剪切荷载作用下方孔应力集中问题的解析解

An Analytical Study on the Stress Concentration Problem of Plates with a Rectangular Hole under Shearing Loads

用解析的方法对孔口的应力集中问题进行研究,讨论在剪切荷载作用下矩形小孔附近的应力分布情况。首先建立剪切荷载作用下带矩形小孔薄板的能量方程,并把小孔看作外部荷载之一,从而得到具有循环周期性的有限元方程。然后采用U变换法对能量方程进行解耦。U变换法是一种能对循环周期结构进行精确分析的解析方法,应用该方法可以得到矩形小孔附近各点位移的解析形式解,进而对各点应力进行分析,得到应力集中系数。

Stress concentration is a familiar problem in structural analyses.By using ordinary method to solve the stress concentration problem of rectangular hole,only the approximate solutions can be found.The U-transformation method is an analytical method which can be used to analyze the structures with cyclic periodicity.This paper applies the U-transformation method to study the stress concentration problem in the plates with a square hole under shearing loads.At first,the governing equation with cyclic periodicity must be established.In order to bring periodicity to the potential energy equation,the interior disfigurement made by the rectangular hole may be replaced by an additional loading.By using the U-transformation technique and the finite element method,the potential energy equation can be uncoupled,and the analytical displacement solutions of the finite element equations are derived in series form.Therefore,the stress concentration can then be discussed easily and conveniently.For plate problem the bilinear rectangular element with four nodes is taken as examples to demonstrate the application of the proposed method,and the stress concentration factor is obtained.