Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet - Risk and Reliability in Geotechnical Engineering - page 326

distribution with a mean and standard deviation of 31.2 and 6.31 kPa, respectively).
Although the failure samples still more or less follow a normal distribution, its mean,
standard deviation, and range are obviously different from those of the nominal distri-
bution. All 1134 failure samples have S uL values smaller than 21.7 kPa, leading to a his-
togram peaking in the lower tail of the nominal distribution. The S uL distribution from
failure samples and the S uL nominal probability distribution are quite different. This sug-
gests that the effect of S uL on failure probability is significant. This result agrees well with
the hypothesis test results (see the ranking in Table 7.8 ) . In contrast, Figure 7.16 shows
a T cr histogram of the 1134 failure samples and a nominal probability distribution of T cr
(i.e., a Normal probability distribution with a mean and standard deviation of 4.0 and
0.48 m, respectively). The T cr histogram of the 1134 failure samples has a pattern simi-
lar to its nominal probability distribution. This suggests that the effect of T cr on failure
probability is rather minimal. The results shown in Figure 7.16 are consistent with the
hypothesis test results.
7.8 SuMMarY anD ConCluDIng reMarkS
This chapter presented an MCS-based practical framework for reliability analysis and
design of geotechnical structures in a commonly available spreadsheet environment. Two
drawbacks of MCS were highlighted: (1) lack of resolution and efficiency, particularly at
small probability levels; and (2) no insight into the relative contributions of various uncer-
tainties to the reliability analysis results. These two drawbacks of MCS were addressed by
introducing an advanced MCS called “Subset Simulation” and probabilistic failure analy-
sis, respectively. Subset Simulation significantly improves the computational efficiency of
MCS at small probability levels. 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. Subset Simulation can also be used
together with the probabilistic failure analysis to improve the efficiency of generating fail-
ure samples.
To further remove the hurdle of reliability algorithms for geotechnical practitioners, the
MCS-based reliability analysis and design approaches were implemented in an Excel spread-
sheet platform. The implementation is deliberately divided into three uncoupled modules:
uncertainty modeling, deterministic modeling, and uncertainty propagation. The reliability
analysis (including uncertainty modeling and propagation) is decoupled from the conven-
tional deterministic geotechnical analysis so that the reliability analysis can proceed as an
extension of the deterministic analysis in a nonintrusive manner. This allows the determin-
istic and reliability analysis to be performed separately by personnel with different expertise
and in a parallel manner.
The MCS-based approaches and their implementations in the spreadsheet were illus-
trated through a drilled shaft design example and reliability analysis of a James Bay
Dike scenario. The results obtained from the drilled shaft example showed that Subset
Simulation significantly improves the computational efficiency at small probability levels
and substantially reduces the sample numbers or computational efforts required in the
expanded RBD approach. The reliability analysis results of the James Bay Dike scenario
showed that the effects of various uncertainties on slope failure probability are properly
prioritized and quantified by the probabilistic failure analysis approach. In addition, the
probabilistic failure analysis approach gives results equivalent to those from sensitivity
studies using repeated MCS runs, and hence, saves additional computational time and
efforts for sensitivity studies.