Relaxing methods applied to engineering problems VIII A. Problems relating to large transverse displacements of thin elastic plates

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Abstract
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 'membrane 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 Kármán 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.

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