Nonlinear oscillations of laminated plates using an accurate four-node rectangular shear flexible material finite element

Abstract

The objective of the present paper is to investigate the large amplitude vibratory behaviour of unsymmetrically laminated plates. For this purpose, an efficient and accurate four-node shear flexible rectangular material finite element (MFE) with six degrees of freedom per node (three displacements (u, v, w) along the x, y and z axes, two rotations (θ_x and θ_y ) about y and x axes and twist (θxy )) is developed. The element assumes bi-cubic polynomial distribution with sixteen generalized undetermined coefficients for the transverse displacement. The fields for section rotations θ_x and θ_y, and in-plane displacementsu andv are derived using moment-shear equilibrium and in-plane equilibrium equations of composite strips along the x- and y-axes. The displacement field so derived not only depends on the element coordinates but is a function of extensional, bending-extensional coupling, bending and transverse shear stiffness as well. The element stiffness and mass matrices are computed numerically by employing 3×3 Gauss-Legendre product rules. The element is found to be free ofshear locking and does not exhibit any spurious modes. In order to compute the nonlinear frequencies, linear mode shape corresponding to the fundamental frequency is assumed as the spatial distribution and nonlinear finite element equations are reduced to a single nonlinear second-order differential equation. This equation is solved by employing thedirect numerical integration method. A series of numerical examples are solved to demonstrate the efficacy of the proposed element.