New benchmark free vibration solutions of non-Lévy-type thick rectangular plates based on third-order shear deformation theory

Abstract

Higher-order shear deformation theories are generally adopted to study the dynamic behavior of substantially thick plates, of which Reddy’s third-order deformation theory (TSDT) is most extensively used in view of its conciseness and high accuracy. However, due to the intractability in solving the governing higher-order partial differential equations, analytic solutions of TSDT-based thick plate problems are limited to very few special cases, i.e., Navier-type or Lévy-type solutions. Since analytic solutions can not only serve as benchmarks for validation of various methods, but are also useful for structural designs, especially at the early design stage, developing new analytic methods that are applicable to more common non-Lévy-type thick plates is of much importance. In this paper, for the first time, a novel symplectic superposition method, which was proposed originally for thin/moderately thick plate problems, is extended to free vibration problems of thick rectangular plates based on the TSDT. Besides yielding Lévy-type solutions, new non-Lévy-type analytic free vibration solutions are obtained. Comprehensive benchmark natural frequencies and mode shapes are tabulated/plotted, all of which are well validated by the 3D theory-based solutions from the finite element method (FEM) and the literature, if any. The present method is implemented in the symplectic space under the Hamiltonian system framework, rather than in the Euclidean space under the Lagrangian system framework as implemented in conventional methods, providing a new route to more analytic solutions for complex plate and shell problems that have not been explored due to mathematical challenge.