The non-linear buckling analysis of cross-ply laminated orthotropic truncated conical shells

In this study, the non-linear buckling behavior of cross-ply laminated orthotropic truncated conical shells under axial load has been investigated. The basic relations of the cross-ply laminated orthotropic truncated conical shells are derived using the von Karman–Donnell-type of kinematic non-linearity. Then modified Donnell type non-linear stability and compatibility equations are obtained and are solved. Finally, the influences of the number and ordering of layers and the variations of the conical shell characteristics on the non-linear axial buckling load are investigated. Comparison with available results is satisfactorily good.     

Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load

Dynamic buckling of functionally graded materials truncated conical shells subjected to normal impact loads is discussed in this paper. In the analysis, the material properties of functionally graded materials shells 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. Geometrically nonlinear large deformation and the initial imperfections are taken into account. Galerkin procedure and Runge–Kutta integration scheme are used to solve nonlinear governing equations numerically. From the characteristics of dynamic response obtain critical loads of the shell according to B-R criterion. From the research results it can be found that gradient properties of the materials have significant effects on the critical buckling loads of FGM shells.     

Analytical investigation on the free vibration behavior of rotating FGM truncated conical shells reinforced by orthogonal eccentric stiffeners

This article presents an analytical investigation on the free vibration behavior of rotating functionally graded truncated conical shells reinforced by stringers and rings with the change of spacing between stringers. Using the Donnell shell theory, smeared stiffeners technique, and taking into account the influences of centrifugal force and Coriolis acceleration, the governing equations are derived. These variable coefficient partial differential equations are studied by the Galerkin method. The sixth-order polynomial equation of natural frequency is obtained. Numerical results show effects of stiffener and input parameters on the frequency of shell.     

On the vibration and stability of shear deformable FGM truncated conical shells subjected to an axial load

The aim of present study is to investigate the vibration and stability of functionally graded (FG) conical shells under a compressive axial load using the shear deformation theory (SDT). The basic equations of shear deformable FG conical shells are derived using Donnell shell theory and solved using Galerkin's method. The novelty of this study is to achieve closed-form solutions for the dimensionless frequencies and critical axial loads for freely-supported FG truncated conical shells on the basis of the SDT. Parametric studies are made to investigate effects of shear stresses, compositional profiles and conical shell characteristics on the critical parameters. Some comparisons with the various studies have been performed in order to show the accuracy of the present study.     

Supersonic flutter prediction of functionally graded conical shells

Aero-thermoelastic analysis of a simply supported functionally graded truncated conical shell subjected to supersonic air flow is performed to predict the flutter boundaries. The temperature-dependent properties of the FG shell are assumed to be graded through the thickness according to a simple rule of mixture and power-law function of volume fractions of material constituents. Through the thickness steady-state heat conduction is considered for thermal analysis. To perform the stability analysis, the general nonlinear equations of motion are first derived using the classical Love’s shell theory and the von Karman–Donnell-type of kinematic nonlinearity together with the linearized first-order piston theory for aerodynamic loading. Then the nonlinear equations of motion are linearized to obtain the linear equilibrium and aeroelastic equations. The equilibrium equations are solved using power series method to obtain the initial stresses induced by aerodynamic and thermal loadings. The results are then used as an input to the aeroelastic stability equations which are finally solved with the generalized Galerkin method. The flutter boundaries are obtained for the FG conical shells with different semi-vertex cone angles, different temperature distributions, and different volume fraction indices.     

Flutter of high-dimension nonlinear system for a FGM truncated conical shell

The nonlinear flutters of a truncated conical shell, which is subjected to aerodynamic pressure and aerodynamic heating, are researched. Material properties with gradient features along the radial direction depend on the temperature. The supersonic aerodynamic force is obtained by applying the first-order piston theory, including the correction factor for curvature. The temperature in the external surface of the functionally graded material truncated conical shell rises as a result of viscous aerodynamic heating, and the temperature distribution along the thickness can be described by polynomial series. Hamilton's principle is utilized to obtain the nonlinear partial differential equilibrium equation of the system. Using Galerkin's method, a high-dimensional nonlinear system can be derived. Without considering the parts of nonlinear terms and the external forcing excitation, the flutter boundaries are obtained by solving the eigenvalues problem. The influences of ratios of top radius to thickness, semi-vertex angle, and volume fraction index on nonlinear dynamic characteristics of functionally graded material truncated conical shell are studied in detail by the fourth-order Runge–Kutta algorithm.     

