Third moment method for reliability analysis with uncertain moments characterized as interval variables

Traditional reliability analysis aims to compute the failure probability based on probability distribution functions, which are constructed using the moments of random parameters. In practice, however, appropriate samples may be insufficient to obtain deterministic values of the moments of all random variables and the exact value of failure probability cannot be obtained. To be consistent with the reality, the uncertainties in moments can be measured as interval variables, and then the bounds of failure probability should be evaluated. In this study, an idealized case is considered, where there is at most one imprecise moment associated with any given input random variable. A third moment method is proposed with uncertain moments measured as interval variables, and is named as TMI method. The proposed TMI method is straightforward including only four steps. Firstly, the derivative of performance function to random variables having uncertain moments is calculated, with the random variables set to be their mean values. Secondly, the values of uncertain moments for computing the bounds of failure probability are determined. Then, with inverse normal transformation defined based on the moments, the performance function at the bounds in Gaussian space is directly constructed. Finally, bounds of failure probability can be evaluated by two times of classical reliability analysis corresponding to the constructed performance functions. The application of TMI method is validated by numerical examples, including high-dimensional and strong nonlinear problems.