Pseudo Analytical Method for Nonlinear Structural Reliability

在随机振动中，系统性能函数在持续时间内变为负值的概率是讨论系统安全性的一种手段。这一概率被称为首次漂移概率。目前，如何发展非线性系统的评价方法已成为众多研究人员关注的问题，分析非线性系统对随机载荷响应最有力的工具是马尔可夫向量法。在这种情况下，状态向量分量的联合概率密度服从福克-普朗克方程。不幸的是，只有极少数封闭形式的解是已知的。即使对于单自由度情况，也只有一些定常解可用。这意味着对于更一般的情况，必须使用诸如等价线性化之类的近似方法。在此背景下，本研究还提出了一种计算一类随机激励下非线性单自由度系统的概率的方法。该方法是一种借助数值模拟响应数据的混合分析方法，即拟分析方法。该方法将非线性状态向量方程近似线性化为离散的高斯型和马尔可夫状态向量方程，并利用卡尔曼滤波算法估计k和k+1时刻的最优状态向量和协方差，得到一组数值响应数据。通过这种方式，可以评估性能的概率。最后，给出了一个非线性结构系统的数值算例，以验证该方法的有效性。

In random vibration, probability of an event that system performance function becomes negative during the time duration is a means to discuss the system safety. This probability is well known as first excursion probability. Presently, how to develop the evaluation method for nonlinear systems has been a major concern among many researchers.The most powerful tool available for the analysis of the response of nonlinear systems to random loading is the Markovian vector approach. In this case the joint probability density of the state vector components is governed by Fokker-Plank equation. Unfortunately, only very few closed-form solutions are known. Even for the SDOF case only some stationary solutions are available. This means that for more general cases approximated methods such as equivalent linearization have to be utilized.In this context, this research also proposes a method to evaluate the probability for nonlinear SDOF systems under a class of stochastic excitations. The method is a hybrid analytical procedure with the help of numerically simulated response data, namely Pseudo Analytical. This method is a class of linearization approaches that linearize approximately a nonlinear state vector equation to a discrete Gaussian and Markovian state vector equation, and the optimal state vector and covariances at times k and k+1 are estimated by the Kalman Filter algorithm with a set of numerically obtained response data. In this way, the probability of performance may be evaluated. Finally, a numerical example of a nonlinear structural system is demonstrated in order to examine the efficiency of the method.