Exact solutions for the free vibrations of rectangular plates having in-plane moments acting on two opposite simply supported edges

An exact solution procedure is formulated for the free vibration analysis of rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying normal stresses causing pure in-plane moments. The other two edges may be clamped, simply supported or free, or they may be elastically supported. The transverse displacement (w) is assumed as sinusoidal in the direction of loading (x), and a power series is assumed in the lateral (y) direction (i.e., the method of Frobenius). Applying the boundary conditions yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to obtain adequate convergence for accurate vibration frequencies, as is demonstrated by two convergence tables. Some interesting and useful results for vibration frequencies and contour plots of their mode shapes are presented for plates having all nine possible combinations of clamped, simply supported or free unloaded edges. Particularly interesting is that for some of the edge conditions, applying opposite in-plane edge moments causes the fundamental frequency to increase initially.