A method for enhancing computational efficiency in Monte Carlo calculation of failure probabilities by exploiting FORM results

The First-Order Reliability Method (FORM) is by far the most widely employed method for structural reliability computation. Its accuracy, however, is fair only when the curvature of the limit state function is very mild. In this paper, a reliability method built on the design point vector given by FORM is proposed. It is based on the computation of the limit state function of a relatively small selected set of Monte Carlo samples that have the highest similarity to the design point in a statistical sense. The similarity is measured by two nonlinear features extracted from the mass of input variable realizations. The bi-dimensionality of the mapping allows a simple visual selection of the highly relevant samples and the relevant sample selection is facilitated by the fact that the failure domain possesses a standard shape in the space spanned by those nonlinear features and also by the feasibility of mapping several possibilities of a second-order approximation of the limit state function onto the plot, which embrace the most relevant samples. The method yields the same estimate of the failure probability as the simple Monte Carlo, with a computational cost limited to the evaluation of a subset of the selected samples. Besides, the reduction of the variability of the probability estimates is easily accomplished. Its application is illustrated with some numerical examples derived from actual structural engineering practice. They show that the method is simple, efficient and elegant, as it allows visualizing the reliability problem in the plot constituted by the two nonlinear features.

► FORM is improved by using the design point vector as clustering device of Monte Carlo samples.
► A standard plot in two dimensions showing safe and unsafe samples is produced.
► The failure samples naturally accommodate in the upper-right sector.
► By selecting the upper-right samples with a secant line, the probability of failure is computed.
► The variance reduction is easily performed.