# Nondestructive Prediction of the Buckling Load of Imperfect Shells

Anaïs Abramian1,2 Emmanuel Virot2,3 Emilio Lozano2,3 Shmuel M. Rubinstein2
Tobias M. Schneider3 1Institut Jean le Rond d’Alembert, Sorbonne Université,
CNRS, UMR 7190, 75005 Paris, France  2John A, Paulson School of Engineering
and Applied Sciences, Harvard University, Cambridge, MA 02138, USA  3Emergent
Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale
de Lausanne, CH 1015 Lausanne, Switzerland

(Received 17 May 2020; revised 7 July 2020; accepted 13 October 2020;
published 25 November 2020)

###### Abstract

From soda cans to space rockets, thin-walled cylindrical shells are abundant,
offering exceptional load carrying capacity at relatively low weight. However,
the actual load at which any shell buckles and collapses is very sensitive to
imperceptible defects and can not be predicted, which challenges the reliable
design of such structures. Consequently, probabilistic descriptions in terms
of empirical design rules are used and reliable design requires to be very
conservative. We introduce a nonlinear description where finite-amplitude
perturbations trigger buckling. Drawing from the analogy between imperfect
shells which buckle and imperfect pipe flow which becomes turbulent, we
experimentally show that lateral probing of cylindrical shells reveals their
strength non-destructively. A new ridge-tracking method is applied to
commercial cylinders with a hole showing that when the location where buckling
is nucleated is known we can accurately predict the buckling load of each
individual shell, within $\pm 5\%$. Our study provides a new promising
framework to understand shell buckling, and more generally, imperfection-
sensitive instabilities.

DOI: 10.1103/PhysRevLett.125.225504

Compress an empty soda can from its top and it will remain stable over a
considerable range of loads. However, at a critical load, the can eventually
buckles, violently collapses, and irreversibly deforms. A classical linear
stability analysis greatly overestimates the buckling load of thin cylinder
shells and domes Seide1960 ; Yamaki1984 ; NASA_SP8032 , because the system is
extremely sensitive to defects Koiter1945 ; Weingarten1968 ; Horton1965 ;
Bushnell981 ; Davis1987 ; Singer2002 .

Classical theory mostly cannot capture the influence of defects and thus, in
general dramatically overestimates the carrying-load of engineered shells,
infamously affecting structural reliability Hilburger2015 . The ratio between
the actual measured buckling load and the theoretical prediction is termed the
knock-down factor. The knock-down factor tends to decrease with the ratio of
the shell’s radius of curvature to its thickness, $R/t$, but there are extreme
stochastic variations between nominally identical shells, as shown in Fig. 1.
Fifty years ago, NASA SP-8007 proposed a conservative phenomenological “design
rule” Weingarten1968 , which estimates a lower bound for the distribution of
knock-down factors and remains the basis for US and European design codes even
today.

Any sound prediction of the knock-down factor of a specific manufactured shell
currently requires complete knowledge of the shell’s imperfections. A full
characterization of imperfections is extremely challenging Hilburger2015  and
practically impossible for real engineered structures. Consequently, the NASA
design rule which on average severely underestimates buckling-loads remains
still in use. Here we introduce a non-destructive method that accurately
predicts the buckling load of an individual defected shell without prior
knowledge of imperfections.

Figure 1:  Knockdown factor $\sigma/\sigma_{c}$ vs $R/t$ collected by NASA
with the empirical design rule indicated by the dashed line. (Data from Seide
et al. (1960) Seide1960 )

Characterizing defect-sensitivity as observed in shell buckling is notoriously
difficult because in any experimental setup a multitude of details need to be
controlled practically perfectly. Notably, a similarly challenging situation
arose in fluid mechanics, where the concepts of finite-amplitude perturbations
in combination with very careful experiments lead to a complete rethinking of
the transition from laminar flow to turbulence in pipe flows Darbyshire1995 ;
Grossmann2000 ; Kerswell2005 ; Eckhardt2007 . Just as there is no universal
buckling load at which real imperfect shells buckle, there is no universal
Reynolds number at which turbulence develops in real, imperfect pipes; and the
critical flow-speed at which turbulence occurs is highly sensitive to
imperfections.

