Analytical solution for the buckling of rectangular plates under uni-axial compression with variable thickness and elasticity modulus in the y-direction

Abstract

The present study introduces a highly accurate numerical calculation of buckling loads for an elastic rectangular plate with variable thickness, elasticity modulus, and density in one direction. The plate has two opposite edges ( x = 0 and a) simply supported and other edges ( y = 0 and b) with various boundary conditions including simply supported, clamped, free, and beam (elastically supported). In-plane normal stresses on two opposite simply supported edges ( x = 0 and a) are not limited to any predefined mathematical equation. By assuming the transverse displacement to vary as sin( mπ x/ a), the governing partial differential equation of plate motion will reduce to an ordinary differential equation in terms of y with variable coefficients, for which an analytical solution is obtained in the form of power series (Frobenius method). Applying the boundary conditions on ( y = 0 and b) yields the problem of finding eigenvalues of a fourth-order characteristic determinant. By retaining sufficient terms in power series, accurate buckling loads for different boundary conditions will be calculated. Finally, the numerical examples have been presented and, in some cases, compared to the relevant numerical results.