Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution

Basing on the First-order Shear Deformation Theory (FSDT), this paper focuses on the dynamic behaviour of moderately thick functionally graded parabolic panels and shells of revolution. A generalization of the power-law distribution presented in literature is proposed. Two different four-parameter power-law distributions are considered for the ceramic volume fraction. Some symmetric and asymmetric material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power-law distributions. The governing equations of motion are expressed as functions of five kinematic parameters. For the discretization of the system equations the Generalized Differential Quadrature (GDQ) method has been used. Numerical results concerning four types of parabolic shell structures illustrate the influence of the parameters of the power-law distribution on the mechanical behaviour of shell structures considered.     

Free vibration analysis of functionally graded panels and shells of revolution

The aim of this paper is to study the dynamic behaviour of functionally graded parabolic and circular panels and shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to study these moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when the materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface is metal rich and on the contrary for the second one. The governing equations of motion are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equations by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. Numerical results concerning eight types of shell structures illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behaviour of parabolic and circular shell structures. Preliminary results were presented by the authors at the XVIII° National Conference of Italian Association of Theoretical and Applied Mechanics (AIMETA 2007) (Tornabene and Viola 27).     

2-D solution for free vibrations of parabolic shells using generalized differential quadrature method

The Generalized Differential Quadrature (GDQ) procedure is developed for the free vibration analysis of complete parabolic shells of revolution and parabolic shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. The results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Several examples of parabolic shell elements are presented to illustrate the validity and the accuracy of GDQ method. Numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Femap/Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions. Different typologies of non-uniform grid point distributions are considered. The effect of the distribution choice of sampling points on the accuracy of GDQ solution is investigated. New numerical results are presented.     

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads is studied in the paper. Through deriving accurate expressions of inertial force and initial hoop tension, a rotating conical shell model is presented based upon the Love's thin shell theory. Considering the periodic axial loads, equations of motion of the system with periodic stiffness coefficients are obtained utilizing the generalized differential quadrature (GDQ) method. Hill's method is introduced for parametric instability analysis. Primary instability regions for various natural modes are computed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are examined.     

Vibration analysis of FG annular sector in moderately thick plates with two piezoelectric layers

Free vibration of functionally graded(FG) annular sector plates embedded with two piezoelectric layers is studied with a generalized differential quadrature(GDQ)method. Based on the first-order shear deformation(FSD) plate theory and Hamilton's principle with parameters satisfying Maxwell's electrostatics equation in the piezoelectric layers, governing equations of motion are developed. Both open and closed circuit(shortly connected) boundary conditions on the piezoelectric surfaces, which are respective conditions for sensors and actuators, are accounted for. It is observed that the open circuit condition gives higher natural frequencies than a shortly connected condition. For the simulation of the potential electric function in piezoelectric layers, a sinusoidal function in the transverse direction is considered. It is assumed that properties of the FG material(FGM) change continuously through the thickness according to a power distribution law.The fast rate convergence and accuracy of the GDQ method with a small number of grid points are demonstrated through some numerical examples. With various combinations of free, clamped, and simply supported boundary conditions, the effects of the thicknesses of piezoelectric layers and host plate, power law index of FGMs, and plate geometrical parameters(e.g., angle and radii of annular sector) on the in-plane and out-of-plane natural frequencies for different FG and piezoelectric materials are also studied. Results can be used to predict the behaviors of FG and piezoelectric materials in mechanical systems.     

Generalized differential and integral quadrature and their application to solve boundary layer equations

Based on the work of generalized differential quadrature (GDQ), a global method of generalized integral quadrature (GIQ) is developed in this paper for approximating an integral of a function over a part of the closed domain. GIQ approximates the integral of a function over the part of the whole closed domain by a linear combination of all the functional values in the whole domain with higher order of accuracy. The weighting coefficients of GIQ can be easily determined from those of GDQ. Applications of GDQ and GIQ to solve boundary layer equations demonstrated that accurate numerical results can be obtained using just a few grid points.     

Transient response of laminated plates with arbitrary laminations and boundary conditions under general dynamic loadings

The dynamic analysis of laminated plates with various loading and boundary conditions is presented employing generalized differential quadrature (GDQ) method. The first-order shear deformation theory is considered to model the transient response of the plate. The GDQ technique together with Newmark integration scheme is employed to solve the system of transient equations governing dynamics of the plate. Different symmetric and asymmetric lamination sequences together with various combinations of clamped, simply supported, and free boundary conditions are considered. Particular interest of this study regards to asymmetric orthotropic plates having free edge and mixed boundary conditions. It is shown that the method provides reasonably accurate results with relatively small number of grid points. Comparison of the results with those of other methods demonstrates a very good agreement. It is also revealed that the present method offers similar order of accuracy for all variables including displacements and stress resultants.     