The challenging problem of the transition to turbulence is tackled within a
nonlinear framework by considering the response of a perfect system, an ideal
pipe, to a jet perturbation of finite-amplitude, $A_{\text{jet}}$, that can
trigger turbulence Darbyshire1995 . In the context of shell buckling, an
analogous approach implies revisiting the NASA data, shown in Fig. 1, by
considering the axial load as an externally-imposed control parameter, like
the Reynolds number. Since controlling manufacturing imperfections is harder
for thinner shells, the geometrical quantity $R/t$ (radius over thickness)
should be treated as a proxy for the vulnerability to defects of a given size.
With this interpretation, the phenomenology of the transition to turbulence in
pipe flow and the buckling of cylindrical shells appears strikingly similar,
as shown in Fig. 2(a,b).

Inspired by the hydrodynamic analogy, we recently developed an experimental
and theoretical framework Virot2017  to study shell buckling. The jets, which
trigger turbulence in pipes correspond to a lateral probe, or a “poker”, in
our system, as shown on Fig. 2(c-d). Similar ideas were recently pursued by
Hutchinson, Thompson and coworkers Thompson2015 ; Thompson2016 ;
Hutchinson2017a ; Hutchinson2017b  as well as others Neville2018 ; Groh2019a ;
Groh2019b . Analogous experiments were performed by Reis and coworkers for
pressurized hemispherical shells Lee2016 ; Marthelot2017  and cylindrical
shells with radially applied perturbation loads were studied in the context of
aerospace applications Craig1973 ; Wullschleger2006 ; Huhne2008 ;
Kriegesmann2012 ; Haynie2012 ; Arbelo2014 ; Hao2015 ; Wagner2018 ; Jiao2018 .

In contrast to previous work, in our approach the poker-induced deformation is
not interpreted as a defect; instead, the poker is a probe exploring the
system’s nonlinear response to finite-amplitude perturbations. By measuring
the poker force applied on the shell $F_{P}$ as a function of displacement,
$D_{P}$, we recently quantified the critical perturbations that triggers
buckling for varying axial loads $F_{A}$. Mapping out the conditions under
which buckling occurs, we identified a universal stability landscape for
cylindrical shells, represented as a surface in a ($F_{A}$, $F_{P}$, $D_{P}$)
three-dimensional space Gerasimidis2018 ; Virot2017 .

Figure 2:  (a) $A_{\text{jet}}$ vs Re for experiments by Darbyshire $\&$
Mullin Darbyshire1995  (b) $R/t$ vs knockdown factor $\sigma/\sigma_{c}$ for
Nasa’s data (Fig. 1). (c/d) Lateral poking of the shell corresponds to the
transverse jet perturbation and probes the stability of the linearly stable
shell to well-controlled perturbations.

The landscape, which can be probed experimentally, characterizes the stability
of a shell, which, in the most general case, is dictated by the complex
correlation structure of its many defects. The stability landscape features a
valley, a lake and a ridge (as described in Virot2017  and Fig. 4). Advancing
the poker beyond the ridge, defined by the peak poker force, will ultimately
bring the system to the lake shore, where the shell buckles. As $F_{A}$ is
increased, the ridge descents, asymptoting towards a zero maximal poker force
where the axial load for spontaneous buckling is reached. It is therefore
tempting to track the path of the ridge and attempt to infer the critical
buckling load without reaching it. Implementing ridge tracking as a non-
destructive method to predict the load-carrying capacity of a given shell and
rethinking buckling as a nonlinear finite-amplitude instability would enable
to finally move beyond the very conservative NASA design rule and develop a
fundamental and scalable approach to confront imperfection sensitivity in the
design and control of thin shells.

