非均匀Reissner板弯曲的精确元法

An Exact Element Method for the Bending of Nonhomogeneous Reissner's Plate

本文在阶梯折算法和精确解析法的基础上,提出构造有限元的新方法——精确元法.该方法不用变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法,得到Reissuer板弯曲的一个非协调单元,它具有十五个自由度.由于节点位移参数仅含有挠度和转角,因此处理任意边界条件非常容易.文中给出证明,位移和内力均收敛于精确解.由精确元法所得到的单元不仅能用于厚板,也可用于薄板.文末给出四个算例.算例表明,利用本文的方法,可获得满意的结果,并有较高的数值精度.

In this paper, based on the step reduction method and exact analytic method, a new method, the exact element method for constructing finite element, is presented. Since the new method doesn't need variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficients. By this method, a triangle noncompatible element with 15 degrees of freedom is derived to solve the bending of nonhomogenous Reissner's plate. Because the displacement parameters at the nodal point only contain deflection and rotation angle, it is convenient to deal with arbitrary boundary conditions. In this paper, the convergence of displacement and stress resultants is proved. The element obtained by the present method can be used for thin and thick plates as well. Four numerical examples are given at the end of this paper, which indicates that we can obtain satisfactory results and have higher numerical precision.