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.     

Buckling of thin-walled orthotropic cylindrical shells under uniform external pressure.: Application to corrugated tin cans

The general instability of thin-walled orthotropic circular cylindrical shells under external pressure is investigated. The buckling pressure can be predicted with the use of simple analytical formulae derived from an asymptotic analysis of the corresponding eigenvalue problems. The results predicted by these formulae are compared with finite element solutions and the four types of experimental models investigated by Ross (Thin Walled Structures 1996;26(3):179–93). The comparison proved to be accurate enough for practical purpose except for experimental model 1.     

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 of super ellipsoidal shells under uniform pressure

This paper is concerned with the elastic buckling of super ellipsoidal shells under external uniform pressure. The middle surface of a super ellipsoidal shell is defined by the following equation: (x/a)²ⁿ+(y/b)²ⁿ+(z/c)²ⁿ=1, where n is an integer varying from unity to infinity. It is clear from the equation that the range of shell shapes covered sphere (n=1, a=b=c) to cube () and ellipsoid (n=1) to cuboid (n=∞). By adopting a recently proposed solid shell element for the buckling analysis, the critical buckling pressures of thin to thick super ellipsoidal shells are obtained and tabulated for engineers. The shell element allows for the effect of transverse shear deformation which becomes significant in thick shells. Their buckling shapes are also examined. In addition, a simple approximate formula for predicting the critical buckling pressure of thick spherical shells is proposed.     

Design charts for the general instability of ring-stiffened conical shells under external hydrostatic pressure

The paper describes experimental tests carried out on three ring-stiffened circular conical shells that suffered plastic general instability under uniform external hydrostatic pressure. In this mode of failure, the entire ring–shell combination buckles bodily in its flank. The cones were carefully machined from EN1A mild steel to a very high degree of precision.Using the results obtained from these three vessels, together with the results obtained from elsewhere, the paper also provides two-design charts, which are much easier to use than older design charts. The design charts allow the possibility of obtaining a plastic knockdown factor, so that the theoretical elastic buckling pressures for perfect vessels, can be divided by the plastic knockdown factor, to give the predicted buckling pressures. Although similar design charts have been produced in the past, the design charts presented here are based on using the simpler ring-stiffened circular cylinder, which has been made equivalent to the much more complex ring-stiffened circular conical shell. The advantage of using this method is that it is simpler and the design time is reduced by a factor of about 10 with little loss of precision. This method can also be used for the design of full-scale vessels.     

Buckling of super ellipsoidal shells under uniform pressure

This article is concerned with the elastic buckling of super ellipsoidal shells under external uniform pressure. The middle surface of a super ellipsoidal shell is defined by the following equation: (x/a)²ⁿ +(y/b)²ⁿ + (z/c)²ⁿ  = 1 where n is an integer varying from unity to infinity. It is clear from the equation that the range of shell shapes covered sphere (n = 1, a = b = c) to cube (n = ∞, a = b = c) and ellipsoid (n = 1) to cuboid (n = ∞). By adopting a recently proposed solid shell element for the buckling analysis, the critical buckling pressures of thin to thick super ellipsoidal shells are obtained and tabulated for engineers. The shell element allows for the effect of transverse shear deformation, which becomes significant in thick shells. Their buckling shapes are also examined. In addition, a simple approximate formula for predicting the critical buckling pressure of thick spherical shells is proposed.     

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.     

Experiments on conical shell reducers under uniform external pressure

In previous studies, a proliferation of research work has been undertaken on the buckling behavior of shell structures particularly conical shells. Nonetheless, no experimental studies are found in the literature on the buckling of a full and real configured model of a slender shell reducer with two cylindrical end boundaries. To this end, buckling behavior of three conical shell reducers under uniform peripheral pressure was investigated and evaluated experimentally in this paper. In addition, relevant FE simulations as well as theoretical predictions were taken into account to compare buckling load and modes of deformation. Derived results were aimed at generalizing the data for conical reducers in full scale within the range of this study.     

Experiments on dented cylindrical shells under peripheral pressure

In spite of numerous papers in the literature on the buckling behavior of cylindrical shell structures, the effect of local large imperfections caused by physical contacts has not been exhaustively examined yet. To this end, this paper reports on an experimental program on the buckling and post-buckling response of thin cylindrical shells with local dent imperfections under uniform external pressure. The results of this study can be used in practical structures with similar geometric features, i.e. D/t ratio.     

