Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment

The free vibration analysis of rotating functionally graded (FG) cylindrical shells subjected to thermal environment is investigated based on the first order shear deformation theory (FSDT) of shells. The formulation includes the centrifugal and Coriolis forces due to rotation of the shell. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermo-mechanical stresses are obtained by solving the thermoelastic equilibrium equations. The equations of motion and the related boundary conditions are derived using Hamilton’s principle. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the thermoelastic equilibrium equations and the equations of motion. The convergence behavior of the method is demonstrated and comparison studies with the available solutions in the literature are performed. Finally, the effects of angular velocity, Coriolis acceleration, temperature dependence of material properties, material property graded index and geometrical parameters on the frequency parameters of the FG cylindrical shells with different boundary conditions are investigated.     

Hybrid layerwise-differential quadrature transient dynamic analysis of functionally graded axisymmetric cylindrical shells subjected to dynamic pressure

In this paper, two computationally efficient and accurate solution methods for transient dynamic analysis of functionally graded (FG) cylindrical shells subjected to internal dynamic pressure are presented. In order to accurately account for the thickness effects, the layerwise theory is employed to approximate the displacement components in the radial direction. In the first solution method, differential quadrature method (DQM) is implemented to discretize the resulting equations in the both spatial and time domains. In the second approach, DQM is applied to discretize equations in the axial direction while Newmark’s time integration scheme is used to solve the problem in the time domain. The fast convergence rate of the methods is demonstrated and their accuracy is verified by comparing the results with those obtained using ANSYS and also with available exact solution of a particular problem. Considerable less computational efforts of the proposed approaches with respect to the finite element method is observed. Furthermore, comparative studies are performed between two approaches in different cases and it is found that the two techniques give very close results. The effects of geometrical parameters and boundary conditions on the transient behavior of shells are also investigated.     

Transient dynamic and free vibration analysis of functionally graded truncated conical shells with non-uniform thickness subjected to mechanical shock loading

This paper is focused on the transient dynamic and free vibration analysis of functionally graded (FG) axisymmetric truncated conical shells with non-uniform thickness. Two numerically efficient and accurate solution methods are presented to study the transient dynamic responses of FG shells subjected to either internal or external mechanical shock loading. Employing the displacement-based layerwise theory in conjunction with the Hamilton’s principle, the transversely discretized equations of motion are obtained. The differential quadrature method (DQM) is used to discretize the resulting equations in the axial direction. To solve the developed time-dependent equations, either DQM (named LWDQ) or Newmark’s time integration scheme (named LWDQN) is employed. The material properties are graded continuously in the thickness direction according to a volume fraction power-law distribution. The developed results are successfully compared with those obtained by ANSYS and also with the available results in the literature. The comparisons demonstrate the accuracy and effectiveness of the aforementioned methods on achievement of fast convergence rate with relatively low computational cost. Finally, the effects of different geometric and material parameters on the dynamic behavior of the FG shells are investigated. Due to high accuracy of the method, the results can be used as benchmarks for future research.     

Application of third order shear deformation theory in buckling analysis of 2D-functionally graded cylindrical shell reinforced by axial stiffeners

This paper presents buckling analysis of a two-dimensional functionally graded cylindrical shell reinforced by axial stiffeners (stringer) under combined compressive axial and transverse uniform distributive load. The shell material properties are graded in the direction of thickness and length according to a simple power law distribution in terms of the volume fractions of the constituents. Primarily, the third order shear deformation theory (TSDT) is used to derive the equilibrium and stability equations. Since there is no closed form solution, the numerical differential quadrature method, (DQM), is applied for solving the stability equations. Initially, the obtained results for an isotropic shell using DQM were verified against those given in the literature for simply supported boundary conditions. The effects of load, geometrical and stringer parameters along with FG power index in the various boundary conditions on the critical buckling load have been studied. The study of results confirms that, stringers have significant effects on critical buckling load.     

