Non-linear static bending and forced vibrations of rectangular plates retaining non-linearities in rotations and thickness deformation

Abstract

Geometrically non-linear static bending and forced vibrations of rectangular plates are studied allowing full non-linear terms associated with Green–Lagrange strain–displacement relations, second-order thickness stretching, third-order shear deformation and rotary inertia by using seven independent parameters to describe the shell kinematics. In particular, in addition to non-linearities in membrane and transverse deflection, non-linear terms associated with rotations and thickness deformation parameters are also included. In order to obtain the governing equations of motion, the three-dimensional constitutive equations are used, removing the assumption of zero transverse normal strain. The boundary conditions of the plate are assumed to be simply supported immovable and the equations of motion are derived by using a Lagrangian approach. The numerical solutions are obtained by using pseudo arc-length continuation and collocation scheme. In order to compare the non-linear static response, another analysis has also been carried out by using the finite element code ANSYS and three-dimensional solid modeling. Results reveal that the new theory with full geometric non-linearities provides significant accuracy improvement for rotational and thickness deformation parameters, and, unlike other shear deformation theories, predicts the correct thickness stretching along the plate.