A variational Ritz formulation for vibration analysis of thick quadrilateral laminated plates

• A variational model for the dynamic study of quadrilateral plates is proposed.
• The kinematic corresponds to the trigonometric shear deformation theory (TSDT).
• Approximate analytical solutions for symmetric laminated plate are obtained.
• New results are presented that can be useful for validation purposes.
• Moderately thick and thick laminated plates can be analyzed with this formulation.

In the present study, a variational Ritz approach for vibration analysis of thick arbitrarily quadrilateral laminated plates, based on the trigonometric shear deformation theory (TSDT) is developed. In this theory, shear stresses are vanished at the top and bottom surfaces of the laminate and shear correction factors are no longer required. A general straight-sided quadrilateral domain is mapped into a square domain in the computational space using a four-node master plate, employing a geometric transformation. The displacement field components are approximated by sets of beam characteristic orthogonal polynomials generated using the Gram–Schmidt procedure. The use of Ritz method allows a high spectral accuracy and faster convergence rates than local methods such as finite element. The algorithm developed is quite general, free of shear locking and can be used to obtain natural frequencies and modal shapes of laminated plates having various material parameters, geometrical planforms, length-to-thickness ratios and any combinations of free, simply supported and clamped edge support conditions.Through several numerical examples, the capability, efficiency and accuracy of the formulation are demonstrated. Convergence studies and comparison with other existing solutions in the literature suggest that the present algorithm is robust and computationally efficient.