Buckling of circular cylindrical shells partially subjected to external liquid pressure

Theoretical analyses are presented for the buckling of circular cylindrical shells partially subjected to external liquid pressure. The shells are assumed to stand vertically with the lower end clamped, and the upper end clamped or free. In the analyses, the Donnell equations are used for the basic equation, and prebuckling deformation as well as the membrane state of stress of the shell are taken into account. The Galerkin method is used, and the critical pressures at various liquid heights as well as the wave numbers, are obtained for a wide range of the geometrical parameters of the shell Z. A convenient chart which indicates the buckling liquid height for a given shell and liquid are presented. Experimental studies are also conducted by using test cylinders made of polyester film, and water. The theoretical and experimental results for the buckling liquid height, are in excellent agreement.     

网状球壳的连续化分析方法

本文对四向杆系、三向杆系、两向杆系组成的各种网状球壳,探讨了基于连续化计算模型的拟壳分析法,给出了网壳等代薄膜刚度和抗弯刚度的表达式,建立了轴对称网状球壳拟壳法的一般性基本方程式,并采用薄膜理论加边界效应的分析方法进行计算。文中给出了如何由壳体内力反算各种网状球壳杆件内力的计算公式,文末附有算例。计算表明,这种连续化的拟壳分析法比较方便,手算电算均可。此外,本文还就常用网状球壳的形式和分类作了讨沦。     

Free vibrations of fluid-filled cylindrical shells on elastic foundations

The free vibration characteristics of fluid-filled cylindrical shells on elastic foundations are presented by a semi-analytical finite element method. A shell is discretized into cylindrical finite elements where shell governing equations based shape functions in the longitudinal direction are used instead of the usual simple polynomials. Non-uniformities of the foundations in the circumferential and longitudinal directions are handled by the Fourier series and an element mesh strategy, respectively. The fluid domain is described by the potential flow theory. The hydrodynamic pressure acting on shells is derived from the condition for dynamic coupling of the fluid-structure. The effect of fluid in a shell, shell geometries, and foundation parameters on the dynamic behavior of fluid-containing shells is investigated. Numerical results based on the present method converge more rapidly than those obtained by the simple polynomial formulation. The method is suitable for the problem considered due to its generality, simplicity, and potential for further development.     

网状扁壳与带肋扁壳组合结构的拟三层壳分析法

本文对网状扁壳与带肋扁壳共同工作的组合结构(可简称组合网状扁壳),采用连续化的拟三层壳的计算模型,按弹性小挠度薄壳理论进行分析计算,推导建立了混合法的基本方程式。由于这种构造上的拟三层壳在一般情况下不存在中面,因而壳体的薄膜内力、弯矩与薄膜应变,弯曲应变是耦合的,存在一个耦合矩阵,使得基本方程式比单层光面的符氏扁壳方程要复杂得多。对于周边简支的组合网状扁壳可求得基本方程式的解析解。文中对三向、四向组合网状扁壳进行了详细讨论,并指出了在特定条件下,可退化为一个当量的各向同性单层扁壳。对于一般网状扁壳的拟壳分析法及带肋扁壳的拟壳分析法分别属于本文的两种特殊情况。文中附有计算例题。     

Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia

The governing strain-displacement and curvature-displacement equations for paraboloidal shells including shear deformation and rotary inertia are solved for free vibration of closed shells. The finite element method is used to obtain three-dimensional frequency of vibration solutions for a variety of boundary conditions, free, fixed and simply supported. Assumptions concerning the circumferential vibrational behavior are incorporated that reduce the analysis to a single coordinate and the element shape function is formulated using the meridional coordinate. The results for frequency of vibration compare favorably with the available literature. Selected results for frequency of vibration are presented in tabular form for several shell parameters, including free, pinned and fixed boundary conditions. Representative mode shapes are plotted for a fixed boundary condition.     

