Shell buckling,without ‘imperfections’

Abstract

The buckling behaviour of thin shell structures under load has been a persistent challenge to engineering designers and researchers over many decades. In this article I consider two unusual experimental studies on the buckling of thin-walled elastic cylindrical shells, each of which sheds intriguing light on the buckling phenomena. The classical theory of buckling of thin cylindrical shells under axial compression predicts that the buckling stress will be proportional to t/R– the ratio of thickness to radius – other things being equal. But collected results of experimental studies from many laboratories, when plotted on log–log scales, show clearly that the buckling stress is actually proportional to (t/R)^1.5, with the measured buckling stresses being scattered through a factor of about 4 for shells with R/t > 200. Such scatter is commonly judged to be in accord with Koiter’s theory of ‘imperfection sensitivity’. But that theory lays no claim to an understanding of the empirical 1.5-power law. I claim that a key to this situation is the experimental performance of some small-scale open-topped silicone rubber shells, buckling under their own weight, which clearly demonstrates a 1.5-power law, but with very little scatter. The buckling mode of these shells involves almost entirely inextensional deformation, with a single small dimple growing near the base, separated from the rest of the shell by a narrow boundary layer that accounts for almost all of the dimple’s elastic strain energy. A straightforward, simple analysis of the mechanics of the dimple is consistent with the experimental 1.5-power law. As noted above, experimental buckling loads of shells that are closed at both ends also show the empirical 1.5-power law, but now with significant statistical scatter. A second aim of the paper is to throw light on that phenomenon. I venture to attribute it to the effect of the boundary conditions of the shell. I adduce support for this view from experimental observations on the buckling of a shell with special, frictional end-fittings. That feature produces significantly higher collapse loads, and with much smaller scatter, than for corresponding shells with fixed boundaries; and it permits striking pre-buckled deformations to occur, of a kind not previously noted. It will be appreciated that neither of the two parts of this article depends on the widely accepted theory of imperfection-sensitivity; hence my choice of title. It is a pleasure for me to submit this article to a special publication in honour of Michael Rotter, with whom I have discussed matters of this sort through three decades.

Keywords

‘1.5-power law’experiments geometric imperfections mechanics of dimples self-weight buckling shell buckling thin cylindrical shells

Introduction

The classical formula (equation (1)) for the critical buckling stress, σ_cl, of a thin elastic cylindrical shell under axial compression was proposed in about 1910 (see Timoshenko and Gere, 1961: §11.1):

\frac{σ_{c l}}{E} \approx \frac{0.6 t}{R}

(1)

Here σ_cl is the (compressive) buckling stress, E is the Young modulus of elasticity of the material, t is the thickness of the shell wall and R is the radius of the shell.

The analysis leading to this equation presupposes that the shell is not so short as to buckle like a short, wide plate, and thus that its length L satisfies (see Calladine, 1983: Chapter 15)

\frac{L}{\sqrt (R t)} > 10

(2)

But it became clear from 1928 onwards (see Timoshenko and Gere, 1961: §11.4) that experimentally determined buckling loads were significantly lower than those given by prediction (1). Figure 1, assembled by Harris et al. (1957) for shells satisfying (2), shows clearly:

That for any given value of R/t, there is considerable scatter in the buckling performance;

That the measured strengths become smaller in comparison with the classical prediction as R/t increases.

Figure 1.

Experimental buckling loads for cylindrical shells, normalised with respect to the classical buckling prediction (1) and plotted versus radius/thickness ratio. Circles are data for R/t > 100 from Harris et al. (1957): see Brush and Almroth (1975) Figure 5.18. The continuous, broken and chain-dotted curves correspond, respectively, to the mean line (3), the lines at ±1 and ±2 standard deviations of Figure 2(a). The vertical bars at R/t = 1800, well above the mean line (3), are data from specimens having special, frictional end-fittings, tested by Lancaster et al. (2000); they are described towards the end of the paper. The longer, open bar represents the mean and ±1 standard deviation of 30 cases where imperfections were deliberately introduced, while the shorter, black bar relates to cases where there were no such imperfections.

For decades the conventional wisdom, following Koiter (1945), has been that this experimental shortfall below the classical prediction is due to the unavoidable presence, in shell specimens as manufactured, of random geometrical imperfections.

The data of Figure 1 (some 190 observations, from 12 different studies) have been re-plotted on logarithmic scales in Figure 2(a). Here it is clear that the experimental data lie in a well-defined band corresponding to the buckling stress level being proportional to (t/R)^1.5. The mean buckling stress is given by

\frac{σ_{mean}}{E} \approx 5 {(\frac{t}{R})}^{1.5}

(3)

Figure 2.

