Experimental studying of buckling of stringer cylindrical shells under axial compression

Results of tests on axial compression of small-sized quality steel cylinder shells strengthened by 24 and 36 longitudinal thin-walled stiffeners are presented. The shell length was varied. Shells both with inside and outside stiffening were tested at simply supported and clamped edges. The shell carrying capacity that was governed in the tests by overall buckling in the elastic range was compared with the estimated critical loads based on structural-orthotropic theory. The satisfactory quantitative correlation has been received only for the long simply supported shells with 36 inner stiffeners, which demonstrated insignificant effect of local undulation that preceded overall deflections. The experimental and the theoretical results differed significantly (twice as much) when the actual mechanism of lateral deflection caused by the intensive local undulation differed from the adopted model.     

Elastic buckling of an axially compressed cylindrical panel with three edges simply supported and one edge free

The paper is devoted to a circular cylindrical panel with three edges simply supported and one edge free. This panel is axially compressed by a load uniformly distributed at both ends. The Donnell equations for linear buckling of shells are assumed and the adequate boundary conditions are defined. The deflection function and the force function–the Airy function are formulated with respect to the boundary conditions. The Donnell equations are reduced to a generalized eigenvalue problem with the use of Galerkin method. The results of numerical investigation are presented in figures.     

An analytical method for the buckling analysis of cylindrical shells with non-axisymmetric thickness variations under external pressure

This article presents an analytical method for the buckling analysis of laterally pressured cylindrical shells with non-axisymmetric thickness variations. The previous results for thickness variations under external pressure are reviewed firstly. Then, a general analytical method that combines the perturbation method and Fourier series expansion is developed to derive buckling load formulas, which is in terms of thickness variation parameter up to arbitrary order. A classical non-axisymmetric thickness variation is discussed in detail by the presented analytical method. When non-axisymmetric modal thickness variation becomes axisymmetric, the buckling loads degenerate to the known results. Furthermore, the influence of circumferential modal thickness variation with mode corresponding to twice the circumferential buckling mode on the buckling of laterally pressured cylindrical shells is analytically investigated and the results show a great agreement with previous numerical ones by Gusic et al. Thus we confirm the presented method. In addition to theoretical analysis, calculations and comparisons are also performed. The general analytical method presented in the article can be utilized to determine the buckling loads of shells with general thickness variations.     

Numerical and experimental investigations on buckling of steel cylindrical shells with elliptical cutout subject to axial compression

The effect of cutouts on load-bearing capacity and buckling behavior of cylindrical shells is an essential consideration in their design.In this paper, simulation and analysis of thin steel cylindrical shells of various lengths and diameters with elliptical cutouts have been studied using the finite element method and the effect of cutout position and the length-to-diameter (L/D) and diameter-to-thickness (D/t) ratios on the buckling and post-buckling behavior of cylindrical shells has been investigated. For several specimens, buckling test was performed using an INSTRON 8802 servo hydraulic machine and the results of experimental tests were compared to numerical results. A very good correlation was observed between numerical simulation and experimental results. Finally, based on the experimental and numerical results, formulas are presented for finding the buckling load of these structures.     

Plastic buckling of axially compressed circular cylindrical shells

For elastic-plastic cylindrical shells with initial axisymmetric imperfections bifurcation into a non-axisymmetric shape is analysed. The shell material is represented by a phenomenological plasticity theory that accounts for the formation of a vertex on subsequent yield surfaces. The influence of various geometric and material parameters is investigated for a wide range of radius-to-thickness ratios. It is shown that for the thicker shells bifurcation generally occurs beyond the maximum axial compressive load. A few analyses for shells with additional non-axisymmetric imperfections show the unstable post-bifurcation behaviour and the sensitivity to imperfections of more general shapes.     

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.     

Sample size dependent probabilistic design of axially compressed cylindrical shells

The fact that the buckling load of cylindrical shells depends on imperfections raised the idea of applying probabilistic design methods for these structures. Whether a probabilistically motivated design load may be regarded as a representative for a type of shell depends, among others, on the samples that build the data basis for the probabilistic methodology. In the current paper a methodology is presented that takes into account the samples size within a fast probabilistic design. The method presented leads to a lower design load, as the sample size decreases. The methodology is applied to a set of 33 beer cans, which have been measured and tested at the Delft University of Technology. The 33 cans are subdivided into groups which are then analyzed probabilistically in order to observe how the probabilistically motivated design load varies for different samples sizes. The results indicate that the method presented provides a useful tool for designers to ensure that a small data basis does not yield a unsafe probabilistic design.     

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.     

