Buckling Of Cylindrical Shells Research Articles

A study of elastic-plastic buckling of cylindrical shells under torsion

The elastic-plastic buckling of cylindrical shells under torsion is analysed with a deep thick-shell model under various boundary conditions. The word ‘deep’ means that in the general equations of equilibrium the three non-linear terms that involve the torsional force are all retained for the buckling analysis. In the Donnell-type shallow-shell theory, however, only one of such terms is retained. The word ‘thick’ means that in calculating strains and stress resultants the factor (1+ z/ R) is retained. This factor results from the trapezoid-like shape of the cross-section and is usually neglected in the thin-shell theory. For boundary conditions, not only the conventional geometrical boundary conditions, which are in terms of displacements and rotations, but also the mechanical boundary conditions, which are in terms of forces and moments, are considered. The numerical results of examples assess the effect of the additional non-linear terms, the effect of the factor (1+ z/ R), and the effect of the mechanical boundary conditions.

TRANSIENT DYNAMIC RESPONSE ANALYSIS OF ORTHOTROPIC CIRCULAR CYLINDRICAL SHELL UNDER EXTERNAL HYDROSTATIC PRESSURE

Following Flügge's exact derivation for the buckling of cylindrical shells, the equations of motion for transient dynamic loading of orthotropic circular cylindrical shells under external hydrostatic pressure have been formulated. The normal mode theory is used to provide transient dynamic response for the equations of motion. The effect of shell's parameters, external hydrostatic pressure and material properties on the shell response has been studied in detail. A part of tables and figures are given in this paper.

Cylindrical shell buckling: a characterization of localization and periodicity

A hypothesis for the prediction of the circumferential wavenumber of buckling of the thin axially-compressed cylindrical shell is presented, based on the addition of a length effect to the classical (Koiter circle) critical load result. Checks against physical and numerical experiments, both by direct comparison of wavenumbers and via a scaling law, provide strong evidence that the hypothesis is correct.

Influence of stress waves on the dynamic progressive and dynamic plastic buckling of cylindrical shells

A new concept is presented for the dynamic elastic–plastic axisymmetric buckling of circular cylindrical shells under axial impact. The phenomena of dynamic plastic buckling (when the entire length of the shell wrinkles before the development of large radial displacements) and dynamic progressive buckling (when the shell folds form sequentially) are analysed from the viewpoint of stress wave propagation resulting from an axial impact. The conditions for the development of dynamic plastic buckling are obtained. A numerical analysis of the buckling phenomena reveals that the material properties together with the geometrical characteristics of the shell determine the particular type of response for high velocity impacts. It is concluded that shells made of strain rate insensitive materials can respond either by dynamic plastic buckling or dynamic progressive buckling, depending on the inertia properties of the shell, while those shells made of strain rate sensitive materials respond always by dynamic progressive buckling. It is shown that the prediction for the peak load, which can develop in a shell for a high velocity impact, depends on the particular yield criterion used in the analysis.

Hydrostatic buckling of shells with various boundary conditions

Eigenvalue buckling of cylindrical shells with various boundary conditions under hydrostatic load is examined, using an energy method. Results are compared to known solutions, where these solutions exist. It is found that, for shells of intermediate length, buckling loads for different end conditions may be determined by applying a simple, scalar multiplier to the pin-ended case. This does not apply to long shells, where the circumferential wave number n≤3. For n=2, the ring equation may be applied to all cases, as the boundary conditions no longer influence the solution. It is seen for the case of a shell with one end pinned and the other end free that the buckling solution collapses to the long shell solution, for geometries of practical interest. The effect of radial elastic restraint at the open end is also examined, as an intermediate case between pinned and free ends. The work has application to the design of suction caissons, where cylinder dimensions are usually in the range of intermediate length shells.

Finite Element simulation of dynamic buckling of cylinders subjected to periodic shear

This paper deals with the buckling of cylindrical shells under a dynamic shear load. The aim of our study is to compare static buckling load and buckling load during a sweep frequency excitation. First, we describe the special experimental device and the two Finite Element codes used in this study. In a second part static tests and corresponding Finite Element calculations are presented in order to have a reference buckling load and to understand the effect of initial imperfections. Then, a vibration analysis is performed in order to investigate the effect of geometric imperfection and a preload. In the last part, we discuss dynamic results. When we reach the first eigen frequency, the buckling load drops and the buckling deformations increase due to a parametric resonance. There is a coupling between vibration and buckling modes.

The influence of circumferential thickness variations on the buckling of cylindrical shells under external pressure

Imperfection sensitivity of thin shells of constant thickness was widely studied during the last 50 years (J.G. Teng, Appl. Mech. Rev. 1996:49(4)). Recently, Koiter et al. (Int. J. Solids Struct. 1994:31(6):797–805) investigated the influence of axisymmetric modal thickness variation on the buckling load of an axially compressed shell. In this paper, the influence of harmonic thickness variation in the circumferential direction on thin cylindrical shell buckling under external pressure is analysed by means of FE bifurcation analysis. Two different FE codes were used, one with quasi-axisymmetrical multimodal Fourier analysis (A. Combescure, Etude de la stabilité des coques minces sous chargements complexes dans INCA. Rapport DMT/94-460, in French, 1994; Modélisation des coques axisymétriques avec imperfection quelconque: L’élément COMI (in French). Rapport DMT/97-189, 1997; Schauder B. Flambage des Coques Cylindriques en Fléxion. Thèse de doctorat, INSA de Lyon, in French, 1997), and the second one with 3D shell elements. A new quasi-axisymmetric element (COMI) is presented. It uses Fourier series expansion in the circumferential direction coupled with a full numerical integration around the circumference, which allows any variation of thickness or of initial imperfections.

