Analytical, three-dimensional elasticity solutions to some plate problems, and some observations on Mindlin's plate theory

Abstract

Solutions are obtained, using the exact three-dimensional theory of elasticity, to (i) the eigenvalue problem or buckling under biaxial compression or the free lateral vibration of a simply supported rectangular plate with orthotropic stress-strain properties, and (ii) the static response of the same plate to a lateral load that varies sinusoidally in two directions. The eigenvalue in problem (i) and the lateral deflections of both the surface and middle plane of the plate, as well as the bending strains, in problem (ii) are obtained in the form of series expansions in even powers of the plate thickness. Exact algebraic expressions are presented for the first two coefficients in the case of orthotropic plates; additional coefficients can be obtained if required, and are given for the much simpler case of isotropic plates. In all cases the first term agrees with classical plate theory. The solutions are compared with those obtained from Mindlin's plate theory. In neither problem is it found to be possible, in general, to choose values for Mindlin's effective shear moduli to make the Mindlin solution agree with the first two terms of the exact solution. There are, however, two exceptions to this, namely a restricted class of orthotropic materials, embracing all isotropic ones, in which the elastic constants satisfy a certain condition, and the case of cylindrical bending when the Mindlin plate reduces to a Timoshenko beam of wide rectangular cross-section. In both these exceptional cases appropriate values for the effective shear moduli are obtained.