Transient responses of magnetostrictive plates by using the GDQ method

We used the generalized differential quadrature (GDQ) method to compute the transient response of thermal stresses and center displacement in laminated magnetostrictive plates under thermal vibration. We obtained the GDQ solutions in a three-layer (0°^m/90°/0) and a 10-layer (0°^m/90°/0°/90°/0)_s laminated magnetostrictive plate with four simply supported edges. We presented the transient responses of thermal stress and center displacement with and without velocity feedback control, respectively. The advantage of the GDQ method used provide us with an efficient method to compute the results including shear deformation effect with a few grid points. These GDQ results had its potential that could be used and considered as basic data in the future magnetostrictive laminate studies.     

An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells

We analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three-dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material, although the analytical solution is also valid for isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flügge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.     

Thermal bending analysis of laminated orthotropic plates by the generalized differential quadrature method

The interlaminar stresses in a thin laminated rectangular orthotropic plate with four sides simply supported edges under bending was determined by using the generalized differential quadrature (GDQ) method involving the effects of thermal expansion strain and transverse load. The approximate stress and displacement solutions are obtained under the effects of thermal expansion force and uniform pressure load for eight-layer unidirectional laminates, symmetric cross-ply laminates. Numerical results on the dominant interlaminar stresses and displacement of bending analysis are compared to the Navier solution. The thermal induced forces have significant effect on the bending of plates.     

Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips

In this paper, the Generalized Differential Quadrature (GDQ) method is used to obtain bending solution of moderately thick rectangular plates. The plate is resting on two-parameter elastic (Pasternak) foundation or strips with a finite width. Various combinations of clamped, simply supported and free boundary conditions are considered. According to the first-order shear deformation theory, the governing equations of the problem consist of three second-order partial differential equations (PDEs) in terms of displacement and rotations of the plate. The governing equations and solution domain is discretized based on the GDQ method. It is demonstrated that the method converges rapidly while providing accurate results with relatively small number of grid points. Accuracy of the results is examined using available data in the literature for Pasternak foundation. Furthermore, due to lack of data for Pasternak strips, all predictions are verified by finite element analysis which can be used as benchmark in future studies.     

Analysis for the thermoelastic bending of a functionally graded material cylindrical shell

An analytic study for thermoelastic bending of a functionally graded material (FGM) cylindrical shell subjected to a uniform transverse mechanical load and non-uniform thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived out. An arbitrary material property of the FGM cylindrical shell is assumed to vary through the thickness of the cylindrical shell, and exact solution of the problem is obtained by using an analytic method. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load and the power law exponent on the deformation of the FGM cylindrical shell are analyzed and discussed.     

Nonlinear stability and dynamics of composite skew plates under nonuniform loadings using differential quadrature method

The buckling, postbuckling and postbuckled vibration behaviour of composite skew plates subjected to nonuniform inplane loadings are presented here. The skew plate is modelled using first order shear deformation theory accounting for von-Kármán geometric nonlinearity and initial geometric imperfections. The different types of nonuniform loads that have been considered in this study are concentrated load, partial load and parabolic load. The explicit analytical expressions for prebuckling stress distributions within composite skew plate subjected to three different types of nonuniform inplane loadings are developed by solving plane elasticity problem using Airy's stress function approach. It is observed that the inplane normal stress distributions within the skew plate due to above nonuniform loadings do not become uniform even at mid-section. The generalized differential quadrature (GDQ) method is used to solve the nonlinear governing partial differential equations. It is observed that the postbuckled load carrying capacity of skew plate under concentrated loading is the lowest compared to other nonuniform and uniform loadings.     

The analysis of functionally graded shallow and non-shallow shell panels with piezoelectric layers under dynamic load and electrostatic excitation based on elasticity

Elasticity solution is presented for finitely long, simply-supported, functionally graded shallow and non-shallow shell panel with two piezoelectric layers under pressure and electrostatic excitation. The functionally graded panel is assumed to be made of many sub panels. Each sub panel is considered as an isotropic layer. Material’s properties in each layer are constant and functionally graded properties are resulted by suitable arrangement of layers in multilayer panel. In each interface between two layers, stress and displacement continuities are satisfied. The highly coupled partial differential equations (p.d.e.) are reduced to ordinary differential equations (o.d.e.) with variable coefficients for non-shallow panel and constant coefficients for shallow shell panel by means of trigonometric function expansion in circumferential and longitudinal directions. The resulting ordinary differential equations are solved by Galerkin finite element method and Newmark method is used to march in time. Numerical examples are presented for functionally graded shell panel with a piezoelectric layer as an actuator in external surface and a piezoelectric layer as a sensor in internal surface and the results of the shallow and non-shallow panels are discussed.     

