Three-dimensional vibration analysis of thick functionally graded conical, cylindrical shell and annular plate structures with arbitrary elastic restraints

Abstract

In the present work, a three-dimensional vibration analysis of thick functionally graded conical, cylindrical shell and annular plate structures with arbitrary elastic restraints is presented. The last two structures are obtained as special cases of the conical shell. The effective material properties of functionally graded structures vary continuously in the thickness direction according the general four-parameter power law distributions in terms of volume fraction of constituents, and are estimated by Voigt’s rule of mixture. The exact solution is obtained by means of variational principle in conjunction with modified Fourier series which is composed of a standard Fourier series and some auxiliary functions. Validity and accuracy of the current method are demonstrated by comparing the present solutions with existing results. Numerous new results are given for functionally graded conical, cylindrical shells and annular plates with various boundary conditions including classical and elastic boundary conditions. Parametric investigations are carried out to study the effects of geometrical parameter, boundary conditions and material profiles on free vibration of functionally graded structures.

Introduction

Functionally graded materials (FGMs) are special composite materials with gradient compositional variation of constituents which results in continuously varying material properties. Such materials possess various advantages over conventional composite laminates, such as smaller stress concentration, higher fracture toughness, and improved residual stress distribution. Therefore, FGMs have been utilized in various engineering application, such as mechanical, aerospace, marine and civil engineering.

The shells and panels of revolution as structure elements occupy a leadership position in a variety of engineering fields due to their mechanical merits and structural efficiencies. In recent years, those structures made of FGMs have been applied widely. In order to use them effectively and successfully, a good understanding of the vibration of the shells and panels of revolution is necessary. The purpose of this paper is to study the free vibration of conical, cylindrical shells and annular plates with three-dimensional elasticity theory derived from shells of revolution.

Extensive investigations have been carried out to study the vibration problems of those FGM structures based on two-dimensional (2-D) theories, such as classic shell theory (SCT), first-order shear deformation theory (FSDT) and higher-order deformation theory (HSDT). The classic shell theories are based on Kirchhoff–Love hypothesis. Many researchers analyzed thin shells using the CSTs for its simplicity, such as Loy et al. [1], Pradhan et al. [2], Haddadpour et al. [3], Shah et al. [4], Arshad et al. [5], Sofiyev [6], Sofiyev and Schnack [7] and Sofiyev and Kuruoglu [8]. Since the shear deformation effect is neglected, the CSTs cannot be applied to moderately thick and thick shells. Shear deformation theories eliminate deficiency of CSTs by accounting for transverse shear deformation of shells. The first-order shear deformation theory assumes constant shear deformation across the shell thickness which results in that the accuracy of solutions strongly depends on the shear correction factors. There are some investigations on FGM shells based on FSDT [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. The higher-order shear deformation theories assume nonlinear strain variation through the shell thickness, which can avoid the shear correction factors and give a better prediction for thick shells. A number of literature concerning vibration problems of shell based on HSDTs can be fund, and among these the one by Najafizadeh and Isvandzibaei [26], Matsunaga [27], Viola et al. [28], Zozulya and Zhang [29], Najafizadeh and Heydari [30], Saidi [31], Ma and Wang [32], Shariyat [33] and Sofiyev and Kuruoglu [34] are mentioned herein. It should be mentioned all 2-D theories are approximate models developed on the basis of certain kinematic assumption which result in relatively simple expression and derivation of solutions. Actually, three-dimensional (3-D) elasticity theory which does not rely on any hypotheses about the distribution field of deformations and stress not only provides realistic results but also allows further physical insight [35]. Therefore, in order to provide accurate solutions the 3-D elasticity is adopted in this paper.

In the recent decades, attempts have been made for three-dimensional vibration analysis of FGM conical, cylindrical shells and annular plates [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]. Chen et al. [36] investigated the free vibration of simply supported, fluid-filled cylindrically orthotropic functionally graded cylindrical shells based on 3D equations of elasticity using a state space method and a laminate approximate model. Santos et al. [37], [38] studied thermoelastic behavior and free vibration of functionally graded cylindrical shells employed a semi-analytical finite model based on the 3-D linear elastic theory. The free, steady-state and transient vibrations of functionally graded cylinders, cones and spheres with arbitrary boundary conditions was investigated by Qu and Meng [39], [40], in which the theoretical modal is formulated by means of a modified variational principle combined with a multi-segment partitioning procedure in the context of 3-D elasticity theory. A three-dimensional free vibration analysis of the functionally graded truncated conical shells subjected to thermal environment was presented by Malekzadeh et al. [41], the differential quadrature method (DQM) is adopted to solve the thermal and thermo-mechanical governing equations. Dong [43] obtained the three-dimensional solution for the free vibration of functionally graded annular plates using the Chebyshev–Ritz method, in which a set of duplicate Chebyshev polynomial series multiplied by the boundary functions satisfying the boundary conditions are chosen as the trial functions of the displacements. Nie and Zhong [44], [45] employed a semi-analytical method, which makes use of the state space method and differential quadrature method, to study the vibration and bending of the functionally graded circular and annular plates with various boundary conditions. Tajeddini et al. [47] analyzed the three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation using the polynomial-Ritz method.

The above review clearly indicates that the available 3-D solutions of FGM conical, cylindrical shells and annular plates are limited and most of the previous studies are confined to vibration problems of FGM shells and plates of revolution with classical boundary conditions. It is noted that there are non-classical boundary conditions such as elastic boundary conditions in practical engineering applications. In this paper, a three-dimensional vibration analysis of thick functionally graded conical, cylindrical shell and annular plate structures with arbitrary elastic restraints is presented. The last two structures are obtained as special cases of the conical shell. The effective material properties of functionally graded structures vary continuously in the thickness direction according the general four-parameter power law distributions in terms of volume fraction of constituents, and are estimated by Voigt’s rule of mixture. The exact solution is obtained by means of variational principle in conjunction with modified Fourier series which is constructed as a standard Fourier series with auxiliary functions. Validity and accuracy of the current method are demonstrated by comparing the present solutions with existing results. Numerous new numerical solutions are given for functionally graded conical, cylindrical shells and annular plates with various boundary conditions including classical and elastic boundary conditions. Parametric investigations are carried out to study the effects of the geometrical parameter, boundary conditions and material profiles on free vibration of functionally graded structures.