Open Access

ASTER Shell: a simple concept to significantly increase the plastic buckling strength of short cylinders subjected to combined external pressure and axial compression

Advanced Modeling and Simulation in Engineering Sciences20152:26

DOI: 10.1186/s40323-015-0047-3

Received: 11 June 2015

Accepted: 1 October 2015

Published: 13 October 2015

Abstract

This paper proposes a new type of shell, similar to a cylindrical shell, which has significantly higher buckling strength when subjected to an arbitrary combination of uniform external pressure and axial compression. The underlying principle consists in a slight modification of the perfect cylinder in order to counteract the natural deformations which get larger and larger and lead to the collapse of the structure. Such shells are called ASTER shells. The concept has been validated through experiments, then analyzed numerically in order to explain what was observed and to propose avenues for improvements. The shells were made of electrodeposited nickel. The material was characterized. The chosen specimens were carefully measured to characterize their thickness and initial imperfections, then tested under the various types of loading. Then they were analyzed using finite elements. Thus, we were able to compare the finite element predictions with the experimental results. This comparison shows that plasticity has a decisive influence on the critical load and that linear elastic dimensioning leads to a serious overestimation of the experimental critical load. Contrary to perfect cylindrical shells, this type of shell is not significantly affected by geometric imperfections: this is another advantage of this type of design. Finally, we propose a numerical analysis in order to optimize the choice of the shape and propose shapes which resist buckling much better than a smooth cylinder when subjected to uniform external pressure, axial compression or a combination of both.

Keywords

Plasticity External pressure Axial compression Imperfections Self-stiffening Buckling Finite elements Tests Plasticity External pressure Axial compression

Background

The buckling of thin cylinders under axial compression or external pressure has been studied for many years [120] and is now well-understood. Geometric imperfections play a major role in a shell’s resistance to axial compression and, generally, a lesser role in its resistance to external pressure [2137]. Fewer works can be found in the literature regarding the effect of plasticity on buckling [3845]: plasticity plays an important role in the case of thicker cylinders and, coupled with initial imperfections, is often responsible for the significant decrease in the critical loads observed experimentally. This type of structure is also sensitive to the care with which the boundary conditions are applied [19, 46]. The effect of the model used for the boundary condition on the critical load is also well-known, but it is more difficult to determine, in each practical case, the exact boundary condition which should be chosen. The literature on the buckling strength under combined axial compression and external pressure is less extensive [47]. The objective of this work is to propose a metallic shell which can be manufactured easily and which, on the one hand, avoids the high sensitivity to imperfections of ordinary shells under axial compression while, on the other hand, having a much better resistance to external pressure. This leads to an alternative to the shell defined in [48, 49] by optimizing a NURBS surface. NURBS shells have better resistance to axial compression, but their resistance to external pressure has not been assessed. There is no theoretical study or experiment available for such shells when the material has the characteristic elastic-plastic behavior of metals.

The paper is organized as follows: the first section describes the ASTER shell concept, the experiments and the results. The second section describes the numerical prediction of these experiments and compares the results with the experimental results. The third section explains how these shells can be optimized to achieve a high resistance to external pressure and a good resistance to axial compression.

ASTER shells: concept and experiments

This section presents the experimental buckling obtained with three types of shells subjected to uniform external pressure, axial compression or a combination of the two. The shells were either smooth cylinders (the reference shells) or ASTER shells.

The ASTER shell concept

The ASTER shell concept derives from the observation of the evolution of the surface of cylindrical shells under external pressure loading and axial compression during lab tests. It has been known for many years that geometric imperfections have a decisive effect on the buckling load. However, it is imperative to know the type of imperfection, how it originates and then evolves during the loading, and its effect on the critical load in order to determine a “shape”, based on the cylindrical geometry of the shell, which would be less sensitive or more effective. The quality of the experiments on the buckling of cylindrical shells carried out at INSA Lyon [5058] has led to the ability to obtain highly-reproducible experimental critical loads of over 85 % of the theoretical critical load under axial compression and 95 % of the theoretical critical load under external pressure. It is possible to greatly reduce the presence of geometric manufacturing imperfections in the shell, for example by using the technique of metal electrodeposition on a machined support which, afterwards, can be discarded without affecting the shell. However, the development of geometric imperfections due to the application of boundary conditions is unavoidable. These systematic imperfections have a typical shape and are identical to those which result from the application of a circumferential membrane stress. They lead to the development of a modal circumferential geometry which is distributed more or less homogeneously over the surface of the shell. In the best-case scenario, the maximum amplitude of the imperfections thus generated is 10–20 % of the shell’s thickness. Subsequently, these initial imperfections cause the whole shell to buckle according to a perfect shell buckling circumferential mode (n) which corresponds to the mode under external pressure. The amplitude of these imperfections increases with the solicitation of this mode (n). The underlying idea of the ASTER shell, which has the same characteristic dimension (mean radius R, thickness h, height L) as the cylindrical reference shell, consists, based on this cylindrical geometry, in developing a deliberately “undulated” shape by means of p small vaults. One chooses p = 2n. These 2n small vaults all have a concavity which points inward in order to prevent the inception and then the amplification of the modal geometry. The choice of these outward vaults was an a priori choice driven by the observation that the curvature of an “incoming” imperfection (pointing toward the center of the shell) tends to be more active than the curvature of an “outgoing” imperfection. Figure 1 illustrates the ASTER shell concept.
Fig. 1

