Exact Benchmark Solutions of Stochastic Responses and Reliabilities of Structures via the Probability Density Integral Equation

Exact analytical benchmarks of structural stochastic responses and reliability are desirable but have intractable issues. To avoid calculations of inverse mappings and its Jacobian in a change-of-variable formula, this study introduces a rigorous framework of the probability density integral equation (PDIE). The new derivation of PDIE is established based on the probability conservation without the calculation of Jacobian of inverse mappings, exhibiting a wider scope than the change-of-variable formula, especially for the mappings of multiple inputs and a single output. The closed-form probability density functions (PDFs), joint PDFs, high-order moments, correlation coefficients, and failure probabilities are achieved by analytically solving the PDIE whatever the existence of inverse mappings and Jacobian. The proposed analytical method is applicable to nonlinear, nonmonotonic, and unequal-dimensional input–output random vectors. The results are demonstrated by several examples, i.e., cantilever beams, rectangular plates, and frame buildings, which show that the obtained analytical solutions for response statistics and rare-event failure probabilities are superior to those of existing studies and Monte Carlo simulation. It is indicated that the obtained analytical solutions can be utilized to not only verify the accuracy of numerical algorithms but also reveal the effects of load variability and material and geometric parameters on system response and reliability.