Drawing from the analogy between imperfect shells which buckle and imperfect
pipe flow which become turbulent, we show that lateral probing of cylindrical
shells can indeed reveal their strength. By examining the safe regions of the
stability landscape and extrapolating to the critical conditions, we use
lateral probing to non-destructively predict the strength of individual
imperfect shells. We direct buckling nucleation, by intentionally introducing
a guiding defect in the shell. Doing so, we show, as a proof of concept, that
probing near this guiding defect yields an accurate estimate of the shell’s
buckling load. Our results show that when the guiding defect is identified,
tracking the ridge and extrapolating to where it vanishes works extremely well
and predicts the critical buckling load to within $5\%$ precision. This result
considerably surpasses the NASA design rule which estimates a lower bound for
the distribution of nominally identical shells while we accurately predict the
strength of individual shells. Ridge-tracking thereby allows for a
deterministic rather than a probabilistic prediction, and thereby opens
avenues towards new design and control strategies for thin shell structures.

Figure 3:  (a) Experimental setup with close-ups of hole and poker tip. (b) A
typical example of ridge tracking. The star indicates the prediction for the
spontaneous buckling load. Inset: $F_{p}$ vs $D_{p}$ for $F_{A}=641\,$N. (c)
Five examples of successfull ridge tracking. Each spontaneous buckling load is
indicated by a star. Inset: Ridge of one can for four different poking
locations. (d) Predicted vs. measured spontaneous buckling load for all 38
tested shells. Green triangles indicate failure initiating far from the
guiding hole. Inset: Relative error $\eta$.  Figure 4:  Schematic 2D
projection of the system’s state-space for a perfect shell (a) and one with
imperfections (b). The stability landscape (c) probes the basin boundary along
a path of dimple-shaped deformations induced by lateral poking. The point on
the ridge (blue dot) is located between the basin boundary and the unbuckled
state. It varies with $F_{A}$ creating the ridge (blue line in (c)) that
asymptotes to zero when the basin of attraction vanishes and spontaneous
buckling is induced. Shell imperfections distort the basin boundary and modify
its experimentally accessible representation, the stability landscape. Ridge-
tracking reveals the influence of imperfections while non-destructively
probing the ridge.

The experimental setup consists of a custom-made biaxial mechanical tester
(ADMET, Inc.) described in detail previously Virot2017 . A vertical actuator,
equipped with a load cell, applies an axial load $F_{A}$ on the sample by
moving at a constant speed of $5\,$mm/min. The lateral actuator has a steel
marble of diameter $4.7\,$mm at its tip and serves as the poker. Importantly,
axial and lateral loading are always displacement-controlled, and lateral
poking is performed at set end-shortening; the axial load varies by less than
$2\%$ during poking. An acquisition card (National Instrument DAQ USB-6001)
records forces and displacements, and controls the actuators through an
analogue signal, enabling autonomous ridge tracking. The samples are empty
$7.5\,$oz mini Coke cans, made of aluminum. These are cylindrical shells of
radius $R=28.6\,$mm, mid-plane thickness $t=104\,\pm 4\,\mu$m (radius-to-
thickness ratio $R/t=274$) and height $L=107\,$mm. The buckling load of each
shell is dictated by the complex correlation structure of its many defects; it
is thus practically impossible to predict where buckling will nucleate. We
therefore intentionally introduce a guiding defect that fixes the preferential
nucleation point to a predefined location close to the probe: A hole of
$1\,$mm in diameter, is drilled $1\,$mm above the mid-plane of the cylinder as
shown in Fig. 3(a). The edge of the hole is circular and slightly curved
inwards. At the chosen parameters, with a radius below the characteristic
scale of $\sqrt{Rt}=1.7\,$mm the strength of the can is not determined by the
hole starnes70 ; toda83 . Instead the distribution of spontaneous buckling
loads remains broad indicating that the stability properties remain dominated
by the unknown defects of each specific can.

