关于圆簿板大挠度问题的正交条件解法

本文重新考察了钱伟长教授求解圆薄板大挠度问题的系统近似法,发现此法实质上可视为奇异摄动理论中的变形参数法.以无量纲中心挠度为小参数,将挠度、中面薄膜力和载荷参数作渐近展开,我们对所得的递推方程给出了正交条件(可解性条件),据此可确定圆薄板的刚度特性.本文指出,利用圆薄板小挠度解和正交条件,可以不经求解方程而导得载荷参数与中心挠度关系的三阶近似以及中心点、边缘处的薄膜力的首项近似.文中对若干特例(均布载荷、复合载荷、各种边界条件)进行了具体计算,所得的结果与钱伟长、叶开沅、黄黔等人在文[1～4]中给出的结果完全相符.

On the Method of Orthogonality Conditions for Solving the Problem of Large Deflection of Circular Plate

In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zang for solving the problem of large deflection of a circular plate, and find that the method could be regarded as the method of strained parameters in the singular perturbation theory. In terms of the parameter representing the ratio of the center deflection to the thickness of the plate, we make the asymptotic expansions of the deflection, membrane stress and the parameter of load as in Ref.[1], and then give the orthogonality conditions (i. e. the solvability conditions) for the resulting equations, by which the stiffness characteristics of the plate could be determined. It is pointed out that with the solutions for the small deflection problem of the circular plate and the orthogonality conditions, we can derive the third order approximate relations between the parameter of load and the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases (i. e. under uniform load, under compound load, with different boundary conditions), we deduce the specific expressions and obtain the results in agreement with the previous oaes given by Chien wei-zang, Yen Kai-yuan and Hwang Chien in Refs.[1-4].