Buckling Of Thin Cylindrical Shells Research Articles

Application of artificial neural network to predict buckling load of thin cylindrical shells under axial compression

The buckling of thin cylindrical shells under axial compression is long-standing problem due to significant difference between experimental and theoretical buckling load predicted by classical buckling theory. The knockdown factors predicted by present design recommendations are very conservative and predictions are less accurate. Artificial Neural Networks (ANN) are used in this study to accurately predict buckling load using experimental data from 38 previous studies. The buckling load was predicted using nine input parameters. The experimental data was divided into two sets having similar distributions of input parameters: training dataset (90%) and validation dataset (10%). The buckling loads predicted by ANN are in good agreement with experimental buckling loads and predictions are within ±10% with few exceptions. The specimens with parameters falling in the range of input parameters were predicted well by ANN, and accuracy of the prediction depends on number of similar parameters in a specific range used for training. The predictions by five design recommendations including NASA and EC-3 were compared with experimental buckling load, the percentage errors of predictions compared to experimental data for more than 50% specimens were within ±50%. The trained ANN models predict buckling loads with higher accuracy compared to design recommendations and can be used for practical designs.

Shell buckling, without ‘imperfections’

The buckling behaviour of thin shell structures under load has been a persistent challenge to engineering designers and researchers over many decades. In this article I consider two unusual experimental studies on the buckling of thin-walled elastic cylindrical shells, each of which sheds intriguing light on the buckling phenomena. The classical theory of buckling of thin cylindrical shells under axial compression predicts that the buckling stress will be proportional to t/ R– the ratio of thickness to radius – other things being equal. But collected results of experimental studies from many laboratories, when plotted on log–log scales, show clearly that the buckling stress is actually proportional to ( t/ R)1.5, with the measured buckling stresses being scattered through a factor of about 4 for shells with R/ t > 200. Such scatter is commonly judged to be in accord with Koiter’s theory of ‘imperfection sensitivity’. But that theory lays no claim to an understanding of the empirical 1.5-power law. I claim that a key to this situation is the experimental performance of some small-scale open-topped silicone rubber shells, buckling under their own weight, which clearly demonstrates a 1.5-power law, but with very little scatter. The buckling mode of these shells involves almost entirely inextensional deformation, with a single small dimple growing near the base, separated from the rest of the shell by a narrow boundary layer that accounts for almost all of the dimple’s elastic strain energy. A straightforward, simple analysis of the mechanics of the dimple is consistent with the experimental 1.5-power law. As noted above, experimental buckling loads of shells that are closed at both ends also show the empirical 1.5-power law, but now with significant statistical scatter. A second aim of the paper is to throw light on that phenomenon. I venture to attribute it to the effect of the boundary conditions of the shell. I adduce support for this view from experimental observations on the buckling of a shell with special, frictional end-fittings. That feature produces significantly higher collapse loads, and with much smaller scatter, than for corresponding shells with fixed boundaries; and it permits striking pre-buckled deformations to occur, of a kind not previously noted. It will be appreciated that neither of the two parts of this article depends on the widely accepted theory of imperfection-sensitivity; hence my choice of title. It is a pleasure for me to submit this article to a special publication in honour of Michael Rotter, with whom I have discussed matters of this sort through three decades.

Analytical research on dynamic buckling of thin cylindrical shells with thickness variation under axial pressure

This paper focuses on the dynamic buckling of thin cylindrical shells with arbitrary axisymmetric thickness variation under time dependent axial pressure. Based on the derivation of stability and compatibility equations of variable thickness cylindrical shells under dynamic external pressure by Aksogan and Sofiyev, the corresponding stability and compatibility equations of thin cylindrical shells with arbitrary axisymmetric thickness variation under dynamic axial pressure are obtained and expressed in nondimensional form. Combining the small parameter perturbation method, Fourier series expansion and the Sachenkov–Baktieva method, analytical formulas of the critical buckling load of thin cylindrical shells with arbitrary axisymmetric thickness variation under axial pressure that varies as a power function of time are obtained. Two cases of thickness variation are introduced to research the critical dynamic buckling load with the present formulas. Effects of thickness variation parameters and loading speed of dynamic axial pressure on the critical buckling load are discussed. The method is also applied to determine the critical dynamic buckling load of thin cylindrical shells with a classical cosine form thickness variation. Results revel that the thickness variation and pressure parameters play a major role in dictating the buckling capacity of thin cylindrical shells under dynamic axial pressure.

