具有随机刚度薄板小挠度问题的数值分析

Numerical Analysis on Thin Plate under Small Deflection with Random Stiffness

从薄板小挠度问题的控制方程出发，将结构的随机参数统一到刚度随机场中，采用局部平均法对刚度随机场进行离散，并将其融于二阶差分方法的列式中，给出了求解刚度随机时薄板上各点位移的数字特征的随机有限元法。该方法不但可以解决复杂边界下的薄板小挠度问题，而且进一步解决了结构的可靠度问题。文中最后对算例进行了计算，用解析解验证了本文方法的正确性。

Problem on thin bending plate is an important object of structural random analysis. Numerical methods are widely employed to solve the problem at present. The stiffness of thin bending plate is considered as a two dimensional random field, which is discretized by the local average method. Based on the common governing equation of a thin bending plate, a general stochastic finite element method is developed, which is used to obtain the statistical characteristics of the displacement field of a thin plate with random stiffness under complex boundary conditions, and provides with a better way to study further the structural safety and reliability as well. Structural random parameters are included in the random stiffness field. The second difference method is employed. A simply supported square plate with uniform load is calculated as an example, the expectation and the variance of the maximum deflection are given, and an analytic solution serves as a benchmark to test the correctness of the numerical solution. The result shows that the stochastic finite element method is reasonable.