Classical and homogenized expressions for the bending solutions of FGM plates based on the four variable plate theories

Exact correspondence relations between the bending solutions of a simply supported rectangular functionally graded material plate based on the four variable refined higher-order shear deformation theory and those of the corresponding reference homogenous Kirchhoff plate based on the classical plate theory are derived analytically for the material properties varying continuously in the thickness direction. The deflection, stress components, the resultant forces and bending moments of a thick functionally graded material plate are expressed analytically in terms of the deflection of the reference homogenous Kirchhoff plate with the same geometry, loadings and boundary constraints. Consequently, the bending solution of a functionally graded material plate based on the higher-order shear deformation theory is simplified as calculations of three scaling factors which can be easily determined analytically for the specified material gradient profile, the shear stress shape function and the aspect ratio of the functionally graded material plate, because the solution of the reference homogenous Kirchhoff plate can be easily found even in the text book. As examples, particular solutions for a functionally graded material plate subjected to both uniformly and sinusoidally distributed loads are presented, which illustrate the validity of this new approach. Accuracy of the present solutions are demonstrated by comparing them with those obtained by different palate theories with different shear stress shape functions available in the literature. The analytical solutions can be used as benchmarks to check numerical solutions of static bending of functionally graded material plates based on different higher-order shear deformation theories.