均布载荷下矩形板大挠度问题的摄动变分解

A Perturbation-Variational Solution of the Large Deflection of Rectangular Plates under Uniform Load

摘要: 本文以中心挠度为摄动参数,将矩形板大挠度问题的非线性偏微分方程组转化成几个线性偏微分方程,然后用变分法求解,得出了具有任意长宽比的板的解答,给出了位移、挠度及各内力的解析表达式;并给出中心点和边界中心的应力数值计算公式。本文还以长宽比λ为参数,作出了最大挠度——载荷曲线及最大应力曲线。其结果与实验进行了比较,表明二者是一致的。

Abstract: In this paper, von Kármán's set of nonlinear equation for large deflection of rectangular plates is at first converted into several sets of linear equations by taking central dimensionless deflection as perturbation parameter, and then, the sets of linear equations for plates with various ratios of length to width are solved with application of variational method, The analytical expressions for displacements and stresses as well as formulas for numerical calculation are worked out, The figures of maximum deflection-load and maximum stress with ratio R of length to width as a parameter are given in this paper, Through comparison, it is found that the results of this paper are quite in accord with experiments.

[1]	Way, S.,Uniformly loaded, clamped, rectangular plates with large deflection,Proc, Fifth Int, Cong, Appl, Mech.(Cambridge, Mass, U, S.A.1938),(1939),123-128.
[2]	Levp, S,Bending of rectangular plates with large deflections, NACA, TR,737(1942).Square plate with clamped edges under normal pressure producing large deflection,NACA, TR, 740 (1942).
[3]	Wang, C, T,Nonlinear large deflection boundary value problems of rectangular plates, NACA, TN,1425 (1948).
[4]	Wang, C, T,Bending of rectangular plates with large deflection,NACA, TN, 1462(1948).
[5]	Chien, W, Z,Large deflection of a circular clamped plate under uniform pressure,Chinese J.of Physics, 7 (1947),102-113.
[6]	Yeh, K, Y,Large deflection of a circular plate with a circular hole at the centre,Acta Scientia Sinico, 2 (1953),127-144.
[7]	Chien, W, Z, and K, Y.Yeh, On the large deflection of circular plate, Acfa Scientia Sinica, 3 (1954),405-436.
[8]	Chien, W, Z, Prohlem of large deflection of circular plate, Nodbitko Z.Archiwum Meehonki Stosowanej,Warszawa, 8 (1956),1-2
[9]	Chien, W, Z, and K, Y.Yeh, On the large deflection of rectangular plate, Proc.Ninth Intern, Congr, Appl, Mech.,Brussels, 6 (1957),403.
[10]	叶开沉、房居贤、王云,在法向均布载荷下四边固定的矩形板大挠度问题的摄动解,《第六届全国弹性元件会议论文集》(厦门)(1981).
[11]	Ramberg, W.,A, B, Mepherson and S, Levp, Normal-pressure tests of rectangular plates, NACA, Report No,748.(1942).