Unified framework of stochastic structural dynamics: direct probability integral method

Abstract

Stochastic dynamics of structures involves the topics of simulation of random excitation, random vibration analysis, reliability estimation and reliability-based design optimization. However, the existing methods only focus on parts of these topics, resulting in the complicated calculation and the absence of versatility. The limited scope of application of approximation methods, low accuracy and inefficiency of numerical methods and the expensive computational cost of stochastic sampling methods restrict the development of stochastic dynamics. In this study, the novel direct probability integral method (DPIM) as a unified framework is proposed to solve the uncertainty propagation and quantification and reliability-based design optimization problems of static and dynamic structures. This method decouples the computation of probability density integration equation (PDIE) and governing equation (dynamic or static equilibrium equation) of structures, and can achieve the probability density functions of stochastic responses and reliabilities for linear or nonlinear structural systems. Firstly, the PDIE governing the propagation of randomness from input to output is derived based on the principle of probability conservation. The two key techniques, i.e., the partition of probability space and smoothing of Dirac delta function, are introduced to solve the PDIE. Then, the first-passage dynamic reliability based on the equivalent extreme value mapping and design under uncertainty are addressed. Finally, numerical examples of static and dynamic structures illustrate that the DPIM is a unified, efficient and easy implementation methodology for uncertainty quantification and design optimization, especially for nonlinear stochastic dynamic analysis of large-scale complex structures.