(a) Double-logarithmic plot of experimental data on thin cylindrical shells as shown in Figure 1, with buckling stress now normalised with respect to the Young modulus E, versus radius/thickness ratio. The heavy best-fitting line (3) has a slope of −1.5, and broken parallel lines at 1 and 2 standard deviations from the mean are also shown. Classical theory (1) is represented by the line of slope of − 1.0. The five parallel lines were obtained from a least-squares fit of the (double-logarithmic) data set. The exponent 1.5 emerged most clearly when data for R/t < 200 were excluded: in this range, there are no data-points having σ > 0.8 σ_cl, as may be seen. Note that the lines for ±2 standard deviations from the mean fit well the extreme observations. (b) Self-weight buckling data from experiments of Calladine and Barber (1970) and Mandal and Calladine (2000) on open-topped silicone rubber shells, plotted as the normalised self-weight vertical stress at the base of the shell versus radius/thickness ratio. The five parallel lines from (a), but not the points there, are also shown.

Practically, all of these data-points lie within two standard deviations from the mean, and these two practical upper and lower bounds on the data correspond to the mean stress multiplied by 2 and 0.5, respectively. An unprejudiced observer, when confronted with this paradoxical plot, might be forgiven for concluding that the classical formula (1) is irrelevant to the practical buckling of these shells; or, at least, that formula (1) is an unpromising starting-point for a rational explanation of the consistent experimental data. It is clearly desirable to explain the mechanics underlying the empirical result (3). But what tools or methods are available for this task?

T von Kármán ((1940); with Dunn and Tsien (1940); with Tsien (1941)) grappled with shell buckling as an example of a non-linear problem. He described an empirical relationship similar to (3), on the basis of a 90-element experimental data set available in 1940 – but with an exponent of 1.4 rather than 1.5. He set up his famous non-linear governing equations; he demonstrated highly unstable buckling when the classical prediction (1) is reached; and he described dynamical ‘snap-through’ to the diamond-covered forms shown, for example, by Timoshenko and Gere (1961, Figure 11.6).

But although von Kármán had a reputation for creating imaginative simple models for other difficult buckling problems – see for example von Kármán et al. (1932)– he did not provide a convincing explanation of a formula like (3), before he was diverted onto more pressing war-time tasks. Neither have many studies based on the presence of initial geometric imperfections of form in the shells, according to the methods introduced by Koiter (1945), since the aim of such studies has been to understand why empirical buckling loads fall so far short of the classical prediction (1) – rather than to explain collected empirical data, as shown in Figures 1 and 2(a).

Our main tasks in this article are, first to provide a rational explanation for the observed 1.5-power law (3); and second to shed further light on the wide scatter of buckling loads, as seen in Figures 1 and 2(a).

Experiments on self-weight buckling of cylindrical shells

Following the collapse of newly built cooling towers at Ferrybridge, UK during a gale in November 1965, my students and I became interested in self-weight buckling of open-topped shells, since this was evidently one of the many contributory factors involved in that disaster (Owen, 1967). We made, with difficulty, a few small-scale model hyperboloidal shells, about 35 cm high, from silicone rubber, and with progressively decreasing thickness; but none of them collapsed under its own weight. We therefore decided to study instead the simpler problem of self-weight buckling of open-topped cylindrical shells – since they would be easier to make, and would remain cylindrical when progressively shortened, in assays to find the greatest height of shell that would stand unaided.

In 1969 my student Nicholas Barber and I, together with our laboratory technician Roger Denston, developed a simple method for casting uniform, thin silicone rubber shells in a rotating cylindrical mould of diameter about 18 cm (Calladine and Barber, 1970). After curing, the shells were cast with a thick disc closure at one end, but were left open at the other. (We quickly learned that it was important to cast the base disc before the shell was detached from the mould.) When extracted from the mould, and placed on a flat horizontal table with the open end up, these specimens were incapable of standing upright under their own weight. But when short horizontal rings were successively cut off the top, a height, L_cr, was eventually reached at which a given specimen could just stand up. Figure 2(b) shows the results of this study (together with those of Mandal and Calladine (2000), see below).

