Relaxation Methods Applied to Engineering Problems. VIIIA. Problems Relating to Large Transverse Displacements of Thin Elastic Plates
Relaxation Methods Applied to Engineering Problems. VIIIA. Problems Relating to Large Transverse Displacements of Thin Elastic Plates
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DOI：
10.1098/rsta.1945.0005
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发表时间：
1945-11
期刊：
影响因子：
--
通讯作者：
J. R. Green;R. Southwell
中科院分区：
文献类型：
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作者：
J. R. Green;R. Southwell
弹性平板的小横向位移服从单一的线性方程，而大位移引起中面的拉伸和由此产生的张力，与曲率的相互作用(即9膜效应)将非线性项引入平衡条件，使这些方程不再相互独立。二阶项是由冯·卡曼在1910年提出的，但修正后的(“大挠度”)方程只在少数情况下求解过，而且相当困难。本文用一种基于松弛方法的方法近似处理了四个算例。第一个和第二个是相对简单的问题，已经准确地解决了，因此可以作为测试用例，即。(A)具有固定边缘的圆板，其承受均匀的横向压力；以及(B)具有“简支”边缘的圆板，其在均匀的边缘推力作用下以径向对称的方式屈曲。第三和第四个例子给常规分析带来了很大的困难：(C)承受均匀横向压力的方板，其边缘被夹紧；(D)受夹紧其边缘的作用而屈曲的方板，其初始倾向于引起均匀剪切状态。
Small transverse displacements of a flat elastic plate are governed by a single linear equation, but large displacements entail stretching of the middle surface and consequent tensions, which interacting with the curvatures (i.e. by 9membrane effect’) introduce non-linear terms into the conditions of equilibrium and so make those equations no longer independent. The second-order terms were formulated by von Karman in 1910, but the amended (‘large deflexion’) equations have been solved only in a few cases, and then with considerable difficulty. In this paper four examples are treated approximately by a technique based on relaxation methods. The first and second are relatively simple problems which have been solved exactly and so serve as test cases, viz. ( a ) a circular plate, with clamped edge, which sustains a uniform transverse pressure and ( b ) a circular plate, with ‘simply supported’ edge, which buckles with radial symmetry under uniform edge thrust. The third and fourth examples present great difficulties to orthodox analysis: they are ( c ) a square plate, sustaining uniform transverse pressure, of which the edges are clamped, ( d ) a square plate buckled by actions which, clamping its edges, tend initially to induce a state of uniform shear.