Thick Cylindrical Shells Research Articles

Solid or shell finite elements to model thick cylindrical tubes and shells under global bending

This paper explores the use of solid continuum finite elements and shell finite elements in the modelling of the nonlinear plastic buckling behaviour of cylindrical metal tubes and shells under global bending. The assumptions of shell analysis become increasingly uncertain as the ratio of the radius of curvature to the thickness becomes smaller, but the classical literature does not draw a clear line to define when a shell treatment is inappropriate and a continuum model becomes essential. This is a particularly important question for the bending of tubular members, pipelines, chimneys, piles, towers and similar structures. This study is therefore concerned solely with the uniform bending of thin tubes or thick shells which fail by plastic buckling well into the strain-hardening range. The analyses employ finite element formulations available in the commercial software ABAQUS because this is the most widely used tool for parametric research studies in this domain with an extensive and diverse element library. The results are of general validity and are applicable to other finite element implementations. This paper thus seeks to determine the adequacy of a thin or thick shell approximation, taking into account geometric nonlinearity, complex equilibrium paths, limit points and bifurcation buckling, extensive material ductility and linear strain hardening. It aims to establish when it is viable to employ shell elements and when this decision will lead to outcomes that lack sufficient precision for engineering design purposes.The results show that both thin and thick (shear-flexible) shell elements may give a reasonably accurate prediction of the buckling moment under global uniform bending for cylindrical tubes as thick as R/t=10. A finite strain and thick shell formulation is additionally shown to model the ductility of such thick tubes well, even when ovalisation of the cross-section and strain hardening are included. The use of solid continuum elements to model tubes in bending is found to become increasingly uneconomical as the R/t ratio rises above 25 with reduced advantages over shell elements, both in terms of the accuracy of the solution and the computation time.

Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes

Cylindrical tubes and membranes are universal structural elements found in biology and engineering over a wide range of scales. Working in the framework of nonlinear elasticity, we consider the possible deformations of elastic cylindrical shells reinforced by one or two families of fibres. We consider both small and large deformations and the reduction from thick cylindrical shells (tubes) to thin shells (cylindrical membranes). In particular, a number of universal parameter regimes can be identified where the response behaviour of the cylinder is qualitatively different. This include the possibility of inversion of twist or axial strain when the cylinder is subject to internal pressure.

Three dimensional frequency analysis of bidirectional functionally graded thick cylindrical shells using a radial point interpolation method (RPIM)

This paper considers a functionally graded (FG) shell using a meshless radial point interpolation method (RPIM). The material is assumed to be bidirectional FG, where the variation is present in both the radial and the axial directions. Based on the three-dimensional equations of motion, the frequency equations are stated using RPIM. Numerical results are presented for a thick shell for various boundary conditions. These results illustrate the influence from the material variation concerning eigenfrequencies and eigenmodes. In addition, the study shows that the RPIM is an efficient method to solve dynamical shell problems.

Elastic analysis of pressurized thick cylindrical shells with variable thickness made of functionally graded materials

In this paper, using the first-order shear deformation theory (FSDT) and matched asymptotic method (MAM) of the perturbation theory, an analytical solution for deformations and stresses of axisymmetric clamped–clamped thick cylindrical shells with variable thickness made of functionally graded materials (FGMs) subjected to internal pressure are presented. The governing equations are a system of ordinary differential equations with variable coefficients. Using the MAM of the perturbation theory, these equations could be converted into a system of algebraic equations and two systems of differential equations with constant coefficients. The displacements and stresses along the radius and length have been plotted and distribution of these is compared with the solution using finite element method (FEM).

Calculation of natural frequencies and vibration modes of variable thickness cylindrical shells using the Wittrick–Williams algorithm

Dynamic stiffness equations are formulated for variable thickness cylindrical shells, under the assumptions of Donnell, Timoshenko and Flügge theories. Transcendental dynamic stiffness matrices are formed by solving numerically the governing eighth order differential equations using the boundary-value solver COLSYS. Undamped natural frequencies are found using the Wittrick–Williams algorithm. The shell is divided into smaller elements whose clamped end natural frequencies exceed the highest frequency of interest. A parametric study examines the effects of varying the geometry, degree of thickness taper and end conditions on the natural frequencies and mode shapes, providing benchmark results for designers and future researchers.