The ASTER shell concept

The ASTER shell thus defined is designated as VM 2n because it consists of 2n vaults. Thus, if the perfect cylindrical shell buckles along mode 10, the corresponding ASTER shell is denoted ASTER VM20 because it has 20 vaults. A vault consists in a circular arc of radius r defined in relation to the height d of the vault with respect to the perfect cylinder. Figure 2 defines the parameters of the vault. This choice of a shape was made in order to facilitate the manufacturing of such a shell.
Fig. 2

ASTER shell: definition of r and d

Manufacturing of the specimens

The cylinders to be tested were produced by electrodeposition of electrolytic nickel onto a machined aluminum die (for the cylindrical shells) or an epoxy resin die in the desired shape (for the ASTER shell).

The die was cast into a mold which consisted of two machined duralumin half shells, then unmolded (Fig. 3).
Fig. 3

The die manufacturing process

Thus, it is possible to manufacture many dies of complex shapes from a single mold. Then, the external surface of the die is made conductive by spraying a thin layer of graphite. The die thus coated is immersed into a nickel sulfamate bath and subjected to a current. The immersion time and current intensity are the parameters which condition the thickness of the deposit. To ensure uniform thickness, the die is rotated in the bath. In addition, masks whose geometry is adjusted by trial-and-error are applied to complete the process. In order to extract the shell to be tested from the die without contact or mechanical actions which could create mechanical imperfections, the aluminum die is dissolved in caustic soda and the resin die is carved out from the inside and separated from the specimen. Finally, the ends are cut to shape by electroerosion to guarantee the quality of the supports and the perpendicularity with the axis of the shell. The thickness of the shell is quasi-constant, except that it is reduced by one-tenth along the vertical creases. It was assumed to be constant and equal to the average value for the purpose of the simulations. By design, these shells are very stiff with respect to circumferential modes 8–13: consequently, the roundness imperfections were negligible (less than one-hundredth of the thickness). This procedure led to the manufacturing of “laboratory quality” shells.

The test rig

The shells were subjected to external pressure, axial compression or a combination of both. The experimental behavior was monitored throughout the experiment by a computer which recorded and processed the desired readings. The axial compressive load and the external pressure were displacement-controlled. At times, the load was sustained in order to allow the radial and axial deformations to be measured.

The global or partial geometry of the shell was measured in its initial state, at several loading levels and, finally, in its post-critical state. In order to do that, a contactless sensor moving circumferentially and axially was used to measure the evolution of the radius of a point of the internal surface. In addition, the normal displacement of a point of the shell was monitored continuously throughout the loading. The chosen point was that which experienced the largest “incoming” displacement during a preloading step at about 15 % of the expected critical pressure. This measurement was used to analyze the loading—radial displacement function and to anticipate the imminent occurrence of buckling.

The geometry of the cylindrical shells

The smooth shells studied were cylinders of average radius \(R=75\; \text{mm}\) and nominal length \(L=120\;\text{mm}\). Their average nominal thickness was about \(150\;\upmu \text{m}\). The ASTER shells were all generated from the same reference smooth shell (same base radius R, same length L and same theoretical thickness h). Each specimen was measured. The thickness was measured by ultrasound at 30 regularly-spaced points on the surface. The thicknesses were quasi-uniform and so, from now on, we will refer only to the average thickness. The initial and post-critical geometries were obtained through the acquisition of 5000 points on the surface, taken along 34 regularly-spaced parallels. The variations in radius were measured along three parallels at one-degree intervals along the circumference, for each specimen and several loading levels (which remained constant during the measurement). Then, the variations of the average radius were decomposed using Fourier series expansion. The maximum amplitude of the initial shape imperfection of all the smooth shells (measured on Fourier modes 9, 10, 11, 12, 13, which were close to buckling mode 11) did not exceed 20 % of the thickness. A typical post-critical geometry of a smooth shell under external pressure is shown in Fig. 4.
Fig. 4

A typical post-buckling geometry

Material properties

The material properties were measured for each specimen. In order to do that, a tensile test piece was cut out of scrap material from a specimen and used to determine the Young’s modulus and nonlinear properties (see Fig. 5).
Fig. 5

A typical stress–strain curve

One can observe that the material ceases to be linear after a very small strain (0.001), after which the tangent modulus diminishes very progressively. This type of behavior is characteristic of many austenitic steels. We will see further on that taking this effect into account is very important for an accurate prediction of buckling.

Experimental results

A number of shells were tested under external pressure alone and under axial compression alone. Then, we tested their buckling strength under combined loading. We will present the experimental results obtained with cylindrical shells and then with two types of ASTER shells. These shells are usually characterized by their Batdorf parameter Z (\(Z = \frac{L^{2}}{Rh} \sqrt{1-\nu ^{2}}\), where \(\nu\) denotes Poisson’s coefficient).