A generalized Ritz-based method for nonlinear buckling of thin cylindrical shells

A generalized Ritz based approach that accounts for pre-buckling bending deformations as well as geometric non-linearity to find the buckling load for thin cylindrical shells under uniform axial compression is proposed. The approach accommodates within a unified framework, based on a shell and spring restraint system, the entire range of boundary conditions: ranging from free boundaries to compliant boundaries. The classical kinematic boundary conditions are recovered as limiting cases. The method involves choice of a single trial function that is invariant with respect to boundary conditions. The proposed approach is validated using existing analytical results, where available, and numerical solutions, otherwise.

Experimental investigation on the buckling of thin cylindrical shells with two-stepwise variable thickness under external pressure

The buckling capacity of the cylindrical shells depends on two geometric ratios of L/R and R/t. However the effect of thickness variation on the behavior of the shells is more complicated and the buckling strength of them is sensitive to the magnitude and shape of geometric imperfections. In this paper the effects of thickness variation and geometric imperfections on the buckling and postbuckling behavior of cylindrical shells are experimentally investigated. The obtained results are presented under the effect of uniform lateral pressure. It is found in this investigation that the buckling mode can be generated in the whole length of the shell, if the thickness variation is low.

Experimental evidence of imperfection influence on the buckling of thin cylindrical shell under uniform external pressure

The load carrying behaviour of cylindrical thin-walled shell structures under pressure load is strongly dependent on the nature and magnitude of the imperfections invariably caused by various manufacturing processes. The present paper examines instabilities of long homogeneous and isotropic thin elastic tubes, characterized by geometric imperfections like eccentricity or ovality, on the buckling behaviour in conditions for which, at present, a complete theoretical analysis was not found in literature. Moreover, the additional aspect of the influence of the welded joint geometry and position is investigated over a wide range of diameter to thickness ratio, extending the findings of previous works. The problem of buckling for variable load conditions is relevant in the context of NPP applications as, for instance the optimisation of an integrated and innovative LWR Steam Generator (SG) tubes, according to the updated ASME rules. To the purpose, at Pisa University a rather intense research activity is being carried out on the buckling of thin walled metal specimens in the dimensional range suitable for the above mentioned application. Therefore a test equipment (with the necessary data acquisition facility), suitable for carrying out test series on this issue, as well as numerical models implemented on the MARC FEM code, were set up. The experiments were conducted on test specimens with different materials, e.g. A-316 ASTM (with and without seam weld) and Inconel 690 TT, as well as different loading conditions (lateral and hydrostatic external pressure). A validation of numerical evaluations by comparison with test results is also performed. A good agreement has been observed between the experimental data and the elasto-plastic finite element analyses results, highlighting also the different influence of the mentioned imperfections on the buckling loads.

Singular perturbation analysis of the buckling of circular cylindrical shells

This paper deals with the buckling of thin cylindrical shells with very large Batdorf parameters under external pressure. We first perform a simplified analysis from which we obtain explicit formulae for the critical load. An asymptotic analysis is carried out with a view to determining the effects of the boundary conditions on the critical loads and buckling shapes. The inverse of the Batdorf parameter is the convenient small parameter of the analysis. Among the whole set of boundary conditions, the axial boundary conditions are found to be have crucially important at the first order. This analysis also shows the existence of boundary layers in which the remaining boundary conditions are only significant at the second order, at the very most.

Dynamic buckling of thin cylindrical shells under axial impact

The dynamic buckling of thin isotropic thermoviscoplastic cylindrical shells compressed with a uniform axial velocity prescribed at the end faces is investigated analytically and numerically. In the first part of the paper, the stressed/deformed state of a shell is assumed to have buckled if infinitesimal perturbations superimposed upon it grow. Cubic algebraic equations are derived for both the initial growth rate of the perturbation and its wavenumber. The wavenumber corresponding to the maximum initial growth rate of a perturbation introduced at an axial strain of 0.1 is taken to determine the buckling mode. The computed buckling modes are found to match well with those listed in the available experimental data. A thermoviscoplastic constitutive relation is used to delineate the influence of material parameters on the buckling behavior. In the second part of the paper, the finite element method is used to analyze the collapse of an imperfect circular cylindrical tube with axial velocity prescribed at one of its flat end faces with the other end face kept fixed. The influence of initial randomly located imperfections on the buckling behavior is investigated and discussed.

