Analysis of Single Variable Thick Plate Buckling Problems using Galerkin Method
DOI:
https://doi.org/10.37933/nipes/7.2.2025.16Abstract
A single variable shear deformable thick plate buckling equation is developed using systematic first principles approach. The equation is solved in closed form for simply supported boundary conditions using Galerkin method for in-plane uniaxial and biaxial compressive loads on the edges. The equation has one unknown and is similar in form as the thin plate equation, rendering it amenable to solution methods for the thin plate equation. The equation was derived using the total energy minimization method. The Galerkin method used exact sinusoidal shape functions of simply supported boundary conditions as the basis functions to construct the Galerkin variational integral which was minimized with respect to the unknown displacement parameters (amplitudes) to yield the governing equations of buckling. The eigenvalue problem was solved to find the zeros which gave the eigenvalues from which the critical buckling load was determined. Comparison of the critical buckling loads with the previous results in literature showed they were identical to the exact critical buckling results for simply supported plates for both cases of uniaxial and biaxial compressive loads considered for various ratios of dimension, a, to thickness h (a/h) and then dimension b to dimension a (b/a). The orthogonality of the eigenfunctions used as the solution basis simplified the resulting integration problem. The Galerkin methods gave exact results for the simply supported plate buckling problem because exact shape function which satisfied the Dirichlet boundary conditions were used, and the domain equations were also satisfied at all points on the plate. The novelty of this work is the first principle step by step approach used in the equilibrium method deployed in the derivation of the governing deferential equation of equilibrium (DEoE). Another unique feature of the study is the systematic formulation and solution of the Galerkin Variational Equation (GVE) for the plate problem considered.