We were surprised by two aspects of these results. First, the points lay on a line of slope of −1.5, rather than −1.0, which we had anticipated by extension of (1). Second, there was unexpectedly little scatter of the points. We brought these puzzling features to the attention of our shell buckling colleagues. (At that time, neither we nor any of our correspondents were aware of von Kármán et al.’s, (1940, equation (13)) empirical 1.4-power law for axially compressed cylindrical shells.) The only positive response to our enquiries, from John Hutchinson, was a suggestion that we should also perform tests on shells of a different diameter. Thus it was that in 2000, some 30 years later, Partha Mandal conducted a second set of experiments, with a mould 30% bigger in diameter, in order to convince sceptical colleagues that elementary dimensional analysis correctly predicted that a change of radius would not affect the plot.

At this point, we plotted our self-weight buckling data onto a newly made double-logarithmic plot (Figure 2(a)) of the data of Figure 1: see Figure 2(b); and we were astonished to find that they agreed remarkably well with (3). Immediately, the scales fell from our eyes (Acts 9, 18); for we saw, at long last, a possible new way to understand the empirical 1.5-power law (3).

Computational study of self-weight buckling

Our next step was to use the standard ABAQUS finite-element package (Hibbitt, Karlsson & Sorensen, Inc., 1995) to analyse the finite displacements of a particular experimental shell in Mandal’s series, having R=120 mm, t = 0.58 mm and L = L_cr = 230 mm – the experimentally observed value. Here, a nice advantage of the computer over a laboratory experiment is that the actual gravitational acceleration g can be multiplied at will by a ‘load factor’G: so G = 1.0 corresponds to the experimental ‘just stable’ critical condition. Once a suitable imperfection pattern had been found (see Mandal and Calladine, 2000 for details), the computations ran smoothly, with the help of the ‘Riks’ algorithm for following descending loads.

Figure 3 shows a plot of G against radial displacement at two different material points P and Q, whose locations on the shell are indicated in the sketch of Figure 4(a). There was hardly any sign of buckling as G increased from zero, until it reached a value of around 1.8, corresponding to the self-weight vertical stress at the base reaching almost the classical buckling value (1). But then the load fell very sharply, and deflections only began to increase significantly when the load factor had fallen to G ≈ 1.3. Thereafter, the radial displacement at the two chosen points increased steadily while G fell slowly to a value of around 1.0.

Figure 3.

Plot of gravity load factor G against normal deflection at two points on the central generator of an open-topped shell, in an ABAQUS computation: see Mandal and Calladine (2000) for details. Material points P and Q on the symmetry meridian are identified in Figure 4(a).

Figure 4.

Schematic representations of the post-buckling modes of three thin-walled cylindrical shells, showing a common dimple motif. All shells are built-in at the base. (a) Open-topped shell loaded by gravity, as described by Mandal and Calladine (2000). (Later it is postulated that the weight of portion ABBA of the shell provides the force that holds the dimple in place.) (b) Shell with its top closed by a diaphragm, loaded vertically by a localised force F at the edge. This situation has been investigated by Guggenberger (2006) in the context of localised support systems for silo structures. (c) As (b), but with an idealised small circular dimple of radius r, whose surface has been inverted to the same radius of curvature as that of the parent shell. Inextensional deformation within the dimple allows F to move vertically by a small distance u.

We may describe this behaviour as a ‘post-buckling plateau’, although as a plateau it is not absolutely flat. The plateau extends to radial displacements of around 10 wall thicknesses, with relatively little change in load factor G. Further computations showed that the initial buckling load was sensitive to the amplitude of an initial geometric imperfection only if that imperfection featured a ‘dimple’ near the base, of the kind to be described below. But in any case, the level of the post-buckling plateau was insensitive to the pattern or amplitude of any initial geometric imperfection.

Of particular relevance is the fact that the post-buckling plateau occurs at around the experimentally observed critical height L_cr. And computations on geometries corresponding to other experimental specimens at their critical heights showed essentially the same feature. Thus, we may conclude that, somehow, our self-weight experimental buckling assay was picking out the almost neutrally stable behaviour of this post-buckling plateau. The path by which the plateau is actually reached evidently depends on the pattern of assumed initial geometric imperfections. But it is not hard to imagine that our practical experimental process of supporting the shell by hand, in attempts to get it to stand up, would introduce a wide range of imperfection patterns.

Our computations also revealed a simple geometric feature of the ‘plateau mode’, as seen in contour plots of the radial deflection of the shell (not shown here, but see Mandal and Calladine (2000)). This can be described adequately by the schematic cartoon of Figure 4(a). The dimple near the base of the shell grows larger and deeper as deflections increase, and as it does so, the highest point of the dimple descends, vertically, by a relatively small amount. If we consider the moiety of the shell above this level as an open-topped storage-tank standing on a level foundation, which is ‘sinking’ in the small region over the dimple, we can see the outward tilting of the shell wall above the dimple as an elementary example of inextensional deformation of the shell wall, as sketched in Figure 4(a): cf. Kamyab and Palmer (1989) or Calladine (1983: §6.5.1).

