Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet - Risk and Reliability in Geotechnical Engineering - page 300

conditional samples are generated for m = 1. Therefore, a total of 95 samples are obtained
from these two levels of simulation. Note that, for illustration purposes, a relatively small
number (i.e., N = 50) of samples are used in each simulation level. Such a small sample num-
ber in each simulation level may not be sufficient to ensure the accuracy of results (i.e., P ( F ))
estimated from the simulation.
Subset Simulation has been applied to various geotechnical engineering problems, such
as slope stability and foundations (Santoso et al., 2009; Wang et al., 2011b; Ahmed and
Soubra, 2011; Wang and Cao, 2013). The next section integrates Subset Simulation with the
expanded RBD approach (Wang, 2011, 2013; Wang et al., 2011a) to improve the efficiency
and resolution of MCS at small probability levels.

7.3 eXPanDeD rbD WIth SubSet SIMulatIon

7.3.1 expanded rbD approach
The expanded RBD approach formulates the design process of geotechnical structures as
an augmented reliability problem (Au, 2005) or expanded reliability problem (Wang et al.,
2011a). The expanded RBD problem refers to a reliability analysis of a system, in which a set
of system design parameters are artificially considered as uncertain with probability distribu-
tions specified by the user for design exploration purposes (Wang, 2011, 2013; Wang and Cao,
2013; Wang et al., 2011a). For example, consider designing a drilled shaft with a diameter
B and depth D . The design process is one of finding a set of B and D values that satisfy both
the ultimate limit state (ULS) and serviceability limit state (SLS) requirements and achieve the
design target failure probability p T or target reliability index β T . In the context of expanded
RBD, the design parameters B and D are considered to be independent discrete random vari-
ables with uniformly distributed probability mass function P ( B , D ), which is given by

1
n BD

PBD ( , ) = (7. 3)

in which n B and n D are the number of possible discrete values for B and D , respectively.
Note that P ( B , D ) does not reflect the uncertainty in B and D , because B and D represent
design decisions and no uncertainty is to be associated with them. Instead, it is used to yield
desired design information. The drilled shaft design process is then considered as a process
of finding failure probabilities corresponding to designs with various combinations of B and
D (i.e., conditional probability P ( F | B , D )) and comparing them with p T . Feasible designs are
those with P ( F | B , D ) ≤ p T . Failure herein refers to events in which the load exceeds capacity.
In other words, failure occurs when the factor of safety (FS) (i.e., FS uls or FS sls for ULS or SLS
requirements, respectively) is less than 1. Using Bayes' Theorem (e.g., Ang and Tang, 2007),
the conditional probability P ( F | B , D ) is given by

PBDF P
PBD

( , | )( ) F

(7.4)

PFBD (| , ) =

( , )

in which P ( B , D | F ) is the conditional joint probability of B and D given failure and P ( F ) is the
probability of failure. The expanded RBD approach employs a single run of MCS to esti-
mate P ( B , D | F ) and P ( F ), which are further used in Equation 7.4 to obtain P ( F | B , D ) (Wang,
2011, 2013; Wang et al., 2011a). Since MCS is simply a repetitive computer execution of tra-
ditional deterministic design calculation with different combinations of input parameters,