Exact solutions for the buckling of rectangular plates having linearly varying in-plane loading on two opposite simply supported edges

An exact solution procedure is formulated for the buckling analysis of rectangular plates having two opposite edges (x = 0 and a) simply supported when these edges are subjected to linearly varying normal stresses σx = −N0[1−α(y/b)]/h, where h is the plate thickness. The other two edges (y = 0 and b) may be clamped, simply supported or free, or they may be elastically supported. By assuming the transverse displacement (w) to vary as, sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (i.e., the method of Frobenius). Applying the boundary conditions at y = 0 and b yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to retain sufficient terms in the power series in calculating accurate buckling loads, as is demonstrated by a convergence table for all nine possible combinations of unloaded clamped, simply supported or free edges at y = 0 and b. Buckling loads are presented for all nine possible edge combinations over the range of aspect ratios 0.5 ⩽ a/b ⩽ 3 for loading parameters α = 0, 0.5, 1, 1.5, 2, for which α = 2 is a pure in-plane bending moment. Some interesting contour plots of their mode shapes are presented for a variety of edge conditions and in-plane moment loadings. Because the nondimensional buckling parameters depend upon the Poisson's ratio (ν) for five of the nine edge combinations, results are shown for them for the complete range, 0 ⩽ ν ⩽ 0.5 valid for isotropic materials. Comparisons are made with results available in the published literature.