For each shell studied, let us define a reference shell which is the smooth cylindrical shell of radius \(R=75\;\text{mm}\), length \(L=120\;\text{mm}\) and thickness h equal to the thickness measured on the shell being considered. This shell has the same Young’s modulus as that measured on the specimen, is built-in at the base and has all its degrees of freedom (translation and rotation) along the upper circle set to zero, except for the axial displacement. We carried out a Fourier series analysis of this reference cylinder using truncated conical finite elements [58] (a mesh of 40 truncated conical elements is more than sufficient) in order to determine two reference critical loads: the critical elastic buckling pressure \(P_{E}\) and the elastic buckling load under uniform axial compression \(F_{E}\). These two quantities were used systematically to normalize all the results.

The case of cylindrical shells

The characteristics of each shell and the test results for the various smooth shells subjected to the simple load cases are given in Table 1.
Table 1

Characteristics of the cylinders and experimental buckling loads

External pressure

A1

A2

A3

A4

h (\(\upmu \text{m}\))

150

150

154

158

R/h

500

500

487

475

Z

1220

1220

1190

1160

E (GPa)

162

155

158

152

\(P_{exp}\) Buckling pressure (MPa)

0.021

0.019

0.020

0.0223

\(P_{E}\) (MPa)

0.0213

0.0204

0.0222

0.0227

\(\frac{P_{exp}}{P_{E}}\)

0.986

0.931

0.900

0.982

Fourier mode n

11

10

11

11

Axial compression

A5

A6

h (\(\mu m\))

150

148

R/h

500

500

Z

1220

1933

E (GPa)

163

155

\(F_{exp}\) Buckling load (N)

11,000

9275

\(F_{E}\) (N)

14,193

13,125

\(\frac{F_{exp}}{F_{E}}\)

0.775

0.707

Fourier mode n

11

12

The case of ASTER VM22 shells

The experimental observations showed that the critical circumferential mode was mainly mode 11. Therefore, we manufactured a VM22-type shell consisting of 22 vaults with their concavity pointed inward. The depth d of the vaults was equal to 2.32 mm, which corresponds to a radius of curvature r equal to 20 mm. Such an amplitude leads to a “hump” amplitude between 4 and 5 times the shell’s nominal thickness. Table 2 below summarizes the various tests and gives the buckling loads of the elementary loading cases. The loading cases are normalized to the linear elastic buckling load of the smooth cylinder with the same radius, height, thickness and material. We found that the ASTER VM22 shells were four times more resistant to external pressure than the corresponding smooth shells and that the theoretical critical load of the reference cylinder associated with the shell being studied could be reached thanks to the vaults. Thus, these vaults have a beneficial effect for all the loading cases and are very effective against external pressure.

Remark the critical loads of shells B3 and B4 under external loading are marked with an * because these two shells were subjected to about ten external pressure cycles at two-thirds of the expected critical load prior to continuing all the way through failure by buckling. This pre-cycling reduces the critical load without cycling by about 25 %. We will return to this point in the discussion of the numerical simulation and explain this observation.
Table 2

ASTER VM22 shells: characteristics and experimental buckling loads

External pressure

B1

B2

B3*

B4*

h (\(\mu m\))

152

155

150

150

R/h

493

484

500

500

E (GPa)

147

161

164

167.5

\(P_{exp}\) Buckling pressure (MPa)

0.091

0.092

0.070

0.072

\(P_{E}\) (MPa)

0.0190

0.0230

0.0216

0.0220

\(\frac{P_{exp}}{P_{E}}\)

4.57

4.00

3.24

3.27

Axial compression

B5

B6

h (\(\upmu \text{m}\))

153

150

R/h

490

500

E (GPa)

162

163

\(F_{exp}\) Buckling Load (N)

14,400

13,200

\(F_{E}\) (N)

14,200

14,067

\(\frac{F_{exp}}{F_{E}}\)

1.01

0.94

Figure 6 shows a post-critical geometry in the case of external pressure loading alone.
Fig. 6

A typical VM22 post-buckled state

The case of ASTER VM14 shells

In order to study the effect of the choice of the ASTER geometry on the increase in the critical load, we also manufactured ASTER VM14 shells with p = 14 vaults along the circumference. The depth d of these vaults was 2.35 mm, which corresponds to a radius of curvature r equal to 35 mm. These shells were tested only under simple loading. Table 3 summarizes the experimental results. One can immediately see that this type of vault is less effective under external pressure and cuts the critical load under axial compression in half.
Table 3

ASTER VM14 shells: characteristics and experimental buckling loads

External pressure

C1

C2

h (\(\upmu \text{m}\))

149

147

R/h

503

510

E (GPa)

154

143

\(P_{exp}\) Buckling pressure (MPa)

0.054

0.045

\(P_{E}\) (MPa)

0.020

0.018

\(\frac{P_{exp}}{P_{E}}\)

2.7

2.6

Axial compression

C3

C4

h (\(\mu m\))

150

148

R/h

500

507

E (GPa)

151

156.5

\(F_{exp}\) Buckling Load (N)

6200

5100

\(F_{E}\) (N)

12937

13,012

\(\frac{F_{exp}}{F_{E}}\)

