An approach to evaluation of EVD and small failure probabilities of uncertain nonlinear structures under stochastic seismic excitations

Efficient assessment of small first-passage failure probabilities of nonlinear structures with uncertain parameters under stochastic seismic excitations is an important but still challenging problem. In principle, the first-passage failure probabilities can be evaluated once the extreme value distribution (EVD) of studied structural response becomes available. With this in mind, this study presents a novel approach, termed as moment-generating function based mixture distribution (MGF-MD), for evaluation of the EVD. In this method, the MGF is firstly introduced to characterize the EVD, and the advantages of this characterization are highlighted. To calculate the MGF defined by a high-dimensional expectation integral, a low-discrepancy sampling technique, named Latinized partially stratified sampling (LPSS), is employed with a small sample size. Besides, the unbiasedness of the estimator is proven and the confidence interval is given. Then, a mixture of two generalized inverse Gaussian distributions (MTGIGD) with a closed-form MGF is proposed to approximate the EVD from the knowledge of its estimated MGF. The parameter estimation is conducted by matching the MGF of MTGIGD with seven values of the estimated one. Three numerical examples, including the EVD of random variables and reliability evaluations of two uncertain nonlinear structures subjected to fully non-stationary stochastic ground motions, are studied. Results indicate that the proposed approach can provide reasonable accuracy and efficiency and is applicable to very high-dimensional systems with small failure probabilities. The source code is readily available at: https://github.com/Chao-Dang/Moment-generating-function-based-mixture-distribution.