Environmental Engineering Reference
In-Depth Information
2002, 2005; Wang, 2012), retaining structures (e.g., Chalermyanont and Benson, 2004,
2005; Fenton et al., 2005; Goh et al., 2009; Wang, 2013), and foundations (e.g., Fenton
and Griffiths, 2002; Phoon et al., 2006; Wang and Kulhawy, 2008; Wang, 2011; Wang
et al., 2011a).
Although MCS provides a simple and robust way to assess failure probability or obtain
other reliability analysis results, it is well recognized that it suffers from two obvious and
significant drawbacks: (1) lack of resolution and efficiency, particularly at small probability
levels, and (2) no insight into the relative contributions of various uncertainties to the reli-
ability analysis results. This chapter addresses the first drawback of MCS by introducing
an advanced MCS method called “Subset Simulation” (Au and Beck, 2001, 2003; Au and
Wang, 2014). The Subset Simulation is integrated with MCS-based reliability analysis and
design approaches to improve the efficiency and resolution of estimating failure probability
at small probability levels. This chapter also deals with the second drawback of MCS using
probabilistic failure analysis. The probabilistic failure analysis approach makes use of the
failure samples generated in MCS and analyzes these failure samples to assess the effects
of various uncertainties on failure probability. It may be further integrated with Subset
Simulation to improve the efficiency of generating failure samples.
This chapter starts with a brief review of Subset Simulation, followed by integration of
Subset Simulation with an MCS-based reliability approach, called expanded reliability-
based design (expanded RBD) approach, and probabilistic failure analysis approach. Then,
these approaches are implemented in a Microsoft Excel spreadsheet environment. A drilled
shaft design example and a slope stability analysis example are used to illustrate such
implementation.
7.2 SubSet SIMulatIon
7.2.1 algorithm
Subset Simulation is an advanced MCS method that makes use of conditional probabil-
ity and Markov chain Monte Carlo (MCMC) method to efficiently compute a small tail
probability (Au and Beck, 2001, 2003; Au and Wang, 2014). It expresses a rare event
E
with a small probability as a sequence of intermediate events {
E
1
,
E
2
, …,
E
m
} with larger
conditional probabilities and employs specially designed Markov chains to generate con-
ditional samples of these intermediate events until the target sample domain is achieved.
Let
Y
be the output parameter that is of interest and increases monotonically and defines
the rare event
E
as
E
=
Y
>
y
, in which
y
is a given threshold value for determining whether
E
occurs. The choice of
Y
is pivotal to the efficient generation of conditional samples
of interest in Subset Simulation. As
Y
increases, Subset Simulation gradually drives the
sampling space to the target sample domain (e.g., failure domain in reliability analysis)
of interest. Hence,
Y
is referred to as “driving variable” in this chapter. Since the condi-
tional samples of interest are different for different problems (e.g., expanded RBD and
probabilistic failure analysis in this chapter), different driving variables shall be adopted
for different applications.
Let
y
=
y
m
>
y
m
−1
>
…
>
y
2
>
y
1
is a decreasing sequence of intermediate threshold values.
Then, the intermediate events {
E
i
,
i
= 1, 2, …,
m
} are defined as
E
i
= {
Y
>
y
i
,
i
= 1, 2, …,
m
}.
They are a sequence of nested events, that is,
EE
…
⊃⊃⊃⊃
−
E
E
m
. Hence, the prob-
1
2
m
1
ability of event
E
, that is,
P
(
E
) =
P
(
Y
>
y
), can be written as
…
PE
()
=
P
()
E
=
P
(
E EE
)
(7.1)
m
mm
1
−
1
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