DETERMINATION OF DISPLACEMENTS AND STRESSES IN PRESSURIZED THICK CYLINDRICAL SHELLS WITH VARIABLE THICKNESS USING PERTURBATION TECHNIQUE

This article presents a closed form analytical solution for clamped-clamped thick cylindrical shells with variable thickness subjected to constant internal pressure. Regarding the problem which could not be solved through plane elasticity theory (PET), the solution based on the first-order shear deformation theory (FSDT) is suggested. Based FSDT on, and the virtual work principle, the differential equations governing axisymmetric thick cylindrical shells with variable thickness have been derived. The governing equations are a system of ordinary differential equations with variable coefficients. Using the matched asymptotic method (MAM) of the perturbation technique, these equations could be converted into a system of algebraic equations with variable coefficients and two systems of differential equations with constant coefficients.DOI: http://dx.doi.org/10.5755/j01.mech.18.1.1274

Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel

Free vibration of Levy-type thick functionally graded (FG) circular cylindrical shell panels is investigated to identify the validity range of two common shell theories namely Donnell and Sanders theories. FG material properties change through the thickness direction according to a power law distribution. The state space approach is applied to solve the problem. The present results are compared with those of the literature and a 3D finite element model for isotropic and FG materials. The effects of various geometry and material parameters on the validity range of these theories are studied for different boundary conditions. The results show that unlike Sanders theory, Donnell one cannot accurately capture natural boundary conditions such as force and moment resultants.

Free vibration of thick laminated cylindrical shells with different boundary conditions using general differential quadrature

Equations of motion with required boundary conditions for deep and thick cylindrical composite shells are shown using two First-order Shear Deformation Theories. The difference between them is the inclusion of the effects of curvature in the evaluation of stiffness parameters. Equations of motion with stress resultants lead to a system of fifteen first-order differential equations for dynamics of cylindrical shells. These equations are solved numerically using the General Differential Quadrature method for free vibration of isotropic, cross-ply, angle-ply and general lay-up cylindrical shells with six types of different boundary conditions using the aforementioned theories. The first five frequency parameters and mode shapes which are obtained from both theories are compared with the available results in the literature and those obtained using a three-dimensional finite element analysis to test the accuracy of the shell theories presented here.

Analyzing the stress–strain state of thick cylindrical shells

Two approaches to the analysis of the stress–strain state of thick cylindrical shells are elaborated. The shell is divided by concentric cross-sectional circles into several coaxial cylindrical shells. Each of these shells has its own curvature determined on its midline. The stress–strain state of the original shell is described by satisfying the interface conditions between the component shells. The distribution of unknown functions throughout the thickness is determined by finding the analytic solution of a system of differential equations in the first approach and is approximated by polynomial functions in the second approach. The stress–strain state of thick shells is analyzed. It is revealed that the effect of reduction becomes stronger with increasing curvature

A study on dynamic behaviour of functionally graded thick hollow circular cylinders

This article presents the analysis of functionally graded hollow circular cylinders with finite length under axisymmetric dynamic loads. It is assumed that the functionally graded cylinder is comprised of metal-phase and ceramic-phase, whereas material properties are graded in the thickness direction of the cylinder according to power law distribution. Two-dimensional finite element method in conjunction with the Newmark method is used to solve the system of time-dependent coupled equations that govern the dynamic responses. By introducing especial elements, it is possible to distribute the material properties through the thickness of cylinder exactly according to the power law distribution. Dynamic loads applied on the cylinder are axisymmetric in the hoop direction and can vary in the radial and axial directions. As examples, the transient responses of functionally graded cylinders, which are excited in radial direction by increasing and sudden internal pressure and line load and also in axial direction by longitudinal incident wavelet, are calculated, and the characteristics of waves are discussed. The results in the present study are obtained for thick cylinder and cylindrical shell and compared with the results for isotropic cylinders.

