The Stressed State of the “Boundary Layer” Type in Cylindrical Shells Investigated according to a Nonclassical Theory

Two variants of a refined theory for calculating the stress–strain state in the boundary zones of cylindrical shells are presented. The relevant mathematical models are based on equations of the three-dimensional elasticity theory and the increase over the thickness of the shell in the orders of the polynomials that approximate the sought displacements. The Lagrange variational principle is applied to the value of the shell’s total energy functional defined more exactly with respect to the classical Kirchhoff–Love theory. The formulated boundary problems allow determination with different degrees of accuracy of additional stressed states of the “boundary layer” type. The calculated results obtained in this work are compared with the results obtained according to the classical theory. It has been established that the above stresses make a significant contribution to the total stressed state of the shell and should be considered when designing and testing machine structures for strength and longevity.     

Vibration analysis of submerged thin FGM cylindrical shells

This study gives a brief work on vibration characteristics of cylindrical shells submerged in an incompressible fluid. The shell is presumed to be structured from functionally graded material. The effect of the fluid is introduced by using the acoustic wave equation. Love’s first order thin shell theory is utilized in the shell dynamical equations. The problem is framed by combining shell dynamical equations with the acoustic wave equation. Fluid-loaded terms are associated with Hankel function of second kind. Wave propagation approach is employed to solve the shell problem. Some comparisons of numerical results are performed for the natural frequencies of simply supported-simply supported, clamped-clamped and clamped-simply supported boundary conditions of isotropic as well as functionally graded cylindrical shells to check the validity of the present approach. The influence of fluid on the submerged functionally graded cylindrical shells is noticed to be very pronounced.     

Non-linear theory of laminated shells accounting for continuity conditions of displacements and tractions at layer interfaces

The present paper is devoted to the formulation of the theory of laminated anisotropic shells. Firstly, a unified representation of displacement variation across the thickness of arbitrary shaped laminated shells is derived on the basis of rigorous kinematical analysis, and then a Kármán-type non-linear theory of such shells is established after taking some approximations for the displacement variation. The theory satisfies the continuity conditions of displacements and tractions at layer interfaces as well as the external boundary conditions on the bounding surfaces, and the governing equations contain only five independent variables. In order to assess the accuracy of the theory, numerical results for some special cases are illustrated and compared with the corresponding exact solutions of three-dimensional elasticity.     

Energy-consistent theory of cylindrical shells

On the basis of the energy-consistent direction in the theory of shells, the deflected mode of circular cylindrical shells, considered as three-dimensional bodies, is studied. At that, two-dimensional equations of the boundary problem, obtained on the basis of the Lagrange principle by means of expansion of the sought relocations in polynomial series regarding the normal coordinate, are used. Equations are presented for the case when the approximation of the relocations in respect of the shell thickness retains summands that are by an order of magnitude higher than that in the Kirchhoff-Love classical theory of shells. The boundary problem formulated is solved with the help of the Laplace transform. As an example, a shell under the action of local loads is considered, for which the deflected mode is determined and a comparison is performed with the results obtained in accordance with the classical theory.     

Analysis of the vibration behavior of FGM cylindrical shells including internal pressure and ring support effects based on Love-Kirchhoff theory with various boundary conditions

This paper presents the study on vibration behavior of functionally graded material (FGM) cylindrical shell with the effects of internal pressure and ring support. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with internal pressure and ring support are established based on strain-displacement relationship using Love-Kirchhoff shell theory. The governing equations of motion were solved by using energy functional and by applying Ritz method. The boundary conditions represented by end conditions of the FGM cylindrical shell are simply supported-simply supported (SS-SS), clamped-clamped (C-C), free-free (F-F), clamped-free (C-F), clamped-simply supported (C-SS), free-simply supported (F-SS), free-sliding (F-SL) and clamped-sliding (C-SL). To check the validity and accuracy of the present method, the results obtained are compared with those available in the literature. The influence of internal pressure, ring support position and the effect of the different boundary conditions on natural frequencies characteristics are studied. These results presented can be used as important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with internal pressure and ring support.     

Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions

This paper presents the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quadrature method (LaDQM), which employs both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion are formulated using the Goldenveizer–Novozhilov shell theory. Appropriate management of exterior grid points is presented to couple the discretized boundary conditions with the governing differential equations instead of using the interior points. The use of compactly supported interpolating basis functions leads to banded and well-conditioned matrices, and thus, enables large-scale computations. The treatment of boundary conditions with exterior grid points avoids spurious eigenvalues. Detailed formulations are presented for the treatment of various shell boundary conditions. Convergence and comparison studies against existing solutions in the literature are carried out to examine the efficiency and reliability of the present approach. It is found that accurate natural frequencies can be obtained by using a small number of grid points with exterior points to accommodate the boundary conditions.     

