Vibration analysis of FGM truncated and complete conical shells resting on elastic foundations under various boundary conditions

This article deals with vibration analysis of clamped (C?CC) and freely supported (Fs?CFs), truncated and complete conical shells on elastic foundations with continuously graded volume fraction. The functionally graded material (FGM) properties are assumed to vary continuously through the thickness of the conical shell. First, the basic relations, i.e., the dynamic stability and compatibility equations, of FGM truncated conical shells on the Pasternak-type elastic foundation are obtained. The displacement and Airy stress function are sought depending on a new parameter ??. The parameter ?? depends on the geometry of the shell and the loading and boundary conditions. By applying the Galerkin method to the foregoing equations, the dimensionless frequency parameters of FGM conical shells on the Pasternak-type elastic foundation for two boundary conditions are obtained. Furthermore, the parameter ?? which is included in the formulae is obtained from the minimization of the dimensionless frequency parameters. Finally, the effects of the stiffness of the foundation, boundary conditions, variations of the conical shell characteristics, and composition profiles on the values of the dimensionless frequency parameters are studied. The results are validated through comparison of obtained values with those in the literature.     

Static response and free vibration of two-dimensional functionally graded metal/ceramic open cylindrical shells under various boundary conditions

The free vibration and static response of a two-dimensional functionally graded (2-D FGM) metal/ceramic open cylindrical shell are analyzed using 2-D generalized differential quadrature method. The open cylindrical shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. This paper presents a novel 2-D power-law distribution for ceramic volume fraction of 2-D FGM that gives designers a powerful tool for flexible designing of structures under multifunctional requirements. Various material profiles in two radial and axial directions are illustrated using the 2-D power-law distribution. The effective material properties at a point are determined in terms of the local volume fractions and the material properties by the Mori–Tanaka scheme. The 2-D generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated, and to validate the results, comparisons are made with the available solutions for FGM cylindrical shells. The interesting results indicate that a graded ceramic volume fraction in two directions has a higher capability to reduce the mechanical stresses and natural frequency than conventional 1-D FGM. The achieved results confirm that natural frequency and mechanical stress distribution can be modified to a required manner by selecting an appropriate volume fraction profile in two directions.     

Vibration of functionally graded cylindrical shells based on higher order shear deformation plate theory with ring support

Summary In this paper, a study on the vibration of thin cylindrical shells with ring supports made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which impose a zero lateral deflection. The study is carried out based on third order shear deformation shell theory (T.S.D.T). The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configurations of the constituent materials on the frequencies. The properties are graded in the thickness direction according to the volume fraction power-law distribution. The analysis is carried out with strains-displacement relations from Love's shell theory. The governing equations are obtained using an energy functional with the Rayleigh-Ritz method. Results are presented on the frequency characteristics, influence of ring support position and the influence of boundary conditions. The present analysis is validated by comparing results with those available in the literature.     

On the stability of FGM shells subjected to combined loads with different edge conditions and resting on elastic foundations

In this study, the stability analysis of functionally graded material (FGM) cylindrical, truncated and complete conical shells subjected to combined loads and resting on elastic foundations for two boundary conditions is investigated. The functionally graded material properties are assumed to vary continuously through the thickness of the conical shell. At first, the basic relations, the stability and compatibility equations of the FGM truncated conical shell on the Pasternak-type elastic foundation are obtained. By applying the Galerkin method to the foregoing equations, the critical combined loads of clamped–clamped and sliding–sliding FGM shells on the Pasternak-type elastic foundation are obtained. Finally, carrying out some computations, effects of the elastic foundation, boundary conditions, the variation of shell characteristics and material composition profiles on the values of critical combined loads have been studied.     

Magnetothermoelastic Bending Analysis of a Functionally Graded Material Cylindrical Shell

An analytic method for bending analysis of a cylindrical shell composed by a functionally graded material (FGM), placed in a uniform magnetic field, subjected to mechanical and thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived, and exact solution of the magnetothermoelastic bending problem is obtained. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load, magnetic intensity, volume exponent, and geometric parameters on the bending deformation of the FGM cylindrical shell are discussed.     

Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory

Size-dependent dynamic stability response of higher-order shear deformable cylindrical microshells made of functionally graded materials (FGMs) and subjected to simply supported end supports is investigated. Material properties of the microshells vary in the thickness direction according to the Mori–Tanaka scheme. The modified couple stress elasticity theory in conjunction with the classical higher-order shear deformation shell theory is utilized to develop non-classical shell model containing additional internal length scale parameter to interpret size effect. The differential equations of motion and boundary conditions are derived by using Hamilton’s principle. The governing equations are then written in the form of Mathieu–Hill equations and then Bolotin’s method is employed to determine the instability regions. Selected numerical results are given to indicate the influences of internal length scale parameter, material property gradient index, static load factor and axial wave number on the dynamic stability behavior of FGM microshells. It is found that the width of the instability region for an FGM microshell increases with the decrease of the value of dimensionless length scale parameter. Moreover, it is shown that the classical shell model has an overestimated prediction for the width of instability region corresponding to the FGM microshells especially with lower values of material property gradient index.     

Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels

In this paper, natural frequencies characteristics of a continuously graded carbon nanotube-reinforced (CGCNTR) cylindrical panels based on the Eshelby–Mori–Tanaka approach is considered. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The CGCNTR shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The novelty of the present work is to exploit Eshelby–Mori–Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various mid radius-to-thickness ratio, shell angle, length-to-mean radius ratio and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR cylindrical panels. The interesting and new results show that continuously graded oriented CNTs volume fractions can be utilized for the management of vibrational behavior of structures so that the frequency parameters of structures made of such material can be considerably improved than that of the nanocomposites reinforced with uniformly distributed CNTs.     

非均匀热载荷作用下功能梯度梁的非线性静态响应

由于功能梯度材料结构沿厚度方向的非均匀材料特性,使得夹紧和简支条件的功能梯度梁有着相当不同的行为特征。该文给出了热载荷作用下,功能梯度梁非线性静态响应的精确解。基于非线性经典梁理论和物理中面的概念导出了功能梯度梁的非线性控制方程。将两个方程化简为一个四阶积分-微分方程。对于两端夹紧的功能梯度梁,其方程和相应的边界条件构成微分特征值问题;但对于两端简支的功能梯度梁,由于非齐次边界条件,将不会得到一个特征值问题。导致了夹紧与简支的功能梯度梁有着完全不同的行为特征。直接求解该积分-微分方程,得到了梁过屈曲和弯曲变形的闭合形式解。利用这个解可以分析梁的屈曲、过屈曲和非线性弯曲等非线性变形现象。最后,利用数值结果研究了材料梯度性质和热载荷对功能梯度梁非线性静态响应的影响。     

The modified couple stress functionally graded cylindrical thin shell formulation

In this article, the functionally graded (FG) cylindrical thin shell formulation is developed by using modified couple stress theory. The equations of motion and classical and nonclassical boundary conditions are extracted based on Hamilton's principle. As a special case, the equations of motion in conjunction with the boundary conditions for simply supported FG cylindrical shell are obtained, and then Navier solution procedure is used for analysis free vibration of nano shell. Afterwards, the influences of different parameters like length scale parameter, distribution of FG properties, and length to radius ratio on dimensionless natural frequency are investigated and compared with classical theory.     

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.     

Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects

This paper presents an analytical approach to investigate the nonlinear static and dynamic unsymmetrical responses of functionally graded shallow spherical shells under external pressure incorporating the effects of temperature. Governing equations for thin FGM spherical shells are derived by using the classical shell theory taking into account von Karman–Donnell geometrical nonlinearity. Approximate solutions are assumed and Galerkin procedure is applied to determine explicit expressions of static critical buckling loads of the shells. For the dynamical response, motion equations are numerically solved by using Runge–Kutta method and the criterion suggested by Budiansky–Roth. A detailed analysis is carried out to show the effects of material and geometrical parameters, boundary conditions and temperature on the stability and dynamical characteristics of FGM shallow spherical shells.     

Dynamic response of functionally graded circular cylindrical shells subjected to radial impulse load

Dynamic response of simply supported circular cylindrical shell made of functionally graded material (FGM) subjected to lateral impulse load is investigated. The effective material properties are assumed to vary continuously along the thickness direction according to a volume fraction power law distribution. First order shear deformation theory and Love’s first approximation theory are utilized in the equilibrium equations. Finally time response of displacement components is derived using mode superposition method. The influence of material composition, FGM configuration and geometrical parameters (length-to-radius and thickness-to-radius ratios) on the dynamic response is investigated. The results show that even though the shell is thin, the value of power law exponent has a considerable effect on the natural frequencies as well as the dynamic response of the functionally graded shell. Verification of the results of natural frequencies and time response of the FGM shell is achieved by making comparison with those available in the literature and those obtained using commercial software.     

