Analysis Of Circular Cylindrical Shells Research Articles

An extended separation-of-variable method for the eigenbuckling of orthotropic open thin circular cylindrical shells

There have been numerous researches on buckling analysis of circular cylindrical shells, but still few researches on closed-form solutions for eigenbuckling of open circular cylindrical shells with non-Levy boundary conditions . This work develops an extended separation-of-variable (eSOV) method to obtain closed-form eigenbuckling solutions for orthotropic open circular cylindrical Donnell–Mushtari thin shells with arbitrary homogeneous boundary conditions. In this eSOV method, buckling mode functions are assumed to be the products of eigenfunctions of tangential and axial coordinate directions, and critical buckling loads corresponding to the two-direction eigenfunctions are assumed to be independent of each other. By employing Rayleigh’s principle, two-direction eighth-order characteristic differential equations are derived, and the two-direction eigenfunctions are expressed in terms of the eigenvalues of the characteristic differential equations, then the closed-form mode functions and explicit equations for critical buckling loads are achieved for arbitary homogeneous boundary conditions. Besides, the solutions presented in this work for open circular cylindrical shells can be simplified to those of closed circular cylindrical shells. The results of the proposed method agree well with those of FEM and other analytical methods in literatures, validating the present solutions.

Free vibration analysis of circular cylindrical shell on elastic foundation using the Rayleigh-Ritz method

A general approach is presented for the free vibration analysis of circular cylindrical shell resting on elastic foundation and subjected to classical boundary conditions of any type. The Winkler/Pasternak model is utilized to simulate the elastic foundation imposed on the cylindrical shell, and then it is easily to derive the potential energy of the elastic foundation. Based on the Flugge shell theory, explicit expressions for the mass and stiffness matrices are obtained. By taking the characteristic beam modal functions as the admissible functions, the Rayleigh-Ritz method is employed to derive the frequency equations of circular cylindrical shell with all the classical boundary conditions and resting on elastic foundation. Once the frequency equation has been determined, the frequencies can be calculated numerically. The excellent accuracy and validity of the present approach are demonstrated by numerical examples and comparisons with the results available in the literature. Finally, some further numerical results are given to illustrate the comprehensive effect of geometric properties and foundation coefficients on the frequencies of circular cylindrical shell in contact with elastic foundation.

Vibration analysis of circular cylindrical shells made of metal foams under various boundary conditions

This study investigates the free vibration of metal foam circular cylindrical shells under various boundary conditions. The elasticity modulus and mass density of the shells vary gradually and continually in the thickness direction. Two types of porosity distribution are taken into account including symmetrical and unsymmetrical distributions. Love’s shell theory is employed to formulate the governing equations and then the Rayleigh–Ritz method is utilized to solve natural frequencies of the system. The results show that the porosity coefficient has important effect on the natural frequencies of metal foam shells. Its effect also relates to the boundary conditions of the shells. Moreover, different porosity distributions make the metal foam shells possess different vibration characteristics, which is quite obvious at large porosity coefficient. As the circumferential wave number increases, the natural frequencies of the metal foam shells tend to the same under various boundary conditions. Additionally, the present results are verified by the comparison with the published ones in the literature.

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.

Application of Vectorial Wave Method in Free Vibration Analysis of Cylindrical Shells

AbstractThe vectorial form of the Wave Propagation Method (VWM), regarding the dispersion of harmonic plain (elasto-dynamic) waves within certain wave-guides, is developed for the vibration analysis of circular cylindrical shells. To obtain this goal, all plain waves are divided into positive-negative going wave vectors along with the shell axis. Based on the Flügge thin shell theory, the shell continuity as well as boundary conditions are well satisfied by introducing the propagation and reflection matrices. Furthermore, all elements of the reflection matrix are derived for certain classical supports. As an example, for demonstrating the feasibility of VWM in the shell vibration analysis, a circular cylindrical shell with two ended flexible support is adopted. The natural frequencies of the systemaswell asmode shapes are obtained using VWM. The aquired results are compared with those of the previous works and found in excellent agreement. It is also found that VWM could mathematically provide a reduced dimensional matrix (dominant matrix) to calculate the natural frequencies of the system. Accordingly, the proposed method can provide high computational efficiency and remarkable accuracy, simultaneously.

Chebyshev Collocation Solutions for Vibration Analysis of Circular Cylindrical Shells with Arbitrary Boundary Conditions

This paper applies the Chebyshev collocation method to finding accurate solutions of natural frequencies for circular cylindrical shells. The shells with different boundary conditions are considered in the parametric study. By using the method to solve the coupled differential equations of motion governing the vibration of the shell, numerical results are obtained from the algebraic eigenvalue equation using the Chebyshev differentiation matrices. And the results satisfy both the geometric and force boundary conditions. Based on the numerical examples, the proposed method shows its capacity and reliability in predicting accurate frequency results for circular cylindrical shells with various boundary conditions as compared to some exact solutions available in the literature.

