Regional importance measures based on failure probability in the presence of epistemic and aleatory uncertainties

For the structural systems with both epistemic and aleatory uncertainties, in order to analyze the effects of different regions of epistemic parameters on failure probability, two regional importance measures (RIMs) are firstly proposed, i.e. contribution to mean of failure probability (CMFP) and contribution to variance of failure probability (CVFP), and their properties are analyzed and verified. Then, to analyze the effect of different regions of the epistemic parameters on their corresponding first-order variance (i.e. main effect) in the Sobol's variance decomposition, another RIM is proposed which is named as contribution to variance of conditional mean of failure probability (CVCFP). The proposed CVCFP is then extended to define another RIM named as contribution to mean of conditional mean of failure probability, i.e. CMCFP, to measure the contribution of regions of epistemic parameters to mean of conditional mean of failure probability. For the problem that the computational cost for calculating the conditional mean of failure probability may be too large to be accepted, the state dependent parameter (SDP) method is introduced to estimate CVCFP and CMCFP. Several examples are used to demonstrate the effectiveness of the proposed RIMs and the efficiency and accuracy of the SDP-based method are also demonstrated by the examples.