Thick Cylindrical Shells Research Articles

Dynamic Analysis of 2D FG Porous Thick Cylindrical Shells Under Thermal Shock

In this research, a thick hollow cylindrical shell made of bidirectional functionally graded open cell porous materials under internal thermal shock according to the classical theory of linear thermo-elasticity is examined for the first time. The cylinder is made of a porous cellular material and its porosity varies along both radial and axial directions continuously. The governing motion equations are obtained by using 2D-axisymmetric theory of linear thermo-elasticity rather than shell theories. This theory represents thickness stretching and gives more precise results. Graded finite element method is employed to model the problem. Applying this method rather than conventional FEM leads to more accurate results, especially for dynamic analyses. The cubic higher order element is used for dividing the solution domain. To obtain transient temperatures, Crank–Nicolson algorithm is used and then the Newmark procedure is used to derive time responses of displacement and stress components. The time history of displacements and stress components for different radial and axial power law exponents, porosity coefficient, boundary conditions, length-to-thickness ratio and two different porosity patterns are investigated in detail. The obtained results show that thermal-induced vibration is generally caused by hoop stress, and frequency and amplitude of vibrations and velocity of stress waves are considerably influenced by the porosity distribution, porosity coefficient and power law exponents in both directions.

Multimodal Resonance Response of Incompressible Hyperelastic Moderately Thick Cylindrical Shells

Based on the modified third-order shear deformation theory, the harmonic balance method, and the pseudo-arclength continuation method with two-point prediction, the nonlinear forced vibration response of incompressible hyperelastic moderately thick cylindrical shells subjected to a concentrated harmonic load at mid-span and simply supported boundary conditions at both ends is investigated. The algorithmic procedure for solving steady-state periodic solutions of strongly nonlinear systems of differential equations is presented. The structural response characteristics of shells under different excitation amplitudes and structural parameters are analyzed. The numerical results indicate that the aspect ratio of moderately thick hyperelastic cylindrical shells has a significant effect on the natural frequency ratio. Different frequency ratios lead to varying nonlinear mode coupling effects. The coupling effects among modes result in complex nonlinear behavior in the vibration response of each mode, leading to abundant multi-valued phenomena in the response curve.

Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies

In spite of extensive studies on the nonlinear vibrations of cylindrical shells, the significant influences of some effective modes and the in-plane nonlinearity on the nonlinear coupled multi-mode vibrations are not clear. In this work, which is based on Donnell’s nonlinear shell theory and Amabili–Reddy’s third-order shear deformation theory, the nonlinear differential equations of motion of both the thin-walled and moderately thick cylindrical shells under the simply-supported boundary condition are established via Lagrange equations. These high-dimensional differential equations with the quadratic and cubic nonlinearities are solved by using an iteration procedure that is a combination of incremental harmonic balance method, pseudo-arclength method and extrapolation techniques. Turning and bifurcation points of the system are determined with the help of the direct method, the stability of solution of the frequency–response is examined by using the multi-variable Floquet theory. The numerical responses obtained by adopting two present shell theories are compared to investigate the influence of ignoring the in-plane nonlinearity on nonlinear vibrations. In the coupled multi-mode vibrations with respect to the fundamental mode m,n, apart from the regular modes i×m,j×ni=1,3,5,…,andj=0,1,2,3,… that include the axisymmetric and asymmetric modes, some irregular modes are taken into account to study the frequency–responses. Results show that present iteration procedure is efficient and successful to get the frequency–responses of the coupled multi-mode vibrations, and the influence of the irregular mode on the coupled multi-mode vibration is dependent on the relationship between the natural frequency of the irregular mode and that of the fundamental mode.