Free Vibration Analysis of Circular Cylindrical Shells with Arbitrary Boundary Conditions by the Method of Reverberation-Ray Matrix

An analytical procedure for free vibration analysis of circular cylindrical shells with arbitrary boundary conditions is developed with the employment of the method of reverberation-ray matrix. Based on the Flügge thin shell theory, the equations of motion are solved and exact solutions of the traveling wave form along the axial direction and the standing wave form along the circumferential direction are obtained. With such a unidirectional traveling wave form solution, the method of reverberation-ray matrix is introduced to derive a unified and compact form of equation for natural frequencies of circular cylindrical shells with arbitrary boundary conditions. The exact frequency parameters obtained in this paper are validated by comparing with those given by other researchers. The effects of the elastic restraints on the frequency parameters are examined in detail and some novel and useful conclusions are achieved.

Dynamic stiffness method in the vibration analysis of circular cylindrical shell

In this paper the dynamic stiffness method is used for free vibration analysis of a circular cylindrical shell. The dynamic stiffness matrix is formulated on the base of the exact solution for free vibration of a circular cylindrical shell according to the Flugge thin shell theory. The matrix is frequency dependent and, besides the stiffness, includes inertia and damping effects. The derived dynamic stiffness matrix is implemented in the code developed in a Matlab program for computing natural frequencies and mode shapes of a circular cylindrical shell. Several numerical examples are carried out. The obtained results are validated against the results obtained by using the commercial finite element program Abaqus as well as the available analytical solutions from the literature.

Free vibration analysis of circular cylindrical shell with non-uniform elastic boundary constraints

As far as the study of circular cylindrical shell is concerned, most of the existing works are limited to classical homogeneous boundary conditions. However, complicated boundary conditions, such as elastic and non-uniform boundary conditions are encountered in many engineering applications. Thus, it is of important practical significance to investigate the vibrations of cylindrical shell with this kind of boundary conditions. This paper is mainly concerned with the free vibrations of cylindrical shell with non-uniform elastic boundary constraints. The exact solution for the problem is obtained using improved Fourier series method, in which each of three displacements of the shell is represented by a Fourier series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions. The unknown expansions coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the four sets of boundary springs at each end of the shell without the need of making any change to the solution procedure. The excellent accuracy of the current results is validated by comparison with the finite element method (FEM) results. Numerical results are presented to illustrate the effects of some interesting and practically important boundary restraints on free vibrations of circular cylindrical shell, including varying stiffness of boundary springs, point supported and partially supported boundary conditions.

An exact series solution for the vibration analysis of cylindrical shells with arbitrary boundary conditions

In this paper, an exact series solution for the vibration analysis of circular cylindrical shells with arbitrary boundary conditions is obtained, using the elastic equations based on Flügge’s theory. Each of the three displacements is represented by a Fourier series and auxiliary functions and sought in a strong form by letting the solution exactly satisfy both the governing differential equations and the boundary conditions on a point-wise basis. Since the series solution has to be truncated for numerical implementation, the term “exactly satisfying” should be understood as a satisfaction with arbitrary precision. One of the important advantages of this approach is that it can be universally applied to shells with a variety of different boundary conditions, without the need of making any corresponding modifications to the solution algorithms and implementation procedures as typically required in other techniques. Furthermore, the current method can be easily used to deal with more complicated boundary conditions such as point supports, partial supports, and non-uniform elastic restraints. Numerical examples are presented regarding the modal parameters of shells with various boundary conditions. The capacity and reliability of this solution method are demonstrated through these examples.

Dynamic Analysis of Circular Cylindrical Shells With General Boundary Conditions Using Modified Fourier Series Method

Dynamic behavior of cylindrical shell structures is an important research topic since they have been extensively used in practical engineering applications. However, the dynamic analysis of circular cylindrical shells with general boundary conditions is rarely studied in the literature probably because of a lack of viable analytical or numerical techniques. In addition, the use of existing solution procedures, which are often only customized for a specific set of different boundary conditions, can easily be inundated by the variety of possible boundary conditions encountered in practice. For instance, even only considering the classical (homogeneous) boundary conditions, one will have a total of 136 different combinations. In this investigation, the flexural and in-plane displacements are generally sought, regardless of boundary conditions, as a simple Fourier series supplemented by several closed-form functions. As a result, a unified analytical method is generally developed for the vibration analysis of circular cylindrical shells with arbitrary boundary conditions including all the classical ones. The Rayleigh-Ritz method is employed to find the displacement solutions. Several examples are given to demonstrate the accuracy and convergence of the current solutions. The modal characteristics and vibration responses of elastically supported shells are discussed for various restraining stiffnesses and configurations. Although the stiffness distributions are here considered to be uniform along the circumferences, the current method can be readily extended to cylindrical shells with nonuniform elastic restraints.