THE ANALYSIS OF SHALLOW SHELLS BASED ON MULTIVARIABLE WAVELET FINITE ELEMENT METHOD

Based on the generalized variational principle and B-spline wavelet on the interval (BSWI), the multivariable BSWI elements with two kinds of variables (TBSWI) for hyperboloidal shell and open cylindrical shell are constructed in this paper. Different from the traditional method, the present one treats the generalized displacement and stress as independent variables. So differentiation and integration are avoided in calculating generalized stress and thus the precision is improved. Furthermore, compared with commonly used Daubechies wavelet, BSWI has explicit expression and excellent approximation property and thus further guarantee satisfactory results. Finally, the effciency of the constructed multivariable shell elements is validated through several numerical examples.     

Application of GDQ Method for Study of Mixed Convection in Horizontal Eccentric Annuli

A generalized differential-integral quadrature (GDQ) discretization technique was used to solve a mixed heat convection problem in a body-fit coordinate system in its primitive variables form. A special treatment of the boundary condition to satisfy the continuity and momentum equations along the boundaries with the implementation of the GDQ method was investigated. Comparisons with the experimental and numerical results of other investigators are presented and discussed. In contrast with the existing published results, this highly accurate method was able to reveal extremely weak net circulation around the outer cylinder. In the horizontal annulus with the mixed heat convection problem the combination of unbalance of buoyancy and centrifugal forces causes net circulation. The net circulation decreases and approaches to zero with the rise of Rayleigh number, and it reaches its minimum value with high eccentricity when the inclination angle of eccentricity is π.     

Stress analysis of functionally graded shells using a 7-parameter shell element

In this paper, finite element stress analysis of functionally graded structures using a high-order spectral/hp shell finite element is presented. The shell element is based on a seven-parameter first-order shear deformation theory in which the seventh parameter, in addition to the usual six degrees of freedom, is the thickness stretch. The continuum shell element is utilized for the numerical simulations of the fully geometrically nonlinear response of functionally graded elastic shell structures. Several nontrivial shell problems are considered to report deflections and stresses, the latter being the main focus of the current paper. It is found that the stresses computed in the current study agree only in some cases with those of ANSYS and/or ABAQUS and thus requires additional study to determine the cause of the disagreement.     

Thermomechanical vibration analysis of a functionally graded shell with flowing fluid

This paper reports the results of an investigation into the vibration of functionally graded cylindrical shells with flowing fluid, embedded in an elastic medium, under mechanical and thermal loads. By considering rotary inertia, the first-order shear deformation theory (FSDT) and the fluid velocity potential, the dynamic equation of functionally graded cylindrical shells with flowing fluid is derived. Here, heat conduction equation along the thickness of the shell is applied to determine the temperature distribution and material properties are assumed to be graded distribution along the thickness direction according to a power-law in terms of the volume fractions of the constituents. The equations of eigenvalue problem are obtained by using a modal expansion method. In numerical examples, effects of material composition, thermal loading, static axial loading, flow velocity, medium stiffness and shell geometry parameters on the free vibration characteristics are described. The new features in this paper are helpful for the application and the design of functionally graded cylindrical shells containing fluid flow.     

一种特殊梯度分布的功能梯度圆柱壳的二维分析

功能梯度材料是一种新型材料,其结构分析已成为当今力学研究的热点。本文对一种特殊梯度分布的功能梯度材料圆柱壳进行了二维精确分析。从弹性力学平面应变问题的基本方程出发,引入应力函数,导出功能梯度材料圆柱壳受静载作用下的控制微分方程。假设材料的杨氏模量沿半径方向呈幂函数分布,泊松比为常数,利用分离变量法,导出了简支边界情况下功能梯度圆柱壳的精确解。通过算例分析了不同梯度变化时,功能梯度圆柱壳内的应力和位移变化规律。计算结果表明不同梯度分布的圆柱壳结构中的应力、位移沿厚度方向的变化规律是不同的,有时甚至差别很大。因此对于材料性质梯度变化的功能梯度材料圆柱壳,必须针对其自身特点,建立相应的理论分析模型。     

Estimates of the influence of the transverse strain in a contact problem with free boundary

We use the contact problem with free boundary for a “heavy cylindrical shell-over-support reinforcement ring” system as an example to study the influence of transverse shear on the stress state in the shell. To obtain the equilibrium equations for the shell and the reinforcement ring, we apply an express algorithm developed by one of the authors to take into account transverse strains in Kirchhoff versions of the theory of shells. The contact problem with free boundary is solved by the generalized reaction method proposed earlier. We use numerical examples to show that the corrections introduced in the stress state by taking into account the transverse shears are by an order of magnitude larger than the traditional estimate of errors in the Kirchhoff hypotheses.