  Performance-based engineering for natural hazards facilitates the design and
appraisal of structures with rigorous evaluation of their uncertain structural
behavior under potentially extreme stochastic loads expressed in terms of
failure probabilities against stated criteria. As a result, efficient
stochastic simulation schemes are central to computational frameworks that aim
to estimate failure probabilities associated with multiple limit states using
limited sample sets. In this work, a generalized stratified sampling scheme is
proposed in which two phases of sampling are involved: the first is devoted to
the generation of strata-wise samples and the estimation of strata
probabilities whereas the second aims at the estimation of strata-wise failure
probabilities. Phase-I sampling enables the selection of a generalized
stratification variable (i.e., not necessarily belonging to the input set of
random variables) for which the probability distribution is not known a priori.
To improve the efficiency, Markov Chain Monte Carlo Phase-I sampling is
proposed when Monte Carlo simulation is deemed infeasible and optimal Phase-II
sampling is implemented based on user-specified target coefficients of
variation for the limit states of interest. The expressions for these
coefficients are derived with due regard to the sample correlations induced by
the Markov chains and the uncertainty in the estimated strata probabilities.
The proposed stochastic simulation scheme reaps the benefits of near-optimal
stratified sampling for a broader choice of stratification variables in
high-dimensional reliability problems with a mechanism to approximately control
the accuracy of the failure probability estimators. The practicality of the
scheme is demonstrated using two examples involving the estimation of failure
probabilities associated with highly nonlinear responses induced by wind and
seismic excitations.
