등변분포 평면응력을 받는 SS-C-SS-C 직사각형 판의 진동과 좌굴의 엄밀해

Abstract

Exact solutions are presented for the free vibration and buckling of rectangular plates haying 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 <TEX>$\sigma$</TEX><TEX>$_{x}$</TEX>=- <TEX>$N_{0}$</TEX>[1-a(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement ( w) to vary as sin(m<TEX>$\pi$</TEX>x/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 byields 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 a= 0, 0.5, 1, 1.5. 2, for which a=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for a= 0, 1, 2 obtained by the method of integration of the differential equation (a=0) or the method of energy (a=1, 2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b =0.5, 1, 2 when a=2, with load intensities <TEX>$N_{0}$</TEX> / <TEX>$N_{cr}$</TEX> =0, 0.5, 0.8, 0.95, 1. where <TEX>$N_{cr}$</TEX> is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes ate also shown.shown.