Dynamic instability of three-layered cylindrical shells containing an FGM interlayer

In this study, the dynamic instability of three-layered cylindrical shells containing a functionally graded (FG) interlayer subjected to static and time dependent periodic axial compressive loads are investigated. The governing relations, modified Donnell type dynamic stability and deformation compatibility equations are derived. The governing equations are reduced Mathieu–Hill equation by using Galerkin׳s method and the expressions for boundaries of unstable regions of three-layered cylindrical shell with an FG interlayer are found. Finally, the effects of variations of volume fractions of FG interlayer and shell characteristics on the magnitudes of boundaries of unstable regions are studied numerically.     

Non-linear in-plane stability analysis of FG circular shallow arches under uniform radial pressure

In this paper, non-linear stability behavior of functionally graded (FG) circular shallow arches subjected to a uniform radial pressure is investigated by an analytical method. For this purpose, the classical single layer assumption is used to approximate the displacement field through the arch. Donnell׳s non-linear model for shallow shells is employed to derive the strain–displacement relations. The material properties vary smoothly through the thickness of the arch according to a power-law distribution. The governing equilibrium equations and the complete set of boundary conditions are extracted employing the principle of virtual displacements and variational calculus. Because of considerable pre-buckling deformations of shallow arches, the stability analysis is accomplished considering the pre-buckling deformations. The non-linear equilibrium paths are presented for two symmetric types of boundary conditions. Results show the influences of properties dispersion, geometrical characteristics, and boundary conditions on the stability behavior of the FG circular shallow arches. Also, the results of the paper are compared with the known data in literature.     

Thermal and mechanical instability of functionally graded truncated conical shells

Thermal and mechanical instability of truncated conical shells made of functionally graded material (FGM) is studied in this paper. It is assumed that the shell is a mixture of metal and ceramic that its properties changes as a function of the shell thickness. The governing equations are based on the first-order shell theory and the Sanders nonlinear kinematics equations. The results are obtained for a number of thermal and mechanical loads and are validated with the known data in the literature.     

Response of functionally graded cylindrical shells under moving thermo-mechanical loads

The transient thermoelastic analysis of functionally graded (FG) cylindrical shells under moving boundary pressure and heat flux is presented. The material properties are assumed to be temperature-dependent and graded in the radial direction. The hyperbolic heat conduction equations are used to include the influence of finite heat wave speed (i.e., the non-Fourier effect). To benefit from the high accuracy and low computational efforts of the differential quadrature method (DQM) in conjunction with the effectiveness of the finite element method (FEM) in general geometry, loading and systematic boundary treatment, a combination of these methods is employed to discretize the governing equations in the spatial domain. The resulting system of differential equations is solved using Newmark's time integration scheme in the temporal domain. The presented formulation and method of solution are validated by showing their fast rate of convergence and by comparing the results, in the limit cases, with those obtained using the commercial finite element package ANSYS and some other available solutions in the literature. Then, the effects of different geometrical, material and load parameters on the transient thermoelastic behavior of the FG cylinders under moving mechanical and thermal loads are studied.     

Thermal buckling of FGM shells resting on a two-parameter elastic foundation

This paper focuses on the thermal buckling analysis of FGM shells resting on the two-parameter elastic foundation. Material properties of the constituents are graded in the thickness direction according to the power-law distribution. The surrounding elastic medium is modeled as an elastic foundation of the Pasternak-type. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell subjected to thermal load and resting on a two-parameter elastic foundation have been derived. Critical temperature differences of FGM truncated conical shells with or without elastic foundations subjected to non-linearly distributed temperature across the thickness of the shells are obtained by solving eigenvalue problems. The appropriate formulas for FGM cylindrical shells with or without elastic foundations are found as a special case. In order to assure the accuracy of the present study, convergence properties of the critical temperature are examined in detail.     

