一個用於分析異向彈性彎曲問題的新邊界積分法

Abstract

本文發展一個分析對稱性複合疊層板彎曲問題的新的邊界積分方程式。此積分方程式的推導係運用一個分析古典板之方法。該法將板彎曲問題的解以類似二維異向彈力問題的史蹉解的形式來表示。傳統邊界積分方程式均利用Betti的互換功定理及適當的格林函數)來導得，但本文則使用柯西積分定理來推導。本研究所將發展之邊界積分方程式的優點之一是可同時得到線性相依之兩對偶積分方程組，無論邊界條件的形式為何，待解的方程式均可透過兩對偶以適定的第二型Fredholm積分方程式表示。本法的另一優點是邊界上所有的應力或彎矩分量均可直接得到，無須另作數值微分。 本文並計算方形板受板緣彎矩作用，板面受集中力或分布力，及含孔洞之無限板在遠處受彎矩等之算例，數值結果與解析解比較顯示本法之準確性極高。

A new boundary integral formulation for the numerical solution of bending problems of anisotropic plates is proposed in this work. The formulation is based on a Stroh-like formalism developed for the classical plate theory. In contrast to the conventional formulation, which is derived from Betti’s reciprocal work theorem with appropriate Green’s functions, the proposed formulation makes use of Cauchy’s integral theorem. An advantage of the new formulation is that it provides dual sets of boundary integral equations, which are linearly dependent. With the dual sets, the integral equations to be solved can always be cast into the form of well-posed Fredholm integral equations of the second type regardless of the types of boundary conditions. Another advantage is that all stress or moment components can be obtained directly without additional numerical differentiations. Numerical examples are given to demonstrate the effectiveness and efficiency of the proposed boundary integral formulation.