Vibration characteristic analysis of three-dimensional sandwich cylindrical shell based on the Spectro-Geometric method

This paper presents the numerical investigations on the free and random vibration properties of thick cylindrical shells under the three-dimensional (3-D) elasticity theory. The analysis of these properties is conducted using the Rayleigh-Ritz method in conjunction with the Spectro-Geometric method (SGM). The cylindrical shell considered in this study is composed of fiber-reinforced composite and functionally graded porous graphene platelet reinforced composites (FGP-GPLRC). To simulate different boundary conditions, an artificial spring technique is utilized. The displacement components, as determined by SGM, are incorporated into the energy expression of the cylindrical shell. By applying the Rayleigh-Ritz method, the intrinsic frequencies and vibration modes of the cylindrical shell are obtained. Additionally, the power spectral density (PSD) function of the cylindrical shell under random excitation is obtained by the Pseudo-Excitation method (PEM). The accuracy and effectiveness of the present method are verified through comparative studies with existing literature and finite element method (FEM). Furthermore, an investigation on the effects of various parameters on the intrinsic frequency and random response of the cylindrical shell is conducted.

Analytical Modeling of Shrink-Fitted FGM Thick-Walled Cylinder

One off the most powerful assembly technique is the shrink-fitting process.It is found in many fields such us mechanics, petroleum, military industries as well as in nuclear power plants etc. This article developed an analytical formulation of shrink-fitted Functionally Graded Material axisymmetric thick-walled cylinder based on the linear plane elasticity theory. The stresses and displacement fields in the thick cylindrical shells are calculated using the laws of linear elasticity. The resulting displacements and stresses are analyzed, and particularly the residual contact pressure and her relationship with the interference values. The results show that the variation of the FGM material composition has a clear effect on the fit pressure in the intersection area of the two fitted cylinders. The value of this pressure affects the distribution of radial and tangential stresses in the FGM cylinder walls. Subsequently, we highlighted the influence of the interference value, on the residual contact pressure which increases with the increase of the interference value. The stresses are modeled for a case study using MATLAB software. keywords. shrink-fit, FGM, Interference, residual stress, Elasticity.

Exact solutions for the symmetric elastoplastic response of functionally graded pressure vessels

With the development of functionally graded (FG) composite material technology, mechanical analyses of FG pressure vessels have attracted attention. Due to the non-uniform stress distributions inside such vessels, irreversible plastic deformation first occurs in some zones of the structure under the action of external loads. It is necessary to perform an elastoplastic analysis of FG structures to determine their load-bearing capacities. In this paper, exact solutions are obtained for the purely elastic, partially plastic, and fully plastic deformation of FG cylindrical and spherical shell pressure vessels under mechanical loads. The material properties (elastic modulus and yield strength) are assumed to vary nonlinearly in the radial direction of the vessel and the Poisson's ratio is assumed to be constant due to Poisson's ratio slight variations in the engineering materials. The novelty of the current work lies in the presentation of complete exact elastoplastic solutions for FG thick cylindrical and spherical shell pressure vessels, taking into account all deformation zones. Tresca's yield criterion is used to formulate four different deformation zones for an ideal plastic material. All of these states of deformation are studied in detail. It is shown that the elastoplastic response of the FG pressure vessels is notably affected by the radial variation of the material properties. With an increase in the FG parameter β, the first yielding position of the FG pressure vessels changes from the inner surface to the outer surface. The conclusions of this work should help with developing designs of FG pressure vessels to prevent yielding under excessive circumferential stresses.

The modified CUF-EFG method for the dynamic analysis of GPLs-CNTs-reinforced FG multilayer thick cylindrical shells under shock loadings: A modified meshless implementation