热环境中的功能梯度扁球壳非线性稳定性分析

基于经典壳理论和von Karman几何非线性理论,导出了功能梯度圆底扁球壳的位移型几何非线性控制方程及简支边界条件,推导过程考虑了均匀变温场及均布外侧压力。用打靶法计算了由控制方程和边界条件提出的两点边值问题,得到了壳体轴对称变形的数值结果。考察了壳体几何参数、材料横向梯度特性、组份材料体积分数指数和弹性模量以及均匀变温场对壳体屈曲平衡路径、上/下临界荷载和平衡构形的影响。数值结果表明:随组分材料体积分数指数的增加和弹性模量的减小,壳体上临界荷载均会显著减小;体积分数指数对壳体下临界荷载影响规律较复杂;均匀升温使壳体上/下临界荷载显著增加/减小。材料横向梯度特性对简支边功能梯度圆底扁球壳屈曲平衡路径和后屈曲稳态构形有显著影响。该文末给出了便于工程设计的两个数表和一些数值曲线。     

Thermal effect on free vibration of ring-stiffened rotating functionally graded conical shell with clamped ends

An approximate solution is presented to investigate the effects of thermal load on the frequency of ring-stiffened rotating functionally graded conical shell. Material properties and the temperature field are assumed to be graded and varied in the thickness direction. The shell is reinforced by equal interval rings. The equations of motion are derived by the Hamilton's principle. Approximate analytical solutions are assumed to satisfy clamped boundary conditions, and then Fourier decomposition and Galerkin method are applied to achieve relations of frequencies. To validate, the comparisons are made with a number of particular cases in literature and with the FEM solutions.     

Large deflection analysis of thermo-mechanical loaded annular FGM plates on nonlinear elastic foundation via DQM

The effects of three-parameter elastic foundations and thermo-mechanical loading on axisymmetric large deflection response of a simply supported annular FGM plate are investigated. An annular FGM plate, resting on a three-parameter elastic foundation under a transverse uniform loading and a transverse non-uniform temperature, is considered. The mechanical and thermal properties of the FGM plate 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. The mathematical modeling of the plate and the resulting nonlinear governing equations of equilibrium are derived based on the first-order shear deformation theory (FSDT) in conjunction with nonlinear von Karman assumptions. A polynomial-based differential quadrature method is used as a simple but powerful numerical technique to discretize the nonlinear governing equations and to implement the boundary conditions. Finally, the effects of certain parameters, such as nonlinear foundations stiffness, volume fraction index, and temperature, on the axisymmetric large deflection response of the FGM plate are obtained and discussed in detail.     

Static and free vibration analysis of sandwich cylindrical shell based on theory of elasticity and using DQM

In this paper, static and free vibration analysis of a sandwich cylindrical shell is performed using theory of elasticity formulation. The core layer is made of functionally graded material with material properties varying along the thickness direction according to a simple power law. For the case of simply supported boundary conditions, equations of motion and equilibrium equations are solved analytically by applying a state-space technique along the radial direction and Fourier series expansion along the axial and circumferential direction. When boundary conditions are not simply supported, a semi-analytically solution is performed by using the differential quadrature method along the axial direction. The present approach is validated by comparing the obtained numerical results with those published in the available literature. Moreover, effects of boundary conditions, graded direction, mid-radius to thickness and length to mid-radius ratios on bending and vibration behavior are considered.     

热冲击下轴向运动FGM梁的自由振动分析

基于Timoshenko梁理论研究两端夹紧、一端夹紧一端简支、两端简支三种不同边界条件下的轴向运动功能梯度材料(FGM)梁在热冲击载荷作用下的自由振动响应。利用Hamilton原理推导热冲击下轴向运动FGM梁的自由振动控制微分方程,并采用分离变量法求解一维热传导方程。通过微分求积法(DQM)在梁的长度方向进行离散,将原方程转化为四阶广义特征值问题,求解FGM梁自由振动的无量纲固有频率并进行特性分析。考虑了不同热冲击载荷,不同梯度指数和不同轴向运动无量纲速度对FGM梁自振频率的影响。结果表明:热冲击载荷越大,对降低FGM梁的固有频率的效果越明显;在轴向运动速度和热流输入不改变的情况下,逐渐增大材料梯度指数会使FGM梁的固有频率随之减小;FGM梁对热冲击短时间内有减缓作用,相对于均匀材料一阶失稳所需时间更长,受到热冲击的FGM梁在轴向运动时也更快达到失稳状态。     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.     

Local Collocation Method for Prediction of Natural Frequency of Functionally Graded Cylindrical Shells

In this article, the local thin plate spline collocation method and first-order shear deformation shell theory are used to predict the natural frequency of functionally graded cylindrical shells. The local collocation method does not require special treatment for essential boundary condition nor time-consuming integration of weak form. Natural frequency of functionally graded cylindrical shells with clamped and simply supported boundary conditions are solved and compared with available published results to assess the performance of the present method.