Reliability-based design - Risk and Reliability in Geotechnical Engineering - page 383

9.11 CouPlIng oF StanD-alone DeterMInIStIC PrograM
anD SPreaDSheet-autoMateD relIabIlItY ProCeDureS
VIa reSPonSe SurFaCe or SIMIlar MethoDS
Programs can be written in the spreadsheet to compute the factor of safety or settlement
(e.g., Low and Tang 2004, p. 87, Low et al. 2007, and the VDrainSt program in Low
2003b). However, there are situations where serviceability or ultimate limit states can only
be evaluated using stand-alone finite element or finite difference programs, or one may
already have a preferred or more accurate deterministic program in hand. In these cir-
cumstances, reliability analysis and RBD using the spreadsheet-automated FORM proce-
dure can still be performed, provided one first obtains a response surface function (via
the established response surface methodology) that closely approximates the outcome of
the stand-alone finite element or finite difference programs. Once the closed-form response
functions have been obtained, performing RBD for a target reliability index is straightfor-
ward and fast. Performing MCS on the closed-form approximate response surface function
also takes little time. The response surface method (or other surrogate methods) was used in
Li (2000) for consolidation analysis of a Singapore land reclamation project, Tandjiria et al.
(2000) and Chan and Low (2012b) for laterally loaded single piles, Xu and Low (2006) for
embankments on soft ground, and Lü and Low (2011) on underground rock excavations,
among others.
9.12 SuMMarY anD ConCluSIonS
This chapter discussed RBD/analysis for a spread footing, a rock slope, three soil slopes,
a laterally loaded single pile, an anchored sheet pile wall, a rockbolt-reinforced tunnel,
and soft clay consolidation accelerated by prefabricated vertical drains, using the practical
FORM procedures of Low and Tang (2004, 2007) and an x -via- n -via- u procedure that is a
very simple variation of the 2007 x -via- n procedure. The efficient spreadsheet-based FORM
procedures can be coupled with stand-alone numerical packages via bridging techniques
such as response surface methods and artificial neural networks.
The expanding dispersion ellipsoid perspective in the original space of the random vari-
ables was presented, as a useful alternative perspective of reliability index and the design
point.
Various insights and interesting features and subtleties of RBD as revealed in the different
RBD cases were discussed, testifying to the ability of RBD to locate the design point for a
target risk level without presuming any partial factors and to automatically reflect paramet-
ric sensitivities and correlations from case to case. Links between RBD and EC7 design were
shown, and the complementary roles that RBD can play to EC7 and LRFD were explained
in Section 9.2.1 and further commented in the different RBD cases of subsequent sections.
RBD analysis involving spatially autocorrelated soil properties was illustrated and dis-
cussed for a Norwegian slope (Section 9.5) and for a laterally loaded single pile (Section
9.7). System FORM with multiple failure modes was discussed in Section 9.6 for a soil slope.
RBD with respect to serviceability limit state was implemented for a laterally loaded pile
in Section 9.7 and for consolidation settlement of soft clay accelerated by vertical drains in
Section 9.10.
It was shown that for some geotechnical problems, performing SORM after FORM
can improve the accuracy of the estimated failure probability, but a pragmatic stand on
the adequacy of the approximate failure probability estimated from FORM β index was
also suggested and discussed in the context of the results of Figure 9.16a . This pragmatic