Buckling of a Holey Column

D. Pihler-Puzović,1 A. L. Hazel,2 and T. Mullin1

1Manchester Centre for Nonlinear Dynamics and School of Physics and Astronomy,University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

2Manchester Centre for Nonlinear Dynamics and School of Mathematics,University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

We report the results of an experimental and numerical investigation of instabilities in a novelvariant of an elastic column under load. Buckling of a standard Euler column under compression is atextbook example of symmetry breaking in physics. We find that introducing a regular line of holesinto the column can prevent lateral symmetry-breaking by allowing the column to instead undergoan internal instability in which the holes are compressed in alternate directions. We investigatethe range of column lengths for which this novel state is triggered and find excellent quantitativeagreement between the computed and observed results.

PACS numbers:

Buckling instabilities are the dominant failure modes ofcompressed thin-walled structures. Their control or sup-pression is vital in applications at all scales [1] rangingfrom large-scale structures such as aircraft to the engi-neering of DNA [2]. An everyday example of bucklingcan be realised using a coffee stirrer compressed uniaxi-ally between the forefinger and thumb. When compressedgently, the stirrer remains straight, but above a criticalcompression it deflects laterally in a similar manner tothe leftmost column in figure 1. The stirrer exhibits clas-sic Euler column buckling whose description dates backto the 18th century [3] and is now used as the archetyp-ical symmetry-breaking, pitchfork bifurcation [4].

In this letter we demonstrate that a straightforwardmodification of this canonical buckling problem can sup-press lateral deflections while maintaining the reflec-tion symmetry about the axial centreline of the column.Specifically, the inclusion of a regular line of circular holesinduces the buckled states shown in the central image infigure 1 and experiments in figure 2. The original Eulerstate remains, see the righthand image in figure 1, butin the new alternating state the holes transform into anarray of mutually orthogonal ellipses and the column re-mains straight. Physically, the modification introducesan internal structure that can support localised bucklingin addition to the global Euler buckling. Mathematically,the original continuous (vertical translation) symmetry isconverted into a discrete (permutation) symmetry, lead-ing to a new and exquisitely rich solution structure.

We performed experiments on a set of samples with upto eight holes. Regular samples were moulded using thehyperelastic material extra hard Sid AD Special (Fegu-ramed GmbH). The material was formed by thoroughlymixing quantities of degassed polyvinyl base and siloxanecatalyst in a 1:1 volume ratio. The liquid mixture waspoured into 3D printed moulds and left to set for 15 min-utes. The samples were gently removed from the mouldsto avoid tearing and any samples with visible imperfec-tions were discarded. Four different samples were pro-

local strainenergy/(E/3)

-0.01 0.1

FIG. 1: Centre & right: two buckled states for a column with20 regular holes; left: Euler buckling of the correspondingsolid column. All states are at the same compression of ε =0.093. The color map shows the local strain energies of thestates normalized by the shear modulus E/3.

duced for each geometry and measured from calibratedimages, taken using a still camera (Nikon D300S), with aresolution of 0.05 mm per pixel. An LED light box wasused for uniform back lighting. The average hole radiusR, the distance between hole centres a and the samplelength l, width w and depth d were measured [22].

Each of the samples was compressed using an In-stron 3345 machine with a 5 N load cell. The force re-quired to achieve a given level of compression was mea-sured to within 0.0025 N. The compression rate was 0.1mm/min, but we also performed experiments at a rateof 0.5 mm/min and obtained the same results. Thus,

2

FIG. 2: An illustrative example of the alternating mode in aholey column compressed by a vernier caliper. The equivalentsolid column is shown alongside and undeformed samples areon the left.

we conclude that the loading was sufficiently slow to beconsidered quasi-static. Samples were placed within amachined perspex channel to facilitate good alignmentduring compression; and to ensure that the top and bot-tom surfaces of the samples remained flat, which helpedprevent out-of-plane buckling. The channel was attachedto the Instron machine and lowered onto a perspex inden-ter. The perspex was stiffer than our samples by factorof 104, so its deformation was negligible. We also foundthat lubricating the top and bottom surfaces of the sam-ples with Vaseline helped to make the forcing uniform.