Vibration, buckling and dynamic stability of cracked cylindrical shells

The presence of cracks in a structure can considerably affect its behaviour. This paper presents a finite element study on the vibration, buckling and dynamic stability behaviour of a cracked cylindrical shell with fixed supports and subject to an in plane compressive/tensile periodic edge load. The effects of crack length and orientation are analysed. Under tension load, the results show that the frequency of the shell initially increases with the load, but then decreases as the load further increases leading to buckling due to tension load. The size and the orientation of the crack and the loading parameter can all have a significant effect on the dynamic stability behaviour of the shell under both compressive and tensile loading. The effects of these parameters are discussed in detail.     

Axially compressed cylindrical shells with local settlement

The design of cylindrical metal silos and tanks is often controlled by considerations of buckling under axial compression. Whilst the effects of geometric imperfections on the buckling strength have been extensively explored, few studies have explored the effects of defects in the boundary conditions and the effects of residual stresses have received even less attention.This paper investigates the initiation and development of imperfections caused by local differential settlement at the supported base and their effect on the elastic buckling of a thin cylindrical shell under axial compression. The shells were treated as initially perfect with perfect support, but developing geometric imperfections and residual stresses as a consequence of local displacement at the supported edge and with residual stresses consistent with the induced geometric imperfections.The results raise interesting questions concerning the criteria of failure and appropriate tolerance measurements for constructed cylindrical shells.     

Re-interpreting simultaneous buckling modes of axially compressed isotropic conical shells

Elastic stability of shell structures under certain loading conditions is characterized by a dramatically unstable postbuckling behavior. The presence of simultaneous ‘competing’ buckling modes (corresponding to the same critical buckling load) is understood to be largely responsible for such behavior. In this paper, within the framework of linear bifurcation eigenvalue analysis and Donnell shallow shell theory, the presence of simultaneous buckling modes in axially compressed isotropic cones is determined using the semi-analytical method of Galerkin. The results are presented in the plane of the dimensionless reciprocal meridional and circumferential buckling half wavelengths, and are compared with the locus of simultaneous buckling modes of axially compressed cylinders, described by the so-called Koiter circle. By using an optimizing procedure, it is shown that the cluster of simultaneous buckling modes in cones is well described by the Koiter circle of an equivalent cylinder with appropriate length and radius. Such optimizing values of length and radius allow us to gain some insight into the simplifying treatment of the buckling of cones through the concept of equivalent cylinder.     

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.     

Vibration correlation technique for the estimation of real boundary conditions and buckling load of unstiffened plates and cylindrical shells

Nondestructive experimental methods to calculate the buckling load of imperfection sensitive thin-walled structures are one of the most important techniques for the validation of new structures and numerical models of large scale aerospace structures. Vibration correlation technique (VCT) allows determining equivalent boundary conditions and buckling load for several types of structures without reaching the instability point. VCT is already widely used for beam structures, but the technique is still under development for thin-walled plates and shells. This paper intends to explain the capabilities and current limitations of this technique applied to two types of structures under buckling conditions: flat plates and cylindrical shells prone to buckling. Experimental results for a flat plate and a cylindrical shell are presented together with reliable finite element models for both cases. Preliminary results showed that the VCT can be used to determine the realistic boundary conditions of a given test setup, providing valuable data for the estimation of the buckling load by finite element models. Also numerical results herein presented show that VCT can be used as a nondestructive tool to estimate the buckling load of unstiffened cylindrical shells. Experimental tests are currently under development to further validate the approach proposed herein.     

Non-probabilistic reliability-based design optimization of stiffened shells under buckling constraint

Stiffened shells are affected by numerous uncertainty factors, such as the variations of manufacturing tolerance, material properties and environment aspects, etc. Due to the expensive experimental cost of stiffened shell, only a limited quantity of statistics about its uncertainty factors are available. In this case, an unjustified assumption of probabilistic model may result in misleading outcomes of reliability-based design optimization (RBDO), and the non-probabilistic convex method is a promising alternative. In this study, a hybrid non-probabilistic convex method based on single-ellipsoid convex model is proposed to minimize the weight of stiffened shells with uncertain-but-bounded variations, where the adaptive chaos control (ACC) method is applied to ensure the robustness of search process of single-ellipsoid convex model, and the particle swarm optimization (PSO) algorithm together with smeared stiffener model are utilized to guarantee the global optimum design. A 3 m-diameter benchmark example illustrates the advantage of the proposed method over RBDO and deterministic optimum methods for stiffened shell with uncertain-but-bounded variations.     

Buckling of cylindrical shells with longitudinal joints under external pressure

In this article, the bucking of cylindrical shells with longitudinal joint has been investigated through the experimental and numerical analysis. It was clarified that the buckling behavior of cylindrical shells with longitudinal joints under lateral external pressure is not only related to its dimension, but also longitudinal joint and an imperfection. The buckling of cylindrical shells with rigid joint buckles only once and in multi-lobe buckling, whereas one with flexible joints buckles twice and firstly in single-lobe buckling in the vicinity of the joint, secondly in multi-lobe buckling in remaining un-deformed area. And the more flexible the longitudinal joint, the lower the critical pressure, with respect to the same dimension of jointed cylindrical shells and imperfection condition. Moreover the numerical analysis approaches were also presented and verified, by which the imperfection can greatly enlarge the effect of joint on buckling has been demonstrated.     