Ridge-tracking requires systematically poking the sample under gradually-
increasing axial loads without triggering buckling, as shown for a typical
experiment in Fig. 3(b). In this example, the sample is initially loaded to an
axial load $F_{A}^{0}=600\pm 1$ N, which is less than $25\%$ of the strength
predicted by linear stability analysis. The axial end-shortening is then set,
and the poker slowly advances towards the sample at a constant rate of
$2\,$mm/min, contacting the shell 1 mm below the hole. Once the poker contacts
the shell, the probe force gradually increases. When the force reaches its
peak $F_{p}^{\text{max}}$, the poker automatically stops at a displacement
$D_{p}^{\text{max}}$, and returns to its initial position, as shown in the
inset of Fig. 3(b). In this example, the point ($F_{A}=592$ N,
$D_{p}^{\text{max}}=0.44$ mm, $F_{p}^{\text{max}}=0.75$ N) defines the first
point of the ridge, as plotted in Fig. 3(b). As previously reported, this
process is reversible and does not damage the sample Virot2017 .

The axial load is then sequentially increased in steps of approximately
$25\,$N and the next poker force-displacement curve is measured. These force-
displacement curves gradually build-up the sample’s stability landscape, which
exhibits a distinct ridge. The procedure is then stopped when the peak poker
force $F_{p}^{\text{max}}$ falls below $0.3\,$N. A linear extrapolation of the
ridge to vanishing poker force yields a prediction for the spontaneous
buckling load. In this specific experiment, ridge-tracking predicts a buckling
load of $F_{A}^{\mathrm{predicted}}=893\,$N. To test this prediction, the
sample is compressed until it spontaneously buckles at a load of
$F_{A}^{\mathrm{buckling}}=887\,$N, remarkably close (within 0.5$\%$) to the
predicted value as shown in Fig. 3(b).

In total, 38 samples were tested, with initial loads ranging between
$F_{A}^{0}=522\,$N to $F_{A}^{0}=694\,$N, and an average load increment of
$23\,$N. One exceptionally fragile sample failed at the first loading, before
being probed. Otherwise, samples sustained repeated loading cycles, their
ridges follow a straight line in the plane ($F_{A}$, $F_{p}^{\text{max}}$),
while converging towards a wide range of buckling loads, as shown for 5
examples in Fig. 3(c).

Intriguingly, the ridges are only straight when probing is done near the hole.
Probing further away from it yields ridges that turn towards the buckling-load
prediction, as shown in the inset to Fig. 3(c). When probing beyond a distance
of approximately 1 cm from the hole, the ridges do not point towards the
actual buckling load, and the prediction fails—hence the importance of
introducing a preferred nucleation location and probing close to it.

Overall, for the 38 samples tested, the ridge-tracking protocol accurately
predicts the buckling load over a wide range of critical loads, from
$F_{A}^{\mathrm{buckling}}=761\,$N to $F_{A}^{\mathrm{buckling}}=1108\,$N, as
shown in Fig. 3(d). The relative error of the prediction,
$\eta=(F_{A}^{\mathrm{predicted}}-F_{A}^{\mathrm{buckling}})/F_{A}^{\mathrm{buckling}}$
has a standard deviation of $6.9\%$ for the whole bunch. Even more remarkably,
when buckling does initiate near the hole, this deviation drops to $2.7\%$ and
the average error $\eta$ is $1.6\%$ (blue diamonds in Fig. 3(d)). This
confirms that ridge tracking works extremely well if we probe in the vicinity
of the hole guiding nucleation of buckling. Besides providing a promising non-
destructive method to probe the load-carrying capacity of a shell, these
experiments unambiguously show that the ridge is locally distorted by the
defects controlling the load-carrying capacity. We now propose an
interpretation of the observed behaviour inspired by the analogy with the
onset of turbulence in pipe flow, based on a dynamical-system approach.

The dynamical systems approach suggests an appealing conceptual framing for
the cylinder buckling problem, illustrated schematically in Fig. 4. All
deformations of a given cylinder span the high dimensional state-space of the
system. Specific deformations including the undeformed cylinder, periodic
eigenmodes considered in classical stability analysis Zoelly1915 ; Koiter1945
but also any local deformation induced by a poker at a specific location are
points in the state-space. Thus, deforming the shell with an advancing poker
corresponds to tracing out a continuous path in state-space. When the shell is
compressed at a sub-critical axial load, the unbuckled state is linearly
stable, but non-linearly unstable. Small deformations will relax back to the
unbuckled state while large amplitude deformations may trigger buckling. The
state-space region around the unbuckled state in which deformations decay
forms the finite basin of attraction of the unbuckled state. It can be
figuratively thought as a deep concave bowl, the unbuckled state at its base,
and all stable deformations roll towards it.

