Abstract Viscoelastic equilibrium problems of K irchhoff plates can be solved in a closed form only under special geometric assumptions, loading conditions and kinematic constraints on the boundary. A new solution procedure, based on a correspondence principle between a linearly elastic, homogeneous and orthotropic S aint -V enant beam under torsion and an isotropic linearly viscoelastic and functionally graded K irchhoff plate with no kinematic constraints on the boundary, is proposed. The methodology is adopted to evaluate displacement, bending–twisting curvature and moment fields of an elliptic plate, with viscoelastic constitutive behavior and loading conditions described by convolution integrals, assessing thus new benchmarks for computational mechanics. The analysis is specialized to periodic fiber-reinforced composites with polymeric matrix described by a four-parameter viscoelastic model.