Buckling strength of the cylindrical shell and tank subjected to axially compressive loads

This paper aims to develop practical design equations and charts estimating the buckling strength of the cylindrical shell and tank subjected to axially compressive loads. Both geometrically perfect and imperfect shells and tanks are studied. Numerical analysis is used to evaluate buckling strength. The modeling method, appropriate element type and necessary number of elements to use in numerical analysis are recommended. According to the results of the parametric study of the perfect shell, the buckling strength decreases significantly as the diameter-to-thickness ratio increases, while it decreases slightly as the height-to-diameter ratio increases. These results are different from those in the case of columns. The buckling strength of the perfect tank placed on an extremely soft foundation and a stiff foundation increases by up to 1.6% and 5.6%, respectively, compared with that of the perfect shell. The buckling strength of the shell and tank decreases significantly as the amplitude of initial geometric imperfection increases. Convenient and sufficiently accurate design equations and charts used for estimating buckling strength are provided.     

Buckling of cylindrical shells with stepwise variable wall thickness under uniform external pressure

Cylindrical shells of stepwise variable wall thickness are widely used for cylindrical containment structures, such as vertical-axis tanks and silos. The thickness is changed because the stress resultants are much larger at lower levels. The increase of internal pressure and axial compression in the shell is addressed by increasing the wall thickness. Each shell is built up from a number of individual strakes of constant thickness. The thickness of the wall increases progressively from top to bottom.Whilst the buckling behaviour of a uniform thickness cylinder under external pressure is well defined, that of a stepped wall cylinder is difficult to determine. In the European standard EN 1993-1-6 (2007) and Recommendations ECCS EDR5 (2008), stepped wall cylinders under circumferential compression are transformed, first into a three-stage cylinder and thence into an equivalent uniform thickness cylinder. This two-stage process leads to a complicated calculation that depends on a chart that requires interpolation and is not easy to use, where the mechanics is somewhat hidden, which cannot be programmed into a spreadsheet leading to difficulties in the practical design of silos and tanks.This paper introduces a new “weighted smeared wall method”, which is proposed as a simpler method to deal with stepped-wall cylinders of short or medium length with any thickness variation. Buckling predictions are made for a wide range of geometries of silos and tanks (unanchored and anchored) using the new hand calculation method and compared both with accurate predictions from finite element calculations using ABAQUS and with the current Eurocode rules. The comparison shows that the weighted smeared wall method provides a close approximation to the external buckling strength of stepped wall cylinders for a wide range of short and medium-length shells, is easily programmed into a spreadsheet and is informative to the designer.     

Reliable and accurate prediction of the experimental buckling of thin-walled cylindrical shell under an axial load

Reliable and accurate method of the experimental buckling prediction of thin-walled cylindrical shell under an eccentric load is presented. The experimental arrangement and specimens are discussed in detail, including the measurement of the geometric imperfections of the specimen's surface using a coordinate measuring machine. Different FE models, in terms of complexity, are used to simulate the experiment arrangement in an attempt to get a good agreement with the experimental buckling loads and study the effect of measured initial geometric imperfections, load eccentricity, load eccentricity position along the shell's circumferential direction and different experimental arrangement that influence the boundary conditions. It has been demonstrated that FE models with simplified rigid support conditions overestimate the prediction of the experimental buckling load even though these models included the effects of the measured initial geometric imperfections and load eccentricity. By contrast, FE models with realistically modeled support conditions achieved the best result. The average deviation −1.59% from the experimental buckling loads was achieved using the FE model simulating the mounting devices as elastic bodies and with surface-to-surface contact interaction behavior on the support. The presented work also demonstrated the strong influence of the eccentric load position along the imperfect shell's circumferential direction on the buckling of the thin-walled shell.     

Collapse of axially compressed cylindrical shells with random imperfections

The establishment of an International Imperfection Data Bank is discussed. Characteristic initial imperfection distributions associated with different fabrication techniques are shown.

Using a first-order, second-moment analysis, a stochastic method is presented, whereby the stability of isotropic, orthotropic and anisotropic nominally circular cylindrical shells under axial compression, external pressure and/or torsion possessing general nonsymmetric random initial imperfections can be evaluated. Results of measurements of initial imperfections are represented in Fourier series and the Fourier coefficients are used to construct the second-order statistical properties needed. The computation of the buckling loads is done with standard computer codes and includes a rigorous satisfaction of the specified boundary conditions.

It is shown that the proposed stochastic approach provides a means to combine the latest theoretical findings with the practical experiences spanning about 75 years in an optimal manner via the advanced computational facilities currently available.