Influence of internal liquid on buckling of circular cylindrical shells partially submerged in a liquid

Theoretical analysis has been presented for the effect of internal liquid pressure on the buckling of circular cylindrical shells subjected to external liquid pressure. The shell is assumed to stand vertically with the lower end clamped, and the upper end clamped or free. In the analysis, the Donnell equation was used for the basic equation, and the prebuckling deformation of the shell was taken into account. The Galerkin method was used, and the critical pressures at various internal liquid heights as well as the buckling wave numbers, were obtained for a wide range of the geometrical parameters Z a given shell. The effect of internal liquid on the buckling mode was also clarified. To confirm the validity of the analysis, experimental studies were also conducted by using test cylinders made of polyester film and water. Theoretical and experimental results were in good agreement.     

Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods

In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e. rings with non-uniform stiffeners eccentricity and unequal stiffeners spacing is investigated using analytical, experimental and finite elements (FE) methods. Ritz method is applied in analytical solution while stiffeners treated as discrete elements. The polynomial functions are used for Ritz functions and natural frequency results for simply supported stiffened cylindrical shell with equal rings spacing and constant eccentricity is compared with other's analytical and experimental results, which showed good agreement. Also, a stiffened shell with unequal rings spacing and non-uniform eccentricity with free–free boundary condition is considered using analytical, experimental and FE methods. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes and damping in each mode. In FE method, two types of modeling, including shell and beam elements and solid element are used, applying ANSYS software. The analytical and the FE results are compared with the experimental one, showing good agreements. Because of insufficient experimental modal data for non-uniformly stiffeners distribution, the results of modal testing obtained in this study could be as useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis.     

Dynamic elastic instability of long circular cylindrical shells under pure bending

The present paper studies the responses and instabilities of long circular cylindrical shells subjected to dynamic pure bending. The dynamic instability characteristics of the shells subjected to a sudden step bending load of infinite duration are explored. Analysis is performed using nonlinear finite element numerical methods. Critical dynamic moments are determined through the use of Budiansky and Routh's stability criterion. Numerical predictions for the dynamic instability are compared with those static results given earlier by Brazier. The effects of shell geometry on the dynamic stability of the shells are shown. It is found that the dominating factor to affect the shell stability is the ovalization of the shell cross-section in the centre of the shell.     

An improved treatment of mixed interpolation functions in eight-node assumed natural strain shell element for vibration analysis

In this article, we investigate the vibration analysis of plates and shells, using an eight-node shell element that allows for the effects of transverse shear deformation and rotary inertia. The natural frequencies of plates and shells are presented, and the forced vibration analysis of plates and shells subjected to arbitrary loading is carried out. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To improve the eight-node shell element for free and forced vibration analysis, a new combination of sampling points for assumed natural strain method was applied. The refined first-order shear deformation theory based on Reissner–Mindlin theory, which directly addresses the transverse shear deformation without a shear correction factor, is adopted for the development of a new eight-node assumed strain shell element with rotary inertia effect. In order to validate the finite element numerical solutions, the reference solutions of plates based on the first-order shear deformation theory are presented. Results of the present theory show good agreement with the reference solutions. In addition, the effect of damping is investigated on the forced vibration analysis of plates and shells.     

外压作用下偏心加劲功能梯度圆柱壳的非线性屈曲和后屈曲分析

通过解析法研究外压作用下功能梯度加劲薄圆柱壳的非线性屈曲和后屈曲性能。通过其在内部的偏心环和纵梁对壳体进行加固,假定壳体和加固件的材料性能在厚度方向为连续梯度。根据VonKarman理论中的刚度法和传统的壳理论推导出基本关系和平衡方程,可更准确地选择三种关于挠曲的近似公式,且使用盖勒金法得出的显式表达式可以推测出临界荷载和后屈曲压力-挠曲曲线。数值结果显示了加固件能有效地增强壳体稳定性。     

An extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) for free vibration analysis of cracked Reissner-Mindlin shells

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) was recently proposed and proven to be robust for free vibration analyses of Reissner-Mindlin shell. The method improves significantly the accuracy of the solution due to softening effect of the cell-based strain smoothing technique. In addition, due to using only three-node triangular elements generated automatically, the CS-FEM-DSG3 can be applied flexibly for arbitrary complicated geometric domains. However so far, the CS-FEM-DSG3 has been only developed for analyzing intact structures without possessing internal cracks. The paper hence tries to extend the CS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells by integrating the original CS-FEM-DSG3 with discontinuous and crack−tip singular enrichment functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3). The accuracy and reliability of the novel XCS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells are investigated through solving three numerical examples and comparing with commercial software ANSYS.     

