Abstract This paper presents the governing equations of rectangular simply supported single-layer and two-layer cross-ply plates of bimodulus materials related to the higher order shear deformation terms. Natural frequency parameters and buckling coefficients obtained by the iteration method are compared with published results for ordinary thick plates. The Mindlin plate theory result can be obtained by simplifying the governing equation of the higher-order plate theory formulation. The effects of higher-order shear deformation terms on neutral surface locations, buckling coefficients and natural frequencies are evaluated by observing the difference between the present higher-order theory results and the Mindlin plate theory results. Also, the bending stress is shown to affect significantly the buckling coefficient K cr and the non-dimensional frequency parameter Ω.