Reinforcement minimization of cylindrical shells

In the search for the absolute minimum amount of reinforcement to be provided in a structure to support predefined loading, most attention has been given to problems for which the relationship between unit cost of material and stress relationships is simple—usually linear. Such an assumption is convenient and reasonably realistic when reinforcement percentages are low. However, for higher reinforcement percentages, and when, as in the case of cylindrical shells, there occur axial as well as radial loads, a more refined analysis procedure is desirable. This paper considers the optimal (absolute minimum reinforcement) strength design of closed cylindrical shells subject to uniform pressure and having rigid ends. Two cases are considered: the shell wall rigidly connected to the ends, and the shell wall hinged at the ends. For convenience, only internal pressure loading is considered in detail, although, using the theory given, results for external pressure cases can readily be obtained. It is assumed that buckling is not a critical factor in the design and that serviceability criteria can be met independently.     

Effect of thickness inhomogeneities in internally pressurized elastic-plastic spherical shells

The behaviour of elastic-plastic spherical shells under internal pressure is investigated numerically for thickness-to-radius ratios ranging from cases of thin shells to very thick shells. The shells under consideration are made of strain-hardening elastic-plastic material with a smooth yield-surface. Attention is restricted to axisymmetric deformations, and results are presented for initial thickness inhomogeneities in various axisymmetric shapes. For smooth thickness-variations in the shape of the critical bifurcation mode, the reduction in maximum pressure is studied together with the distribution of deformations in the final collapse mode. Also, the possibility of flow localization due to more localized, initially thin regions on a spherical shell is investigated.     

S型功能梯度扁球壳非线性屈曲的渐近分析

本文利用渐近迭代法获得了边界弹性支撑的功能梯度扁球壳的非线性屈曲问题的理论解．假设材料组分体积分数沿壳体厚度方向呈sigmoid幂函数变化,边界上考虑一般的弹性支撑约束．基于经典的薄壳理论和几何非线性关系,导出了S型功能梯度扁球壳的非线性屈曲问题的控制方程．采用渐近迭代法通过两次迭代得到了无量纲挠度和均布荷载之间的非线性特征关系．通过与已有有限元方法和其他数值方法的结果对比,验证了本文解的有效性和高精度．同时,通过计算阐述了缺陷因子、材料参数、边界约束系数及特征几何参数对壳体临界屈曲荷载的影响．     

Buckling behaviour of imperfect spherical shells

For the design of spherical shells under external pressure relatively few information can be found in corresponding codes and recommendations, e.g. not at all in the new draft of Eurocode 3 ENV 1993-1-6. Under this aspect, new design rules for these shells were developed, which take into account relevant details like boundary conditions, material properties, and imperfections. They are usually based on a large number of systematic numerical simulations to obtain results describing the load carrying behaviour and imperfection sensitivity of thin spherical shells. In addition, previous theoretical and experimental results are discussed. Based on the results, diagrams and design rules have been developed which might be used for new recommendations in the design concept of the Eurocode.     

Buckling under the external pressure of cylindrical shells with variable thickness

This study focuses on the buckling of cylindrical shells with small thickness variations under external pressure. Asymptotic formulas in terms of the thickness non-uniformity parameter are derived by the combined perturbation and Bubnov–Galerkin methods. In addition to the analytic investigation based on the thin shell theory, a numerical analysis is also performed. Results from these formulas are discussed and compared with those obtained by other authors.     

Nonlinear axisymmetric thermomechanical response of piezo-FGM shallow spherical shells

Thermomechanical instability of shallow spherical shells made of functionally graded material (FGM) and surface-bonded piezoelectric actuators is studied in this paper. The governing equations are based on the classical shell theory of shells and the Sanders nonlinear kinematics equations. It is assumed that the property of the FGMs varies continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituent materials. The constituent materials of the functionally graded shell are assumed to be mixture of ceramic and metal. The analytical solutions are obtained for uniform external pressure, thermal loading, and constant applied actuator voltage.     

Numerical implementation of composite Koiter shells including membrane/bending coupling coefficientsImplémentation numérique des coques composites de Koiter prenant en considération les coefficients de couplage membrane/flexions

We propose a way for determining the generalized coefficients of rigidity – some of which are membrane/bending coupling coefficients – which appear in the deformation energy of the Koiter model of thin shells. This is concerned with a heterogeneous material in the thickness direction. A new program to compute these coefficients is implemented in the finite element code Modulef, in order to simulate problems of thin multilayered shells with linearly elastic anisotropic layers. We propose an example of an inhibited multilayered thin shell, with hyperbolic middle surface, involving a composite material with unidirectional fibres. To cite this article: H. Ranarivelo, C. R. Mecanique 330 (2002) 273–278.     