0.48

0.39

A post-critical geometry under external pressure is shown in Fig. 7.
Fig. 7

A typical VM14 post-buckled state

The effects of loading combinations on buckling

This section presents the effects of an interaction between loading cases on buckling for the first two shell types. In the case of combined loads, we increased the applied pressure and the axial compression load simultaneously. These two loading cases were apportioned to the reference linear buckling loads for the cylinder being considered. The results for smooth cylinders are given in Table 4.
Table 4

Buckling under combined loading: characteristics of the cylinders and experimental buckling loads

Specimen

h (μm)

E (GPa)

\(P_{exp}\) (MPa)

\(P_{exp}\) (N)

\(P_E\) (MPa)

\(F_E\) (N)

Load ratio \(\frac{F P_{E}}{F_{E} P}\)

A1

150

163

0.021

0.000

0.0213

14,111

0

A7

142

150

0.015

2000

0.0173

11,718

4

A8

145

163

0.016

4500

0.0197

13,276

8

A9

138

159

0.0131

3900

0.0171

11,737

8

A10

148

155

0.012

6700

0.0197

13,150

16

A11

139

156

0.0075

6800

0.0171

11,681

26

A12

148

155

0.0078

7200

0.0197

13,150

26

A13

137

160

0.0045

7800

0.0169

11,643

50

A5

150

163

0.000

11,000

0.0214

14,193

\(\infty\)

The results for VM22 shells are given in Table 5.
Table 5

Buckling under combined loading: experimental buckling loads for VM22 ASTER shells

Specimen

h (μm)

E (GPa)

\(P_{exp}\) (MPa)

\(F_{exp}\) (N)

\(P_E\) (MPa)

\(F_E\) (N)

Load ratio \(\frac{F}{F_{E}}\frac{P_{E}}{P}\)

B2

150

161

0.0920

0

0.023

14,004

0

B7

142

160

0.0543

2500

0.0184

12,314

0.7

B8

140

148

0.0380

4500

0.0164

11,053

1.4

B9

140

160

0.0326

7900

0.0177

11,951

1.8

B10

152

163

0.0205

14,840

0.0221

14,337

3.5

B5

153

162

0.000

14,200

0.0224

14,200

\(\infty\)

These results will be interpreted later with the presentation of the calculation results.

Discussion of the experimental results

The sum of these experimental results shows the interest of ASTER type shells in increasing buckling strength. One can see that the choice of the number of vaults plays an important role in the quality of the buckling strength. This type of shell, which can be manufactured easily, enables the resistance to buckling under external pressure to be quadrupled and makes the shell quasi-insensitive to imperfections under axial compression. This result is verified experimentally provided one chooses a number of vaults p equal to twice the critical number of circumferential waves in buckling (n). We will now attempt to understand why by means of a numerical simulation of these tests. Then, we will try to find a way to improve this design even further: in order to do that, we will attempt to predict the critical loads for a range of values of p through calculations. Thus, we will seek the existence of an optimum p for this type of shell subjected to combined load cases.

The finite element model

In this section, we undertake to model these tests using finite elements and compare the experimental results with these numerical simulations.

Perfect cylindrical shells and ASTER shells

First, we carried out two-node axisymmetric or quasi-axisymmetric finite element calculations of the smooth shells (using COMU axisymmetric elements with non-axisymmetric modal imperfections), then 3D calculations of the ASTER shells using DKT elements. In the latter case, a half-shell was meshed for each calculation. The base was built-in and the top of the cylinder was left free to dilate vertically. The remaining degrees of freedom of the circle were fixed. All the calculations were performed using the nonlinear finite element code Stanlax [58].

The types of analyses performed

For each cylinder, we first performed an elastic buckling analysis. Then we carried out different analyses depending on the case. For the smooth cylinders, we performed incremental, geometrically nonlinear axisymmetric analyses with elastic or elastic-plastic behavior. We tested the possible loss of stability of the equilibrium thus obtained through a plastic bifurcation analysis on the Fourier modes [58]. For the 3D shells, we simply carried out a linear stability analysis followed by incremental, geometrically nonlinear, elastic and elastic-plastic analyses. We also tested the stability after each step of the nonlinear calculations. For all these calculations, we performed a convergence analysis. We found that 40 axisymmetric elements are more than sufficient for the critical loads to converge, and that 40 elements along the axis and 20 elements along the circumference are needed for the 3D calculation of each ASTER vault (which comes to 1600 DKT elements for each vault). The choice of the model for the nonlinear traction curve was important. We chose the same yield stress \(\sigma _{y}=150 \text{MPa}\) for all the calculations. The elastic-plastic behavior was given by the reference curve of Fig. 5. The initial plastic strain was calculated by dividing the yield stress by the Young’s modulus. The base of the cylinder was built-in for all the calculations. All the degrees of freedom of the upper circle were fixed, except for the axial displacement. This choice is consistent with the control of the experiments which nullified the end load associated with the pressure.

Results of the calculations and comparison with the experimental results

This section presents the results of the simulations and compares them to the experimental results.

