矩形边界平面问题在边界条件作用下应力解

Stress Solution for Plane Problem with Rectangular Boundary Under Boundary Condition Action

边界条件作用下平面问题应力解为双调和方程解,由通解和特解组成.应力函数通解为含有8个待定系数的双向单三角级数,与矩形边界8个边界条件相对应;级数曲线波形符合相应方向边界法向面力或法向位移固有分布特点.应力函数特解有边界法向面力、边界法向位移和边界常量剪应力3种;将任意分布的边界法向面力和边界法向位移进行格式化处理,由给定构造规则确定前2个特解.在特定边界条件下常量剪应力特解可不予考虑.随法向边界条件的变化有16种应力函数表达式,以处理256种矩形边界平面问题,推导了常见边界的特解表达式,分析了求解过程中的几个问题,并附有算例.

The plane problem stress solution of boundary condition is the classical bi-harmonic equation solution and consists of the homogeneous solution and particular solution.The stress function homogeneous solution is the bi-directional single trigonometric series including eight undetermined coefficients in correspondence with the eight boundary conditions.The curve form of the series must be in accord with the intrinsic character of the normal surface force and normal displacement at the edges.There are three particular solutions： it is the normal surface force,normal edge displacement and edge constant shear stress.Translating different the surface force and the displacement into the form of series,and using the given regulations construct the first two particular solutions form.The third particular solution may be not considered under the specified boundary conditions.The stress function solution has 16 kinds of expression with the change of edge normal supported condition;it can solve the 256 types of plane problem with rectangular boundary.In this paper,the particular solutions of common boundary condition are presented,some problems in the evaluation process are discussed and some examples are provided.