Free vibration analysis of functionally graded conical shell panels by a meshless method

A free vibration analysis of metal and ceramic functionally graded conical shell panels is presented using the element-free kp-Ritz method. The first-order shear deformation shell theory is used to account for the transverse shear strains and rotary inertia, and mesh-free kernel particle functions are employed to approximate the two-dimensional displacement fields. The material properties of the conical shell panels are assumed to vary continuously through their thickness in accordance with a power-law distribution of the volume fractions of their constituents. Convergence studies are performed in terms of the number of nodes, and comparisons of the current solutions and those reported in literature are provided to verify the accuracy of the proposed method. Two types of functionally graded conical shell panels, including Al/ZrO₂ and Ti–6Al–4V/aluminum oxide, are chosen in the study, and the effects of the volume fraction, boundary condition, semi-vertex angle, and length-to-thickness ratio on their frequency characteristics are discussed in detail.     

The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure

The purpose of this paper is to investigate the elastic buckling of FGM truncated thin conical shells under combined axial tension and hydrostatic pressure. Here axial tensions are separately applied to small and large bases of the truncated conical shell, respectively. It is assumed that the cone is a mixture of metal and ceramic, and that its properties changes as the power and exponential functions of the shell thickness. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell, subject to combined axial tension and hydrostatic pressure, have been derived. Applying Galerkin’s method general formulas have been obtained for the critical combined and separate loads of FGM conical shells. The appropriate formulas for homogenous and FGM cylindrical shells are found as a special case. Effects of changing shell characteristics, material composition and volume fraction of constituent materials on the critical combined and separate loads of FGM shells with simply supported edges are also investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Frequency analysis of rotating truncated circular orthotropic conical shells with different boundary conditions

In this paper, a study is made of the influences of orthotropic properties and boundary conditions on the free vibrations of a rotating, truncated, circular orthotropic conical shell. The study includes the effects of the Coriolis and centrifugal accelerations and the initial hoop tension. It is based on the Love first-approximation theory and Galerkin procedure. Results are obtained for the frequency characteristics of different orthotropic parameters, rotating velocities, cone angles and boundary conditions. Influences of orthotropic properties and boundary conditions on the relationships between frequency parameter and rotation velocity are discussed for different cone angles. To validate the present analysis, comparisons are made with those available in the open literature and very good agreements are obtained.     

Buckling of conical composite shells

A semi-analytical approach is proposed to obtain the linear buckling response of conical composite shells under axial compression load. A first order shear deformation shell theory along with linear strain-displacement relations is assumed. Using the principle of minimum total potential energy, the governing equilibrium equations are found and Ritz method is applied to solve them. Parametric study is performed by finding the effect of cone angle and fiber orientation on the critical buckling load of the conical composite shells.     

Buckling of FGM cylindrical shells subjected to pure bending load

In this paper, buckling behaviors of composite cylindrical shells made from functionally graded materials (FGMs) subjected to pure bending load were investigated. The material properties were assumed to be graded along the thickness. The non-uniform bending force on the shell section was considered in the buckling government equation of FGM cylindrical shells based on the Donnell shallow shell theory. The prebuckling deformation of the FGM cylindrical shells was neglected and the buckling mode was assumed to occur non-uniformly in local district along the shell circumferential direction. The eigenvalue method was used to obtain the buckling critical condition. The theoretical results were in excellent agreement with those of ABAQUS code. Results show that the inhomogenity of the materials is significant for buckling of FGM cylindrical shells.     

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.     

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.     