Cylindrical shells

Let \(P_{exp}\) and \(F_{exp}\) be respectively the experimental buckling pressure in MPa and the experimental axial buckling load in N. \(P_{E}\) and \(F_{E}\) denote respectively the calculated linear elastic buckling pressure (taking into account the following forces and the axial compression load leading to Euler buckling). \(P_{NL}\) and \(P_{NLP}\) denote respectively the nonlinear elastic and nonlinear plastic buckling pressure. F is the axial compression load. The plastic buckling loads were calculated using the tangent modulus theory.

The case of perfect shells

This section concerns simple external pressure or axial compression loading. The results of the calculations for perfect structures are compared to the experimental results for smooth cylinders A1–A6 in Table 6.
Table 6

Perfect cylinders: experimental and calculated buckling loads

External pressure

A1

A2

A3

A4

\(P_{exp}\)

0.021

0.019

0.020

0.0223

Fourier mode \(n_{exp}\)

11

10

11

11

\(\frac{P_{exp}}{P_{E}}\)

0.986

0.931

0.900

0.982

Fourier mode \(n_{E}\)

11

11

11

11

\(\frac{P_{exp}}{P_{NL}}\)

0.991

0.920

0.900

0.982

\(\frac{P_{exp}}{P_{NLP}}\)

0.991

0.920

0.900

0.982

Axial compression

A5

A6

\(F_{exp}\)

11,000

9275

Fourier mode \(n_{exp}\)

11

12

\(\frac{F_{exp}}{F_{E}}\)

0.775

0.707

Fourier mode \(n_{E}\)

2–15

2–15

\(\frac{F_{exp}}{F_{NL}}\)

0.648

0.708

\(\frac{F_{exp}}{P_{NLP}}\)

1.01

0.919

First, let us analyze the buckling pressure predictions. The linear buckling calculations predicted the experimental critical loads with less than 10 % error in all cases. In two cases, the experimental load was predicted within less than 3 %. Buckling occurred in the elastic domain. There was no nonlinear pre-buckling effect.

The buckling calculations under axial compression clearly establish that plasticity is responsible for the experimental load being 30 % less than the Euler load. In the case of pure axial compression, we found the same results as in the literature: a large number of Fourier modes correspond to the same critical load. The buckling modes for Fourier modes 0 and 11 are given in Fig. 8. Under uniform axial compression, the cylinder always buckles on mode 11. The corresponding shape is shown in Fig. 9.
Fig. 8

Buckling modes under axial compression

Fig. 9

Buckling modes of a cylinder under external pressure

Influence of initial imperfections

One can also consider the influence of initial imperfections on the critical load. In order to do that, we added an initial imperfection to the perfect geometry in the shape of the elastic buckling mode with amplitude \(\delta\). Three cases were calculated (\(\frac{\delta }{h}=0.2, 0.1\) and 0.01). The largest amplitude corresponds to the maximum imperfection measured on the specimens tested. The calculations were performed with COMU elements, which are specifically adapted to this approach [59]: the imperfection was monomodal on Fourier mode 11, and the response was decomposed according to Fourier modes 0, 11, 22 and 33. Plasticity was evaluated at 21 points along the circumference. Table 7 summarizes the results of the geometric and material nonlinear calculations for the three imperfection amplitudes. Plasticity was not activated in the case of pressure loading, but only for axial compression.
Table 7

Imperfect cylinders: experimental and calculated buckling loads

External pressure

A1

A2

A3

A4

\(\frac{P_{exp}}{P_{NLP}}\)

0.991

0.920

0.900

0.982

\(\frac{P_{exp}}{P_{NLPCOMU}} ; \frac{\delta }{h}=0.01\)

0.986

0.879

0.920

1.00

\(\frac{P_{exp}}{P_{NLPCOMU}} ; \frac{\delta }{h}=0.1\)

1.06

0.937

0.985

1.069

\(\frac{P_{exp}}{P_{NLPCOMU}} ; \frac{\delta }{h}=0.2\)

1.15

0.994

1.04

1.129

Axial compression

A5

A6

\(\frac{F_{exp}}{P_{NLP}}\)

1.01

0.919

\(\frac{F_{exp}}{F_{NLPCOMU}} ; \frac{\delta }{h}=0.01\)

1.076

0.922

\(\frac{F_{exp}}{F_{NLPCOMU}} ; \frac{\delta }{h}=0.1\)

1.080

0.988

\(\frac{F_{exp}}{F_{NLPCOMU}} ; \frac{\delta }{h}=0.2\)

1.087

0.996

These results show that with an amplitude of initial modal imperfections less than 20 % of the thickness (which is consistent with our experiments) the calculations match the experimental results.

