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, bending deformations of lithospheric plates are calculated based on Kirchhoff’s theory of bending of thin plates, which was formulated about 170 years ago. This paper examines the possibility of refining these calculations based on the theory of bending of thick plates developed 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 allows a more detailed and clearer understanding of the differences between the equations and the sense of the simplifications made in these theories. By comparing the analytical solutions of the Timoshenko and Reissner equations with the known test analytical solutions of the exact elasticity equations, quantitative estimates of the accuracy of these theories are given.