Displacement Governing Equations of Moderately Thick Cylindrical Shallow Shells by Transverse Shearing Deformation and the General Solution

Displacement fundamental equations of the moderately thick cylindrical shallow shells concerning five independent variables, i.e. five middle surface displacements are established based on the displacement fundamental equations of the moderately thick shells by transverse shearing deformation and basic hypothesis on shallow shells. Three assistant displacement functions are introduced to solve the equations, which are tenth-order differential equations with variable coefficient; and then five second-order differential equations are converted into a second-order differential equation and two fourth-order transition differential equations using the Cauchy-Riemann condition, afterwards another assistant displacement function is introduced to build its decoupled governing differential equations, finally five displacement components through four assistant displacement functions are obtained.

Postbuckling of shear deformable stiffened anisotropic laminated cylindrical shell under axial compression

A postbuckling analysis is presented for a shear deformable anisotropic laminated cylindrical shell with stiffener of finite length subjected to axial compression. The material of each layer of the shell is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on a higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity and including the extension/twist, extension/flexural and flexural/twist couplings. The ‘smeared stiffener’ approach is adopted for the beam stiffeners. This arrangement allows the beam stiffeners to be assembled directly into the global stiffness matrix. The nonlinear prebuckling deformations and 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. The numerical illustrations concern the postbuckling response of perfect and imperfect, grid, axial, ring stiffened, and unstiffened shells. The results confirm that there exists a compressive stress along with an associate shear stress and twisting when the anisotropic shell is subjected to axial compression. The postbuckling equilibrium path is unstable for the moderately thick cylindrical shell under axial compression and the stiffened shell structure is imperfection-sensitive.

Free vibrations of thick layered cylindrical shells

Two approaches to the calculation of closed thick layered cylindrical shells are developed. They are based on division of the cylindrical shell across its thickness by concentric circumferential surfaces into a series of constituent cylindrical shells. Satisfying the contact conditions on the surfaces between constituent shells, it is possible to determine the frequency of free bending vibrations of the initial shell with a sufficient accuracy. In the first approach, the distribution of unknown functions across the shell thickness is sought on the basis of an analytical solution to the corresponding system of differential equations; in the second one, the distribution is assigned by polynomial approximation functions.

152 種々の境界条件を有する厚肉円筒殼の自由振動特性(ロボティクス・メカトロニクス/機械力学・計測制御)

152 種々の境界条件を有する厚肉円筒殼の自由振動特性(ロボティクス・メカトロニクス/機械力学・計測制御)

POSTBUCKLING OF SHEAR-DEFORMABLE ANISOTROPIC LAMINATED CYLINDRICAL SHELLS UNDER AXIAL COMPRESSION

A postbuckling analysis is presented for a shear-deformable anisotropic laminated cylindrical shell of finite length subjected to axial compression. The material of each layer of the shell is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are based on a higher order shear-deformable shell theory with the von Kármán–Donnell type of kinematic nonlinearity and including the extension/twist, extension/flexural and flexural/twist couplings. The nonlinear prebuckling deformations and 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. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different values of shell parameters and stacking sequence. The results confirm that there exists a compressive stress along with an associate shear stress and twisting when the anisotropic shell is subjected to axial compression. The postbuckling equilibrium path is unstable for the moderately thick cylindrical shell under axial compression and the shell structure is imperfection-sensitive.