On the buckling of cylindrical shells with through cracks under axial load

Presence of cracks or similar imperfections can considerably reduce the buckling load of a shell structure. In this paper, the buckling of cylindrical shells with through cracks has been studied. A general finite element model has been proposed, verified and applied to some novel cracked shell buckling problems for which documented results are not available. A special purpose program has been developed for generating finite elements models of cylindrical shells with cracks of varying length and orientation. The buckling behavior of cracked cylinders in tension and compression has been studied. The results of the analysis are presented in parametric form when it seems to be appropriate. Sensitivity of the buckling load to the crack length and orientation has also been investigated.

Elastic buckling analysis of ring-stiffened cylindrical shells under general pressure loading via the Ritz method

This paper presents the Ritz method for the elastic buckling analysis of shells with ring-stiffeners under general pressure loading. The stiffeners may be of any cross-sectional shape and arbitrarily distributed along the shell length. Using polynomial functions multiplied by boundary equations raised to appropriate powers as the Ritz functions, the method can accommodate any combination of end conditions. As far as it is known, the Ritz method has not been automated in this way for the buckling of ring-stiffened shells. By formulating in a nondimensional form, generic buckling solutions for shells with various end conditions, stiffener distributions and under various pressure distributions, were presented. These new buckling solutions should serve as useful reference sources for checking the validity and accuracy of other numerical methods and software for buckling of cylindrical shells. This paper also shows that the appropriate distribution of ring stiffeners can lead to a significant increase in the buckling capacity over that of a stiffened shell with evenly spaced and identical ring stiffeners.

Influence of localized imperfections on the buckling of cylindrical shells under axial compression

The influence of localized imperfections on the buckling of a long cylindrical shell under axial compression is analysed by using a double scale analysis including interaction modes. This leads to a system of coupled complex non-linear differential equations with discontinuous derivatives. We propose analytical formulas to predict the reduction of the critical buckling load.

Computation of Homoclinic Orbits in Partial Differential Equations: An Application to Cylindrical Shell Buckling

This paper concerns numerical computation of localized solutions in partial differential equations (PDEs) on unbounded domains. The application is to the von Karman--Donnell equations, a coupled system of elliptic equations describing the equilibrium of an axially compressed cylindrical shell. Earlier work suggests that axially localized solutions are the physically preferred buckling modes. Hence the problem is posed on a cylindrical domain that is unbounded axially and solutions are sought which are homoclinic in the axial variable and periodic circumferentially. The numerical method is based on a Galerkin spectral decomposition circumferentially to pose ordinary differential equations (ODEs) in the unbounded coordinate. Methods for location and parameter of homoclinic solutions of ODEs are then adapted, making special use of the symmetry and reversibility properties of solutions observed experimentally. Thus a formally well-posed problem is reduced to a rotational subgroup circumferentially and posed over a truncation of the half-interval axially. The method for location of solutions makes use of asymptotic approximations where available. More generally, an adaptation of the successive continuation shooting method is used in the lowest possible number of circumferential modes, followed by additional homotopies to add more modes by in the strength of nonlinear mode-interaction terms. The method is illustrated step by step to produce a variety of homoclinic solutions of the equations and compute their bifurcation diagrams as the loading parameter varies. Good agreement with experimental data is found. All computations are performed using AUTO. The techniques illustrated here for the von Karman--Donnell equations are applicable to a wider class of PDEs.

Effects of openings of the buckling of cylindrical shells subjected to axial compression

Experimental and numerical methods are used to study the stability problem of cylindrical shells with cut-outs. The paper presents parametric research of the shape (square, rectangular, circular), the dimensions (axial and circumferential sizes, diameter) of the hole. The effect of the location and the number of the holes are also studied. The analysis indicates that the critical load is sensitive to the opening angle or circumferential size of the hole. The function (critical load-opening angle) is linear for large openings and independent of the geometrical imperfections of the shell. However for small openings, it is necessary to take into account the coupling between the initial geometrical imperfections and the openings. The linear approach does not fit because of the importance of the evolution of the displacements near the openings. These results will be used for the development of European rules.

On plastic buckling of cylindrical shells struck axially with a mass

Buckling of elastic-plastic cylindrical shells axially struck with a mass is discussed. The effect of stress wave fronts travelling along the shell is taken into account. This allows us to cast some more light on the mechanism of progressive buckling. Only axisymmetric buckling modes are considered. The analysis is based on the Kirchhoff-Love hypotheses. As a constitutive law, the rate independent elasto-plastic relations with linear strain hardening and von Mises yield condition are adopted. A method for calculating bifurcation times and buckling modes is presented. Numerical examples are given.

