Generalization of Plane Stress and Plane Strain States to Elastic Plates of Finite Thickness

This paper presents a novel method to establish a general solution for an isotropic homogeneous elastic plate of finite thickness. Under the assumption of vanishing out-of-plane shear stresses, a necessary condition of solvability of elastic problems is obtained. Moreover, a general solution dependent on the thickness-wise coordinate is derived, where the unknown function is still governed by a two-dimensional biharmonic equation. In terms of the two-dimensional Airy stress function or the classical solution in plane strain state, exact elastic stresses in an elastic plate of finite thickness are given. The solution for plane strain state can be a special case of the present, whereas that classic plane stress state is reduced by letting thickness approach zero. A comparison of the results between the present paper and plane stress state is made.