双参数弹性地基上的FGM壳体的热屈曲性能

分析双参数弹性地基上的FGM壳体的热屈曲性能。根据幂律分布,从厚度方向对各个构件的材料性能进行分级。周边的弹性介质被模拟为Pasternak弹性地基。在假定基本关系之后,对温度荷载下及位于双参数弹性地基上的FGM截顶圆锥壳体的稳定性和相容方程式进行分析。通过求解特征值,得到沿壳体厚度方向非线性分布的温度荷载下的基于或不基于弹性地基上的FGM截顶圆锥壳体临界温差。作为一个特殊案例,提出基于或不基于弹性地基上的FGM截顶圆锥壳体的方程。为了保证目前研究的正确性,详细地评价了临界温度的收敛性。     

Free vibration analysis of joined conical shells: Analytical and experimental study

Natural frequencies and mode shapes of two joined isotropic conical shells are presented in this study. The joined conical shells can be considered as the general case for joined cylindrical–conical shells, joined cylinder–plates or cone–plates, conical and cylindrical shells with stepped thicknesses and also annular plates. Governing equations are obtained using thin-walled shallow shell theory of Donnell and Hamilton׳s principle. The continuity conditions at the joining section of the cones are appropriate expressions among stress resultants and deformations. The equations are solved assuming trigonometric response in circumferential and series solution in meridional directions and all combinations of boundary conditions can be assumed in this method. The results are compared and validated with the available results in other investigations and also modal testing. The effects of semi-vertex angles and meridional lengths on the natural frequency and circumferential wave number of joined shells are investigated.     

Stress analysis of functionally graded open cylindrical shell reinforced by agglomerated carbon nanotubes

In this paper, stresses due to bending behavior of functionally graded carbon nanotube-reinforced (FGCNTR) open cylindrical shells subjected to mechanical loads is studied. The material properties of FGCNTR shells are assumed to be graded in the thickness direction, and are estimated using a two-parameter micromechanics model in which Eshelby–Mori–Tanaka approach is employed. The primary bending formulation is based on the linear, small-strain, three-dimensional elasticity theory. In addition, the cylindrical shells are analyzed using the third-order shear deformation theory (TSDT). In order to discretize the governing equations, the two-dimensional generalized differential quadrature method (2-D GDQM) in the thickness and longitudinal directions and the trigonometric functions in tangential direction are used. The effects of agglomeration parameters, CNTs volume fraction, and CNTs distribution through the thickness on the bending behavior of FGCNTR open cylindrical shells are studied. In addition, the mechanical stresses obtained from 3-D elasticity are compared with those obtained using TSDT for a different range of geometric and agglomeration parameters.     

The effect of elastic foundations on the nonlinear buckling behavior of axially compressed heterogeneous orthotropic truncated conical shells

The buckling problem of a heterogeneous orthotropic truncated conical shell subjected to an axial load and surrounded by elastic media is analyzed based on the finite deformation theory. Using von-Karman nonlinearity, the governing equations of elastic buckling of heterogeneous orthotropic truncated conical shells surrounded by elastic media are derived. The governing equations are solved using superposition and Galerkin methods and obtained expressions for upper and lower critical axial loads. The influences of elastic foundations, heterogeneity, orthotropy and geometric characteristics on the upper and lower critical loads of conical shells with and without elastic foundations are studied in detail.     

Free vibration analysis of functionally graded cylindrical shells including thermal effects

Free vibration analysis of simply supported FG cylindrical shells for four sets of in-plane boundary conditions is performed. The material properties are assumed to be temperature-dependant and gradually changed in the thickness direction of the shell. The effects of temperature rise are investigated by specifying arbitrary high temperature on the outer surface and the ambient temperature on the inner surface of the cylinder. Distribution of temperature across the shell thickness is found from steady state heat conduction only in the thickness direction. The equations of motion are based on Love's shell theory and the von Karman–Donnell-type of kinematic nonlinearity. The static analysis is first performed to determine the prestressed state induced by the thermal loadings, using the exact solution of the governing equations and then the equations of motion are solved by Galerkin's method. The results are obtained to indicate the effects of power law index on the natural frequencies and corresponding mode shapes in the thermal environment.     

