Direct probability integral method for reliability sensitivity analysis and optimal design of structures

Reliability sensitivity analysis plays an essential role in structural reliability design, especially for reliability-based design optimization (RBDO). However, for some complex and difficult problems, such as the RBDO of time-variant systems with multiple most probable points, there is still a lack of a powerful and versatile method. In this paper, the novel direct probability integral method (DPIM) is proposed for reliability sensitivity analyses to the distribution parameters of random variables and deterministic variables, and the time-invariant and time-variant RBDO problems are resolved in a unified framework. Firstly, in the context of probability density integral, the influences of two types of variables on structural reliability are elaborated. The reliability sensitivities with respect to the distribution parameters of random variables and deterministic variables are derived from probability density integral equation, which significantly improves the computational efficiency and accuracy of reliability sensitivity. Then, the analytical solutions of reliability sensitivity of a typical problem are achieved which are regarded as benchmark solutions. Based on the proposed approach for sensitivity analysis, the time-invariant and time-variant RBDO problems with multiple most probable points can be universally solved. Finally, three examples verify the high accuracy and efficiency of DPIM. It is shown that the proposed approach is capable of addressing RBDO problems with the design variables involving distribution parameters of random variables or/and deterministic variables. Specifically, the numerical results of building structure indicate that the cost of structure increases dramatically with increasing of specified reliability level, and RBDO takes an optimal trade-off between economy and safety in structural design.