Importance analysis on the failure probability of the fuzzy and random system and its state dependent parameter solution

For structure systems with fuzzy input uncertainty as well as random one, the effects of the two kinds of uncertainties on the failure probability of the structure are studied, and an importance analysis model is established to quantitatively evaluate these effects. Based on the fact that the fuzziness of the output is determined by that of the input, the importance measure is defined to evaluate the effect of the fuzzy input variable. For the random input variable, the established model analyzes its importance from two aspects: (1) its effect on the most plausible value of the failure probability, (2) its effect on the imprecision of the failure probability. In the process of computing the importance measures, the conditional failure probability and unconditional one need to be evaluated, which is time demanding for practical engineering problems. For efficiently performing importance analysis in the presence of the fuzzy input variables and the random ones, the importance sampling (IS) method combined with the state dependent parameter (SDP) method is presented. Several examples show that the established importance analysis model can reflect the effects of the two kinds of input variables on the safety of the structure comprehensively and reasonably, and the presented solution can improve computational efficiency considerably with acceptable precision.