Application of modified first order shear deformation theory to vibration analysis of SSSS and CCSS thick anisotropic rectangular plates

Abstract

This research presents a vibration analysis of a thick anisotropic rectangular plate using modified first shear deformation theory. Modified first shear deformation theory, which is not built upon the classical plate theory, was used to develop the kinematic equations and constitutive relations of a deformed section of thick anisotropic rectangular plate from which the generalized stress equations were determined. Using the assumptions of the theory, the strain energy and external work equations were formulated, and by employing the principles of total minimum potential energy, the total potential energy functional of a thick anisotropic plate was developed. Minimization of the total potential energy functional with respect to the deflection function (w) and with respect to the shear rotations (Φx) and (Φy) respectively, resulted in the governing differential equation and the two compatibility equations of the plate. The displacement functions that satisfy the governing and compatibility equations were obtained by solving the governing and compatibility equations. From the general displacement function, the peculiar deflection equations (shape functions) were obtained for the boundary conditions considered, which are simply supported on its four edges (SSSS), clamped on two adjacent edges, and simply supported on the other two (CCSS). Using the displacement equation and the equation for rotation in the x-direction (Φx) and equation for rotation in y-direction (Φy) the direct governing and two direct compatibility equations were obtained, from which coefficients that enable the formula for calculating fundamental natural frequencies to be obtained. For the boundary condition analyzed in this work, the stiffness coefficients (kR, kQ, kRQ, kq, kRRQ , kRQQ , kNR, kNQ, kNRQ and kλ ) were computed and used in determining the fundamental natural frequency parameter values at various span to depth ratios (5,10,20, 25 and 100), aspect ratios (1 to 2 at the increment of 0.1) and angle of fibre orientation (0o, 15o, 45o). The solutions of this study were compared with those from previous researchers. The fundamental natural frequency parameter values obtained in this study were compared with the work of Reddy (1984) for 0o angle of fibre orientation at span to depth ratios of 5,10, 20, 25, 50 and 100 at an aspect ratio of 1. The percentage difference values were 6.073%, 3.197%, 1.132%, 0.788%, 0.255%, and 0.112%, respectively. These differences revealed the closeness of the results of this present study to the results of Reddy (1984). This shows that the present theory provides good and acceptable solutions to the vibration problems of thick anisotropic rectangular plates.