Twin-characteristic-parameters analysis for elastic dynamic buckling of thin cylindrical shells under axial step loading

The relations of the critical stress and transverse inertia effect to the loading duration are investigated for the dynamic buckling of thin cylindrical shells under axial step loading. The critical stress and the inertial exponent are treated as the two characteristic parameters. The criterion of energy conservation is used to derive the supplementary restraint condition for buckling deformations at compression wave front. By use of the Galerkin method, an algebraic eigenvalue problem for the two characteristic parameters is derived from the governing equations and boundary conditions. The solution of the eigenvalue problem, which satisfies the supplementary restraint condition, gives the values of the critical stress and the inertial exponent for the dynamic buckling. The relation of critical stress to loading duration, predicted by the theoretical analysis, is in reasonable agreement with the experimental results.

The Buckling of Thin Cylindrical Shells under Axial Compression

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The Buckling of Thin Cylindrical Shells under Axial Compression

The Buckling of Thin Cylindrical Shells under Axial Compression

Computational p -element method on the effects of thickness and length on self-weight buckling of thin cylindrical shells via various shell theories

This paper is concerned with the development of a global p-element method for the analysis of self-weight buckling of thin cylindrical shells. Such buckling problems occur when cylindrical shells are subject to high-g acceleration, for instance the launching of rockets and missiles under high propulsive power. The cylindrical shells may have any combination of free, simply supported and clamped ends. A p-element computational method has been developed based on various thin shell theories including Donnell, Sanders and Goldenveizer-Novozhilov models. The strain energy for the global element during buckling is formulated and an eigenvalue equation is derived. Unlike the conventional buckling problem where the eigenvalue is directly solved, a pre-determined buckling parameter is fixed at the outset for a geometric-dependent stiffness and a recursive numerical procedure is developed to compute the effect of critical buckling length. The critical buckling length is found to be proportional to thickness to a power of approximately 0.9. The effects of shell thickness and length on buckling parameter are also investigated. Comparison of results from various shell theories indicates solutions of the Sanders and Goldenveizer-Novozhilov shell theories are in excellent agreement while the Donnel shell theory is good for buckling of short cylindrical shells.

Buckling of thin cylindrical shells under axial compression

Simple experiments on self-weight buckling of thin, open-top, fixed-base, small-scale silicone rubber cylindrical shells are presented in this article. The buckling heights were found to be proportional to thickness raised to the power of approximately 1.5, compared to 1.0 as in the classical theory. A non-linear finite-element analysis of self-weight buckling showed that there is a ‘post-buckling plateau’ load corresponding to the experimental buckling loads. Moreover, the results of the present experiments showed very little ‘scatter’ in the buckling heights, compared to the large scatter in the experimental data from the literature on the buckling of thin cylindrical shells under axial compressive load (which also have buckling stress proportional to thickness raised to the power of approximately 1.5), although there were measurable imperfections in terms of thickness variations. These observations somehow defy the accepted hypothesis of ‘imperfection-sensitivity’ in shell buckling. The most obvious explanation of the difference is that the open-topped shells of the present study are statically determinate, whereas the usual closed- ended shells of tests reported in the literature are statically indeterminate; the possibility of high initial stresses may explain the scatter in the experimental observations in the literature.

Paradoxical buckling behaviour of a thin cylindrical shell under axial compression

The widely accepted theory of buckling of thin cylindrical shells under axial compressive loading emphasises the sensitivity of the buckling load to the presence of initial imperfections. These imperfections are conventionally taken to be minor geometric perturbations of a shell which is initially stress-free. The original aim of the present study was to investigate the effect on the buckling load of imperfections in the form of local initial stress, which are probably more typical of practice than purely geometric ones. Experiments were performed on a vertical “melinex” cylinder of diameter ∼0.9 m and height ∼0.7 m, with radius /thickness ratio ∼1800. The upper and lower edges of the cylinder were clamped to end discs by means of circumferential belts — an arrangement that allowed states of self-stress to be introduced to the shell readily by means of local “uplift” at the base. The upper disc was made sufficiently heavy to buckle the shell, and it was supported by a vertical central rod under screw control. Many buckling tests were performed. Surprisingly, the buckling loads were generally at the upper end of the range of fractions of the classical buckling load that have been found in many previous experimental studies. Even when the local uplift at the base caused a local “dimple” to be formed before the shell was loaded, the buckling load was relatively high. A surface-scanning apparatus allowed the geometric form of the shell to be monitored, and the progress of such a dimple to be followed; and it was found that a dimple generally grew in size and migrated in a stable fashion up the shell as the load increased, until a point was reached when unstable buckling occurred. These unexpected and paradoxical features of the behaviour of the experimental shell may be attributed to the particular boundary conditions of the shell, which provide in effect statically determinate support conditions. This study raises some new issues in the field of shell buckling, both for the understanding of buckling phenomena and for the rational design of shells by engineers against buckling.

