Local estimation of failure probability function by weighted approach

In the reliability-based design of engineering systems, it is often required to evaluate the failure probability for different values of distribution parameters involved in the specification of design configuration. The failure probability as a function of the distribution parameters is referred as the ‘failure probability function (FPF)’ in this work. From first principles, this problem requires repeated reliability analyses to estimate the failure probability for different distribution parameter values, which is a computationally expensive task. A “weighted approach” is proposed in this work to locally evaluate the FPF efficiently by means of a single simulation. The basic idea is to rewrite the failure probability estimate for a given set of random samples in simulation as a function of the distribution parameters. It is shown that the FPF can be written as a weighted sum of sample values. The latter must be evaluated by system analysis (the most time-consuming task) but they do not depend on the distribution. Direct Monte Carlo simulation, importance sampling and Subset Simulation are incorporated under the proposed approach. Examples are given to illustrate their application.