Vibration analysis of shear deformable plates by a new mixed stress finite element

Plate finite elements are commonly used for static and dynamic structural analyses and a number of these elements appeared in the literature in the last years. Some of them have been shown to be numerically accurate, but accuracy was often obtained at the price of mathematical complexity and/or high computational cost. A simple four-node quadrilateral element for the static analysis of shear deformable elastic plates, named 9betaQ4, has been recently proposed. The element is based on a hybrid stress formulation involving, as primary variables, compatible displacements and elementwise equilibrated stress resultants. The transverse displacement approximation is enhanced via the so called linked interpolation, which includes nodal rotation parameters through higher-order interpolation functions. A suitable approximation for a priori equilibrated stress resultants is selected, ruled by the minimum number of parameters. The element is stable and locking-free. Within this framework, in the present work a new mixed stress finite element for the shear deformable plate bending dynamics is presented. The new element is developed from the 9betaQ4 element, following the consistent approach recently proposed. In the spirit of the original hybrid stress approach, the idea is to separate the stress field into two parts. The first part is required to satisfy a priori the static equilibrium equations, as in the original hybrid stress formulation. The second part is introduced upon the need to balance inertia forces and consistently represented as suggested by dynamic equilibrium equations. In particular, the divergence space for this second stress part is assumed to be the same as the representation space for the inertia forces resulting from assumed displacements. On this regard, notice that displacements within the element are assumed as independent from displacements at the element boundary and continuity is weakly enforced by means of tractions which are assumed to be independent from stresses. Interior displacement, stress and traction parameters can be condensed out at the element level, so operating finally upon nodal displacements only The performance of the proposed element is tested on some numerical examples, involving both thin and thick plates.