The main scope of this paper is to develop the Carrera unified formulation in cylindrical coordinates for two-dimensional meshless modelling of graphene platelet and carbon nanotube reinforced functionally graded multilayer thick cylindrical shells with the same accuracy of three-dimensional modelling. The cylindrical shell is composed of several homogeneous layers with uniform distribution of both GPL and CNT contents in each layer. The effective macroscopic mechanical properties of each GPLs and CNTs reinforced layer are estimated using the modified Halpin-Tsai (H-T) micromechanical model and rule of mixture. The volume fraction of nanofillers gradually changes layer to layer by nonlinear continuous function which leads to functionally graded (FG) distribution of nanofillers. Five grading patterns of layer distribution along the thickness direction of cylindrical shell are considered to investigate the effect of layer arrangement on dynamic behavior of cylindrical shells. The FG multilayer hybrid nanocomposite cylindrical shell is assumed to be under mechanical shock loading. The numerical method for dynamic analysis of multilayer cylindrical shell is based on element free Galerkin (EFG) method with radial basis shape functions in the framework of displacement-based Carrera unified formulation (CUF). The present EFG-CUF method is compared with analytical method for isotropic homogeneous shells and results available in the literature for FG multilayer cylindrical shells. Considering the results related to various expansion orders, proved accuracy and efficiency of proposed method for dynamic analysis of thin to thick FG multilayer cylindrical shells at high expansion orders ‘n > 2’. After verifying the analytical method, the effects of several key factors such as layer distribution patterns, volume fraction index, number of layers, nanofiller weight fraction, aspect ratios of cylinder, and damping ratio on static, free vibration, time history and wave propagation of cylinder are studied in detail.

Buckling analysis of thick cylindrical shells using micropolar theory

Modeling complex materials with internal structure such as porous solids is challenging as in some cases the classical theory cannot provide precise responses. By considering the scale effects through additional kinematic descriptors and six constants for isotropic materials, the micropolar theory can accurately model complex materials like bone. This paper studies the buckling of a thick cylindrical shell using classical and non-classical theories. The cylinder’s material is considered bone and isotropic, and the critical load value for different geometries and boundary conditions has been obtained. Finally, the size-effect and importance of micropolar theory in micro dimensions are investigated. The micropolar equations are solved by a numerical solution using the 3D finite element method. The results show that decreasing the macroscopic length increases the stiffness of the cylinder more than that predicted by classical theory; In addition, by increasing the thickness of the cylinder and the importance of shear stresses, the micropolar theory predicts a more critical load than the classical one, and the result differences become more significant between micropolar and classical theories. Also, the characteristic length of the micropolar is investigated. The results show that the change of the critical load increased by moving toward the micro dimensions.

Free vibration analysis of cylindrical sandwich panel with electro-rheological core and FG-GPLRC facing sheets based on First order shear deformation theory referred by Qatu

In this study, free vibration and damping behavior of cylindrical sandwich panel with electro-rheological core and graphene platelets reinforced composite (GPLRC) facing sheets based on first-order shear deformation theory of thick cylindrical shells referred by Qatu (FSDTQ) is investigated. Effective material properties of graphene platelets reinforced composite are determined according to the Halpin–Tsai micromechanical model. The governing equations of motion with required boundary conditions are derived using Hamilton’s principle. Afterward, these equations are solved analytically using the Fourier series solution for simply-supported boundary conditions, and for the other edges boundary conditions, we used the generalized differential quadrature method. The exactness and correctness of the present formulation are validated by comparing the results with those of existing literature. Finally, the influences of various parameters such as the electric fields, geometrically parameters, boundary conditions, the volume fraction of GPL, and GPL distributions pattern on vibration behavior and modal loss factor is examined. According to the results, the influence of the GPL volume fraction and electric field on the frequencies and modal loss factor is significant. This achievement can help design intelligent controlling structures for various applications.