In the theoretical model we assume a Neo-Hookean,incompressible hyperelastic material [5, 6] under planestrain. Hence, the second Piola–Kirchhoff stress tensorσij = (E/3) gij−pGij , where p is the solid pressure, E isthe Young’s modulus, gij and Gij are the contravariantundeformed and deformed metric tensors respectively.The principle of virtual displacements and the incom-pressibility constraint yield the required governing equa-tions:∫

σij δεij dV0 = 0 and∫(detG− 1) δpdV0 = 0,

where εij is the Green–Lagrange strain tensor, detG isthe determinant of the deformed metric tensor, V0 is theundeformed volume and the Einstein summation conven-tion is used. The C++ library oomph-lib [7] was usedto solve the variational form above via a finite-elementapproximation in two-dimensions with quadratic interpo-lation for the displacements and linear interpolation forthe pressure.

We used an average experimental geometry in themodel and each sample comprised N identical elemen-tal cells, with hole radius, R = 4.21 mm, distance be-tween the hole centres, a = 9.84 mm, and sample width,w = 11.19 mm. The results for samples with three or

fewer holes were sensitive to the precise details of thegeometry at the constrained boundaries and an extra0.69 mm of elastic material was added at each end ofthe model sample to mimic the experiments. A struc-tured mesh of quadrilateral elements was used to discre-tise each elemental cell and this discretisation was re-peated throughout the domain to ensure that the appro-priate symmetries were present. The side boundaries ofthe samples were traction-free; the bottom boundary waspinned to ensure zero displacement in any direction; andthe compression was induced by prescribing the displace-ment of the upper boundary, which was constrained toremain horizontal.

The stress-strain curves from the experiments andmodel calculations are shown in figure 3. The engineeringstrain, ε, is defined to be the compression of a sample rel-ative to its initial length l and the engineering stress, σ, isthe ratio of the applied force to the initial cross-sectionalarea in contact with the loader d× w.

For ε < 0.04, all samples in figure 3 follow a linearstress-strain relationship with a gradient equivalent to aneffective Young’s modulus of 213.8 ± 0.5 kPa. Bucklingoccurs through a symmetry-breaking bifurcation, accom-panied by a change in gradient, at εcr(σcr). Both εcr andthe corresponding σcr decrease with increasing samplelength. The error bars in figure 3 represent the stan-dard deviation over the different experimental runs withthe different samples. They are only significant in thevertical direction because the critical stress is sensitiveto precise details of sample geometry and end loadingconditions but the critical strain is not. We confirmedthat the critical strains for buckling were independent ofthe discretisation, but the critical loads were sensitive tothe third significant figure, indicating a similar sensitiv-ity to that found in the experiments. Indeed, providedthe appropriate local critical strain can be induced thealternating buckling should occur for any elastic mate-rial. This is because the buckling occurs when it is en-ergetically favourable for regions of the material to bendrather than further compress, i.e. buckling is primarilydetermined by the local geometry of the sample.

The agreement between the experimental and the nu-merical results in figure 3 is striking. Buckling occurs inan Euler-like state for the sample with two holes, whichis strongly dominated by the boundary conditions, butin the alternating state for all other samples. The insetsin figure 3 also show typical post-buckled shapes of thesamples, obtained experimentally and in the model. Byexamining the local strain energy, σijεij , of a buckled col-umn in the Euler and the alternating states at the samelevel of compression, it is possible to predict the statethat will be observed after buckling. For example, in thecase of the sample with six holes, shown in the inset in fig-ure 4, the total strain energy of the Euler state is greaterthan that of the alternating state and consequently theholey column buckles in the latter lower-energy state.

