Finite element analysis and evaluation of an extended two‐dimensional elasticity theory: Application to thick plate vibrations

An extended two‐dimensional (2D) elasticity theory based on the weighted residual concept is developed in an attempt to capture thickness effects in vibrating thick plates. Plate theories can only provide approximate solutions due to certain kinematics assumptions. The weighting functions are expressed in terms of Legendre orthogonal polynomials. A formulation in rectangular Cartesian coordinates serves as the basis for an implementation of the finite element method. The applicability of the extended 2D elasticity theory is demonstrated through examples of rectangular, thick plates with different aspect ratios, thickness ratios, and boundary conditions. The accuracy of the extended 2D elasticity theory is assessed through a three‐dimensional finite element analysis. The frequencies and modal stresses show excellent agreement with three‐dimensional finite element calculations.