A unified analysis framework of static and dynamic structural reliabilities based on direct probability integral method

•A unified framework of static/dynamic reliability analysis is established based on direct probability integral method (DPIM).•New formula to determine smoothing parameter of Dirac function is suggested.•Two DPIM-based approaches for dynamic reliability analysis are proposed.•Example of nonlinear dynamic structure indicates superiority of unified framework. Generally, the static and dynamic reliabilities of structures are addressed separately in the existing methods except the computationally expensive stochastic sampling-based approaches. This study establishes a unified framework of reliability analysis for static and dynamic structures based on the direct probability integral method (DPIM). Firstly, the probability density integral equations (PDIEs) of performance functions for static and dynamic structures are presented based on the principle of probability conservation. The DPIM decouples the physical mapping (i.e., performance function) of structure and PDIE, and involves the partition of probability space and the smoothing of Dirac delta function. This study proposes a new adaptive formula of smoothing parameter based on kernel density estimation. Then, the improved DPIM is utilized to obtain the probability density function (PDF) of performance functions by solving the corresponding representative values and the PDIE successively. Furthermore, the reliability of static structure is calculated by integrating the PDF of performance function within safety domain. To overcome the difficulty of evaluating first passage dynamic reliability, the two approaches, namely the DPIM-based absorbing condition (DPIM-AC) and the DPIM-based extreme value distribution (DPIM-EVD), are also proposed. Finally, three engineering examples with stochastic parameters and random excitation indicate the desired efficiency and accuracy of the established framework for unified reliability analysis. Specifically, the challenging issue of dynamic reliability assessment for nonlinear structural system is attacked based on DPIM rather than Monte Carlo simulation or other sampling-based method. The proposed method is beneficial for propagation analysis of aleatory or/and epistemic uncertainties, as well as for stochastic model updating.