System reliability analyses of static and dynamic structures via direct probability integral method

Abstract Structural system reliability analysis is of important significance for evaluating the safety of structures with many components. Since a structural system can be acted by static or dynamic load, a unified and efficient method is required to assess the system reliability of static or dynamic load induced structures. In this study, the direct probability integral method (DPIM) is proposed to uniformly attack system reliability problems of static and dynamic structures. Firstly, the static and first-passage dynamic reliability formulas of the series, parallel and mixed systems are established by the joint probability density function (PDF) of multiple performance functions. Based on the probability density integral equation (PDIE) of performance functions, the DPIM is proposed along the two approaches, i.e., DPIM-S and DPIM-H. The former computes the system reliability using the PDF of extreme value mapping of performance functions, which is obtained by smoothing Dirac delta function . In the latter, the system reliability formulas with Heaviside function are analytically derived by the PDIE of multiple performance functions. Specially, the role of smoothing of Dirac delta function in DPIM for stochastic response and reliability analyses is revealed. Finally, five typical examples, including two mathematical examples and three static and dynamic structural systems, demonstrate high efficiency and accuracy of the DPIM for system reliability computation. Because of omitting the smoothing of Dirac delta function, the DPIM-H takes less CPU time for solving system reliabilities of static and dynamic structures than DPIM-S, while the DPIM-S has a significant advantage of obtaining the PDF of performance function.