Reduced stiffness buckling of sandwich cylindrical shells under uniform external pressure

A reduced stiffness lower bound method for the buckling of laterally pressure loaded sandwich cylindrical shell is proposed. Also, an attempt is made to assess the validity of the proposed reduced stiffness lower bound with FEM numerical examples. In addition, the proposed method is compared with classical and Plantema's approaches of the buckling of the laterally pressure loaded sandwich cylindrical shell. Comparison of the proposed reduced stiffness lower bound with that obtained from non-linear FEM analysis verifies that it indeed provides a safe lower bound to the buckling of laterally pressure loaded sandwich cylindrical shells. The attractive feature of the proposed reduced stiffness method is that it can be readily used in designing laterally pressure loaded sandwich cylindrical shells without being concerned about geometrical imperfections.     

Short-term behaviour of shallow thin-walled concrete dome under uniform external pressure

Thin-walled spherical concrete shells or domes find widespread use in many applications, including in many iconic engineering structures of historical and religious significance. Despite this, very few experimental investigations have been reported in the open literature of shallow spherical concrete domes which allow for the effects of geometric and material non-linearities and of imperfections to be identified. This information is essential, however, in order to validate sophisticated numerical treatments, as well as to calibrate practical design and construction guidelines and is therefore much-needed. This paper reports an experimental study of a shallow thin-walled concrete dome under short-term loading, without the use of reinforcement in the concrete. The dome is 30 mm thick and has a base diameter of 3 m, being supported on a steel ring beam. The testing of the dome to failure under a uniform external pressure is described in the paper, and it is shown that it failed in a non-axisymmetric buckling mode well before the concrete reached its compressive strength. The failure pressure is compared with the ‘theoretical’ buckling results and the analytical results based on finite element analyses. In particular, this paper presents a comprehensive set of experimental data for the load–displacement and load–strain relationships and their distributions across the spherical dome throughout the loading regime.     

Buckling of cylindrical shells under transverse shear

This work concerns with experimental studies on buckling of thin-walled circular cylindrical shells under transverse shear. The buckling loads are also obtained from finite element models, empirical formulae and codes and are compared. Experiments are conducted on 12 models made of stainless steel by rolling and longitudinal seam welding. In situ initial geometric imperfection surveys are carried out. The tests are conducted with and without axial constraint at the point diametrically opposite the loading. Theoretical analyses are carried out using ABAQUS finite element code. Two finite element models considered are: (i) geometry with real imperfection (FES-I) and (ii) critical mode imperfect geometry (FES-II). In the former, the imperfections are imposed at all nodes and in the latter, the imperfection is imposed by renormalizing the eigen mode, using the maximum measured imperfection. General nonlinear option is employed in both the cases for estimating the buckling load. Galletly and Blachut’s expressions, design guidelines of Japan for LMFBR main vessel expressions (empirical formulae), ASME and aerospace structural design codes are used for comparing with experimental loads.The comparisons of experimental, numerical and analytical buckling loads reveal the following. The numerical results are always higher than the experimental values; the percentage difference depends on the wall thickness. FES-II predicts somewhat a lower load than that of the FES-I. The Japanese guidelines predict the lowest load, which is conservative. Experimental loads are lower than that predicted by both ASME and aerospace structural design codes.     

Elastic buckling of barrelled shell under external pressure

The aim of the paper is to present a procedure for design of a family of shells of revolution of constant mass and, as a next step, of constant volume. As a reference a cylindrical shell is taken into consideration. By decreasing the value of the meridional radius of curvature R₁, which for cylindrical shell equals infinity, barrelled shells are created up to the spherical shell for which both meridional and circumferential radii of curvature are equal (R₁=R₂). A numerical example of using the presented procedure is considered. Then for the family of shells of revolution of constant mass a buckling analysis using FEM method is carried out. Results of the analysis show the relationship between the radius of curvature of the shell R₁ and the critical load p_cr in the case of uniform external pressure.     

An improved formulation for the assessment of the capacity load of circular rings and cylindrical shells under external pressure. Part 1. Analytical derivation

In this two-part set of articles the capacity load of circular rings under external pressure is investigated assuming as a starting point the classic Levy–Timoshenko approach, which is still at the bases of most design codes for cylindrical shells. This first paper presents a perfected analytical formulation of the problem, which accurately accounts for the onset of plasticity and incorporates the effect of various geometrical imperfections. A rigorous non-linear formulation is first derived and, subsequently, an algebraic expression which avoids the recourse to numerical solution strategies is established.     

