Analogies between Bending of Plates and Torsion Problem

Abstract

Various analogies between the Saint Venant‐type torsion of prismatic bars with solid cross sections of polygonal planform and the problem of bending of simply supported, thin plates are discussed. The analogous plate has the same planform as the cross section of the bar and is bent by imposed thermal curvatures, or by lateral force loadings, respectively. Those plate loadings are proportional to the rate of torsional twist. The analogies are based on a reduction of the fourth‐order problem of bending of thin plates to a second‐order Poisson differential equation with Dirichlet boundary conditions, which is possible in the case of simply supported, polygonal plates. As a main result, a direct correspondence between the torsional shear stresses and the shear forces in the analogous plate loaded by uniformly distributed forces is established. In this sense, numerical routines for fourth‐order plate‐bending problems can be used, or checked, respectively, in the context of the second‐order Saint Venant torsion problem.