The contact problem of two coaxial cylindrical shells

The contact problem of two coaxial cylindrical shells is investigated. A shear deformation theory is used in setting up the governing differential equations. The inner shell is subjected to a constant pressure, and a prescribed initial separation between the inner and outer shells is allowed. Subsequently, the redistribution of tractions and the nature of variation of contact pressure are discussed for different shell parameters and boundary conditions. A possible limitation of the shear deformation theory is discussed.     

Frequency analysis of multiple layered cylindrical shells under lateral pressure with asymmetric boundary conditions

Natural frequency characteristics of a thin-walled multiple layered cylindrical shell under lateral pressure are studied. The multiple layered cylindrical shell configuration is formed by three layers of isotropic material where the inner and outer layers are stainless steel and the middle layer is aluminum. The multiple layered shell equations with lateral pressure are established based on Love＇s shell theory. The governing equations of motion with lateral pressure are employed by using energy functional and applying the Ritz method. The boundary conditions represented by end conditions of the multiple layered cylindrical shell are simply supported-clamped（SS-C）, free-clamped（F-C） and simply supported-free（SS-F）. The influence of different lateral pressures, different thickness to radius ratios, different length to radius ratios and effect of the asymmetric boundary conditions on natural frequency characteristics are studied. It is shown that the lateral pressure has effect on the natural frequency of multiple layered cylindrical shell and causes the natural frequency to increase. The natural frequency of the developed multilayered cylindrical shell is validated by comparing with those in the literature. The proposed research provides an effective approach for vibration analysis shell structures subjected to lateral pressure with an energy method.     

Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics

In the present study, the free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. Eringen's nonlocal elasticity equations are incorporated into the classical Donnell shell theory accounting for small scale effects. The Rayleigh-Ritz technique is applied to consider different sets of boundary conditions. The displacements are represented as functions of polynomial series to implement the Rayleigh-Ritz method to the governing differential equations of nonlocal shell model and obtain the natural frequencies of DWCNTs relevant to different values of nonlocal parameter and aspect ratio. To extract the proper values of nonlocal parameter, molecular dynamics (MD) simulations are employed for various armchair and zigzag DWCNTs, the results of which are matched with those of nonlocal continuum model through a nonlinear least square fitting procedure. It is found that the present nonlocal elastic shell model with its appropriate values of nonlocal parameter has the capability to predict the free vibration behavior of DWCNTs, which is comparable with the results of MD simulations.     

Vibration of cylindrical shells with ring support

In this paper, a study on the vibration of thin cylindrical shells with ring supports is presented. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using Sanders' shell theory. The governing equations are obtained using an energy functional with the 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.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

Natural frequencies and buckling of pressurized nanotubes using shear deformable nonlocal shell model

The small-scale effect on the natural frequencies and buckling of pressurized nanotubes is investigated in this study. Based on the firstorder shear deformable shell theory, the nonlocal theory of elasticity is used to account for the small-scale effect and the governing equations of motion are obtained. Applying modal analysis technique and based on Galerkin’s method a procedure is proposed to obtain natural frequencies of vibrations. For the case of nanotubes with simply supported boundary conditions, explicit expressions are obtained which establish the dependency of the natural frequencies and buckling loads of the nanotube on the small-scale parameter and natural frequencies obtained by local continuum mechanics. The obtained solutions generalize the results of nano-bar and -beam models and are verified by the literature. Based on several numerical studies some conclusions are drawn about the small-scale effect on the natural frequencies and buckling pressure of the nanotubes.     

Vibration of functionally graded cylindrical shells based on different shear deformation shell theories with ring support under various boundary conditions

In the present work, study of the vibration of thin cylindrical shells with ring supports made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. Effects of boundary conditions and ring support on the natural frequencies of the FGM cylindrical shell are studied. The cylindrical shells have ring supports which are arbitrarily placed along the shell and which imposed a zero lateral deflection. The study is carried out using different shear deformation shell theories. The analysis is carried out using Hamilton’s principle. The governing equations of motion of a FGM cylindrical shells are derived based on various shear deformation theories. 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. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin M. M. Najafizadeh received his BS degree in 1995 from Azad University (Arak) and the Ms Degree in 1997 from Azad University (Arak), and his Ph.D. degree in 2003 from Science and Research Branch Islamic Azad University (Tehran, Iran), all in mechanical Engineering. He is member of faculty in Islamic Azad University (Arak) since 1998. He teaches courses in the areas of dynamics, theory of plates and shells and finite element method. He has published more than 20 articles in journals and conference proceeding. Mohammad Reza Isvandzibaei received his Ms Degree from Azad University (Arak), and now he is the student of Ph.D. in university of Pune, (India) all in mechanical Engineering. He is member of faculty in Islamic Azad University (Andimeshk).     

