Bending of Thin Elastic Plates of Variable Thickness

Abstract

Southwell's equations for the bending of thin elastic plates are derived from a new point of view and are extended to plates of variable thickness and mixed boundary conditions. The bend­ ing and twisting moments and the transverse shear in the plate are expressed in terms of two functions U and V. The boundary conditions are such that, on an unsupported edge, the boundary values of U and V are known and that, on a clamped edge, the first derivatives of U, V are known. Thus, the new formulation is particularly adaptable to plates with free edges, for which the classical KirchhofT-Love formulation expressed in terms of the deflection function is inconvenient because the boundary conditions involve the second- and third-order deriv­ atives. The theory is applied to a square plate with linear thickness variation (cross sections in the form of a double wedge) simply supported at two diagonal corners and loaded by a pair of equal forces at the other two corners. This loading condition is chosen for its simplicity in experimental verification that is carried out. The stresses and deflection of this plate are found by the relax­ ation method. The profound influence of the thickness variation on the stress distribution is shown by comparing the results with the exact solution for a plate of uniform thickness under the same loading condition. The theoretical results agree with the ex­ periments within the estimated error of both, indicating that the formulation of the bending problems for plates of variable thick­ ness is sound. The influence of the thickness variation and the power of the method are further illustrated by the results for the bending of the square plate by uniform edge moments and also for a fullspan solid swept wing.