Extended importance sampling and its application in reliability analysis of the rotating shaft of some horizontal tailplane

An extended importance sampling is presented to calculate the failure probability of structure system with multiple limit states involving some different basic variables, which can not be solved by the conventional importance sampling. The extended design point is proposed on the design point of the conventional importance sampling. And the sampling probability density function of the extended importance sampling is constructed at the extended design point. Formulae to estimate failure probability of the structure system P_f, variance and the coefficient of variation of P_f are derived. The extended importance sampling is applicable to the structure system reliability with one and multiple extended design points. While one extended design point of one failure mode is much more significant than others, sampling probability density function of the extended importance sampling is constructed at the extended design point. Otherwise it should be constructed at multiple extended design points. The present method is applied to analyze the reliability of the rotating shaft of some horizontal tailplane. There are thirteen limit states equations involving lots of basic variables in the rotating shaft. The advantage of the present method is illustrated by the reliability analysis of the rotating shaft. Comparing with the Monte-Carlo method, this method has higher precision and less computational cost. And the precision of the extended importance sampling based on the multiple extended design points is higher than that based on one extended design points. But the extended importance sampling based on multiple extended design points requires more computational cost. In application, the two extended importance sampling methods should be tradeoff.