Comparative Analysis and Unified Derivation of Reissner’s Equations For 2D Bending of Thick Plates and Timoshenko’s Equations for Bending of Beams

Abstract

Currently, calculations of flexural deformations of lithospheric plates are carried out on the basis of Kirchhoff’s theory of bending of thin plates formulated about 170 years ago. The paper examines the possibility of refining these calculations based on the theory of bending of thick plates by Timoshenko and Reissner. A new unified derivation is presented of the Timoshenko equations for 2D bending of beams and the Reissner equations for bending of slabs by direct transformation of the general elasticity equations with a simple approximate replacement of power cubic functions with effective linear ones. This derivation offers a simpler and more detailed understanding of the difference between the equations and the meaning of the simplifications made in these theories. By comparing the analytical solutions of the Timoshenko and Reissner equations with the existing test analytical solutions of the exact elasticity equations, quantitative estimates of the accuracy of these theories are presented.