Free vibration of composite circular cylindrical shells with random material properties. Part II: Applications

In the second part of this study the approach developed in Part I has been used to analyse free vibration of three composite circular cylindrical shells with random scatter in the material properties. The cases considered are – specially orthotropic symmetric shells in axisymmetric and asymmetric oscillations, and antisymmetric cross ply laminated shell in axisymmetric oscillations. With known statistics of the material properties the mean and the variance of the natural frequencies have been obtained. Numerical results have been presented for graphite–epoxy composite shells.     

Buckling of composite circular cylindrical shells with random material properties

Composite materials exhibit a scatter in their properties. This is generally ignored in conventional structural analysis leading to results that may be non-conservative. The present study discusses the critical buckling analysis for circular cylindrical shells of laminated composites incorporating the effects of randomness in the material properties. A perturbation approach has been employed to develop expressions for the mean and variance of the critical buckling load in terms of material property statistics. Working of the approach has been illustrated with an example.     

Free vibration of composite cylindrical panels with random material properties

Virtually in all structural systems, and in particular composites, there are uncertainties in the system parameters because of practical bounds on the quality control. In the present work the effect of variations in the mechanical properties of laminated composite cylindrical panels on its natural frequency has been obtained by modeling these as random variables. The transverse shear and rotatory inertia effects have been included in the governing equations. A perturbation approach is presented to obtain the mean and variance of the random natural frequencies. The effects of thickness ratios, edge support conditions and standard deviation of material properties on response of shallow square panels have been investigated. Results have been obtained by employing the finite element method. The approach has been validated by comparison of results with other approaches.     

Free vibration of laminated composite conical shells with random material properties

In the present study, the sensitivity of randomness in material parameters on linear free vibration response of conical shells is presented. Higher order shear deformation theory is used to model system behavior and uncertain lamina material properties are modeled as basic random variables. A finite element method is successfully combined with first-order perturbation technique to obtain the response statistics of the structure. The solution methodology is validated with the results available in the literature and an independent Monte Carlo simulation. Typical numerical results for second-order statistics of linear free vibration response of simply supported laminated composite conical shells are obtained for different lamination schemes and thickness to radius ratios.     

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.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part I: Axially-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to axial compression in thermal environments. Two kinds of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of axially-loaded, perfect and imperfect, FG-CNTRC cylindrical shells under different sets of thermal environmental conditions. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling load as well as postbuckling strength of the shell under axial compression. The results reveal that the CNT volume fraction has a significant effect on the buckling load and postbuckling behavior of CNTRC shells.     

Free vibration of composite cylindrical polygonal ducts

A simple numerical method using the Rayleigh-Ritz principle is presented for computing the natural frequencies and mode shapes of cylindrical polygonal ducts constructed from composite materials. The duct creases are assumed to be parallel to one of the principal axes of orthotropy. Numerical results are presented for triangular and square cylindrical ducts with various degrees of orthotropy.     

Free vibration response of composite sandwich cylindrical shell with flexible core

This study deals with the free vibration analysis of composite sandwich cylindrical shell with a flexible core using a higher order sandwich panel theory. The formulation uses the classical shell theory for the face sheets and an elasticity theory for the core and includes derivation of the governing equations along with the appropriate boundary conditions. The model consists of a systematic approach for the analysis of sandwich shells with a flexible core, having high-order effects caused by the nonlinearity of the in-plane and the vertical displacements of the core. The behavior is presented in terms of internal resultants and displacements in the faces, peeling and shear stresses in the face–core interface and stress and displacement field in the core. The accuracy of the solution is examined by comparing the results obtained with the analytical and numerical results published in the literatures. The parametric study is also included to investigate the effect of geometrical properties such as radius of curvature, length and sector angle of the shell.     

Free vibration of axisymmetric and beam-like cylindrical shells,partially filled with liquid

A theory is presented for the determination of the free vibration characteristics of anisotropic thin cylindrical shells, partially or completely filled with liquid, for two circumferential wave numbers, n = 0, axisymmetric and n = 1, beam-like. The method used was a combination of finite element analysis and classical shell theory. The shell was subdivided into cylindrical finite elements and the displacement functions were obtained using the shell equations. Expressions for the mass and stiffness matrices for a finite element and for the whole structure were obtained. A finite element was developed for the liquid in cases of potential flow. The natural frequencies of the shell, both empty and partially filled, were obtained and compared with existing experiments and other theories.     

