FREE TRANSVERSE VIBRATIONS OF RECTANGULAR AND CIRCULAR ORTHOTROPIC PLATES

Abstract

The work presented in this thesis, is an attempt to study the vibrational behavior of orthotropic plates. It consists of nine chapters. Chapter I provide an up-to-date survey of the literature on the vibration of rectangular plates with various complicating effects such as non-homogeneity, thickness variation, elastic foundation, in-plane force, and elastically restrained edge. It reveals that almost no work has been done dealing with non-homogeneous orthotropic rectangular plates. In Chapters II to V, the effect of non-homogeneity with other complicating effects has been analyzed on the vibration characteristics of orthotropic rectangular plates. Further, in chapters VI and VII, the impact of varying in-plane forces has been studied on the vibration behavior of rectangular plates while chapters VIII and IX deal with circular plates of rectangular orthotropy. The numerical results have been computed employing four different numerical techniques for the first three modes of vibration on the basis of classical plate theory correct to four decimals. Mode shapes for specified plates have been illustrated in each chapter. The results would be of great interest to design engineers. The chapter-wise summary has been given as follows: Chapter II analyses the free transverse vibrations of non-homogeneous orthotropic rectangular plates of non-uniform thickness and resting on a Winkler type elastic foundation. The non-homogeneity of the plate material is assumed to arise due to the exponential variation in Young's moduli and density along one direction. Following Levy approach i.e. the two parallel edges are simply supported, the fourth order differential equation governing the motion of such plates of exponentially varying thickness in one direction, has been solved by using the quintic splines interpolation technique for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. Effect of the non-homogeneity and elastic foundation together with other plate parameters such as aspect ratio and thickness variation on the natural frequencies of vibration is illustrated for all the three modes. Comparison of results with the published literature demonstrates the computational efficiency of the method. Chapter III deals with the buckling and vibration behavior of non-homogeneous orthotropic rectangular plates of variable thickness having two opposite edges (y = 0 and b) simply supported and these are subjected to constant in-plane force. Assuming the variations for non-homogeneity and thickness of the plate as in chapter II, quintic splines technique has been used to obtain the frequency equations for three different combinations of clamped, simply supported and free boundary conditions at the other two edges (x = 0 and a). The effect of in-plane force parameter together with non-homogeneity, aspect ratio and thickness variation on the natural frequencies of vibration has been analyzed. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed. In chapter IV, an analysis for buckling and vibration characteristics of non-homogeneous orthotropic rectangular plates of varying thickness subjected to uniform biaxial in-plane forces has been presented. The two opposite edges (y = 0 and b) have been taken as simply supported while the other two edges (x = 0 and a) may have any combination of clamped and simply supported edge conditions. The governing differential equation of such plates has been ii solved using quintic spline technique with exponential variation for non-homogeneity and thickness of the plate as in chapter II. The effect of various parameters on the natural frequencies and critical buckling loads has been presented. In Chapter V, the effect of Pasternak type foundation has been investigated on the free transverse vibrations of non-homogeneous orthotropic rectangular plates with exponential variation in Young's moduli, density and thickness of the plate along x direction. The two opposite edges (y = 0 and b) are assumed to be simply supported while the other two edges (x = 0 and x = a) may have any combination of clamped and simply supported edge conditions. By expressing the displacement mode as a sine function of the variable between simply supported edges the fourth order partial differential equation governing the motion of such plates get reduced to an ordinary differential equation with variable coefficients which is then solved numerically by using Chebyshev collocation technique. The lowest three frequencies have been computed to study the behavior of foundation parameters together with other plate parameters for two different combinations of boundary conditions. A comparison of results with those obtained by other methods shows the computational efficiency of the approach. Chapter VI is concerned with the buckling and vibration of orthotropic rectangular plates having two opposite edges (y = 0 and b) simply supported, with these edges subjected to linearly varying in-plane forces Ny =—Noll— ax / The other two edges (x = 0 and a) may be clamped, simply supported or free. A semi-analytical approach has been used for the solution. Assuming the transverse displacement w to vary as sin pay 1 b , the partial differential equation governing the motion is reduced to an ordinary differential equation in x 111 with variable coefficients. The resulting ordinary differential equation is then solved numerically by the method of differential quadrature for three different combinations of clamped, simply supported and free boundary conditions. The effect of in-plane force parameter, loading parameter, together with the aspect ratio on the natural frequencies for the first three modes of vibration is obtained. The critical buckling loads in compression for various values of aspect ratio and loading parameter have been computed. The computational efficiency of the technique has been verified by comparing the results. In Chapter VII, the analysis of chapter VI has been extended to study the effect of Pastemak foundation for two different combinations of clamped and simply supported boundary conditions. The effect of foundation parameters together with in-plane force parameter, loading parameter and aspect ratio on the vibration characteristics of orthotropic rectangular plates has been studied for the first three modes. The critical buckling loads have been computed. Modes shapes have been presented. In chapter VIII, the effect of two-parameter foundation and thickness variation on the vibrational .behavior of circular plates of rectangular orthotropy has been presented. The thickness of the plate is assumed to vary quadratically in the radial direction. Ritz method has been used to obtain the frequency equations for clamped and simply supported plates. The lowest three roots of these equations have been obtained as the first three natural frequencies using bisection method. The effects of foundation parameters together with thickness variation on the natural frequencies for both the plates have been analyzed. Comparison of frequencies with those existing in the literature shows a close agreement. iv Chapter IX deals with the transverse free vibrations of circular plates of rectangular orhtotropy of quadratically varying thickness with elastically restrained edge. Ritz method has been employed to obtain the first three natural frequencies for different values of flexibility conditions. The classical edge conditions, namely, clamped, simply supported and free have been obtained as special cases. The effect of edge conditions and thickness variation on the natural frequencies has been investigated for all the three plates. Mode shapes have been computed.