Buckling of thick cylindrical shells under external pressure: A new analytical expression for the critical load and comparison with elasticity solutions

In this paper a set of stability equations for thick cylindrical shells is derived and solved analytically. The set is obtained by integration of the differential stability equations across the thickness of the shell. The effects of transverse shear and the non-linear variation of the stresses and displacements are accounted for with the aid of the higher order shell theory proposed by [Voyiadjis, G.Z. and Shi, G., 1991, A refined two-dimensional theory for thick cylindrical shells, International Journal of Solids and Structures, 27(3), 261–282.]. For a thick shell under external hydrostatic pressure, the stability equations are solved analytically and yield an improved expression for the buckling load. Reference solutions are also obtained by solving numerically the differential stability equations. Both the full set that contains strains and rotations as well as the simplified set that contains rotations only were solved numerically. The relative magnitude of shear strain and rotation was examined and the effect of thickness was quantified. Differences between the benchmark solutions and the analytic expressions based on the refined theory and the classical shell theory are analysed and discussed. It is shown that the new analytic expression provides significantly improved predictions compared to the formula based on thin shell theory.

Plastic Buckling of Cylindrical Shells Under Transverse Loading

Thick cylindrical shells under transverse loading exhibit an elephant foot buckling mode, whereas moderately thick cylindrical shells show a diamond buckling mode. There exists some intermediate geometry at which the transition between buckling modes can take place. This behavior is significantly influenced by the radius-to-thickness ratio and the material yield strength, rather than the length-to-radius ratio and the axial force. This paper presents a critical value at which the transition of buckling modes occurs as a function of the radius-to-thickness ratio and the material yield strength. The result shows that the circumferential wave number of the diamond buckling mode increases with decreasing wall thickness. The strain concentration is also intensified for the diamond buckling modes compared with the elephant foot buckling modes.

Three-dimensional free vibration analysis of thick cylindrical shells resting on two-parameter elastic supports

Three-dimensional free vibration of a circular cylindrical shell in contact with an elastic medium will be studied. The response of the elastic medium is formulated by the Winkler/Pasternak model. The layerwise theory, in conjunction with a three-dimensional form of Hamilton's principle, is used to obtain the transversely discretized equations of motion, and the related boundary conditions. The differential quadrature method is employed to discretize the resulting equations in the axial direction and for the solution procedure. To validate the formulation, the results are compared with the available exact solutions and also with ANSYS finite elements solutions. By examining the results for thick circular shells under various boundary conditions and on different elastic foundations, the influence of such parameters, and in particular, those due to elastic foundations are studied.

ANALYTICAL AND NUMERICAL STRESS ANALYSIS OF THICK CYLINDER SUBJECTED TO INTERNAL PRESSURE

The present work is an attempt to investigate the vibrations characteristics and effect of static stresses and deformation in partially pressurized thick cylindrical shells, such as the gun barrels. The method used cover analytical investigation developed to determine static stresses and deformation along the thick cylindrical shell using LAME'S equation. The numerical investigation is developed using the finite element method with axisymmetric element (Plane 42) four nodes to determine the static response and solid element (Solid 45) eight nodes for vibration analysis by using the ANSYS package. The obtained results show a good agreement with the other investigators. It's found that the natural frequency of the selected models almost equal (150. Hz) and these results indicate that the frequency of powder gasses pressure more than (150 Hz) to be far away from resonance phenomena.

Analysis of wave propagation in carbon nanotubes via elastic shell theories

This paper investigates wave propagation in both single-walled carbon nanotubes (SWNTs) and double-walled carbon nanotubes (DWNTs) via two developed elastic shell theories: Love’s thin cylindrical shell theory and the Cooper–Naghdi thick cylindrical shell theory. In studying DWNTs, the van der Waals effect is accounted for and modeled with the two theories. The elastic thick shell theory, in which the shear and inertia effects are taken into account, is developed first to investigate the wave propagations of CNTs to provide more accurate wave dispersions for higher modes. The material properties of the CNTs that are used in the two shell theories are proposed, and the expression of the inertia moment of the cross area in the thick shell theory is recommended. The dispersion results that are derived via the two theories are compared to show the feasibility of those theories in studying CNTs. Radius-dependent wave propagation results in SWNTs and DWNTs are also studied via the two theories.