The probability density evolution method for dynamic reliability assessment of nonlinear stochastic structures
非线性随机结构动力可靠度的密度演化方法

An original approach for dynamic reliability assessment of nonlinear stochastic structures is proposed.In the past few years,a new method,named the probability density evolution method,has been developed,showing versatile capability in engineering stochastic mechanics such as the stochastic response analysis of either linear or nonlinear structures in either static or dynamic occasions.In the method,the probability density evolution equation,a first order quasi-linear partial differential equation in terms of the joint probability density,is deduced and uncoupled to a one-dimensional partial differential equation,which is easy to be numerically solved combining the deterministic dynamic response analysis of structures,such as the precise integration method or Newmark-Beta time integration method,and the finite difference method.Therefore,the instantaneous probability density function,rather than the second order statistical characteristics such as the mean,the covariance function and the power spectrum density and so on which are focused on by traditional stochastic finite element methods,of the response quantities of interest can be obtained.In the present paper,the probability density evolution method is applied to assess the dynamic reliability of stochastic structures.To achieve the purpose,an absorbing boundary condition corresponding to the failure criterion of the first passage problem is imposed on the probability density evolution equation.Solving the initial-boundary-value partial differential equation problem with a numerical algorithm will give the "remaining" probability density function of the response.The dynamic reliability can then be assessed through integrating the "remaining" probability density function over the safe domain.The numerical algorithm is studied in detail where an adaptive TVD difference scheme is presented.In contrast to the widely used level-crossing process based reliability assessment approach,in the proposed method the mean out-crossing rate,which is usually obtained through the Rice formula,is not needed,nor the properties of the level-crossing process such as the Poisson and Markovian assumption.Therefore the proposed method is expected to have a high accuracy.An 8-story frame structure with bilinear hysteretic restoring force,which is subjected to seismic excitation,is investigated.The results of dynamic reliability assessment are compared with those evaluated by the Monte Carlo simulation.The investigation shows that the proposed approach is of high accuracy and efficiency.