Evaluation of modal stress resultants in freely vibrating plates

In the dynamic analysis of a very large floating structure (VLFS), it is crucial that the stress resultants are accurately determined for design purposes. This paper highlights some problems in obtaining accurate modal stress-resultant distributions in freely vibrating rectangular plates (for modeling box-like VLFSs) using various conventional methods. First, it is shown herein that if one adopts the classical thin plate theory and the Galerkin's method with commonly used modal functions consisting of the products of free-free beam modes, the natural boundary conditions cannot be satisfied at the free edges and the shear forces are completely erroneous, even when the eigenvalues have already converged. Second, it is shown that the problem still persists somewhat with the adoption of the more refined plate theory of Mindlin and the use of both (a) NASTRAN (that employs the finite element method) and (b) the Ritz method. The former method requires extremely fine mesh designs while the latter requires very high degrees of polynomial functions to achieve some form of satisfaction of the natural boundary conditions. Third, it is demonstrated that a modified version of the Ritz method, involving the use of a penalty functional for enforcement of the natural boundary conditions, also did not solve the problem when the plate is relatively thin. In fact, the method produces artificial stiffening to the plate. It is hoped that this paper will inspire researchers to develop an efficient technique for determining accurate stress resultants in a freely vibrating plate, apart from taking the brute force approach in having an extremely fine finite element mesh or using a very high polynomial degree