An analytical method for free vibration analysis of multi-layered transversely isotropic cylindrical shells

This work presents an effective analytical method based on displacement potential functions (DPF) for solving 3-D free vibration problem of thick and multi-layered transversely isotropic cylindrical shells with simply supported end boundary conditions. By using the DPF method, the three-dimensional elasticity equations are simplified and decoupled into two linear partial differential equations of fourth and second order as governing differential equations. These equations are analytically solved using the separation of variables method in terms of fields that exactly satisfy end boundary conditions and the continuity of a closed cylinder in the hoop direction. One of the advantages of the present work is the analysis of thick cylindrical shells using an exact method without any approximations and simplifying assumptions in the distribution of stress or strain along the thickness of the shells. The analysis is extended to multi-layered cylindrical shells by applying appropriate boundary conditions at the interfacial surfaces of the layers. Numerical comparisons to other analytical works shows the accuracy of the present work and also due to the lack of study for multi-layered transversely isotropic cylindrical shells, 3-D finite element analysis has been performed for comparison. In addition, various more involved problems are studied and solved analytically for single and multi-layered shells of transversely isotropic material and the effect of various parameters such as shell dimensions, mechanical properties, number of layers and other parameters on shell behavior are evaluated. The results of the present study can be used as benchmark solutions for other studies.

Exact static axisymmetric solutions of thick functionally graded cylindrical shells with general boundary conditions

An analytical solution is presented for the axisymmetric static responses of thick functionally graded (FGM) cylindrical shells with general boundary conditions, which strictly satisfies the boundary conditions along the thickness. The illustrative examples show that the presented solution of thick cylindrical shells under various boundary conditions is of high precision and good convergence. The exact distributions of displacement and stress components along the radial and axial directions are revealed, and the influences of thickness-to-radius ratio, span-to-radius ratio and material gradient index on the static responses of the thick FGM cylindrical shell are analyzed.

A semi-analytical approach for instability analysis of composite cylindrical shells subjected to harmonic axial loading

In this article, nonlinear vibration and dynamic stability analyses of simply supported laminated composite circular cylindrical shells subjected to periodic edge loading are carried out. A third-order shear deformation shell theory that considers all the nonlinear terms in all five kinematic parameters and rotary inertia is used to develop the present mathematical model so that the model is also valid for thick cylindrical shells. Hamilton’s principle, an energy-based approach, is used to obtain the governing partial differential equations (PDEs) of motion of the cylindrical shell. Further, these equations are reduced into ordinary differential equations by employing Galerkin’s method. The incremental harmonic balance (IHB) method in conjunction with the pseudo-arc-length method is used to obtain the frequency-amplitude response of the system. For obtaining the zone of instability regions, Bolotin’s method is adopted. For more practical significance, analysis of results is also extended by considering damping into account for the composite cylindrical shells. Time history response and phase portrait are plotted by adopting Newmark-beta method. The effects of the static load factor, dynamic load factor, modal damping coefficient, and stacking sequence on nonlinear vibration, instability regions and time history responses are also examined.

Semi-analytical and experimental studies on travelling wave vibrations of a moderately thick cylindrical shell subject to a spinning motion

Spinning cylindrical shells are the indispensable basic components of engineering structures and exhibit complicated dynamic responses, the experimental studies on travelling wave vibrations are rare. This work numerically and experimentally investigates the travelling wave vibrations of a spinning cylindrical shell, i.e. a cylindrical aluminium shell with the integrated flanges. A new third-order shear deformation theory (TSDT) is adopted for the moderately thick cylindrical shell. Chebyshev polynomials are utilized to accurately simulate the circumferential-wave-dependent mode functions and the classical boundary conditions of the shell. The natural frequencies of the travelling waves of the spinning cylindrical shell are derived via Ritz method. The bifurcation of the natural frequencies of the spinning cylindrical shell is experimentally observed and discussed. Experimental results validate the mode shapes and the natural frequencies of the travelling waves, where the movable simply-supported ends and the sliding ends are chosen to approximate the boundary condition of the integrated flanges. In addition, the numerical comparisons among the TSDT, the first-order shear deformation theory and Donnell’s shell theory against the natural frequencies are presented.