Elastic, Plastic, and Creep Buckling of Imperfect Cylinders Under Mechanical and Thermal Loading

Based on the concept of secant and tangent modulus, the nonlinear equilibrium and stability equations of thin cylindrical shells under axial compression, external pressure, or external fluid pressure are derived. The resulting equations are applicable to shells without length limitation as the rotations and transverse shears are included in the derivations. The reduction factors for plastic and creep buckling are then obtained. A procedure for determining secant and tangent modulus in the very general case of elastic, plastic, or creep stress in the presence of temperature gradient is proposed. Then, using Donnell’s nonlinear theory of shells, the effect of initial imperfection on the strength of the elastic shell is discussed. The foregoing results are extended to plastic and creep buckling of cylindrical shells of arbitrary length and temperature gradient. Some design curves are proposed using the obtained equations. Finally, the present results are compared with available results in the literature and the accuracy of the method is examined.

Elastic pulse buckling of cylindrical shells under radial impulsive loading

When a cylindrical shell is subjected to dynamic implusive loading in radial direction and the ratio of radius-to-thickness exceeds a special value, the cylindrical shell will produce elastic dynamic buckling. This paper which is based on the results of some relative experiments, assumes deformation mode and utilizes Lagrange method to analyse the elastic pluse buckling of a thin cylindrical shell with a finite length under a cosine impulse. The dynamic buckling equations are derived and solved by numerical method. The results of calculation are compared with some relative calculation results which were obtained by other authors.

On internal stability problems in structured materials

Internal instability of structured materials and elements is occasionally observed and has been verified by experiments. The specific problems in this contribution treated analytically and numerically consist of internal instability phenomena of prestressed elastic materials and the so-called ‘membrane buckling’ of thin elastic plates and shells. A stability theory taking into consideration independent local rotations is outlined; this theory is then used to treat the membrane buckling of cylindrical shells and the in-plane buckling of rectangular plates. It is shown that under certain circumstances, the in-plane buckling mode may precede the out-of-plane buckling deformation. To simulate the internal stability phenomena numerically, a number of discrete models of structured materials are considered; based on these models numerical Finite Element (FE) buckling analyses are carried out, including linear analyses for the membrane buckling of a circular cylindrical shell model and the in-plane buckling analysis of a flat plate. The FE simulations effectively afford buckling loads and buckling modes.

Buckling of circular cylindrical shells subject to uniform lateral pressure

A theoretical investigation of the buckling of cylindrical shells under uniform external lateral pressure loading is presented, based on Flügge's stability equations in coupled form; these lead to great accuracy. The numerical process gives the buckling pressure for a selected circumferential buckling mode, material, geometry and boundary conditions. The influence of 17 different homogeneous boundary conditions placed on the displacements u, v and w, and on the slope d w/d x is investigated. A wide range of geometries (0.5 ≤ L/ R ≤ 5 and 300 ≤ R/ h ≤ 3000) is considered. Comparisons are made with some analyses in the literature. It is also found that, contrary to the widespread understanding that the critical pressure for a free cylinder is the same as for a ring, the present model obtains a slightly lower buckling pressure which depends on the length.

Dynamic Pulse Buckling of Cylindrical Shells Subjected to External Impulsive Loading

The dynamic plastic buckling of cylindrical shells (rings) subjected to general initial impulsive velocity and subjected to impulsive loading is studied based on the energy criterion. A simple analysis method is given and the formulas of critical mode numbers ncr and critical impulsive velocity νcr are obtained.

An evaluation method for elastic-plastic buckling of cylindrical shells under shear forces

Since vessels of fast breeder reactors are relatively thin-walled, the prevention of buckling against seismic loading is one of the key issues in their structural design. Buckling of cylindrical vessels under shear forces occurs with a shear and/or bending mode and in the elastic-plastic region. In this paper, we propose a buckling strength equation for cylindrical vessels under static shear loads, which is developed on the basis of theoretical considerations for plasticity and shape imperfections. The effects caused by nonlinear distribution of stresses along the circumference in the elastic-plastic region are also considered as correction factors. The equation is validated by the existing and the present buckling test data, as well as FEM analyses. The safety factors to be employed in the design are proposed by evaluating the reliability of the proposed equation in the light of available test data.

Elastic buckling of cylindrical shells with elastic cores—I. Analysis

Thin walled cylindrical shell structures are widespread in nature; examples include plant stems, porcupine quills and hedgehog spines. All have an outer shell of almost fully dense material supported by a low density, cellular core. In nature, all are loaded in some combination of axial compression and bending; failure is typically by buckling. Natural structures are often optimized. Here we have analysed the elastic buckling of a thin cylindrical shell supported by an elastic core to show that this structural configuration achieves significant weight saving over a hollow cylinder. Biomimicking of natural cylindrical shell structures may offer the potential to increase the mechanical efficiency of engineering cylindrical shells. The results of the analysis are compared with data in the following, companion paper.