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A system of equations is developed for the theory of bending of thin elastic plates which takes into account the transverse shear deformability of the plate. This system of equations is of such nature that three boundary conditions can and must be prescribed along the edge of the plate. The general solution of the system of equations is obtained in terms of two plane harmonic functions and one function which is the general solution of the equation Δψ − (10/h2)ψ = 0. The general results of the paper are applied (a) to the problem of torsion of a rectangular plate, (b) to the problems of plain bending and pure twisting of an infinite plate with a circular hole. In these two problems important differences are noted between the results of the present theory and the results obtained by means of the classical plate theory. It is indicated that the present theory may be applied to other problems where the deviations from the results of classical plate theory are of interest. Among these other problems is the determination of the reactions along the edges of a simply supported rectangular plate, where the classical theory leads to concentrated reactions at the corners of the plate. These concentrated reactions will not occur in the solution of the foregoing problem by means of the theory given in the present paper.

1. Introduction. There are in the theory of elasticity two well known variational theorems or principles, one of them being for displacements and the other one being for stresses [1]. The former is also called the principle of minimum potential energy while the latter is often referred to as the principle of minimum complementary energy. With reference to the differential equations of the theory of elasticity, which consist of equations of equilibrium for the components of stress and of the stress-displacement relations, a possible characterisation of the difference between the two variational theorems is as follows. In the theorem for displacements the stress-displacement relations are taken as equations of definition for the components of stress in terms of appropriate displacement derivatives and the variational equation is equivalent to the system of differential equations of equilibrium. In the theorem for stresses the differential equations of equilibrium serve to restrict the class of admissible stress variations and the variational equation is equivalent to the system of stress-displacement relations.Both variational principles have been found valuable for the purpose of obtaining approximate solutions of boundary value problems. When viewed in the above light these approximate solutions are such that part of the complete system of differential equations (either the stress-displacement relations or the equations of equilibrium) is satisfied exactly while the remaining equations are S'atisfied approximately only.It is natural to ask whether it might not be possible to use the calculus of variations for the purpose of obtaining approximate solutions in such a manner that there is no preferential treatment for either one of the two kinds of differential equations which occur in the theory. In what follows this question is answered in the affirmative. A variational problem will be formulated which has both the equation of equilibrium and the stress-displacement relations as appropriate Euler equations.It is also shown that application of the theorem to a problem which had previously been treated by means of the method of complementary energy leads to the results obtained by the complementary-energy method in a manner which represents some simplification over the earlier derivations.

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