Mathematical Modeling of Anisotropic Hyperelastic Cylindrical Thick Shells by Incorporating Thickness Deformation and Compressibility with Application to Arterial Walls

This paper is devoted to mathematical modeling of anisotropic hyperelastic thick cylindrical shells like arteries by taking into account the volume compressibility and through-the-thickness deformation. To describe the hyperelastic behavior of this kind of shells and extract their constitutive relations, the modified anisotropic (MA) model is employed, which is able to characterize compressible behavior of hyperelastic materials like soft tissues. By considering the arterial segment as a thick cylindrical shell, the higher order thickness deformation shell theory together with nonlinear Green’s strains are exploited to express its deformations and capture the thickness stretching effect. The shell is then discretized via Lagrange equations of motion to achieve the responses of the arterial wall. Numerical results are given in two sections. First, the presented formulation is validated by conducting a comparative study on a single-layer compressible pressurized shell. Close agreement between the outcomes of the developed model with those provided by finite element (FE) simulation signifies the necessity of incorporating thickness deformation in the higher order shell theory. In the next step, by considering the artery as a two-layer cylindrical shell comprising the media (the middle layer of the artery) and the adventitia (the outer layer), mechanical responses of an arterial wall subjected to an internal pressure are appraised through a number of parametric studies.

Polynomial mode approximations for longitudinal wave dispersion in solid and hollow circular cylinders

The characteristic equation for wave dispersion in solid or hollow circular cylinders is a transcendental equation: no closed-form solution exists and it can only be solved numerically. The characteristic equation for a hollow circular cylinder is substantially longer and more complex to solve than the equation for a solid cylinder. This paper examines, at different orders of approximation, two polynomial approximations (Jacobi mode approximation and an approximation by Widehammar, Gradin and Lundberg) which simplify the computation of dispersion curves. We focus on the approximation error for the first four modes of axially symmetric longitudinal waves, for both solid and hollow circular cylinders. The two polynomial approximations yield accurate dispersion curves for a solid cylinder, provided a sufficient order is used. For a hollow circular cylinder, the numerical limits of thinner cylinders prevent the use of the highest orders of approximation, limiting the accuracy of the dispersion curves. Classic theories for thin and thick cylindrical shells (by Naghdi and Cooper, and Herrmann and Mirsky), remain useful alternatives when computing dispersion curves for the first propagation mode for very thin hollow circular cylinders. We find that Widehammar’s approximation yields more accurate dispersion curves than Jacobi mode approximation for hollow cylinders, for a given order of approximation, as well as being more straightforward to program.

Eigenfrequencies and eigenmodes of anisotropic thick cylindrical shell

Thick cylindrical shells are used in various applications, and there are many studies on strength and vibration analysis. In vibration analysis of isotropic and orthotropic thick cylindrical shells, it is commonly assumed that displacements are affected by rotational inertia and shear deformation and the shell thickness does not change during deformation. On the other hand, vibration analysis of orthotropic cylinders has been studied based on three-dimensional analysis theory without these assumptions. However, vibration analysis of thick cylindrical shells based on three-dimensional analysis theory has not been conducted. Previously, we studied an orthotropic parallelepiped as a single continuum and reported the relationships between dimensions and natural angular frequencies and vibration modes. In this report, a thick cylindrical shell with deformation in thickness direction and orthotropy is investigated. The natural angular frequencies and modal functions of the axisymmetric vibration are derived using three-dimensional analytical theory. In the calculation example, the relationship between dimensionless axial wavenumber and natural angular frequencies and modal functions of the model with varying thickness ratios are illustrated. The frequency veering of natural angular frequencies occurs, and the wavenumber where the frequency veering occurs depends on the thickness ratio. The number of radial nodes varies independently of the frequency veering, and the sign of the slope of radial node vs. dimensionless wavenumber changes around the wavenumber where veering occurs. The number of axial nodes and their positions vary only around the value of the wavenumber where veering occurs.