Vibration control of laminated truncated conical shell via magnetostrictive layers

Abstract

In this study feedback control is applied to control the free vibration response of an isotropic truncated conical shell embedded with magnetostrictive layers. Classical shell theory is applied to derive the shell vibration equations. The results are derived based on the Galerkin method and the results are compared with published results and the results of finite element software in order to determine the accuracy of using method. The influence of several parameters such as the thickness of magnetostrictive layers, control gain, length and radius of the large edge of the shell on the vibration suppression of fundamental frequency is determined.     

The stability of a thin three-layered composite truncated conical shell containing an FGM layer subjected to non-uniform lateral pressure

This paper examines the stability of thin three-layered truncated conical shells containing a functionally graded (FG) layer subjected to non-uniform lateral pressure varying with the longitudinal coordinate. The material properties of the functionally graded layer are assumed to vary continuously through the thickness of the shell, and the variation of properties follows an arbitrary distribution in terms of the volume fractions of the constituents. Further, the fundamental relations for stability and compatibility equations of three-layered truncated conical shells containing an FGM layer have been obtained. These equations, ascertained via Galerkin’s method, have been transformed into a pair of time-dependent differential equations. Then, critical non-uniform lateral pressure has been conclusively obtained. This paper is the result of a detailed parametric study conducted to determine the influences of thickness variations in the FG layer, radius-to-thickness ratio, lengths-to-radius ratio, and the material composition and material profile index on the critical parameters of three-layered, truncated, conical shells. Finally, the results will be validated through the comparison of obtained values with those in the existing literature.     

不同边界条件下截顶正交各向异性圆锥壳的振动和声辐射研究

采用波传播法和Galerkin法,研究了两端简支和固支边界条件对截顶正交各向异性圆锥壳的固有频率和声辐射的影响,通过与现有文献及数值仿真结果相比较,验证了理论分析的正确性。结果表明,简支和固支边界对正交各向异性圆锥壳固有频率的影响与圆锥壳半顶角的大小和母线与环向方向的弹性模量比的大小相关。当正交各向异性圆锥壳的半顶角较大时,对与不同的弹性模量比,固支圆锥壳的固有频率都大于简支圆锥壳的固有频率;而对于小半顶角的正交各向异性圆锥壳,当弹性模量比较小时,简支圆锥壳在大周向波数处的固有频率略大于固支圆锥壳的固有频率。随着圆锥壳半顶角的增大,两种边界对正交各向异性圆锥壳的声压级的影响也逐渐增大。     

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.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

On the solution of nonlinear vibration of truncated conical shells covered by functionally graded coatings

This paper deals with the linear and nonlinear vibrations of a truncated conical shell; both internal and external surfaces are covered by functionally graded coatings (FGCs). The theoretical formulation is based on the von Karman–Donnell-type nonlinear kinematics. The material properties of FGCs 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 fundamental relations, the modified Donnell-type nonlinear motion, and compatibility equations of the truncated conical shell with FGCs are derived. The basic equations are reduced to the ordinary differential equation depending on time with geometric nonlinearity using the Superposition and Galerkin methods. By applying the homotopy perturbation method to the foregoing equation, the relation between nonlinear frequency parameters with the dimensionless amplitude of a truncated conical shell with FGCs is obtained. Parametric studies are performed to illustrate the effect of different values of thickness and material composition of the FGCs on the frequency-amplitude relationships. The validity of the present solution is demonstrated by comparison with solutions available in the literature.     

Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper investigates the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments. The surrounding elastic medium is modeled as a Pasternak foundation. Two kinds of micromechanics models, namely, Voigt model and Mori-Tanaka model, are considered. The motion equations are based on a higher order shear deformation shell theory that includes shell-foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM shells. Numerical results demonstrate that in most cases the natural frequencies of the FGM shells are increased but the nonlinear to linear frequency ratios of the FGM shells are decreased with increase in foundation stiffness. The results confirm that in most cases Voigt model and Mori-Tanaka model have the same accuracy for predicting the vibration characteristics of FGM shells.