Application of the subset simulation approach to spatially varying soils - Risk and Reliability in Geotechnical Engineering - page 549

application of the subset simulation
approach to spatially varying soils
Ashraf Ahmed and Abdul-Hamid Soubra
15.1 IntroDuCtIon
The probabilistic analysis of geotechnical structures presenting spatial variability in the
soil properties is generally performed using Monte Carlo simulation (MCS) methodology.
This method is not suitable for the computation of the failure probabilities encountered in
practice (especially when using a computationally expensive finite-element/finite-difference
deterministic model) due to the large number of simulations required to calculate a small
failure probability. For this reason, only the mean value and the standard deviation of the
system response were extensively investigated in literature when using this method.
As an alternative to MCS methodology, Au and Beck (2001) proposed the subset simula-
tion (SS) approach to calculate the small failure probabilities. The SS method was mainly
applied in the literature to problems where the uncertain parameters are modeled by random
variables. In this chapter, the SS method is employed in the case of a spatially varying soil.
The Karhunen-Loève (K-L) expansion is used to discretize the random field (i.e., to trans-
form the random field into a finite number of random variables) and to generate random
field realizations that respect a prescribed autocorrelation function.
After a brief overview of the K-L expansion method, both the classical SS approach
(involving the case where the uncertain parameters are modeled by random variables) and
its extension to the case of a spatially varying soil are presented in the form of a step-by-step
procedure for practical use. Three application examples are provided in this chapter. They
aim at showing the practical implementation of (i) the method of generation of a random
field by K-L expansion, (ii) the SS approach in case where the uncertain parameters are
modeled by random variables, and (iii) the SS approach in case where the uncertain soil
properties are modeled by random fields.
15.2 karhunen-loèVe eXPanSIon MethoDologY For the
DISCretIzatIon oF a ranDoM FIelD
A random field is typically described by (i) a probability density function (PDF) and (ii)
a covariance function (or an autocorrelation function, which is the covariance function
divided by the variance of the random field). There are several types of autocorrelation func-
tions (e.g., white noise, linear, exponential, squared exponential, and power autocorrelation
functions). For more details, one can refer to Baecher and Christian (2003). In the present
chapter, an exponential covariance function was used to represent the correlation structure
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