Poisson’s Theory for Analysis of Bending of Isotropic and Anisotropic Plates

Sixteen-decade-old problem of Poisson-Kirchhoff’s boundary conditions paradox is resolved in the case of isotropic plates through a theory designated as “Poisson’s theory of plates in bending.” It is based on “assuming” zero transverse shear stresses instead of strains. Reactive (statically equivalent) transverse shear stresses are gradients of a function (in place of in-plane displacements as gradients of vertical deflection) so that reactive transverse stresses are independent of material constants in the preliminary solution. Equations governing in-plane displacements are independent of the vertical (transverse) deflection . Coupling of these equations with is the root cause for the boundary conditions paradox. Edge support condition on does not play any role in obtaining in-plane displacements. Normally, solutions to the displacements are obtained from governing equations based on the stationary property of relevant total potential and reactive transverse shear stresses are expressed in terms of these displacements. In the present study, a reverse process in obtaining preliminary solution is adapted in which reactive transverse stresses are determined first and displacements are obtained in terms of these stresses. Equations governing second-order corrections to preliminary solutions of bending of anisotropic plates are derived through application of an iterative method used earlier for the analysis of bending of isotropic plates. 1. Introduction Kirchhoff’s theory [1] and first-order shear deformation theory based on Hencky’s work [2] abbreviated as FSDT of plates in bending are simple theories and continuously used to obtain design information. Kirchhoff’s theory consists of a single variable model in which in-plane displacements are expressed in terms of gradients of vertical deflection so that zero face shear conditions are satisfied. is governed by a fourth-order equation associated with two edge conditions instead of three edge conditions required in a 3D problem. Consequence of this lacuna is the well-known Poisson-Kirchhoff boundary conditions paradox (see Reissner’s article [3]). Assumption of zero transverse shear strains is discarded in FSDT forming a three-variable model. Vertical deflection and in-plane displacements are coupled in the governing differential equations and boundary conditions. Reactive (statically equivalent) transverse shears are combined with in-plane shear resulting in approximation of associated torsion problem instead of flexure problem. In Kirchhoff's theory, in-plane shear is combined with transverse shear implied