Some analytical solutions of functionally graded Kirchhoff plates

The elastostatic problem of a functionally graded Kirchhoff plate, with no kinematic constraints on the boundary, under constant distributions of transverse loads per unit area and of boundary bending couples is investigated. Closed-form expressions are provided for displacements, bending-twisting curvatures and moments of an isotropic plate with elastic stiffness and boundary distributed shear forces, assigned respectively in terms of the stress function and of its normal derivative of a corresponding Saint-Venant beam under torsion. The methodology is adopted to solve circular plates with local and Eringen-type elastic constitutive behaviors, providing thus new benchmarks for computational mechanics. The proposed approach can be used to obtain other exact solutions for plates whose planform coincides with the cross-section of beams for which the Prandtl stress function is known in an analytical form.