Load interaction diagrams

Let us now address the interactions between loading cases. In order to do that, we consider a reference shell with Young’s modulus 160,000 Mpa and thickness 150 μm. With this cylinder, elastic buckling occurs at an external pressure \(P_{E}\) equal to 0.02103 MPa, and at an axial compression load equal to 13,961 N. We calculated the linear, elastic nonlinear and elastic-plastic nonlinear critical loads for six loading combinations. We also calculated the elastic-plastic nonlinear response with a modal imperfection parallel to the buckling mode shape under external pressure and an amplitude equal to 10 or 1 % of the thickness for each case. Very similar results were obtained with a imperfection of the same amplitude, but parallel to the buckling mode under axial compression. Table 8 summarizes all the calculations. In this Table, \(\Lambda _{E}\) is the multiplying coefficient of the applied loading combination which leads to Euler buckling. Similarly, \(\Lambda _{NL}\) and \(\Lambda _{NLP}\) are the multiplying coefficients which lead to elastic and plastic nonlinear buckling, respectively. Coefficients \(\Lambda _{defa=0.01}\) and \(\Lambda _{defa=0.1}\) are the multiplying coefficients leading to the elastic-plastic buckling load for an initial imperfection of amplitude 0.01 and 0.1 % of the thickness, respectively. These are compared with the experimental results obtained for shells A1, A5 and A7–A13. All these results are normalized to the Euler pressure in abscissa and the Euler axial compression load in ordinate. Table 9 gives the normalized experimental results. Figure 10 shows a comparison between the interaction diagram obtained from the calculations and the experimental data.
Fig. 10

Cylinders under combined loading (experiments and calculations): interaction diagram

Table 8

Buckling under combined loading: calculated ratios under various modeling assumptions

Pressure load ratio

Axial load ratio

\(\Lambda _{E}\)

\(\Lambda _{NL}\)

\(\Lambda _{NLP}\)

\(\Lambda _{\delta\;=\;0.01}\)

\(\Lambda _{\delta\;=\;0.1}\)

0.0

1.0

1.00

0.994

0.756

0.724

0.71

0.2

0.8

1.17

1.171

0.944

0.960

0.895

0.4

0.6

1.18

1.186

1.186

1.161

1.06

0.6

0.4

1.14

1.146

1.146

1.107

1.102

0.8

0.2

1.07

1.078

1.078

1.042

0.97

1.0

0.0

1.00

1.01

1.01

0.99

0.92

Table 9

Buckling under combined loading: Euler loads and experimental ratios

Specimen

\(P_{E}\) (MPa)

\(F_{E}\) (N)

\(\frac{P_{exp}}{P_{E}}\)

\(\frac{F_{exp}}{F_{E}}\)

A1

0.0210

14,111

0.99

0.021

A7

0.01725

11,718

0.87

0.17

A8

0.01972

13,276

0.81

0.34

A9

0.01707

11,737

0.76

0.33

A10

0.01971

13,150

0.61

0.51

A11

0.01704

11,680

0.44

0.58

A12

0.01971

13,150

0.40

0.55

A13

0.0169

11,643

0.27

0.67

A5

0.0214

14,193

0.00

0.77

This figure shows good agreement between calculations and experimental results concerning the interaction diagram. In the case of significant axial compression loads, as noted previously, plasticity plays an important role. Conversely, it does not affect external pressure loading. All the interaction curves are convex. Therefore, a linear interaction is conservative, provided the critical load reduction effects on axial compression (plasticity) and on external pressure (geometric imperfections) are taken into account. For these tests, the calculations with an amplitude of initial modal imperfection equal to 10 % of the thickness lead to a good estimate of the experimental critical load in all the cases.

The case of ASTER VM22 shells

In this section, we focus on the prediction of the critical load for ASTER VM22 shells (22 outward vaults). The results of the calculations for the “perfect” structure under simple pressure loading for shells B1 and B2 and under uniform axial compression for shell B5 are compared to the experimental results in Table 10.
Table 10

ASTER VM22 shells: comparison of the calculated and experimental buckling loads

Specimen

Loading type

Experimental buckling load

\(\frac{\Lambda _{exp}}{\Lambda _{E}}\)

\(\frac{\Lambda _{exp}}{\Lambda _{NL}}\)

\(\frac{\Lambda _{exp}}{\Lambda _{NLP}}\)

B1

Pressure

0.091 MPa

4.57

0.58

1.01

B2

Pressure

0.092 MPa

4.00

0.512

0.98

B5

Axial load

14,200 N

0.99

0.84

1.03

The calculated predictions match the experimental results very well when plasticity is taken into account. The nonlinear elastic calculations systematically overestimate the critical loads. Two calculated buckling deformed shapes (under axial compression and under external pressure) are presented in Fig. 11. Now let us address combined loading. These cases concern the experiments on shells B1, B2, B7, B8, B9, B10 and B5. The theoretical loads are normalized to the Euler loads of the corresponding perfect cylindrical shells. The theoretical critical loads under combined loading were calculated with the same properties as the reference shell (Young’s modulus 160,000 Mpa and thickness 150 μm, as in the case of smooth cylinders) and divided by the elastic buckling loads of the same shell (Table 11).
Fig. 11

ASTER VM22 buckling modes

Table 11

ASTER VM22 shells: experimental buckling loads under combined loading

Specimen

\(P_{E}\) (MPa)

\(F_{E}\) (N)

\(\frac{Pcr_{exp}}{P_{E}}\)

\(\frac{Fcr_{exp}}{F_{E}}\)

B1

0.0199

13,171

4.57

0.00

B2

0.023

14,972

4.00

0.00

B7

0.0184

12,315

2.95

0.20

B8

0.0164

11,053

2.32

0.41

B9

0.0177

11,951

1.84

0.66

B10

0.0221

14,337

0.93

1.04

B5

0.021

14,400

0

1.01

B6

0.021

14,119

0

0.99

The calculation results are given in Table 12.
Table 12

ASTER VM22 shell buckling under combined loading: calculated ratios under various modeling assumptions