Dimple mechanics

In an unrelated computational assay, which I came across quite by accident, Guggenberger (2006) has made an interesting and relevant computational study of the development of dimples in thin cylindrical elastic shells, under the action of localised axial forces applied at an edge, as shown schematically in Figure 4(b). Buckling occurs when a peak load has been reached, whereupon a dimple begins to form. Guggenberger found that, after falling below its peak value, the load remains roughly constant as the dimple grows: there is a sort of first-order ‘plateau’ in the force/deflection characteristic. He also found that the response depended primarily on the total force F, and to a negligible extent upon the width of the shell’s circumference over which the load is spread, up to a width of order √(Rt).

The ‘plateau’ values of Guggenberger’s curves are consistent with the formula

F_{h} \approx 0.9 E t^{2.5} / R^{0.5}

(4)

These shells were held circular by a diaphragm at the loaded edge, but were free to rotate there about the local tangent; that is, the edge was ‘hinged’, hence subscript h in equation (4). Zhu et al. (2002) made similar calculations, but now with the loaded edge of the shell restrained from rotation about the local tangent. A typical plot is shown in Figure 5: Zhu likewise found that the force remained constant (to first order) as the dimple grew in size, in accordance with

F \approx 1.4 E t^{2.5} / R^{0.5}

(5)

Figure 5.

ABAQUS results for a shell loaded as in Figure 4(b): load factor versus maximum inward deflection of the dimple. The load is factored from an arbitrary ‘reference load’, and a quantitative formula (5) for the minimum load is indicated. The shell had a radius of 120 mm and thickness of 0.58 mm. The behaviour is similar when the thickness of the shell is altered: see Zhu et al. (2002).

The form of equations (4) and (5) is the same, of course: the difference in the numerical constants doubtless reflects the different boundary conditions at the loaded edge.

These results suggest the beginnings of a simple explanation for the self-weight post-buckling mode, as shown in Figure 4(a). But it will be useful first to introduce another striking result, which helps with the quantification of the physical properties of dimples.

A crude analysis of Guggenberger’s dimple

Several authors have studied the mechanics of the inversion of a thin-walled elastic spherical shell under the action of a localised radial load, directed inwards – see Mescall (1965), Bushnell (1967), Pogorelov (1967), Ranjan and Steele (1977), Nowinka and Lukasiewicz (1991) and Holst and Calladine (1994). The general picture is the same in all of these studies. The situation is most readily described with respect to Figure 6(a), which shows schematically a diametral cross-section of a thin shell of radius a. A central portion of the shell has been inverted, and it has the same geometrical configuration as if an axisymmetric cap had been cut out, turned over and re-united with the parent shell around its edge. The deformed meridian of the shell actually makes a smooth transition between the original and the inverted portions, in the form of a localised ‘boundary layer’, as shown on the left of the diagram – rather than an idealised sharp crease, as shown on the right.

Figure 6.

(a) Cross-section of an elastic thin-walled spherical shell that is being inverted by a central force. To the right is shown a truly inextensional mode of deformation, with a sharp crease of angle ψ; while on the left the actual smooth boundary layer is shown schematically. (b) Perspective general schematic sketch of a curved boundary-layer crease in a surface which has been inverted. The surface may originally have been plane, cylindrical or spherical. The bold line represents the crease, and families of light lines indicate the curvature of the outer and inner portions of the surface. At the point marked on the crease, |Δк| is defined as the jump in curvature (of lines locally parallel to the crease) as one crosses the crease. When the picture represents a portion of the boundary layer ring in Figure 6(a), |Δк| = 2/a.

A localised boundary layer of this type was first described by Horace Lamb, acting as a referee of Love’s famous paper of 1888 on shells (see Calladine, 1988, on the governing equations of thin shell structures). Love had maintained that the modes of free vibration of a thin hemispherical bowl were exclusively extensional. This was in direct conflict with Rayleigh’s description of inextensional modes for the same problem, which had enabled him to predict correctly the ringing tone of a hemispherical bell. Lamb de-fused the controversy by showing that Rayleigh’s solution was correct in essentials, but that it required the addition of a narrow boundary layer at the free edge in order to satisfy in full Love’s equations.

Mallock (1908) described, qualitatively, the competing bending and stretching effects within such a boundary layer or ‘knuckle’, when a ductile tube is being compressed lengthwise, and adopts a ‘concertina’ mode of progressive collapse with, typically, six lobes around the circumference.

