banner.jpg Improvement of FORM/SORM Estimates by Markov Chain Monte Carlo Karl Breitung Dep. of Civil, Geo and Environmental Engineering, Technical University of Munich, Germany. breitu@aol.com ABSTRACT

Failure probability calculations in high dimensional spaces are a basic problem in structural reliability. Due to the increasing dimensions of the problems it was found that the quality of FORM/SORM decreases considerably here. The logical solution would have been to improve the SORM approximations. Instead a new approach, subset simulation (SS), was championed by many researchers. It is claimed that SS does not suffer from the deficiencies of SORM and can easily solve high-dimensional reliability problems for small probabilities. However, this is not true as examples show.
However it is possible to improve FORM/SORM using MCMC. Consider a failure domain F = fx; g(x) < 0g with g(x) the LSF in the standard normal space. With MCMC one can calculate integrals over F of the form h(x) . φn(x), but not the normalizing constant P(F). However, this can be achieved comparing the failure domain with another having a known probability content; so not P(F) is estimated, but the quotient of the two probabilities. A suitable choice here is FL = {x; gL(x) < 0} given by the linearized LSF gL(x), so P(FL) = ϕ(|x*|) with x* the design point. Now running two MCMC’s, one on F and one on FL, one gets by comparison an estimate for the failure probability P(F). In the same way, this can be done using as starting point the quadratic domains defined by SORM estimates. Improving FORM/SORM by MCMC combines the advantages of analytic methods with the flexibility of Monte Carlo approaches. For this approach there are no smoothness requirements for the limit state surfaces. It avoids the costs and deficiencies of the sequential approach of SS style methods.

Keywords: FORM/SORM, Laplace method, Markov chain Monte Carlo, Bennett acceptance ration.

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