Limit analysis of stochastic structures in the framework of the Probability Density Evolution Method

In the present paper a Probability Density Evolution formulation is proposed for the limit analysis of stochastic systems, which can accurately and efficiently evaluate the effect the system's random parameters have on its nonlinear and limit response. The proposed formulation of the classic Probability Density Evolution Method reduces the corresponding Generalized Density Evolution equations, which are partial differential equations, to a system of ordinary differential equations, that can be efficiently solved with the method of characteristics. With this reformulation, the cumulative distribution function of the critical load of the structure can be accurately and efficiently evaluated. The estimation of stochastic limit buckling loads of imperfection sensitive structures is a natural extension of this method. In addition, a methodology is put forward for the estimation of the probabilistic characteristics of the full load-displacement curve for a stochastic nonlinear system in the context of Newton-Raphson incremental-iterative schemes. The main advantage of the proposed approaches is that they allow for a quantification of the effect of uncertainties on the structural capacity, with only a small number of deterministic analyses compared to Monte Carlo simulation. The applicability and validity of the proposed methodology for limit and nonlinear structural analysis is verified through extensive numerical investigations.