Buckling analysis of Mindlin plates under the Green-Lagrange strain hypothesis

In the present paper, the influence of Green-Lagrange nonlinear strain-displacement terms, usually considered negligible under the von Kármán hypothesis, on the buckling of isotropic, moderately thick plates and shells, subjected to uniaxial load in the x or y direction and biaxial in-plane load, is investigated. The first order shear deformation plate theory is applied, and the governing equations are derived using the principle of minimum of the strain energy. If compared with the classical equation adopted in the literature, new nonlinear terms are here presented. They show as disregarded the non linear terms related to the in-plane strain-component can be legitimate, in the Mindlin model, only when the buckling mode involves both negligible in-plane displacements and rotations, if compared with the out-of plane component (Difficult to understand). The general Levy type solution method is employed, and exact buckling loads and mode shapes are derived. To verify the accuracy of the solution obtained, comparisons with existing data are first made. Then, through graphics and tables, the effect of the nonlinear strain-displacement terms for a range of boundary and load conditions, variations of aspect ratio, thickness ratio and changes in geometry is presented, which shows how the von Kármán model can sensibly overestimate the critical load for structures characterized by the modes involving comparable in-plane and out-of-plane displacements.