合成展开法求解圆薄板大挠度问题
The Solution of Large Deflection Problem of Thin Circular Plate by the Method of Composite Expansion
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摘要: 本文用合成展开摄动法,把外场解和内层解结合起来,求解圆薄板大挠度问题.本文把Hencky的薄膜解当作外场解的一级近似解,并求出了外场解的二级近似解.利用边界内层坐标,求得了相应的各级内层解,即边界层解.本文采用最大位移和板厚之比的倒数作为小参数,所得结果大大改进了1948年作者所得的结果.Abstract: In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky's membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.
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[1] Chien Wei-zang,Asymptotic behavior of a thin clamped circular plate under uniform normal pressure at very large deflection,Science Reports(A) of National Tsing Hua University,5,1(1948),1-24. [2] Bromberg,E.,Nonlinear bending of a circular plate under normal pressure,Comm.Pure Appl.Math.,9(1956),633-659. [3] Срубшик Л.С.и В.И.Юдович,Асимлтоика уравнений болъшого лрогиба,круглой симметрично загуженней пластины,ПАН СССР,139,2(1961). [4] O'Malley,R.E.Jr.,Boundary layer methods for nonlinear initial value problems,SIAM Review,13(1971),425-434. [5] Chudov,L.A.,Some shortcomings of classical boundary layer theory,NASA(transl) TTF-360,TT65-50138(1966). [6] Latta,G.E.,Advanced Ordinary Differential Equations,Lecture Notes,Stanford University(1964). [7] 周焕文,奇异摄动法在圆板大挠度问题中的应用,钱伟长主编《奇异摄动理论及其在力学中的应用》,科学出版社(1982), 310-338. -
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