A new approach to static and dynamic analysis of composite plates with different boundary conditions

Abstract

In the present paper, a new analytical method is developed to analyse the response of laminated composite plates subjected to static and dynamic loading. The modal forms are presented in terms of double Fourier series. The derivatives of the double Fourier series are legitimized using Stokes's transformation. The method developed in this work is simpler than the methods adopted by other research workers, the main advantage being the ease with which one finds the modal forces, moments and mode shapes for laminated composite plates with different boundary conditions. The results from the present analysis are compared with those obtained from the FEM code NISA II. The agreement between the results is quite good.

Introduction

The knowledge of the mechanical behaviour of composite plates subjected to static and dynamic loading is of primary importance in engineering applications. The ability to achieve high strength- and high stiffness-to-weight ratios for fibre-reinforced composite materials have led to greater flexibility in the design of advanced structures. The structural behaviour of fibre composite plates subject to different types of loading has been studied by different authors. Sun and Chattopadhyay [1] studied a simply supported orthotropic plate subjected to the concentrated central impact using the equations developed by Whitney and Pagano [2]. In their work the nonlinear integral equation was solved numerically. Dobyns [3] used the same method, but replaced the concentrated impact load by a uniform pressure on a rectangular patch. Laplace transform technique was used to obtain an analytical solution to the impact problem by Swanson and Christoforou [4]. Carvalho and Gliedes Soares [5] developed a numerical approach by using the technique of Fourier series expansion. Also, Mittal et al. [6] developed an exact closed form solution for the dynamic response of clamped circular plates. Mittal and Khalili [7] used the method of Fourier transform and Hertz-Sveklo's contact theory to study the transverse impact of composite plates. Liew [8] carried out vibration analysis of laminated plates using Reissner–Mindlin theory. Using Galerkin method and Fourier series, Shi [9] studied the vibrational behaviour of clamped laminated plates, while Darvizeh et al. [10] studied the vibrational behaviour of laminated circular cylinder. The present study develops a new analytical method by assuming that the functions describing the plate displacement in the z direction and shear rotations in the x and y directions are in the form of double Fourier series. Stokes's transformation [11] has been exploited to legitimize the derivatives of this Fourier series for different sets of boundary conditions. The plate equations developed by Whitney and Pagano [2] for orthotropic laminated plates are used. The method employed in the present work gives an explicit expression for modal forces, moments and mode shapes of multi-layered composite plates with different boundary conditions.

In the dynamic analysis, the effects of boundary conditions and the variation of fibre orientations on natural frequencies are considered. In the static analysis, mode shapes corresponding to static loading are presented in the form of figures. The results of static and dynamic analyses are compared with those obtained from a FEM computer code NISA II. The results are in good agreement.