Natural vibrations of shear deformable rhombic plates with clamped and free edge conditions

Abstract

In this work C 0 continuous finite element assemblages of nine-node Lagrangian isoparametric quadrilateral plates based on a higher order shear deformable thick plate theory are used to analyze the natural vibrations of thick and thin skewed plates with clamped and completely free edges. The shear deformable plate elements used here possess additional nodal displacement degrees of freedom, which are derived by retaining higher order powers of the thickness coordinate in the in-plane displacement fields. These additional degrees of freedom are essential to the proper representation of the transverse shear vibratory strains of thick plates having an increasing degree of skewness. Essential rotary inertia terms are derived and included in the present analysis. An extensive number of nondimensional frequencies and mode shapes are calculated for thick and thin skewed plates having various combinations of clamped and free edge conditions, and arbitrary degrees of skewness. The efficacy of using higher order shear deformable plate finite elements for predicting the in-plane vibration modes of skewed plates is found to increase as the thickness ratio decreases and the skew angle increases. The present work shows that higher order shear deformable finite elements essentially eliminate the over-correction of thick skew plate frequencies that is produced when finite elements based on the widely used first-order Reissner/Mindlin plate theory are utilized. Results using the present finite element analysis are compared with those obtained using three-dimensional elasticity-based analyses.