States outside the basin of attraction will not return to the unbuckled state
but lead to a nonlinear instability and trigger buckling. Separating
deformations that trigger buckling from those that return to the unbuckled
state is the basin boundary, a co-dimension one manifold in state-space
defining the nonlinear stability threshold. As the axial load increases, the
basin of attraction shrinks, until the basin boundary eventually reaches the
unbuckled state, as shown in Fig. 4(a). At this load, the basin vanishes, the
unbuckled state becomes linearly unstable and spontaneous buckling is
triggered by infinitesimal perturbations. For a perfect shell, the basin
collapses at the critical load computed by standard linear stability theory.

For a real shell with imperfections, we expect all state-space structures and
specifically the basin boundary to be distorted compared to the perfect shell,
as shown on Fig. 4(b). Consequently, the basin boundary will reach the
unbuckled state at a different load, namely the spontaneous buckling load of
the defected shell.

Embedded in the basin boundary are unstable saddle points including localized
single dimple deformations Kreilos2017 ; Horak2006 ; Audoly2019 ; Groh2019a
which define the critical amplitude of a localized perturbation triggering
buckling nonlinearly. The advancing poker thus probes a path in state-space
that leads to the saddle point within the basin boundary. Introducing an
additional controlled imperfection, such as slightly curved holes deforms the
basin of attraction, and likely moves the corresponding single dimple saddle
point closer to the unbuckled state. Thus for the real shell, increasing the
axial load will trigger the localized unstable mode at the location of the
this defect first, which rationalizes how the hole controls the location where
buckling is preferentially nucleated.

We suggest the stability landscape is composed of experimentally-accessible
projections of the basin of attraction along one path of deformations in a
direction corresponding to the single dimple deformations and at different
axial loads. The stability landscape thus indicates how the basin of
attraction, measured in the direction of the guiding imperfection shrinks
under increasing axial load and at which load it vanishes. Probing the shell
at one location, passing the ridge and reaching the lake, thus yields the
distance between the unbuckled state and the basin boundary, as shown in Fig.
4(c). A convenient feature of ridge tracking, as opposed to lake tracking, is
that we are tracking the height of the cliff, and not its width, while both
vanish at the critical axial buckling load. This probing protocol keeps the
system safely within the basin of attraction.

The successful prediction of spontaneous buckling loads highlights the utility
of transferring nonlinear concepts from the study of turbulence transition in
subcritical shear flows to shell buckling. Treating buckling as a nonlinear
instability triggered by finite-amplitude perturbations of a subcritically
loaded shell rather than as linear instability problem, rationalizes the
striking sensitivity to defects. Moreover, it provides avenues for predicting
the strength of a shell in the presence of an unknown defect distribution. We
have specifically shown that characterizing the load-dependent critical
perturbation amplitude via ridge tracking can predict the strength of
commercial mini-Coke cans. The method involves introducing an additional
localized imperfection in the form of a hole and probing in the vicinity of
it. Notably, introducing the hole does not determine the strength of the shell
which still varies greatly. In fact the hole does not appear to weaken the can
at all; thus, introducing holes does not trivialize the challenge of buckling-
load prediction. We hypothesize that the additionally introduced imperfection
guarantees that the unstable mode excited first will indeed be a local dimple
buckle near the probing location. Under these conditions, the stability
landscape provides the most relevant projection of the basin boundary. Without
a priori knowledge of where buckling will initiate, ridge tracking will likely
require to probe at numerous locations. The precise conditions under which the
additional controlled defect aids probing yet not affects the strength of the
shell itself remain to be investigated in detail. Interestingly, while the
specific hole does not determine the strength of the can, it does seem to
serve as a nucleus or lightning rod for the nucleation of buckling, forcing a
specific initiation location and allowing ridge tracking to work. Introducing
controlled defects and ridge tracking opens new and exciting approaches for
probing and controlling the stability of thin shell structures.

