Exact solutions for vibration and buckling of a SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

摘要

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x = 0 and a) simply supported and the other two (y = 0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ{sub}x = -N{sub}0[1 - α (y/b)]/h, where h is the plate thickness. 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 (the method of Frobenius). Applying the clamped boundary conditions at y = 0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α = 0, 0.5, 1, 1.5, 2, for which α = 2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α =0, 1, 2 obtained by the method of integration of the differential equation (α = 0) or the method of energy (α = 1, 2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b = 0.5, 1, 2 subjected to three types of loadings (α = 0, 1, 2), with load intensities N{sub}0/N{sub}(cr) = 0, 0.5, 0.8, 0.95, 1, where N{sub}(cr) is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.