3

For the solid column, the buckled states correspondto global sinusoidal modes and the strain energy in-creases with the increasing number of lateral deflections[8]. Thus the mode with a single deflection is always pre-ferred. For the holey column, the buckled states are as-sociated with subgroups of the discrete symmetry groupD2 × SN , which represents the direct product of the re-flection symmetries of the rectangle, D2, and all permu-tations of the N holes, SN . Unlike the solid column,where the buckling modes remain the same irrespectiveof length, in the holey column the number of subgroupsincreases as N ! [9]. We have confirmed that all the pos-sible states predicted by group theory can be found inthe model for columns with two and three holes. Al-though this complexity is an inherent feature of the ho-ley column, a remarkable observation for the samples infigure 3 is that only the buckling states shown in figure 1are seen in experiments. These two states represent dif-ferent extremes in the sense that the strain energy is leastevenly distributed throughout the material in the Euler-like state, whereas in the alternating state the strain en-ergy is more evenly distributed than in any other state.Thus, we can predict the preferred buckling state of anycolumn by computing εcr for these two states keeping thesame void to material ratio as in figure 3 but increasingnumber of holes, see figure 4.

The data were obtained using the model now in the ab-sence of extra end material to preserve the permutationsymmetry and εcr was calculated by solving a standard

Engineering strain, ε0 0.05 0.1 0.15

Engineeringstress,σ(M

Pa)

0

0.005

0.01

0.015

8 holes

3 holes

ε = 0.08ε = 0.09

ε = 0.05

ε = 0.06

5 holes

2 holes

FIG. 3: Stress-strain curves for the compression of the holeycolumns with 2, 3, 5 and 8 holes, respectively, obtained experi-mentally (filled markers) and in the theoretical model (emptymarkers). The errorbars are the standard deviation over 8experimental runs, 2 for each sample. The transition to thebucked state occurs where the data diverges from the initiallinear relationship. Typical buckled states, obtained experi-mentally (left) and in the theoretical model (right), and thecorresponding strain values are shown as the insets.

extended system to find points at which the Jacobian(tangent stiffness matrix) has a zero eigenvalue corre-sponding to the appropriate eigenmode. εcr is plotted asa function of the number of holes N on a log-log scalefor the Euler and the alternating buckling states. Forcomparison, we also show εcr at the onset of buckling forthe corresponding solid columns without holes (in thiscase N corresponds to the ratio of the sample length tothe length of the elemental cell). The theoretical scalingfor the Euler state in a solid column with clamped endsgives εcr = 4π2I/(dwN2a2), where I = dw3/12 is thearea moment of inertia of the cross section of the column[8]. This prediction is demonstrated in figure 4 with thelower dotted line and becomes quantitative once N ≥ 6.

The critical strain for the Euler state in the holey col-umn exhibits the same 1/N2 scaling, suggesting that thesame essential mechanism is at play. Examination of thestrain energy in figures 1 and 4 and the supplementaryvideo material indicates that the deformation is confinedto thin columns whose width is of the order of the nar-rowest cross-section of elastic material. The predictionof the Euler theory in which the area moment of inertiaand column width are replaced by those correspondingto the narrowest cross-section, Imin = (w3 − 8R3)/12,wmin = (w − 2R), is shown as the upper dotted line infigure 4. The numerical data is in excellent accord with

Number of holes, N5 10 20 40

Criticalen

gineeringstrain,ε c

r

10-3

10-2

10-1

Euler mode, solid columnEuler modeAlternating bucking modePeriodic calculationAsymptotic solutions

2

1

ε0cr = 0.0349

ε = 0.11816

local strainenergy/(E/3)-0.01 0.1

FIG. 4: Log-log plot of εcr as a function of N obtained numer-ically for (�) the Euler and (•) the alternating states inthe holey column with the same void to bulk ratio as infigure 3 and (F) the Euler state in the corresponding solidcolumn. The lines between discrete points are added toguide the eye. The horizontal dashed line, εpcr = 0.035,is εcr for the two-hole sample with periodic boundaryconditions. The dotted lines are predictions of the Eulertheory for columns with fixed ends. The insets are thesame states as in figure 1 with 6 holes at ε = 0.11816.