An experimental study on externally pressurized stiffened and thickened cylindrical shells

It has long been identified that stiffening of steel shells is one of the most effective ways of enhancing the capacity of these structures. Stiffeners largely in the form of welded elements have been employed to strengthen shell structures in which the stiffeners generally cover the whole length of the structure. In this research the effect of partial and full length stiffening of shells was studied in which the stiffeners were attached without welding to avoid the adverse effect of the residual stresses. Furthermore, local thickening of the shells by the same stiffening strips was investigated and the results were evaluated against the plain specimen. The effect of strengthening provided by local thickening was slightly less but comparable to that provided by the stiffeners.     

Stability of relatively deep segments of spherical shells loaded by external pressure

The buckling of shells in the form of spherical segment depends strictly on its rise. Determination of full equilibrium paths for shells of higher rise is very laborious and evokes many numerical problems. Spherical caps loaded by the external pressure and clamped along the base circle are the subject of a detailed analysis. The stability analysis for shells of relative slenderness of interval λ=3.5–12 was performed and is presented in the paper. Three critical points, and namely the primary bifurcation point, the primary higher limit point and the primary lower limit point were basis for the plot of relative critical pressure versus slenderness parameter. This plot has big practical significance. One can read off from it the value of critical pressure being the basis of designing procedure, which takes into account stability criterion. The author's program based on FEM and taking into account all singularities characteristic for nonlinear elastic stability, was used in calculations. The correctness of the approach was verified on the example of spherical segment of slenderness λ=8 solved before by other authors.     

Optimal design of shells against buckling under overall bending and external pressure

In this paper, the problem of optimal design of shells against instability is considered. A thin-walled shell is loaded, in general, by an external pressure and lateral forces causing overall bending moment (which varies along the axis of a shell) and the appropriate shearing force. We look for the shape of meridian as well as the thickness of a shell, which ensure the maximal critical value of the loading parameter. As the equality constraints, the volume of material and the capacity of a shell are considered. The concept of a shell of uniform stability is applied.     

Vibration of a thin-walled carbon fibre corrugated circular cylinder under external water pressure

The paper reports on a theoretical and an experimental investigation carried out on a thin-walled corrugated carbon fibre circular cylinder in air and also under external water pressure. This corrugated circular cylinder was invented by Ross in 1987.The theoretical investigation was carried out using the finite element analysis to model both the structure and the fluid. The theoretical investigation used two different programs, one of which was the giant computer program ANSYS and the other was an in-house program developed by Ross and Little. For the shell structure, the ANSYS program used two different doubly curved thin-walled shell elements, while the in-house program used a simpler axisymmetric thin-walled shell element. This axisymmetric element allowed a sinusoidal variation of the displacements in the circumferential direction, thus, decreasing preparation and computational time. Agreement between the different finite elements was found to be quite good. The investigation also found that there was good correlation between experiment and theory for the in-house software, but was a little disappointing when using ANSYS. Errors may, however, have occurred with the experimental results, as the model was hand-made and neither its geometry nor its material properties were perfect. It was found particularly encouraging for the in-house software to give better results than ANSYS, as the in-house software only took a few hours to set up the computer model, and a few seconds to analyse the vessel, whereas the ANSYS software took several weeks to set up the computer model, and several minutes to analyse the shell. The ANSYS software, however, did have the advantage in producing excellent graphical displays in both the pre-processing and post-processing modes.     

Combined axial and pressure buckling of end supported shells of revolution

A reduced stiffness theoretical analysis of the imperfection sensitive elastic buckling for end supported shells of revolution is extended to the case of arbitrary combinations of axial and radial pressure loading. Depending upon the shell and loading parameters, the potential reductions in load capacity due to imperfections are shown to involve two distinct forms of post-buckling loss of stiffness. Lower bounds in each of these regimes are provided by appropriate reduced stiffness models, and shown by comparisons with available test data to be reliable even for relatively perfect test models. By attributing reductions in load carrying capacity to weakened end support conditions, it is suggested that past interpretations of these tests may have underestimated the deleterious effects of initial imperfections.