A relook at Reissner’s theory of plates in bending

Abstract

Shear deformation and higher order theories of plates in bending are (generally) based on plate element equilibrium equations derived either through variational principles or other methods. They involve coupling of flexure with torsion (torsion-type) problem and if applied vertical load is along one face of the plate, coupling even with extension problem. These coupled problems with reference to vertical deflection of plate in flexure result in artificial deflection due to torsion and increased deflection of faces of the plate due to extension. Coupling in the former case is eliminated earlier using an iterative method for analysis of thick plates in bending. The method is extended here for the analysis of associated stretching problem in flexure.

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

“…Higher order correction to 0 uncoupled from torsion is 1.262 so that total correction to the value from Kirchhoff 's theory is about 1.53 [7]. Correction due to coupling with torsion is 1.45 [5] whereas it is 1.423 from FSDT and other sixth order theories [9].…”

“…We use thickness-wise distribution functions ( ) generated from recurrence relations [7] with 0 = 1, 2 +1, = 2 , 2 +2, = − 2 +1 such that 2 +2 (±1) = 0. They are (up to = 5) …”

“…In the present work, reverse process is adapted in which reactive transverse stresses are determined first and the displacements are obtained in terms of these stresses. Equations governing second-order corrections to the preliminary solutions are derived through the application of an iterative method used earlier [7] for the analysis of bending of isotropic plates.…”

On a sequence of approximate solutions: bending of a simply supported square plate

“…Dependence of analysis on vertical deflection w 0 (x, y) is eliminated (one should note that the coupling of analysis with w 0 in energy methods is due to the work done by the applied transverse stresses during deformation and also the root cause for Poisson-Kirchhoff's boundary conditions paradox). Also, the coupling between flexure problem and associated torsion problem is eliminated through an iterative method (Vijayakumar 2011a). The problem at each stage of iteration is defined by a sixth-order system of equations using a solution at the preceding stage of iteration.…”

“…The problem at each stage of iteration is defined by a sixth-order system of equations using a solution at the preceding stage of iteration. In the earlier investigations (Vijayakumar 2009(Vijayakumar , 2011a, in-plane displacements are expressed in terms of gradients of two functions, ψ and φ, in which ψ is related to w and φ is required to decouple bending and torsion problems. The function φ is an auxiliary plane harmonic function from zero rotation (v, x − u, y ) denoted by ω z about z-axis.…”