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

Abstract

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 x =− N 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 0/ N cr =0,0.5,0.8,0.95,1, where N cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.