Buckling Analysis of Circular Cylindrical Shells Partially Subjected to External and Internal Liquid Pressures

Buckling Analysis of Circular Cylindrical Shells Partially Subjected to External and Internal Liquid Pressures

Dynamic instability of a circular cylindrical shell carrying a top mass under base excitation: Experiments and theory

The present paper is focused on the experimental and theoretical analysis of circular cylindrical shells under base excitation. The shell axis is vertical, it is clamped at the base and connected to a rigid body on the top; the base provides a vertical seismic-like excitation. The goal is to investigate the shell response when a resonant harmonic forcing is applied: the first axisymmetric mode is excited around the resonance at relatively low frequency and low amplitude of excitation. A violent resonant phenomenon is experimentally observed as well as an interesting saturation phenomenon close to the previously mentioned resonance. A theoretical model is developed to reproduce the experimental evidence and provide an explanation of the complex dynamics observed experimentally; the model takes into account geometric shell nonlinearities, electrodynamic shaker equations and the shell shaker interaction.

Static Analysis of Circular Cylindrical Shell Under Hydrostatic and Ring Forces

Analysis of circular cylindrical shell under the action of hydrostatic and stiffening ring forces is carried out in this work. The differential equation of equilibrium, similar to that of beam on elastic foundation, was obtained from static principles on the assumptions of P. L. Pasternak. The initial value method was used to solve the obtained fourth order differential equation for both cases of hydrostatic and ring forces. Combined actions of hydrostatic and ring forces were studied as the location of the ring was varied along the height of the reservoir. It appears that the most favourable location for the ring is 2/3 of its height measured from the top. Bending mo-ment, shear force and hoop tension diagrams, essentially necessary for design of the reservoir, were plotted under the action of these forces.Keywords: Static analysis, cylindrical shell, hydrostatic forces, reservoir, initial value.

Analysis of circular cylindrical shells under harmonic forces

A generalized thin shell theory for the stress-deformation analysis of thin-walled circular cylindrical shells is formulated based on the Hamilton's variational principle. The theory is applicable to multiple in-phase and out-of-phase harmonic forces and general boundary conditions. The stationary conditions of the Hamilton functional are then evoked to recover the equations of motion and boundary conditions of the problem. The unknown displacement fields are expanded as infinite complex Fourier series in the circumferential coordinate. The resulting field equations are observed to couple the radial, tangential and longitudinal displacement contributions within each mode, while uncoupling the contributions of a given Fourier mode from those of other Fourier modes. A general solution for the obtained field equations is developed for harmonic loading of general spatial distribution. The method developed is verified against well established solutions and its applicability to practical problems is illustrated through examples.

Study on free vibration analysis of circular cylindrical shells using wave propagation

A wave propagation approach is presented for free vibration analysis of circular cylindrical shell, based on Flügge classical thin shell theory. The validity and accuracy of the wave approach is studied in detail, including aspects of frequencies, vibration shapes and wavenumbers. An exact solution for free vibration of circular cylindrical shell is also given in the present analysis. For the comparisons of these two approaches for natural frequencies, vibration shape of a shell are discussed for shear diaphragm–shear diaphragm (SD–SD), clamped–clamped (C–C) and clamped–shear diaphragm (C–SD) boundary conditions. The results show that wave propagation has high accuracy for long shell and SD–SD boundary conditions.

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.

Finite element creep buckling analysis of circular cylindrical shell under axial compression taking account of creep damage

Cylindrical shells are utilized as structural elements of nuclear power plants, heat exchangers or pressure vessels, which are operated under elevated temperature. Creep buckling is one of the failure modes of structures at elevated temperature. In some experiments conducted by other authors, axially compressive cylindrical shells with a large ratio of radius to thickness were observed to buckle with circumferential waves. It is observed that the circumferential waves occur due to bifurcation buckling. But, the critical time and the minimum loading for bifurcation buckling obtained from calculations of finite element analyses are not in very good agreement with those of the experiments. One of the reasons for the disagreement is considered to be that the creep constitutive equations employed in many previous analyses represent the steady creep. The creep phenomena usually have primary creep period, steady creep one and tertiary creep one. A creep strain - time relation through the three periods can be simulated by using a constitutive equation based on creep damage mechanics. In the present analysis, we analyzed the bifurcation creep buckling of circular cylindrical shells subjected to axial compression by the use of the finite element method taking account of the creep damage mechanics proposeol by of Kachanov-Rabotonov.

Effects of Boundary Conditions on the Parametric Resonance of Cylindrical Shells under Axial Loading

In this paper, a formulation for the dynamic stability analysis of circular cylindrical shells under axial compression with various boundary conditions is presented. The present study uses Love’s first approximation theory for thin shells and the characteristic beam functions as approximate axial modal functions. Applying the Ritz procedure to the Lagrangian energy expression yields a system of Mathieu–Hill equations the stability of which is analyzed using Bolotin’s method. The present study examines the effects of different boundary conditions on the parametric response of homogeneous isotropic cylindrical shells for various transverse modes and length parameters.

Postbuckling analysis of circular cylindrical shell under combined loads

The postbuckling behavior of circular cylindrical shells of finite length under the combined action of external pressure and axial compression is analyzed using the spline finite strip method. The basic formulae and computational strategies proposed by the authors are extended successfully to the present problem. Similar to the results obtained by the authors for the shell under axial compression or pressure, the postbuckling equilibrium path computed is closer to the experimental results than the existing analytical ones and the postbuckling contour map of equal radial deflection computed are in good agreement with the experimental results, not only in shape but also in the deflection amplitudes.