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.