Development of the consistent second-order plate theory for transversely isotropic plates and its analytical assessment from the three-dimensional perspective

Abstract

In this paper, the consistent second-order plate theory is developed for transversely isotropic plates. It is validated against the three-dimensional elasticity theory using a well-known benchmark problem of a simply-supported rectangular plate subjected to symmetric transverse sinusoidal loading. The choice of the benchmark problem is based on the fact that it allows for an exact three-dimensional elasticity solution to be derived in closed form. In this study, a closed-form solution based on Elliot’s displacement potentials for transversely isotropic solids is specifically derived for validation purposes. Its equivalence to other closed-form analytical solutions is established. Expanding the closed-form analytical solution into a power-law series with respect to the non-dimensionalised plate thickness enables a direct term-by-term comparison with the consistent second-order plate theory solution and provides a valuable mechanism to validate the consistent plate theory for transversely isotropic plates in a purely analytical form. The term-by term comparison reveals that the first terms of the above power-law series coincide exactly with the expressions of the consistent second-order plate theory. In addition to the analytical validation, a numerical validation using the finite element method is performed. A comparative analysis of several plate theories for transversely isotropic plates demonstrates that the consistent plate theory can predict displacements and stresses in thick transversely isotropic plates with very high degree of accuracy, such that even for very thick plates with a thickness-to-length ratio of 0.5, the deviation from the three-dimensional elasticity solution is less than 1%.