The uncertainty importance measures of the structural system in view of mixed uncertain variables

The uncertainty importance measure, i.e. global sensitivity analysis, of the basic variable is used for investigating the influence of model input uncertainty on model output uncertainty. There are two kinds of uncertainty importance measures of the structural reliability and response with respect to both random and fuzzy-valued input variables are investigated for problems with aleatory uncertainty and epistemic uncertainty. First, the structural reliability and probability distribution of the response at each membership level are analyzed. Then, the differences between the unconditional and conditional membership functions (MFs) of reliability and the unconditional and conditional probability density functions (PDFs) of response at all membership levels are measured for the presented uncertainty importance measures. The mathematical properties of the presented importance measures are discussed and proven in this study. The defined uncertainty importance measures are easy to apprehend and are not restricted to the distribution form of random variables or to the membership function of fuzzy-valued variables. All evaluations are based on the PDF of random variables and the MF of fuzzy-valued variables, and thus, the established importance measures are global sensitivity indicators that consider the influence of random and fuzzy-valued input uncertainty on the structural reliability and response. The results of the examples show that the proposed sensitivity indicators of uncertainty importance measure can intuitively describe the effects of random and fuzzy-valued input variables on the reliability and the probability distribution of response for single and multiple failure modes.