Numerical developments in solving the buckling and vibration of Mindlin plates

Abstract

Based on the Mindlin's shear deformation plate theory, the strain and kinetic energy functionals of thick isotropic plates on different plate domains have been derived. Applying a newly developed pb-2 Rayleigh-Ritz method to these energy functionals, the governing eigenvalue equations for buckling and vibration of Mindlin plates have been obtained. A special feature of the pb-2 Rayleigh-Ritz method lies in the definition of the Ritz function which consists of the product of a two-dimensional polynomial function and the boundary expressions. Each of the equations of the boundary is raised to the power of 0 or 1 according to the edge restraint conditions (free, simply supported or clamped). Thus, pb-2 Ritz functions automatically satisfy the geometric boundary conditions of plates from the on-set, which makes the Rayleigh-Ritz method readily applicable for plates of arbitrary shape and boundary conditions. Buckling and vibration of Mindlin plates with classic boundary conditions and with internal restraints have been studied. The effects of shear deformation, rotary inertia and plate dimensions on the buckling and vibration behavior of the plates are investigated. A numerical approach has been proposed to study the vibration of Mindlin plates of arbitrary shape with internal point supports. By making use of the point constraint functions, the eigenvalue equation obtained by the pb-2 Rayleigh-Ritz method is modified, which leads to a new eigenvalue equation with fewer degrees of freedom. A study on the vibration of plates under isotropic inplane pressure has been carried out. It has been observed that the frequencies of the plates increase with an increasing tensile pressure and decrease with and increasing compressive pressure. The vibration behavior of rectangular Mindlin plates with intermediate stiffeners has also been investigated. The effects of the torsional rigidity of the stiffeners, the shear deformation and rotary inertia on the vibration frequencies of the stiffened plates have been studied. Finally, the effect of prebuckling inplane deformation on the buckling stress factors of rectangular Mindlin plates has also been examined. It has been seen that the effect of prebuckling inplane deformation on the buckling stress factors is of the same order of magnitude as the effect of shear deformation. The effect of the Poisson ratio on buckling load has also been studied. Using the pb-2 Rayleigh-Ritz method, extensive numerical results have been generated in the present study for the buckling and vibration of Mindlin plates of various plate shapes with different boundary conditions, internal restraints and loading conditions. These results, most of which are not available in the open literature, should serve as useful information for engineers and researchers.