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

Abstract

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 0 ( , ). Coupling of these equations with 0 is the root cause for the boundary conditions paradox. Edge support condition on 0 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.

Review of a Few Selected Theories of Plates in Bending

“…Prescribed τ xy along an edge is in the form of its tangential gradient contributing an artificial additional vertical shear. This additional shear is due to partial nullification of interior τ xy in bending problem [ 10 , 11 ].…”

“…In-plane variables ( u 1 , v 1 ) uncoupled from w 0 are basic variables as in the earlier investigations [ 6 , 11 , 24 ]. Due to the condition ω z = 0 required to decouple bending and torsion problems, one obtains reactive transverse stresses from thickness-wise integration of equilibrium equations Constitutive relation gives ε z = − μf 1 e 1 .…”

“…It shows that the solution for w from the present analysis provides proper correction to the estimation of face deflection. It is underestimated by 2.16% [ 11 , 23 ].…”

Analysis of SSSS and CCCC Thick Anisotropic Rectangular Plate Using Exact Displacement Function

“…Illing, E [16] Analyzed the bending of thin anisotropic plates using complex fourth order polynomial differential equation. Vijayakumar, K [17] used Poisson's theory for the analysis of bending of isotropic and anisotropic plates. Vasilenko, A. T [18] determined the bending of an anisotropic elliptic plate on an elastic foundation using the method of successive approximations.…”