Imperfection sensitivity of an orthotropic spherical shell under external pressure

In the first part of this paper, rib-stiffened thin-walled spherical shells under external hydrostatic pressure are optimized using classical approximate methods and empirical knock-down-factors. In the second part of the paper, the influence of known imperfections is investigated.The thin-walled spherical shells under external pressure are very sensitive to geometrical imperfections. Hoff recognized that for entire isotropic spherical shells the more likely imperfection will be a local circular dent, which for such shells, can always be considered as an axisymmetric one. Hoff's idea has been further investigated by Koga–Hoff, Galletly et al. These results showed that for a given depth of an imperfection a critical size of the corresponding circular dent exists, giving the minimum for the actual load carrying capacity of the shell.This paper suggests to extend Hoff's theory to isogrid and waffle-grid stiffened spherical shells. The issue of these investigations is a set of knock-down-factors plotted versus imperfection amplitude related to the total thickness of the rib-stiffened (isogrid or waffle-grid) shell. These curves fit reasonably with those established for isotropic shells by Hoff et al. or by Koiter, and enable to estimate the jeopardy of measured actual dents.     

Radial oscillations of nonhomogeneous,thick-walled cylindrical and spherical shells subjected to finite deformations

The infinitesimal breathing motions of long cylindrical tubes and hollow spherical shells of arbitrary wall thickness subjected to a finite deformation field caused by uniform internal and/or external pressures are investigated. A neo-Hookean material with a material constant varying continuously along the radial direction is used. The shell is first subjected to finite static deformations and is then exposed to a secondary dynamic displacement field. Based on the theory of small deformations superposed on large deformations, closed form expressions are obtained for the frequency of small oscillations about the highly prestressed state. Frequency versus initial deformation parameter curves are given for several nohomogeneity functions and for various wall thicknesses.     

Experimental Investigation of the Stress Concentration in Axially Compressed Thick Cylindrical Shells with Rectangular Holes

The stress concentration in cylindrical shells of various thicknesses is studied experimentally. The shells are made of epoxy resin, have a hole, and are subjected to axial compression. The shell thickness and the opening angle are shown to affect the stresses near the corners of the hole. The experimental and theoretical values of the stress concentration factor are compared for a thin shell     

Axisymmetric static and dynamic buckling of hollow microspheres

Syntactic foams are particulate composites that are obtained by dispersing thin hollow inclusions in a matrix material. The wide spectrum of applications of these composites in naval and aerospace structures has fostered a multitude of theoretical, numerical, and experimental studies on the mechanical behavior of syntactic foams and their constituents. In this work, we study static and dynamic axisymmetric buckling of single hollow spherical particles modeled as non-linear thin shells. Specifically, we compare theoretical predictions obtained by using Donnell, Sanders–Koiter, and Teng–Hong non-linear shell theories. The equations of motion of the particle are obtained from Hamilton׳s principle, and the Galerkin method is used to formulate a tractable non-linear system of coupled ordinary differential equations. An iterative solution procedure based on the modified Newton–Raphson method is developed to estimate the critical static load of the microballoon, and alternative methodologies of reduced complexity are further discussed. For dynamic buckling analysis, a Newmark-type integration scheme is integrated with the modified Newton–Raphson method to evaluate the transient response of the shell. Results are specialized to glass particles, and a parametric study is conducted to investigate the effect of microballoon wall thickness on the predictions of the selected non-linear shell theories. Comparison with finite element predictions demonstrates that Sanders–Koiter theory provides accurate estimates of the static critical load for a wide set of particle wall thicknesses. On the other hand, Donnell and Teng–Hong theories should be considered valid only for very thin particles, with the latter theory generally providing better agreement with finite element findings due to its more complete kinematics. In this context, we also demonstrate that a full non-linear analysis is required when considering thicker shells, while simplified treatment can be utilized for thin particles. For dynamic buckling, we confirm the accuracy of Sanders–Koiter theory for all the considered particle thicknesses and of Teng–Hong and Donnell theories for very thin particles.     

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.     

Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment

Based on the nonlinear large deflection theory of cylindrical shells, this paper deals with the nonlinear buckling problem of functionally graded cylindrical shells under torsion load by using the energy method and the nonlinear strain–displacement relations of large deformation. The material properties of the functionally graded shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, on the base of taking the temperature-dependent material properties into account, various effects of external thermal environment on the critical state of the shell are also investigated. Numerical results show various effects of the inhomogeneous parameter, the dimensional parameters and external thermal environment on nonlinear buckling of functionally graded cylindrical shells under torsion. The present theoretical results are verified by those in literature.     

Axially compressed buckling of pressured multiwall carbon nanotubes

This paper studies axially compressed buckling of an individual multiwall carbon nanotube subjected to an internal or external radial pressure. The emphasis is placed on new physical phenomena due to combined axial stress and radial pressure. According to the radius-to-thickness ratio, multiwall carbon nanotubes discussed here are classified into three types: thin, thick, and (almost) solid. The critical axial stress and the buckling mode are calculated for various radial pressures, with detailed comparison to the classic results of singlelayer elastic shells under combined loadings. It is shown that the buckling mode associated with the minimum axial stress is determined uniquely for multiwall carbon nanotubes under combined axial stress and radial pressure, while it is not unique under pure axial stress. In particular, a thin N-wall nanotube (defined by the radius-to-thickness ratio larger than 5) is shown to be approximately equivalent to a single layer elastic shell whose effective bending stiffness and thickness are N times the effective bending stiffness and thickness of singlewall carbon nanotubes. Based on this result, an approximate method is suggested to substitute a multiwall nanotube of many layers by a multilayer elastic shell of fewer layers with acceptable relative errors. Especially, the present results show that the predicted increase of the critical axial stress due to an internal radial pressure appears to be in qualitative agreement with some known results for filled singlewall carbon nanotubes obtained by molecular dynamics simulations.     

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.     

A new displacement-type stability equation and general stability analysis of laminated composite circular conical shells with triangular grid stiffeners

In this paper,based on the theory of Donnell-type shallow shell,a new displacement-type stability equations is first developed for laminated composite circular conical shellswith triangular grid stiffeners by using the variational calculus and generalized smeared-stiffener theory.The most general bending stretching couplings,the effect of eccentricity ofstiffeners are considered.Then,for general stability of composite triangular grid stiffenedconical shells without twist coupling terms,the approximate formulas are obtained forcritical external pressure by using Galerkin‘s procedure.Numerical examples for a certainC/E composite conical shells with inside triangular grid stiffeners are calculated and theresults are in good agreement with the experimental data.Finally,the influence of someparameters on critical external pressure is studied.The stability equations developed andthe formulas for critical external pressure obtained in this paper should be very useful in theastronautical engineering design.     

Optimal design of axisymmetric plastic shallow shells of von Mises material

An optimal design technique developed earlier for axisymmetric plates and circular cylindrical shells is accommodated for shallow spherical shells subjected to uniform transverse pressure. Material of the shells is assumed to be rigid-plastic obeying the von Mises yield condition and the associated deformation law. The post-yield behaviour of the shells is taken into account. The weight minimization is performed under the condition that the maximal deflection of the shell of variable thickness coincides with the deflection of the reference shell of constant thickness. The problem is transformed into a non-linear boundary value problem which is solved numerically.     

Large deflections of deep orthotropic spherical shells under radial concentrated load: asymptotic solution

Nonlinear behavior of deep orthotropic spherical shells under inward radial concentrated load is studied. The singular perturbation method is developed and applied to Reissner’s equations describing axially symmetric large deflections of thin shells of revolution. A small parameter proportional to the ratio of shell thickness to the sphere radius is used. The simple asymptotic formulas describing load–deflection diagrams, maximum bending and membrane stresses of the structure are derived. The influence of boundary conditions on the behavior of the shell by large deflections is considered. Obtained asymptotic solution is in close agreement with the experimental and numerical results and has the same accuracy (in the asymptotic meaning) as the given equations of nonlinear theory of thin shells.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

Stability and asymmetric vibrations of pressurized compressible hyperelastic cylindrical shells

Cylindrical shells of arbitrary wall thickness subjected to uniform radial tensile or compressive dead-load traction are investigated. The material of the shell is assumed to be homogeneous, isotropic, compressible and hyperelastic. The stability of the finitely deformed state and small, free, radial vibrations about this state are investigated using the theory of small deformations superposed on large elastic deformations. The governing equations are solved numerically using both the multiple shooting method and the finite element method. For the finite element method the commercial program ABAQUS is used.¹ The loss of stability occurs when the motions cease to be periodic. The effects of several geometric and material properties on the stress and the deformation fields are investigated.