific design situation, but can also be prefer-
able in other situations; for example, when uncertain parameters are statistically dependent
as discussed above. As noted by Kulhawy and Phoon (2002, p. 32), “reliability-based design
(RBD) is the only methodology available to date that can ensure self-consistency from both
physical and probabilistic requirements.”
The computation of the reliability a priori was outlined in Section 5.3.2. We now con-
sider the computation of the posterior reliability conditional on the measurement outcomes
Z
,
expressed in terms of the conditional probability of failure Pr(
F
|
Z
) or the posterior reliability
index β″. Following Section 5.3.2, failure is expressed through the LSF
g
(
x
) as
F
= {
g
(
X
) ≤ 0}.
If the posterior PDF of
X
is available in analytical form, then the computation of Pr(
F
|
Z
) can
be carried out with the classical structural reliability methods. It is only necessary to replace
the prior joint PDF of
X
with the posterior
′′
FZ
|
)
=∫
f
X
xx
( )d
.
′′
g
()
x
≤
0
f
X
is not available in explicit form and can only be
computed numerically, a number of approaches have been proposed that are based on first
computing samples of ′′
For the case that the posterior PDF
′′
f
X
and then applying these samples for determining the reliability
using a Monte Carlo or other simulation approaches (e.g., Jensen et al. 2013; Sundar and
Manohar 2013). These methods can lack efficiency or require problem-specific adjustments
that can make their application cumbersome. An efficient and simple alternative has been
proposed in Straub (2011), which is based on translating the likelihood function into an
LSF, and then performing two reliability computations.
Following Straub (2011), an LSF
h
(
x
,
p
) can be defined as
hp p L
(, )
x
=−
( ).
x
(5.40)
This LSF is defined in the augmented outcome space of
X
and
P
. The latter is a standard
ensure that the product
cL(
x
) is not larger than 1; it can be chosen freely as long as it is
ensured that
cL
(
x
) ≤ 1 for any
x
. This LSF provides an alternative definition of the measure-
ments
Z
described by the likelihood function
L(
x
). It can be shown that the conditional
probability of failure given
Z
is
∫
f
()dd
xx
p
X
( )
≤∩
(
)
≤
g
x
0
h
x
,
p
0
Pr(
FZ
|
)
=
.
(5.41)
∫
f
()dd
xx
p
X
(
)
≤
hp
x
,
0
Both the numerator and the denominator in
Equation 5.41
are structural reliability prob-
lems and can be solved with any of the available structural reliability methods. Note that
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