梯形板弯曲问题的康托洛维奇解
Kantorovich Solution for the Problem of Bending of a Ladder Plate
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摘要: 在康托洛维奇对矩形板弯曲问题的有效近似解的基础上,本文进一步探讨了在不同边界条件下的梯形板弯曲问题的康氏解法.将板的位移用一级近似位移函数ω(x,y)=u(x,y)v(y)表示,式中, 在x方向的位移采用广义梁函数,用最小势能原理建立起对应于不同边界条件下的关于y方向位移函数v(y)的变系数常微分方程,求解微分方程,并利用边界条件,求出v(y)的精确解,从而可得到近似程度较高的梯形板弯曲问题的解.Abstract: Based on the Kantorovich approximation solution for a rectangular plate in bending, this paper deals with the solutions for the ladder plate with various boundary conditions. The deflection of the plate is expressed in a first-order displacement function w(x,y)=(x,y)v(y) where the u(x,y) in x direction is the generalized beam function. By making use of the principle of least potential energy, the variable coefficients differential equations for v(y) may he established. By solving is, these differential euqations and making use of the boundary conditions, the accurate solutions of v(y) in y direction may be obtained. Then the displacement function w(x,y) is the solution for the problem of the bending of the ladder plate with a better degree of approximation.
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[1] (1) 钱伟长,《变分法及有限元》.科学出版社(1980). [2] (2) 胡海昌,《弹性力学的变分原理及其应用》.科学出版社(1981). [3] (3) [美]S.铁摩辛柯.S,沃诺斯基.《板壳理论》.科学出版社(1977).
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