The authors would like to thank John W. Hutchinson and Pedro. M. Reis for
insightful discussions. This work was funded by the National Science
Foundation through the Harvard Materials Research Science and Engineering
Center DMR-1420570 and the Swiss SNSF under Grant No. 200021-165530. S. M. R.
acknowledges support from the Alfred P. Sloan Research Foundation
(FG2016-6925).

## References

  * (1) P. Seide, V. I. Weingarten, E. J. Morgan, The development of design criteria for elastic stability of thin shell structures, Space Technology Laboratories, Report STL/TR-60-0000-19425 (1960).
  * (2) , N. Yamaki, Elastic stability of circular cylindrical shells, Elsevier (1984).
  * (3) V. I. Weingarten, P. Seide, A. L. Braslow, Buckling of Thin-Walled doubly curved shells, NASA Space Vehicle Design Criteria, NASA SP-8032 (1969).
  * (4) V. I. Weingarten, P. Seide, E. J. Morgan, Buckling of Thin-Walled Circular Cylinders, NASA Space Vehicle Design Criteria, NASA SP-8007 (1968).
  * (5) W. H. Horton, S. C. Durham, Imperfections, a main contributor to scatter in experimental values of buckling load, International Journal of Solids and Structures, 59–72 (1965).
  * (6) W. T. Koiter, The Stability of Elastic Equilibrium, Ph. D. thesis, TH Delft (1945)
  * (7) D. Bushnell, Buckling of shells, pitfall for designers, AIAA Journal 19 1183–1226, (1981).
  * (8) R. C. Davis, F. Carder, Buckling tests of a 10-foot diameter stiffened cylinder with rectangular cutouts, NASA Technical Memorandum 88996, (1987).
  * (9) J. Singer, J. Arbocz, T. Weller, Experimental Methods in Buckling of Thin-Walled Structures, John Wiley & sons, New York, Volume 1 and 2 (2002).
  * (10) M. W. Hilburger, W. Allen Waters, W. T. Haynie, Buckling test results from the 8-foot-diameter orthogrid-stiffened cylinder test article TA01, NASA Technical Publication TP-2015-218785, (2015).
  * (11) A. G. Darbyshire, T. Mullin, Transition to turbulence in constant-mass-flux pipe flow, Journal of Fluid Mechanics, 289, 83-114 (1995).
  * (12) S. Grossmann, The onset of shear flow turbulence, Reviews of Modern Physics, 72, 603-618 (2000).
  * (13) R. Kerswell, Recent progress in understanding the transition to turbulence in a pipe, Nonlinearity, 18, R17-R44 (2005).
  * (14) B. Eckhardt, T. M. Schneider, B. Hof, J. Westerweel, Turbulence Transition in Pipe Flow, Annual Review of Fluid Mechanics, 39, 447-469 (2007).
  * (15) E. Virot, T. Kreilos, T. M. Schneider, S. M. Rubinstein, E. Virot, T. Kreilos, T. M. Schneider, S. M. Rubinstein, Stability landscape of shell buckling, Physical Review Letters, 119, 224101 (2017).
  * (16) J. M. T. Thompson, Advances in Shell Buckling: Theory and Experiments, International Journal of Bifurcation and Chaos 25, 1530001 (2015).
  * (17) J. M. T. Thompson, J. Sieber, Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling, International Journal of Bifurcation and Chaos 26, 1630003 (2016).
  * (18) J. W. Hutchinson, J. M. T. Thompson, Nonlinear buckling interaction for spherical shells subject to pressure and probing forces, Journal of Applied Mechanics 84, 061001 (2017).
  * (19) J. W. Hutchinson, J. M. T. Thompson, Imperfections and energy barriers in shell buckling, International Journal of Solids and Structures 148, 157-168 (2017).
  * (20) Groh R. M. J., Pirrera A., On the role of localizations in buckling of axially compressed cylinders, Proc. R. Soc. A 475, 20190006 (2019).
  * (21) Groh R. M. J., Pirrera A., Spatial chaos as a governing factor for imperfection sensitivity in shell buckling, Physical Review E 100, 032205 (2019).
