New algorithm to solve the sensitivity of response's cumulative distribution function with respect to distribution parameters of basic variables

To quantify effects of distribution parameters of the input variables, the sensitivity of the response's cumulative distribution function (CDF) with respect to those parameters is researched. For linear performance response functions solely involving normally distributed variables, analytical expressions of the CDF sensitivity are presented. As for a general structural application which probably involves non-normal variables, nonlinear performance functions or multiple response functions, this contribution puts forwards a new algorithm based on moment estimation to solve the CDF sensitivity. By using the relationship between the CDF and failure probability, and the reliability sensitivity based on moment estimation, the proposed algorithm establishes an analytical expression with the moments of performance function and threshold values for the CDF sensitivity. With the first four moments obtained by point estimate, the CDF sensitivity value at each threshold can be computed. Compared with numerical simulation methods, the most remarkable advantage of the proposed algorithm lies in that no additional evaluation of the performance function is needed at different thresholds, as the CDF sensitivity value at each threshold can be calculated using the moments of performance function obtained at one certain threshold. Furthermore, the proposed algorithm is not limited by the distribution type of basic variables, and is applicable to problems with multiple failure modes, yet its efficiency and precision when applied to problems involving highly dimensional variables or highly nonlinear performance functions still need to be improved. To illustrate the applicability and strength of the proposed algorithm, numerical and engineering examples are provided and discussed.