Pressure load ratio

Axial load ratio

\(\Lambda _{E}\)

\(\Lambda _{NL}\)

\(\Lambda _{NLP}\)

0.0

1.0

1.60

0.95

0.84

0.1

0.9

1.79

1.65

1.29

0.2

0.8

2.03

1.80

1.36

0.3

0.7

2.33

1.98

1.45

0.4

0.6

2.75

2.27

1.57

0.5

0.5

2.95

2.48

1.74

0.6

0.4

3.06

2.83

1.95

0.7

0.3

3.12

3.31

2.23

0.8

0.2

3.16

3.92

2.60

0.9

0.1

3.29

5.00

3.08

1.0

0.0

3.31

5.95

4.01

The results of the various calculations are compared with the experimental results in Fig. 12.
Fig. 12

ASTER VM22 shell under combined loading (experiments and predictions): interaction diagram

One can observe that in most cases a calculation based on linear elastic analysis is not predictive. The comparison with geometrically nonlinear calculations is even worse: in this case, the geometric nonlinearities have a stiffening effect. The only predictive calculation is the one which takes plasticity into account. This effect is drastic in almost all the cases: plasticity reduces the critical load by about one-third. In this type of stiffening effect, plasticity always plays an important role.

The case of ASTER VM14 shells

In this section, we focus on the numerical prediction of the tests for 14-vaulted shells. The results of the 3D calculations are compared to the experimental results in Table 13. Figure 13 shows the two elastic buckling modes. In this case in which the vaults lead to a less significant gain in critical load, the geometrically nonlinear predictions are good for the first three shells. The addition of plasticity slightly improves the results. For shell C4, which is subjected to axial compression, the predictions exceed the experimental critical load by 20 %.
Fig. 13

ASTER VM14 buckling modes

Table 13

ASTER VM14 shells: comparison of the calculated and experimental buckling loads

Specimen

Loading type

Experimental buckling load

\(\frac{\Lambda _{exp}}{\Lambda _{E}}\)

\(\frac{\Lambda _{exp}}{\Lambda _{NL}}\)

\(\frac{\Lambda _{exp}}{\Lambda _{NLP}}\)

C1

Pressure

0.054 MPa

2.7

0.99

0.93

C2

Pressure

0.046 MPa

2.6

0.85

0.87

C3

Axial load

6200 N

0.48

0.97

1.0

C4

Axial load

5100 N

0.39

0.78

0.83

Influence of imperfections

The previous results concerned multi-vaulted shells without imperfections. The next question is that of the influence of possible shape imperfections on the critical load. We performed a series of calculations to evaluate the influence of initial imperfections on buckling. In order to do that, we created an imperfect shell by adding to the initial perfect (multi-vaulted) geometry an imperfection, parallel to the elastic buckling mode under external pressure, with an amplitude of 10 % of the thickness. Then we calculated the elastic-plastic nonlinear critical loads under uniform axial compression and under uniform external pressure for shells VM14 and VM22. The reductions observed in the critical loads did not exceed 3 %. We obtained similar results with an imperfection parallel to the buckling mode under axial compression. This led to the conclusion that these shells are relatively insensitive to initial imperfections, both in the case of axial compression and in the case of external pressure.

Discussion

We found that vaults lead to a significant increase in the buckling strength under uniform external pressure and also improve the resistance to axial compression. This type of shell is relatively insensitive to initial imperfections. The choice of the number of vaults also plays a role. The effect becomes significant once the number of vaults equals twice the Fourier buckling mode of the smooth cylinder under external pressure. The question then arises of the optimal choice of the number and depth of the vaults with regard to buckling strength under combined loading. This is the subject of the next section.

The optimization procedure

We carried out a systematic study of the effect of the number and depth of the vaults on the buckling strength under external pressure alone, under axial compression alone and, finally, under combined loading with 50 % of the critical axial compression and 50 % of the critical external pressure. In order to do that, we first assumed a constant vault depth d. We used the reference shell (Young’s modulus 160,000 MPa and thickness 150 μm).

Influence of the number of vaults p

Under external pressure alone

The results of the linear elastic, nonlinear elastic and elastic-plastic calculations are given in Fig. 14. One can observe that the critical loads increase very rapidly beyond mode 10 (which corresponds to 20 vaults) and reach a factor 4. Then, they continue to increase progressively with the number of vaults to reach a factor greater than 10 for 60 vaults. Let us note that, in this case, the nonlinear elastic estimates are very optimistic. Plasticity plays a significant role once the number of vaults exceeds 20. Beyond 50 vaults, the critical load reaches a maximum of 10 times the Euler load.
Fig. 14

ASTER VM shells: influence of the number of vaults on the buckling pressure

Under axial compression alone

The results of the calculations are given in Fig. 15.
Fig. 15

ASTER VM shells: influence of the number of vaults on the axial compression buckling load

ASTER shells are less effective in terms of resisting axial compression. Nevertheless, beyond 20 vaults, the critical load is multiplied by two. Then it increases progressively with the number of vaults and reaches the Euler load for 80 vaults. In this case, taking nonlinearities into account leads to a significant decrease in the predicted critical load. Plasticity reduces it by about another 10 %.

