The probability density evolution method for dynamic reliability assessment of nonlinear stochastic structures

An original approach for dynamic reliability assessmentof nonlinear stochastic structures is proposed. In the past few years, a newmethod, named the probability density evolution method, has been developed,showing versatile capability in engineering stochastic mechanics such as thestochastic response analysis of either linear or nonlinear structures ineither static or dynamic occasions. In the method, the probability densityevolution equation, a first order quasi-linear partial differential equationin terms of the joint probability density, is deduced and uncoupled to aone-dimensional partial differential equation, which is easy to benumerically solved combining the deterministic dynamic response analysis ofstructures, such as the precise integration method or Newmark-Beta timeintegration method, and the finite difference method. Therefore, theinstantaneous probability density function, rather than the second orderstatistical characteristics such as the mean, the covariance function andthe power spectrum density and so on which are focused on by traditionalstochastic finite element methods, of the response quantities of interestcan be obtained. In the present paper, the probability density evolutionmethod is applied to assess the dynamic reliability of stochasticstructures. To achieve the purpose, an absorbing boundary conditioncorresponding to the failure criterion of the first passage problem isimposed on the probability density evolution equation. Solving theinitial-boundary-value partial differential equation problem with anumerical algorithm will give the ``remaining'' probability density functionof the response. The dynamic reliability can then be assessed throughintegrating the ``remaining'' probability density function over the safedomain. The numerical algorithm is studied in detail where an adaptive TVDdifference scheme is presented. In contrast to the widely usedlevel-crossing process based reliability assessment approach, in theproposed method the mean out-crossing rate, which is usually obtainedthrough the Rice formula, is not needed, nor the properties of thelevel-crossing process such as the Poisson and Markovian assumption.Therefore the proposed method is expected to have a high accuracy. An8-story frame structure with bilinear hysteretic restoring force, which issubjected to seismic excitation, is investigated. The results of dynamicreliability assessment are compared with those evaluated by the Monte Carlosimulation. The investigation shows that the proposed approach is of highaccuracy and efficiency.