  * (22) Neville, R. M., Groh R. M. J., Pirrera A., Schenk, M., Shape Control for Experimental Continuation, Physical Review Letters 120, 254101 (2018).
  * (23) A. Lee, F. L. Jiménez, J. Marthelot, J. W. Hutchinson, and P. M. Reis, The geometric role of precisely engineered imperfections on the critical buckling load of spherical elastic shells, Journal of Applied Mechanics, 83, 111005 (2016)
  * (24) J. Marthelot, F. Lopez Jimenez, A. Lee, J. W. Hutchinson, P. M. Reis, Buckling of a pressurized hemispherical shell subjected to a probing force, Journal of Applied Mechanics 84, 121005 (2017).
  * (25) Kriegesmann, B., Hilburger, M., and Rolfes, R., The Effects of Geometric and Loading Imperfections on the Response and Lower-Bound Buckling Load of a Compression-Loaded Cylindrical Shell, AIAA Paper 2012-1864 (2012).
  * (26) Wagner, H. N. R., Huhne, C., and Niemann, S.Robust Knockdown Factors for the Design of Spherical Shells Under External Pressure: Development and Validation, Int. J. Mech. Sci. 141 58–77 (2018).
  * (27) Haynie, W., Hilburger, M., Bogge, M., Maspoli, M., and Kriegesmann, B., Validation of Lower-Bound Estimates for Compression-Loaded Cylindrical Shells, AIAA Paper 2012-1689 (2012).
  * (28) Arbelo, M. A., Degenhardt, R., Castro, S.G.P., Zimmermann, R., Numerical characterization of imperfection sensitive composite structures Composite Structures 108 295-303 (2014)
  * (29) Hao, P., Wang, B., Tian, K., Du, K., and Zhang, X., Influence of Imperfection Distributions for Cylindrical Stiffened Shells With Weld Lands, Thin-Walled Struct. 93 177–187 (2015).
  * (30) Huhne, C., Rolfes, R., Breitbach, E., and Tessmer, J., Robust Design of Composite Cylinder Shells Under Axial Compression—Simulation and Validation, Thin-Walled Struct. 46 947–962 (2008).
  * (31) P. Jiao, Z. Chen, X. Tang, W. Su, J. Wu, Design of axially loaded isotropic cylindrical shells using multiple perturbation load approach – Simulation and validation Thin-Walled Struct. 133 1-16 (2018)
  * (32) Wullschleger, L., Numerical Investigation of the Buckling Behaviour o Axially Compressed Circular Cylinders Having Parametric Initial Dimple Imperfections, Doctoral thesis, ETH, Zurich, Switzerland (2006).
  * (33) J. I. Craig, M. F. Duggan, Nondestructive Shell-stability Estimation by a Combined-loading Technique, Experimental Mechanics, September, 381-388 (1973)
  * (34) S. Gerasimidis, E. Virot, J. W. Hutchinson, S. M. Rubinstein, On Establishing Buckling Knockdowns for Imperfection-Sensitive Shell Structures, Journal of Applied Mechanics 85, 091010 (2018).
  * (35) R. Zoelly, Über ein Knickungsproblem an der Kugelschale, Thesis, Zürich, 19157).
  * (36) Horák, J., Lord, G. J., Peletier, M.A., Cylinder buckling: the mountain pass as an organizing center, SIAM J. Appl. Math. 66 1793–1824 (2006).
  * (37) T. Kreilos, T. M. Schneider, Fully localized post-buckling states of cylindrical shells under axial compression, Proceedings of the Royal Society of London A, 473 20170177 (2017).
  * (38) B. Audoly, J. W. Hutchinson, Localization in Spherical Shell Buckling, J. Mech. Phys. Solids 136 103720 (2017).
  * (39) See supplementary information SI.
  * (40) Toda, S., Buckling of cylinders with cutouts under axial compression. Experimental Mechanics, 23(4), 414-417 (1983).
  * (41) Starnes, J. H., The effect of a circular hole on the buckling of cylindrical shells (Doctoral dissertation, California Institute of Technology) (1970).