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

- Alain Combescure
^{1}Email author and - Jean-Francois Jullien
^{2}

**2**:26

**DOI: **10.1186/s40323-015-0047-3

© Combescure and Jullien. 2015

**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 [1–20] 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 [21–37]. Fewer works can be found in the literature regarding the effect of plasticity on buckling [38–45]: 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

*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.

### 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).

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

### Material properties

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

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.

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 |

#### The case of ASTER VM14 shells

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 |

### The effects of loading combinations on buckling

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 | \(\infty\) |

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

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.

#### Influence of initial imperfections

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

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 |

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

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 |

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 |

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 |

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

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

#### Under axial compression alone

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)

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

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

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 Access**This 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

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