基于Mathematica的蜂窝梁孔口应力集中问题的复分析

Complex Analysis of Stress Concentrations in Cellular Beam with a Hole Based on the Mathematica

目的研究不同孔型的孔口应力集中问题,为蜂窝梁及其他开孔结构的工程设计应用提供参考.方法利用弹性力学的基本理论,运用复变函数的方法,引入保角变换,以无限大开孔板为例,借助Mathematica的符号计算功能,编制相应的程序对常用的蜂窝梁孔型—六边形、四边形、圆形孔口进行分析求解.运用孔口纯弯状态下有限元分析与理论计算结果进行对比和修正.结果计算推导得出了在几种不同应力边界条件下的孔口处的复应力函数,进而得到对应的孔口应力分布特征,借助有限元分析给出了纯弯状态下孔口应力的理论计算公式.结论孔口的应力集中程度随孔角曲率半径减小而加剧,圆形孔口能有效降低应力集中的影响,是一种理想的开孔形式.

To offer references in engineering designs for the cellular beam and other structure with opening, this paper studied stress concentrations of holes with different shapes in beam. Based on the elastic theory, complex function analysis and the conformal mapping technique, with the symbolic computational software Mathematica, stress concentrations of holes such as hexagon, square and circle types that are often used in the cellular beam were calculated. Theoretic.al calculation results of stresses on the hole under pure bending state were compared and corrected with the finite element analysis. The complex stress functions under axial loading, pure sheafing and pure bending were calculated and the stress distribution behaviors on the hole were obtained. The theoretical calculational formulas for stress concentrations on the hole under the pure bending with the aid of finite element analysis were built. It indicates that stress concentrations on the hole increase rapidly with decrease of curvature radius by the hole angle. The circular hole is a kind of ideal form of opening, which can effectively reduce the stress concentrations.