Static and vibration analysis of orthotropic toroidal shells of variable thickness by differential quadrature

In this study the static and free vibration characteristics of linear elastic orthotropic toroidal shells of variable thickness are considered. A solution based on the Sanders-Budiansky shell equations is developed. The semi-analytical differential quadrature method in which Fourier series are written in the circumferential direction is adopted. This approach reduces the computational work to a series of one-dimensional problems. A novelty in the solution concerns the use of power series as trial functions in a domain exhibiting cyclic periodicity. Using the developed theory numerical results are determined for two separate applications. The results obtained are compared with results from the finite element method, and conclusions are drawn.     

Dynamic stiffness analysis of toroidal shells

The free vibration of a toroidal shell is studied using the dynamic stiffness method. The dynamic stiffness method eliminates both spatial discretization error and mesh generation. Moreover, with a finite number of degrees of freedom, the dynamic stiffness method can predict an infinite number of natural frequencies. The dynamic behavior of the toroidal shell is modeled by DMV (Donnell-Mushtari-Vlasov) linear thin shell theory in the present paper. However, the procedure can be adapted to be used with any other linear thin shell theory without difficulty. Since a close form solution of toroidal shell using DMV theory is not (yet) possible, in order to obtain the desired dynamic stiffness matrix, a finite number of Fourier's series terms are taken in the circumferential direction and the unknown longitudinal displacements are then solved from the reduced governing equations exactly. The solution obtained from the dynamic stiffness method can be regarded as semi-analytical due to the Fourier approximation. With the dynamic stiffness matrices in hands, a toroidal shell with different boundary conditions and connections (to other toroidal shells) can be analyzed. This paper presents the procedure and assumption made in order to obtain the dynamic stiffness matrix of a toroidal shell in harmonic oscillation. Also some numerical examples will be given and discussed.     

Transient dynamic response of composite circular cylindrical shells under radial impulse load and axial compressive loads

Free and forced vibration of composite circular cylindrical shells are investigated based on the first love's approximation theory using the first-order shear deformation shell theory. The boundary conditions (BCs) are considered as clamped-free edges. The dynamic response of the composite shells is studied under transverse impulse and axial compressive loads. The axial compressive load was less than critical buckling loads. The modal technique is used to develop the analytical solution of the composite cylindrical shell. The solution for the shell under the given loading conditions can be found using the convolution integrals. The effect of fiber orientation, axial load, and some of the geometric parameters on the time response of the shells has been shown. The results show that dynamic responses are governed primarily by natural period of the structure. The accuracy of the analysis has been examined by comparing results with those available in the literature and experiments.     

Free vibrations of orthotropic spherical shells

The equations of motion for the free vibrations of orthotropically stiffened open spherical shells are developed and solved using a suitable finite difference model. Variations in both flexural and extensional orthotropic stiffness properties are investigated by means of carefully selected parametric studies. A systematic examination of the contribution to strain energy in each mode, arising from the various components of orthotropic shell stiffness, is shown to assist the interpretation of the effects of orthotropic stiffness changes, and to allow prediction of approximate frequency spectra. Based on the analysis of a related isotropic spherical shell it is shown how a modified form of Rayleigh's method provides approximations of frequency spectra sufficiently accurate to assist the conceptual dynamic design process.     