THERMOELASTIC BUCKLING OF THIN CYLINDRICAL SHELLS BASED ON IMPROVED STABILITY EQUATIONS

The nonlinear strain-displacement relations in general cylindrical coordinates are simplified by Sander's assumptions for the cylindrical shells and substituted into the total potential energy function for thermoelastic loading. The Euler equations are then applied to the functional of energy, and the general thermoelastic equations of nonlinear shell theory are obtained and compared with the Donnel equations. An improvement is observed in the resulting equations as no length limitations are imposed on a thin cylindrical shell. The stability equations are then derived through the second variation of potential energy, and the same improvements are extended to the resulting thermoelastic stability equations. Based on the improved equilibrium and stability equations, the magnitude of thennoelastic buckling of thin cylindrical shells under different thermal loadings is obtained. The results are extended to short and long thin cylindrical shells.

Buckling of Anodic Films on Aluminum: Determination of Plastic Deformation Energy of Coatings

The mechanical instability of oxide layers on anodized aluminum generates a regular arrangement of straight lines parallel to the cylindrical electrode axis. This is analyzed using Euler's equation for buckling of thin cylindrical shells under compressive forces, due to volume expansion concurrent with metal‐oxide formation. The coating critical stress is determined as well as the energy dissipated by plastic deformation of the anodic coating. Most of the energy associated with compressive stress of the oxide film is dissipated by plastic deformation.

A boundary layer theory for the buckling of thin cylindrical shells under external pressure

Based on the boundary layer theory for the buckling of thin elastic shells suggested in ref. [14]. the buckling and postbuckling behavior of clamped circular cylindrical shells under lateral or hydrostatic pressure is studied applying singular perturbation method by taking deflection as perturbation parameter. The effects of initial geometric imperfection are also considered. Some numerical results for perfect and imperfect cylindrical shells are given. The analytical results obtained are compared with some experimental data in detail, which shows that both are rather coincident.

Dynamic buckling of cylindrical shells from oscillating waves following axial impact

An experimental and theoretical investigation of dynamic buckling of thin cylindrical shells under oscillating stress waves following axial impact shows that hoop breathing response induced by the Poisson effect plays a strong role in initiating buckling, and that the stress oscillations allow buckling initially localized near the impacted end to propagate up the shell toward the free end. As a result of this energy spreading, the total compressive impulse of multipulse loading can be substantially larger than the critical impulse for a single pulse. The radial deformation of breathing provides a deterministic mechanism for initiation of symmetric buckling, which dominates early response for superclassical impact loads. For subclassical impact loads, the circumferential stress resultant from the Poisson effect excites hyperbolic growth of a large group of asymmetric modes; a continuous range of impact loads extending to values below the static classical buckling load therefore result in dynamic buckling within the scope of classical buckling theory with inertia terms added.

開口部補強円筒殻の軸圧縮座屈に関する一実験結果

The experimental results on the buckling of thin cylindrical shells with a reinforced circular hole under axial compressive load are presented and discussed. The shell around a hole is reinforced by thickening the annular area. The variation of buckling load with the thickness or the radius of the reinforcing area is measured experimentally for both cases of central loading and eccentric loading.It is found that the reinforcement by thickening the wall thickness is more effective compared with by broading the reinforcing radius in order to recover the buckling value of monocoque cylindrical shell with a circular hole. The specimens of cylindrical shells and reinforcement materials used for these tests were made of Lumirror Polyester Sheet.

Effect of a circular hole on the buckling of cylindrical shells loaded by axial compression.

An experimental and analytical investigation of the effect of a circular hole on the buckling of thin cylindrical shells under axial compression was carried out. The experimental results were obtained from tests performed on seamless electroformed copper shells and Mylar shells with a lap joint seam. These results indicated that the character of the shell buckling was dependent on a parameter which is proportional to the hole radius divided by the square root of the product of the shell radius and thickness. For small values of this parameter, there was no apparent effect of the hole on the buckling load. For slightly larger values of the parameter, the shells still buckled into a general collapse configuration, but the buckling loads were sharply reduced as the parameter increased. For still larger values of the parameter, the buckling loads were further reduced, and the shells buckled into a stable local buckling configuration.