A non-polynomial axiomatic framework for modelling and bending analysis of doubly curved spherical and cylindrical shells: An analytical solution

In the present work, the doubly curved spherical and cylindrical laminated composite shells are modelled and analysed in the framework of non-polynomial axiomatic approach. The inverse hyperbolic shear deformation theory originally developed for the laminated composite plates is extended to model the deformation characteristics of laminated composite shells. The mathematical formulation is developed under the assumption of linear structural kinematics and linear-elastic-orthotropic material behaviour. The governing equations of the model are obtained using the principle of virtual work and solved in exact manner for simply supported boundary conditions following the Navier solution methodology. The bending response of thick and thin spherical and cylindrical shells subjected to different types of transverse loads such as point load, uniform load and sinusoidal load is analysed in the framework of developed methodology. The obtained results due to inverse hyperbolic shear deformation theory are compared with other shell theories and on the basis of this comparison, the validity and applicability of the inverse hyperbolic shear deformation theory for doubly curved spherical and cylindrical shells is ensured.

Vibration Characteristics of Subway Tunnel Structure in Viscous Soil Medium

Based on research on subway tunnels in a viscous soil medium, this paper establishes the vibration equation of a tunnel structure by using the theory of moderately thick cylindrical shells and the method of wave propagation. The soil around the tunnel is represented in simplified form as an isotropic viscoelastic medium to obtain the equation of motion of the soil, and the vibration control equation of the tunnel under the influence of viscous soil is obtained by coupling. By numerical calculations, the variation trends in the natural vibration frequency of the tunnel and attenuation affected by soil viscosity under different modes are given. Furthermore, the influences of the tunnel radius, wall thickness, and length on the vibration characteristics of a tunnel structure in viscous soil are discussed. This study will provide a reference for the design of subway vehicles and the antivibration design of subway tunnel structures.

Effect of porosity distribution on natural frequencies of thin and thick cylinders based on Mirsky–Hermann’s shear deformation theory

In this paper, the effect of the porosity coefficient and its distribution on the natural frequencies and mode shapes of a thin and thick cylindrical shell made of asymmetric porous materials is investigated analytically. The mechanical properties of the porous shell are non-uniform in the thickness and isotropic in the circumferential directions. The displacement field is assumed based on the first-order Mirsky–Hermann theory by considering the effects of transverse normal strain and rotary inertia. For more accurate stress resultants, the trapezoidal shape factor is considered in the formulation. The constitutive relations follow Biot’s theory for the porous materials and the kinematic of the problem is linear. The governing equations include four coupled partial differential equations that are solved using an analytical method and they are compared with the finite element analysis. By the sensitivity analysis, the effects of porosity distribution and geometry of the cylinder on the natural frequencies and mode shapes are studied. The results show that the largest frequency corresponds to the material which has the maximum density inside of the cylinder. Abbreviations: FSDT, first-order shear-deformation theory; 2D, two-dimensional; 3D, three-dimensional; FG, functionally graded; FGM, functionally graded materials; DQ, differential quadrature; MDEW, maximum density in outer (external) wall; MDIW, maximum density in inner wall; C.C, complex conjugate (of proceeding terms)

Vibration frequency of thick functionally graded material cylindrical shells with fully homogeneous equation and third-order shear deformation theory under thermal environment

The effects of third-order shear deformation theory and varied shear correction coefficient on the vibration frequency of thick functionally graded material cylindrical shells with fully homogeneous equation under thermal environment are investigated. The nonlinear coefficient term of displacement field of third-order shear deformation theory is included to derive the fully homogeneous equation under free vibration of functionally graded material cylindrical shells. The determinant of the coefficient matrix in dynamic equilibrium differential equations under free vibration can be represented into the fully fifth-order polynomial equation, thus the natural frequency can be found. Two parametric effects of environment temperature and functionally graded material power law index on the natural frequency of functionally graded material thick cylindrical shells with and without the nonlinear coefficient term of displacement fields are computed and investigated.