Local buckling in infinitely,long cylindrical shells subjected uniform external pressure

This paper presents a unique approach to analyze the buckling of an infinitely long cylindrical shell subjected to the external pressure. Buckling is considered to occur locally in the shell, spreading over a certain length along the longitudinal axis of the shell. A plausible function of the flexural displacement is created according to Timonshenko's ring solution of the transverse collapse mode. The governing equations based on Donnell–МУШТАРИ's shell theory are solved using Ritz method and the equilibrium conditions are educed. Numerical computations are performed for cases when shell thickness/radius ratios are 0.1, 0.05 and 0.03. In general, the pressure decreases sharply with a very slight increase of the normalized radial deflection just at the beginning of the initiation, then falls quite slowly till the two opposite points on the inner surface of the shell contact each other. It is found that the buckling pressure of the shell converges to the critical value given by Donnell–МУШТАРИ's shell theory and the span of the buckling mode in the longitudinal axis of the shell is independent of material properties. Solutions given in this paper can be used to address the problem of steady-state buckle propagation in the shells.     

有多条轴对称焊接凹陷的钢圆柱薄壳在内压力和轴压力共同作用下的强度

钢筒仓中圆柱薄壳承受内压力和轴压力共同作用:在轴压和低内压作用下,壳可发生弹性失稳;在轴压和高内压作用下,壳可发生塑性破坏.内压轴压共同作用下的圆柱薄壳对几何缺陷比较敏感:单条轴对称焊接凹陷可使壳的承载力降低;而多条轴对称凹陷由于相邻凹陷的相互作用,可使壳的强度进一步降低;对于小间距的凹陷,这种相互作用更为明显.本文首次对有多条小间距轴对称凹陷轴压圆柱薄壳的整体结构在不同水平内压作用下的强度进行了有限元分析,并将计算结果与欧洲规范EC3的设计曲线进行比较,对EC3的设计曲线提出了修改意见,从而达到安全设计的目的.     

Plastic buckling of complete toroidal shells of elliptical cross-section subjected to internal pressure

The plastic bifurcation buckling pressures of 60 internally-pressurised, perfect, complete toroidal shells of elliptical cross-section are given in the present paper, assuming elastic, perfectly plastic, material behaviour. The shell buckling programs employed in the computations were BOSOR 5 and INCA. Denoting the major-to-minor axis ratio by k, the numerical results show that the plastic buckling pressures are considerably lower than their elastic counterparts in the range 1.25≤k≤1.5 and are approximately equal to them for k=2.5. A limited study of the effects of non-axisymmetric initial geometric imperfections on the buckling pressures of the shells was also carried out using the INCA code. For the four cases studied the post-buckling behaviour was stable. This means that designers can use the buckling pressures given herein for perfect shells as a basis for their initial designs.     

Buckling of intermediate ring supported cylindrical shells under axial compression

This paper is concerned with the elastic buckling of axially compressed, circular cylindrical shells with intermediate ring supports. The simple Timoshenko thin shell theory and the more sophisticated Flügge thin shell theory have been adopted in the modeling of the cylindrical shells. We used these two representative theories to examine the sensitivity of the buckling solutions to the different degree of approximations made in shell theories. By dividing the shell into segments at the locations of the ring supports, the state-space technique is employed to derive the solutions for each shell segment and the domain decomposition method utilized to impose the equilibrium and compatibility conditions at the interfaces of the shell segments. First-known exact buckling factors are obtained for cylindrical shells of one and multiple intermediate ring supports and various combinations of boundary conditions. Comparison studies are carried out against benchmark solutions and independent numerical results from ANSYS and p-Ritz analyses. The influence of the locations of the ring supports on the buckling behaviour of the shells is examined.     

Vibration analysis of toroidal panels using the differential quadrature method

The free vibration characteristics of linear elastic toroidal shell panels are determined. A solution based on the Mushtari–Vlasov–Donnell shell equations is developed using the Differential Quadrature Method. The work represents the first application of this method to problems in shell theory with variable coefficients in the governing equations. Numerical results are calculated using the method, and these are compared with results found using a Fourier series and a finite element solution.