Three-dimensional free vibration analysis of variable stiffness laminated composite rectangular plates

The free vibration of variable stiffness laminated composite rectangular plates is studied for the first time on the basis of three-dimensional elasticity theory combined with the p-version of the finite element method. Each layer is modeled as one brick p-element with curvilinear fibers. The element stiffness and mass matrices are derived based on the principle of virtual displacements. Inter-element compatibility is achieved by matching the generalized displacements at vertices, edges, and faces shared by elements. Results are obtained for frequencies, modal displacements, and modal stresses of symmetric and anti-symmetric laminates with various combinations of free, simply supported, and clamped boundary conditions. The method is validated through convergence study and comparison with published three-dimensional frequencies for constant stiffness laminated composite plates. The frequencies predicted by the equivalent single-layer classical plate theory and first-order shear deformation theory show deviation from three-dimensional solutions. Phenomena such as the discontinuity of inter-laminar modal flexural stresses, change of sign of modal transverse shear stresses through the thickness, and modal cross-sectional warping are observed and explained. New three-dimensional frequencies for variable stiffness laminated composite plates are provided, which may serve as a benchmark for future studies.