Under combined loading (50 % axial compression, 50 % external pressure)

The results of the calculations are given in Fig. 16.
Fig. 16

ASTER VM shells: influence of the number of vaults on buckling under combined loading

For this loading combination, the linear and nonlinear elastic predictions are quite similar, but very optimistic. Taking plasticity into account plays a decisive role. However, one can observe that beyond 20 vaults the critical load is multiplied by two, then decreases by a maximum of 20 %.

Influence of the vault depth d

In this section, we study how the vault depth affects the improvement brought about by outward vaults. This presentation is limited to ASTER VM22 shells. We considered several ratios of the vault depth to the thickness of the shell. The evolution of the radius of curvature of the vault as a function of this ratio is shown in Fig. 17.
Fig. 17

ASTER VM22 shells: the relative vault depth as a function of the vault radius r

The calculation results for external pressure loading alone are given in Fig. 18. One can observe an improvement factor between 5 and 6 when the ratio d/h becomes greater than 7.5. First, the nonlinear critical loads are greater than the linear critical loads. Plasticity limits the effectiveness of the shell when the ratio exceeds 10.
Fig. 18

ASTER VM22 shells: influence of the relative vault depth on the buckling load under external pressure

Figures 19 and 20 show similar tendencies for axial loading and combined 50 % external pressure and 50 % axial compression loading. The nonlinear elastic and elastic-plastic critical loads become less than the Euler load when the vault depth exceeds 10 times the thickness.
Fig. 19

ASTER VM22 shells: influence of the relative vault depth on the buckling load under axial compression

Fig. 20

ASTER VM22 shells: influence of the relative vault depth on the buckling load under combined loading

Based on these results, we can conclude that the recommended vault depth is about 10 times the thickness.

ASTER VM-22 shells: investigation of inward vaults

In this section, we compare VM22 shells and shells defined with inverted vaults (i.e. with the concavity turned outwards) of the same depth d or radius of curvature r. These new shells are designated as ASTER VM-22. The deformed buckling shape under axial compression is shown in Fig. 21.
Fig. 21

ASTER VM-22: buckling mode under axial compression

The calculated buckling loads are compared in Table 14.
Table 14

Comparison of the calculated buckling loads of ASTER VM22 and VM-22 shells

Shell name

d/h

r (mm)

\(\frac{\Lambda _{E}}{P_{E}}\)

\(\frac{\Lambda _{NL}}{P_{E}}\)

\(\frac{\Lambda _{NLP}}{P_{E}}\)

External pressure

 VM 22

15.47

20

3.35

7.5

4.47

 VM -22

15.47

20

4.96

3.5

4.6

 VM -22

22.7

30

7.19

5.9

4.15

Shell name

d/h

r (mm)

\(\frac{\Lambda _{E}}{PF_{E}}\)

\(\frac{\Lambda _{NL}}{PF_{E}}\)

\(\frac{\Lambda _{NLP}}{PF_{E}}\)

50 % external pressure 50 % axial compression

 VM 22

15.47

20

1.22

1.43

0.97

 VM -22

15.47

20

1.25

1.03

0.76

 VM -22

22.7

30

2.12

1.53

0.90

Shell name

d/h

r (mm)

\(\frac{\Lambda _{E}}{F_{E}}\)

\(\frac{\Lambda _{NL}}{F_{E}}\)

\(\frac{\Lambda _{NLP}}{F_{E}}\)

Axial compression

 VM 22

15.47

20

1.48

1.19

0.92

 VM -22

15.47

20

1.13

0.95

0.77

 VM -22

22.7

30

1.79

1.48

1.15

One can observe that if one keeps the vault depth constant the outward vaults are more effective against buckling. Conversely, if one keeps the radius of curvature r constant, the inward vaults are slightly better, especially in the axial compression case.

Conclusions

In summary, we developed shells which are relatively insensitive to imperfections and which resist buckling under external pressure much better than smooth cylindrical shells. Their axial compression strength is also better. These results require the use of a number of vaults at least equal to twice the Fourier mode number of the critical buckling mode of the smooth cylinder under external pressure. The recommended vault depth d is at least ten times the thickness. In addition, these shells are relatively insensitive to initial imperfections, which is an important advantage over the reference smooth cylinders. This type of shell can be manufactured relatively easily. The choice of vaults with their concavity turned inward seems to be slightly better when external pressures are predominant. The conclusions given in the paper take into account the load reduction effects due to plasticity, which play a significant role (all the more so when the stiffening effect is high). Therefore, one could improve the buckling strength of this type of shell much further by choosing materials with a high yield strength.

Declarations

Authors’ contributions

JFJ as provided the experimental results. AC has developped the FEM software and done all the computations. Both authors have read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
SAFRAN-AREVA-Chair, LaMCoS UMR CNRS 5259, INSA-LYON, Universite de Lyon
(2)
LaMCoS UMR CNRS 5259, INSA-LYON, Universite de Lyon

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© Combescure and Jullien. 2015