Abstract
This paper proposes a unified and efficient direct probability integral method (DPIM) to calculate the probability density function (PDF) of responses for linear and nonlinear stochastic structures under static and dynamic loads. Firstly, based on the principle of probability conservation, the probability density integral equation (PDIE) equivalent to the probability density differential equation is derived for stochastic system. We highlight that, for time dependent stochastic system the PDIE is satisfied at each time instant. Secondly, the novel DPIM is proposed to solve PDIE directly by means of the point selection technique based on generalized F discrepancy and the smoothing of Dirac delta function. Moreover, the difference and connection among the DPIM, the existing probability density evolution method and probability transformation method are examined. Finally, four typical examples for stochastic response analysis, including the linear and nonlinear systems subjected to static and dynamic loads, demonstrate the high computational efficiency and accuracy of proposed DPIM.
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