An improved estimate for the error in the classical, linear theory of plate bending

Abstract

The relative mean square error in the three-dimensional stress field predicted by classical plate theory is shown to be O ( h / L ∗ ) 2 O{\left ( {h/{L_ * }} \right )^2} , where h h is the plate thickness and L ∗ {L_ * } is a mean square measure of the wavelength of the midplane deformation pattern. This improves a recent result of Nordgren who obtained a relative error estimate of O ( h / L ∗ ) O\left ( {h/{L_ * }} \right ) . The improved error estimate, which, like Nordgren’s, is based on the Prager—Synge hypercircle theorem in elasticity, is obtained by constructing a kinematically admissible three-dimensional displacement field that depends on the solution of the classical plate equations but which yields an accurate, nonzero distribution of the transverse shearing strain through the thickness.