基于传统板理论的智能圆形FGM薄板自由振动研究

基于传统板理论（CPT）对圆形功能梯度（FG）薄板的自由振动性能进行分析。这种板具有2个由压电（PZT4）材料构成的均布驱动器。对FG底层板的材料特性进行假定,考虑了材料体积分数的能量分布,将材料沿厚度方向进行分级,并采用二次方程模拟沿压电覆层厚度方向分布的电势场。采用运动微分方程来模拟板的固定端边界条件。介绍了详细的数学推导过程,并采用数值分析来研究FG板的梯度变化对结构自由振动的影响。其分析结果得到了三维有限元分析结果的证实。     

外压作用下偏心加劲功能梯度圆柱壳的非线性屈曲和后屈曲分析

通过解析法研究外压作用下功能梯度加劲薄圆柱壳的非线性屈曲和后屈曲性能。通过其在内部的偏心环和纵梁对壳体进行加固,假定壳体和加固件的材料性能在厚度方向为连续梯度。根据VonKarman理论中的刚度法和传统的壳理论推导出基本关系和平衡方程,可更准确地选择三种关于挠曲的近似公式,且使用盖勒金法得出的显式表达式可以推测出临界荷载和后屈曲压力-挠曲曲线。数值结果显示了加固件能有效地增强壳体稳定性。     

Thermo elastic analysis of a functionally graded rotating disk with small and large deflections

A functionally graded (FG) rotating disk with axisymmetric bending and steady-state thermal loading is studied. The material properties of the disk are assumed to be graded in the direction of the thickness by a power law distribution of volume fractions of the constituents. First-order shear deformation Mindlin plate and von Karman theories are employed. New set of equilibrium equations with small and large deflections are developed. Using small deflection theory an exact solution for displacement field is given. Solutions are obtained in series form in case of large deflection. Mechanical responses are compared small deflection versus large deflection as well as homogeneous versus FG disks. It is observed that for particular values of the grading index n of material properties mechanical responses in FG disk can be smaller than in a homogeneous disk. It is seen that given the non-dimensional maximum vertical displacement w_max/h close to 0.4 for a homogeneous (full-ceramic in this study) disk greater errors in the mechanical responses for FG disks would be introduced if one uses small deflection theory.     

Asymptotic solutions of axisymmetric laminated conical shells

The three-dimensional (3D) solution of laminated conical shells subjected to axisymmetric loads is presented using the method of perturbation. The formulation begins with the basic 3D elasticity equations without making any static or kinematic assumptions in advance. After introducing a set of proper dimensionless variables, asymptotically expanding the field variables and then successively integrating the resulting equations through the thickness direction, we obtain the recursive sets of governing equations for various orders. The edge boundary conditions at each order level are derived as the resultant forms by following the variational approach. The method of differential quadrature (DQ) is used to determine the present asymptotic solution for various orders. For illustration purposes, the simply supported laminated conical shells under uniformly and sinusoidally distributed lateral pressure and the clamped laminated conical shells under edge torsion, are studied. It is shown that the present asymptotic DQ solution can be obtained order-by-order in a hierarchic and consistent manner and asymptotically approaches to the 3D elasticity solution.     

Effects of hinges and deployment angle on the energy absorption characteristics of a single cell in a deployable energy absorber

Numerical simulation is carried out to investigate the crushing characteristics of a single cell in a fan-shaped deployable energy absorber (FDEA) under quasi-static axial loading. FDEA can effectively improve the crashworthiness behavior of aircrafts with the advantages of saving space and deploying actively. Hinges are added to the single cell to meet the need of fan-shaped deployment. The finite element model is established to study the effects of hinge׳s parameters, including material properties such as Young׳s modulus, yield strength and the tube thickness, on the single cell׳s energy absorption characteristics. The relationship between the deployment angle and the specific energy absorption (SEA) of the single cell is also studied. The numerical results indicate that the energy absorption increases rapidly as yield strength and the hinge׳s thickness increase, while it only has minor correlation with Young׳s modulus of the material. Three different modes of the cell appear during its axial crushing as the deployment angle increases. Besides, experiments were conducted to observe the crushing mode of the straight single cell, and the results are compared with the numerical simulation results. Finally, a theoretical model of a straight single cell with hinges is proposed to predict the mean crushing force, which is in good agreement with the numerical simulation.     