Elasticity solution of functionally graded circular and annular plates integrated with sensor and actuator layers using differential quadrature

Based on three-dimensional theory of elasticity axisymmetric static analysis of functionally graded circular and annular plates imbedded in piezoelectric layers is investigated using differential quadrature method (DQM). The plate has various edges boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. This method can give an analytical solution along the graded direction using the state space method (SSM) and an effective approximate solution along the radial direction using the one-dimensional DQM. The method is validated by comparing numerical results with the results obtained in the literature. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers on the mechanical behavior of plate is studied. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the static behavior of FG circular and annular plates are investigated.     

Static analysis of functionally graded cylindrical shell with piezoelectric layers using differential quadrature method

Three-dimensional solution for static analysis of functionally graded (FG) cylindrical shell with bonded piezoelectric layers is presented using differential quadrature method (DQM) and state-space approach. Applying the DQM to the governing differential equations and to the edges boundary conditions, new state equations about state variables at discrete points are derived. The stress, displacement, and electric potential distributions are obtained by solving these state equations. The convergence and accuracy of the present method is validated by comparing numerical results for the hybrid FG cylindrical shell with simply-supported edges with the analytical solution that has been published in the literature. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers and gradient index on the mechanical behavior of shell is studied.     

Temperature-dependent discrete layer-differential quadrature bending analysis of the multi-layered functionally graded annular plates rested on a two-parameter elastic foundation

A discrete layer approach coupled with the differential quadrature method (DQM) is employed to temperature dependent analyze the laminated functionally graded (FG) annular plates under mechanical loading in a thermal environment. The formulations are derived based on the elasticity theory, which includes the effects of the initial thermal stresses and two-parameter elastic foundation. The material properties are assumed to be temperature-dependent and graded in the thickness direction. In order to accurately evaluate the effect of the thermal environment, the initial thermal stresses are obtained by solving the thermoelastic equilibrium equation. Comparison studies with the available solutions in the literature for FG plates are performed. Then, as an application, three common types of FG sandwich plates, namely, the sandwich with homogeneous face sheets and FG core and the sandwich with FG face sheets and homogeneous metal (soft) and ceramic (hard) core are analyzed. The influences of temperature rise, temperature-dependence of material properties, layers lay-out, foundation stiffness parameters, material graded index, and geometrical parameters on the solution are carried out. The new results can be used as benchmark solutions for future researches.     

On a problem of the vibration of functionally graded conical shells with mixed boundary conditions

This paper presents a theoretical approach to solve vibration problems of functionally graded (FG) truncated conical shells under mixed boundary conditions. The material properties of FG shell are assumed to vary continuously through the thickness of the conical shell. The fundamental relations, motion and strain compatibility equations of FG truncated conical shells are derived by means of the Airy stress function method. Two cases of mixed boundary conditions are investigated. The basic equations are solved by using Galerkin method and fundamental cyclic frequencies of FG truncated conical shells are obtained. The results are compared and validated with the results available in the literature. The detailed parametric studies are carried out to investigate the influences of radius-to-thickness ratio, lengths-to-radius ratio, material composition and mixed boundary conditions on the fundamental cyclic frequencies of truncated conical shells.     

Three-dimensional free vibration analysis of thick laminated annular sector plates using a hybrid method

An accurate and efficient solution procedure based on the three-dimensional elasticity theory for the free vibration analysis of thick laminated annular sector plates is presented. Plates with simply supported radial edges and arbitrary boundary conditions on their circular edges are considered. In order to accurately model the variation of material properties across the thickness, the layerwise theory is used to approximate the displacement components in this direction. Then, employing the Hamilton’s principle together with the modal analysis, through-the-thickness and circumferential discretized form of the equations of motion and the related boundary conditions are obtained. Finally, the differential quadrature method (DQM) as an efficient and accurate numerical method is applied to discretize the resulting variable coefficients differential equations in the radial direction. The fast rate of convergence of the method is demonstrated and to show its high accuracy, comparison studies with the available results in the literature are made. Finally, some new results are prepared, which can be used as benchmark solutions for future works.     

