An exact polynomial solution for a uniformly loaded, antisymmetric cross-ply laminated, clamped elliptic plate is combined with the Tsai-Hill failure criterion for an initial flexural failure analysis of GFRP and CFRP plates. Non-dimensionalised initial failure loads and associated plate centre deflections—both of which are significantly dependent on the direction of the applied loading—are presented graphically for the complete range of aspect ratios for plates consisting of varying numbers of laminae. It is also demonstrated that changes in the in-plane boundary conditions produce negligible changes in the initial failure response of the plate.