A semi-analytical approach for linear and non-linear analysis of unstiffened laminated composite cylinders and cones under axial,torsion and pressure loads

A semi-analytical model for the non-linear analysis of simply supported, unstiffened laminated composite cylinders and cones using the Ritz method and the Classical Laminated Plate Theory is proposed. A matrix notation is used to formulate the problem using Donnell׳s and Sanders׳ non-linear equations. The approximation functions proposed are capable to simulate the elephant׳s foot effect, a common phenomenon and a common failure mode for cylindrical and conical structures under axial compression. Axial, torsion and pressure loads can be applied individually or combined, and solutions for linear static, linear buckling and non-linear buckling analyses are presented and verified using a commercial finite element software. The presented non-linear buckling analyses used perturbation loads to create the initial geometric imperfections, showing the capability of the method for arbitrary imperfection patterns. The linear stiffness matrices are integrated analytically and for the conical structures an approximation is proposed to overcome the non-integrable expressions.     

Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers

The free vibration behavior of quadrilateral laminated thin-to-moderately thick plates with carbon nanotube reinforced composite (CNTRC) layers is studied. The governing equations are based on the first-order shear deformation theory (FSDT). The solution procedure is based on transforming the governing differential equations from an arbitrary straight-sided physical domain to a regular computational one, and discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. Four different profiles of single walled carbon nanotubes (SWCNTs) distribution through the thickness of layers are considered, which are uniformly distributed (UD) and three others are functionally graded (FG) distributions. The fast rate of convergence of the presented approach is numerically demonstrated and to show its high accuracy, wherever possible comparison studies with the available results in the open literature are performed. Then, the effects of volume fraction of carbon nanotubes (CNTs), geometrical shape parameters, thickness-to-length and aspect ratios, different kinds of CNTs distribution along the layers thickness and different boundary conditions on the natural frequencies of laminated plates are studied.     

径向荷载作用下功能梯度柱壳非线性后屈曲研究

基于柱壳非线性大变形理论,研究承受径向荷载的功能梯度柱壳的非线性后屈曲性能。依据功能梯度参数,功能梯度材料沿厚度方向的属性有差异。考虑依赖温度的材料属性,比较不同温度的影响。研究了功能梯度参数与空间参数的影响。最后,利用试验结果对理论结果进行修正。     

Buckling of thin-walled conical shells under uniform external pressure

Shells are for the most part the deep-seated structures in manufacturing submarines, missiles, tanks and their roofs, and fluid reservoirs; therefore it is a matter of concern to bring about some basic regulations associated with the existing codes. Above all, truncated conical shells (frusta) and shallow conical caps (SCC) subjected to external uniform pressure when discharging liquids or wind loads are discussed closely in this paper concerning and thrashing out their empirical nonlinear responses along with envisaging numerical methods in contrast. The buckling aptitude of shells is contingent upon two leading geometric ratios of “slant-length to radius” (L/R) and “radius to thickness” (R/t). In this paper, developing six frusta and four shallow cap specimens and their relevant FE models, use is made of laboratory modus operandi to enumerate buckling elastic and plastic responses and asymmetric imperfection sensitivity, whose adequacy has been reckoned through comparisons with arithmetical and numerical data correspondingly. These obtained upshots were aimed at validating and generalizing the data for unstiffened truncated cones and SCC in full scale.     

Buckling of imperfect functionally graded plates under in-plane compressive loading

Buckling behavior of rectangular functionally graded plates with geometrical imperfections is studied in this paper. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable. The plate is assumed to be under in-plane compressive loading. Simultaneous solving of the stability and compatibility equations in conjunction with the equilibrium equations leads to the buckling relation of the plate. The critical buckling load of a sample plate is obtained and compared for different geometrical ratios. The results are reduced and compared with the results of perfect functionally graded and imperfect isotropic plates.