For two different lamination schemes, under appropriate boundary conditions and sinusoidal distribution of the transverse load, the exact form of the spatial variation of the solution is obtained, and the problem is reduced to the solution of a system of ordinary differential equations in time, which are integrated numerically using Newmark’s direct integration method. Numerical results for deflections and stresses are presented showing the effect of plate side-to-thickness ratio, aspect ratio, material orthotropy, and lamination scheme. The results presented herein should be of interest to composite-structure designers, and to experimentalists and numerical analysts in verifying their results.