4

the asymptotic prediction, albeit approaching it for muchlarger N .

The critical strain associated with the Euler state de-creases as the length of the column increases, ultimatelytending to zero. On the other hand, the critical strainfor the alternating buckling state appears to saturate ata constant value as the column length increases. Thecritical buckling strain for an “infinite” column in whicha two-hole sample is subject to periodic boundary condi-tions in the vertical direction but with traction-free sidesis given by εpcr = 0.035, which gives a lower bound forεcr. Although we might expect εcr → εpcp as N increases,it is not clear that the influence of the boundaries canever be totally neglected. For the periodic calculations,we used an unstructured mesh of triangles in our numer-ics and modified the boundary conditions so that mate-rial points on the upper and lower boundaries that wereoriginally aligned vertically remained separated by a con-stant vertical distance. This distance was then decreasedto simulate the compression, while the pressures at theupper and lower boundaries were constrained to be thesame. The different behaviours for the Euler and alter-nating states in figure 4 as the column length increasessuggest that a sufficiently long holey column will alwaysbuckle as an Euler column, unless specifically forced intothe alternating buckling state.

We have shown that a simple modification of thecanonical example of mechanical buckling and symmetry-breaking bifurcation has non-intuitive and potentiallyfar-reaching consequences. The buckling still occurs viaa symmetry-breaking bifurcation, but broken symmetryis not necessarily the left-right reflection symmetry. Asa consequence, it is possible to realise a buckled statethat can accommodate the same mechanical load withina much more confined space than conventional columns.Localised buckling is observed in shell structures, butnot in solid columns; and one interpretation of ourgeometric modification is that it introduces shell-likebehaviour by adding a second, smaller, lengthscale tothe structure. Broadening our approach to includethe effects of void fraction, hole shape and hole sizedistribution will further enhance the diversity of the phe-nomena potentially allowing custom tuning of bucklingstates to specific applications. As for buckling in morecomplex elastic structures [10–14], the reported effectsare scale independent offering the prospect of controllingoptical solitons [15], designing soft robotics [16], Natureinspired photonics [17–19] and even diverting damagingearthquake vibrations [20]. The phenomena also hasimplications for the widely-used Euler–Greenhill formula[21] , suggesting that the height at which an elastic poletopples could be increased by modifying the internalstructure.

The authors thank A. Juel, M. Heil, L. Ducloue and C.Johnson for useful discussions and M. Walker and D. Chorlton

for technical support.

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are: R = 4.27±0.08 mm, a = 9.9±0.05 mm, l = 20.98±0.07 mm, w = 11.14± 0.05 mm and d = 6.87± 0.29 mm;(b) 3 holes are: R = 4.2± 0.05 mm, a = 9.83± 0.05 mm,l = 30.83 ± 0.05 mm, w = 11.25 ± 0.05 mm and d =7.28 ± 0.06 mm; (c) 5 holes are: R = 4.23 ± 0.05 mm,a = 9.85 ± 0.05 mm, l = 50.64 ± 0.05 mm, w = 11.12 ±0.05 mm and d = 7.12 ± 0.11 mm; (d) 8 holes are: R =4.19±0.05 mm, a = 9.83±0.05 mm, l = 80.24±0.15 mm,w = 11.25 ± 0.05 mm and d = 7.12 ± 0.05 mm. Theaverage sample size, also used in the theoretical model,is: R = 4.21 ± 0.03 mm, a = 9.84 ± 0.02 mm, w =11.19 ± 0.07 mm, d = 7.1 ± 0.17 mm and l − N × a =1.38± 0.18 mm.