Free vibration of laminated composite plates with a central circular hole

This paper presents finite element results and some experimental data on the free vibration of symmetric laminated composite plates containing a central hole. The varying parameters in the study are the hole sizes, the boundary conditions along the edges of the plate and the aspect ratio of the plate. Hole diameters ranging from 0.1 to 0.9 times the width of the plate and the aspect ratio varying from 1 to 4 were analysed in the numerical study. The other parameters such as the stacking sequence and the lamination materials are kept constant. The stacking sequence and material properties chosen for the study are those typically found in aircraft structural applications. The close correlation between numerical and experimental data provides some confidence in the finite element method used.     

On the buckling and vibration of antisymmetric angle-ply laminated circular cylindrical shells

The buckling and free vibration problem of a thin composite antisymmetric angle-ply laminated circular cylindrical shell is studied. The Donnell-type equations of motion, in terms of the shell middle surface displacement components, are used and solved approximately by means of Galerkin's method. Numerical results are presented and the expectation that the bending-stretching coupling phenomenon rapidly dies out as the number of layers increases is confirmed.     

Nonlinear thermal vibration of carbon nanotube polymer composite elliptical cylindrical shells

International Journal of Mechanics and Materials in Design - This paper investigated the nonlinear vibration and dynamic response of the carbon nanotube polymer composite elliptical cylindrical...     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part II: Pressure-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to lateral or hydrostatic pressure in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under external pressure is proposed and a singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. Numerical results for pressure-loaded, perfect and imperfect, FG-CNTRC cylindrical shells are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling pressure as well as postbuckling strength of the shell under external pressure. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling pressure and postbuckling behavior of CNTRC shells.     

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.     

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.     

Magnetoelastic modeling of circular cylindrical shells immersed in a magnetic field. Part I: Magnetoelastic loads considering finite dimensional effects

Determination of magnetoelastic loads acting on a perfectly electro-conductive circular cylindrical shell immersed in a uniform applied magnetic field is addressed. The finite dimensional effects related to the finite length and finite thickness of the shell are taken into consideration. Fourier integral method is used to derive the singular integral equations governing the distributed magnetoelastic loads. As special cases, determination of magnetoelastic loads via discarding the thickness effect are obtained from the general formulation, and the magnetoelastic loads of infinitely long shells are derived. Magnetoelastic loads on plate strips or infinite plates are also reduced from the general formulation. To the best of the authors’ knowledge, this represents the first work devoted to the analytical determination of magnetoelastic loads on circular cylindrical shells considering the finite length and thickness effects.     

A high-order theory for the analysis of circular cylindrical composite sandwich shells with transversely compliant core subjected to external loads

A new model based on the high order sandwich panel theory is proposed to study the effect of external loads on the free vibration of circular cylindrical composite sandwich shells with transversely compliant core, including also the calculation of the buckling loads. In the present model, in contrast to most of the available sandwich plate and shell theories, no prior assumptions are made with respect to the displacement field in the core. Herein the displacement and the stress fields of the core material are determined through a 3D elasticity solution. The performance of the present theory is compared with that of other sandwich theories by the presentation of comparative results obtained for several examples encompassing different material properties and geometric parameters. It is shown that the present model produce results of very high accuracy, and it is suggested that the present model, which is based on a 3D elasticity solution for the core material, can be used as a benchmark in future studies of the free vibration and buckling of circular cylindrical composite sandwich shells with a transversely compliant core.     

复杂边界条件圆柱壳自由振动特性分析

提出一种半解析法来分析圆柱壳结构自由振动特性。将圆柱壳壳结构在轴向方向分解为若干壳段,用沿轴向的Jacobi多项式和沿周向的Fourie r级数来表示各个壳段的位移函数,并采用罚参数法对圆柱壳结构的边界条件和壳段间的连续性条件进行模拟;最后,基于Rayleigh-Ritz法求得圆柱壳结构的自由振动频率。研究表明,该方法具有较好的收敛性,与公开发表文献一致性较高,研究成果可为复杂边界条件下圆柱壳结构自由振动特性分析提供数据积累和方法依据。     

Non-linear response statistics of composite laminates with random material properties under random loading

Laminated composite plates find extensive use in many engineering applications. Some of these incorporate large deflections that may not be in the linear range. The external loading may be random in nature. The laminate material properties show an inherent dispersion around a mean value. In this paper the static response of laminated composite flat plates to transverse random loading has been studied. The material properties have been taken as random variables for accurate prediction of the system behaviour. The basic formulation of the problem has been developed based on the classical laminate theory and the Von-Karman non-linear strain–displacement relationship. A first order perturbation technique has been used to obtain the second order response statistics. Typical results have been presented for a plate with all edges simply supported. A comparison has been drawn with Monte Carlo simulation results for validation of the proposed approach. The effects of side-to-thickness ratio, aspect ratio and change in standard deviation of input random variables have been investigated for cross-ply symmetric and anti-symmetric laminates.