On the nonlinear vibration of heated bimetallic shallow shells of revolution

The axisymmetrically nonlinear free vibration of a bimetallic shallow shell of revolution under uniformly distributed static temperature changes is investigated. Based on the nonlinear bending theory of thin shallow shells, the governing equations are established in forms similar to those of classical single-layered shells theory by redetermination of reference surface of coordinate. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich averaging method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude are obtained successfully. A detailed parametric study is conducted involving shell geometry and temperature parameters. The effects of these variables on the frequency-amplitude characteristics are plotted and discussed.     

Stability analysis of whirling composite shells partially filled with two liquid phases

In this paper, the stability of whirling composite cylindrical shells partially filled with two liquid phases is studied. Using the first-order shear shell theory, the structural dynamics of the shell is modeled and based on the Navier-Stokes equations for ideal liquid, a 2D model is developed for liquid motion at each section of the cylinder. In steady state condition, liquids are supposed to locate according to mass density. In this study, the thick shells are investigated. Using boundary conditions between liquids, the model of coupled fluid-structure system is obtained. This coupled fluid-structure model is employed to determine the critical speed of the system. The effects of the main variables on the stability of the shell are studied and the results are investigated.     

Postbuckling of cross-ply laminated cylindrical shells with piezoelectric actuators under complex loading conditions

A postbuckling analysis is presented for a cross-ply laminated cylindrical shell with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric field is assumed to be the transverse component E_z only. The material properties are assumed to be independent of the temperature and the electric field. The governing equations are based on the classical shell theory with a von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators subjected to combined mechanical loading of external pressure and axial compression, and under different sets of thermal and electric loading conditions. The effects played by temperature rise, applied voltage, shell geometric parameter, stacking sequence, as well as initial geometric imperfections are studied.     

An efficient approach for free vibration analysis of conical shells

In this paper, the global method of generalized differential quadrature (GDQ) is applied for the first time to study the free vibration of isotropic conical shells. The shell equations used are Love-type. The displacement fields are expressed as product of unknown functions along the axial direction and Fourier functions along the circumferential direction. The derivatives in both the governing equations and the boundary conditions are discretized by the GDQ method. Using the GDQ method, the natural frequencies can be easily and accurately obtained by using a considerably small number of grid points. The accuracy and efficiency of the GDQ method is examined by comparing the results with those in the literature and very good agreement is observed. The fundamental frequency parameters for four sets of boundary conditions and various semivertex angles are also shown in the paper.     

Thermomechanical postbuckling analysis of moderately thick functionally graded plates and shallow shells

In this paper, an analytical solution is provided for the postbuckling behaviour of moderately thick plates and shallow shells made of functionally graded materials (FGMs) under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell, according to a power law distribution of the volume fraction of the constituents. The fundamental equations for moderately thick rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection and high-order shear deformation theory for moderately thick plates. The solution is obtained in terms of mixed Fourier series and the obtained results are compared with those of the Reissner–Mindlin's theory for moderately thick plates and the classical theory ignoring transverse shear deformation. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behaviour and the associated stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

Influence of the initial imperfection on the non-linear buckling response of FGM truncated conical shells

Non-linear buckling analyses of imperfect functionally graded truncated conical shells with simply supported boundary conditions and subjected to an axial compressive load have been presented in this work. The material properties of functionally graded shells are assumed to vary continuously through the thickness of the shell. The non-linear prebuckling deformations and initial geometric imperfections of an FGM truncated conical shell are both taken into account. The fundamental relations, modified Donnell type non-linear stability and compatibility equations of an imperfect FGM truncated conical shell are obtained and are solved by superposition and Galerkin methods, and the upper and lower critical axial loads has been found analytically. The numerical illustrations concern the non-linear buckling response of FGM truncated conical shells with different values of truncated conical shell parameters, initial imperfections and compositional profiles. Comparing the results of this study with those in the literature validates the present analysis.     

Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness

The free vibration analysis of laminated conical shells with variable stiffness is presented using the method of differential quadrature (DQ). The stiffness coefficients are assumed to be functions of the circumferential coordinate that may be more close to the realistic applications. The first-order shear deformation shell theory is used to account for the effects of transverse shear deformations. In the DQ method, the governing equations and the corresponding boundary conditions are replaced by a system of simultaneously algebraic equations in terms of the function values of all the sampling points in the whole domain. These equations constitute a well-posed eigenvalue problem where the total number of equations is identical to that of unknowns and they can be solved readily. By vanishing the semivertex angle (α) of the conical shell, we can reduce the formulation of laminated conical shells to that of laminated cylindrical shells of which stiffness coefficients are the constants. Besides, the present formulation is also applicable to the analysis of annular plates by letting α=π/2. Illustrative examples are given to demonstrate the performance of the present DQ method for the analysis of various structures (annular plates, cylindrical shells and conical shells). The discrepancies between the analyses of laminated conical shells considering the constant stiffness and the variable stiffness are mainly concerned.