Postbuckling of pressure-loaded FGM hybrid cylindrical shells in thermal environments

A postbuckling analysis is presented for a functionally graded cylindrical shell with piezoelectric actuators subjected to lateral or hydrostatic pressure combined with electric loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation theory with a von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling is extended to the case of FGM hybrid laminated cylindrical shells of finite length. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of pressure-loaded, perfect and imperfect, FGM cylindrical shells with fully covered piezoelectric actuators under different sets of thermal and electric loading conditions. The results reveal that temperature dependency, temperature change and volume fraction distribution have a significant effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells. In contrast, the control voltage only has a very small effect on the buckling pressure and postbuckling behavior of FGM hybrid cylindrical shells.     

Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded cylindrical shells subjected to mechanical loadings. Eight types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded cylindrical shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.     

The elastic response of functionally graded cylindrical shells to low-velocity impact

This paper deals with the problem of functionally graded (FG) cylindrical shells subjected to low-velocity impact by a solid striker. An analytic solution to predict the impact response of the FG cylindrical shells with one layer or multi-layers is presented. The solution includes both contact deformation and transverse shear deformation. The effective material properties of functionally graded materials (FGMs) for the cylindrical shells are assumed to vary continuously through the shell thickness and are graded in the shell thickness direction according to a volume fraction power law distribution. This is implemented in the governing equation of motion and thus included in the present solution. Four types of FG cylindrical shells composed of stainless steel and silicon nitride are configured and their transient responses to impact are computed using the present solution. The effects of the constituent volume fraction and the FGM configuration on the transient response of the laminated cylindrical shell induced by impact are examined.     

Free vibration of functionally graded arbitrary straight-sided quadrilateral plates in thermal environment

As a first endeavor, the free vibration of functionally graded (FG) arbitrary straight-sided quadrilateral plates under thermal environment and based on the first order shear deformation theory (FSDT) is presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermal stresses are evaluated by solving the thermo-elastic equilibrium equations. The solution procedure is based on transformation of the governing equations from physical domain to computational domain and then the discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The accuracy of the present method is demonstrated by studying the free vibration of isotropic and FG plates with various shapes and comparing the solutions obtained against existing results in literature. Then, the effects of thickness-to-length ratio, volume fraction index, temperature rise, geometrical shape and the boundary conditions on the frequency parameters of the plate are studied.     

Postbuckling of functionally graded fiber reinforced composite laminated cylindrical shells, Part I: Theory and solutions

Buckling and postbuckling behavior are presented for fiber reinforced composite (FRC) laminated cylindrical shells subjected to axial compression or a uniform external pressure in thermal environments. Two kinds of fiber reinforced composite laminated shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The governing equations are based on a higher order shear deformation shell theory with von Kármán-type of kinematic non-linearity and including the extension-twist, extension-flexural and flexural-twist couplings. The thermal effects are also included, and the material properties of FRC laminated cylindrical shells are estimated through a micromechanical model and are assumed to be temperature dependent. The non-linear prebuckling deformations and the initial geometric imperfections of the shell are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths of FRC laminated cylindrical shells.     

Three-dimensional nonlinear thermo-elastic analysis of functionally graded cylindrical shells with piezoelectric layers by differential quadrature method

Three-dimensional nonlinear thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers under the effect of asymmetric thermo-electro-mechanical loads is carried out. The strain–displacement relations are based on the nonlinear Lagrangian strain–displacement relations; that is, nonlinear terms containing derivatives of the displacement in the radial direction are included. Material properties of the shell are assumed to be graded in the radial direction according to a power law but the Poisson’s ratio is assumed to be constant. Cylindrical shells are assumed to be under the effect of pressure loading in cosine form, ring pressure loads, electric and temperature fields. Numerical results of stress, displacement, electric and thermal fields are obtained by using two versions of the differential quadrature methods, namely polynomial and Fourier quadrature methods. The convergence of the solution is studied, and results of the axisymmetric loadings are verified with reported results for a cylindrical shell with material properties obeying a power law. Effects of the grading index of material properties, the temperature difference, the ratio of the mean radius to the thickness of the shell, boundary conditions, the thickness of piezoelectric layers and electric excitation on stress, displacement, electric and temperature fields are presented.     

Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution

In this study, based on the three-dimensional theory of elasticity, static and free vibration characteristics of continuously graded fiber-reinforced (CGFR) cylindrical shells are considered by making use of a generalized power-law distribution. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material. The CGFR cylindrical shells have a smooth variation of matrix volume fraction in the radial direction. Symmetric and asymmetric volume fraction profiles are presented in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by a generalized differential quadrature method. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. The main contribution of this work is to illustrate useful results for a cylindrical shell continuously graded fiber reinforced in the radial direction. Finally, these results are compared with a similar discrete laminated composite cylindrical shell.     

Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties

As a first endeavor, the thermal buckling and postbuckling analysis of functionally graded (FG) annular plates with material properties graded in the radial direction is presented. The formulation is derived based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain in conjunction with von Karman’s assumptions. The material properties are temperature-dependent and graded according to the power law distribution. It is assumed that the temperature varies along the radial direction. Using the virtual work principle, the pre-buckling and postbuckling equilibrium equations and the related boundary conditions are derived. Differential quadrature method (DQM) as an efficient numerical technique is adopted to solve the governing equations. The presented formulation and the method of solution are validated by performing convergence and comparison studies with available results in the literature. Finally, the effects of volume fraction index, geometrical parameters, mechanical/thermal properties of the constituent materials and boundary conditions on the thermal buckling and postbuckling behavior of the radially graded annular plate are evaluated and discussed.     

Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory

A two-dimensional (2D) higher-order deformation theory is presented for vibration and buckling problems of circular cylindrical shells made of functionally graded materials (FGMs). The modulus of elasticity of functionally graded (FG) shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal deformations, and rotatory inertia is derived through Hamilton’s principle. Several sets of truncated Mth order approximate theories are applied to solve the eigenvalue problems of simply supported FG circular cylindrical shells. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency for the fundamental mode r=s=1 are examined in detail. A comparison of the present natural frequencies of isotropic and FG shells is also made with previously published results. Critical buckling stresses of simply supported FG circular cylindrical shells subjected to axial stress are also obtained and a relation between the buckling stress and natural frequency is presented. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses are calculated by integrating the three-dimensional (3D) equations of motion in the thickness direction satisfying the stress boundary conditions at the outer and inner surfaces. The 2D higher-order deformation theory has an advantage in the analysis of vibration and buckling problems of FG circular cylindrical shells.     

Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel

Free vibration of Levy-type thick functionally graded (FG) circular cylindrical shell panels is investigated to identify the validity range of two common shell theories namely Donnell and Sanders theories. FG material properties change through the thickness direction according to a power law distribution. The state space approach is applied to solve the problem. The present results are compared with those of the literature and a 3D finite element model for isotropic and FG materials. The effects of various geometry and material parameters on the validity range of these theories are studied for different boundary conditions. The results show that unlike Sanders theory, Donnell one cannot accurately capture natural boundary conditions such as force and moment resultants.     

On the solution of eigenvalue problems of laminated non-homogeneous orthotropic circular shells with clamped edges subjected to hydrostatic pressure

This article presents a method to study the free vibration and stability of laminated homogeneous and non-homogeneous orthotropic cylindrical, truncated and complete conical shells of general staking with clamped edges under a hydrostatic pressure. Based on the Love first approximation theory, the basic relations, the modified Donnell-type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells, the material properties of which vary piecewise continuously in the thickness direction. To solve this problem an unknown parameter λ was included in the approximation functions. Applying Galerkin methods, the buckling pressures and fundamental natural frequencies of laminated homogeneous and non-homogeneous orthotropic conical shells are obtained. The parameter λ which is included in the obtained formulas is obtained from the minimum conditions of critical stresses and frequencies. The different generalized values are obtained for the parameter λ for buckling pressures and frequencies of cylindrical shells, truncated and complete conical shells. The appropriate formulas for single-layer and laminated cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the influences of the degree of non-homogeneity, the number and ordering of layers and the variations of conical shell characteristics on the critical hydrostatic pressure and natural frequencies are investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.