An efficient estimation of failure probability in the presence of random and interval hybrid uncertainty

In the presence of random and interval hybrid uncertainty (RI-HU), the safety degree of the structure system can be quantified by the upper and lower bounds of failure probability. However, there is a lack of efficient methods for estimating failure probability under RI-HU in present. Therefore, a novel method is proposed in this paper. In the proposed method, the interval variables are extended to the random variables by assigning a priori probability density function, in which the conditional density estimation (CDE)–based method and conditional probability estimation (CPE)–based method are proposed, and the failure probability varying with the interval variables can be obtained by only one group Monte Carlo simulation (MCS). Since the computational complexity of CPE is much lower than that of CDE, the CPE-based method is mainly concerned. In the CPE-based method, the conditional failure probability on a realization of the extended interval vector is approximated by that on a differential region adjacent to the corresponding realization; then, the density function estimation required in the CDE can be avoided. In order to ensure the accuracy of the CPE, a strategy is proposed to adaptively select the differential region, in which the MCS can be combined with the CPE (CPE + MCS) and the adaptive Kriging can be nested into the CPE + MCS for improving the efficiency. To improve the efficiency further, the meta-model importance sampling nested Kriging is combined with the CPE-based method. The presented examples illustrate the superiority of the proposed method over the existing methods.