In the elastic boundary layer of Figure 6(a), Dimensional Analysis shows that the total strain energy is minimised when the width of the boundary layer in the meridional direction is of order √(at), and when the ratio of bending to stretching strain energy is 3:1.

Pogorelov (1967) gave a formula equivalent to the following for the elastic strain energy Ω per unit length of boundary layer:

Ω \approx 0.033 E t^{2.5} ψ^{2} {(\frac{2}{a})}^{0.5}

(6)

Here, ψ is the included angle of the idealised crease, as shown in Figure 6(a). Note that t appears in this expression, just as it does in Guggenberger’s formula (4), with an exponent of 2.5.

Holst and Calladine (1994) have shown, from ABAQUS finite-element studies, that formula (6) gives a good account of the total strain energy of the deformed shell, even when radius r (which is used to calculate ψ, as in Figure 6(a)) is of the same order as the width of the boundary layer, and when the strain energy of bending in the inverted portion is discounted.

This enables us to make a crude but illuminating re-derivation of Guggenberger’s result for the force F required to hold a dimple in place in a cylindrical shell of radius R (Figure 4(b); see Calladine, 2000). For the sake of simplicity, we consider a dimple bounded by a circle of radius r, as shown in Figure 4(c), and with the inverted portion in the form of a cylinder of radius R. Just as for the inversion of the spherical shell, we shall assume that the elastic strain energy of distortion resides almost entirely in the boundary-layer circle separating the dimple from the parent shell: apart from that boundary layer, the deformation is inextensional.

In order to evaluate the total elastic strain energy, we need to know values of ψ around the circle; and for small values of r/R (r/R < 1/3, say) we find that ψ is uniform, with

ψ \approx \frac{r}{R}

(7)

At this stage, it is not obvious how to proceed, because formula (6) includes a, the radius of the spherical shell for which it was derived. Further analysis shows that the same formula can be used more generally in cases where the idealised crease, as here, is not planar, provided the term (2/a) in equation (6) is replaced by |Δк|, where Δк is the jump in curvature of the surface, measured parallel to the boundary layer, as we cross the crease in its idealised form, as shown in Figure 6(b). (The resulting formula is precisely equivalent to Pogorelov’s (1967) different general formulae). For the case of the spherical shell, there is a jump from +1/a to −1/a as we cross the crease; so equation (6) is recovered. But for the dimple in the cylindrical shell, there is a jump from |1/R| to 0 at points on axial or transverse diameters of the dimples; indeed

| Δ κ | \approx \frac{1}{R}

(8)

uniformly around the entire circular boundary layer.

For a dimple of radius r, we may thus evaluate U, the total elastic strain energy in the boundary layer:

U \approx 0.033 E t^{2.5} {(\frac{r}{R})}^{2} {(\frac{1}{R})}^{0.5} \times 2 π r \approx 0.2 E {(\frac{t}{R})}^{2.5} r^{3}

(9)

Next we evaluate the axial displacement u of the top of the dimple. We do this by using the kinematic condition that the axial generator in the dimple is inextensional. This gives, to first order, by use of a Taylor expansion:

u = 2 (r - R \sin (\frac{r}{R})) \approx \frac{r^{3}}{3 R^{2}}

(10)

Now the dimple is held in position by an axial force F. Since there is negligible elastic strain energy stored in the inextensionally deformed portions of the shell, U is a good approximation to the total elastic energy stored; hence

F = \frac{d U}{d u} \approx 0.6 E t^{2.5} / R^{0.5}

(11)

This formula is of precisely the same form as Guggenberger’s approximate empirical expressions (4), and (5), and in particular is independent of r.

The low value of the constant in (11) is attributable to the fact that the central diameter of the hypothetical dimple shown in Figure 4(c) is a little shorter than the portion of circumference which it replaces. A simple adjustment to the calculation (see Figure 7), in order to ensure that the length of the central circumference is preserved, shows that the total strain energy of the surrounding boundary layer is roughly doubled, thereby raising the value of the constant in equation (6) to approximately 1.2; which is in fair agreement with equations (4) and (5). But for present purposes the important result is the way in which F comes out as proportional to E t^2.5/R^0.5, and independent of r in our simplified calculation; and is also in broad agreement with Guggenberger’s result.

Figure 7.

The cross-section of the shell in Figure 4(c), through the centre of the dimple, may be represented by the line PABAP. This is a little shorter than the original circular arc PACAP, since line ABA is shorter than arc ACA. In order to remedy this defect, consider a crude adjustment, involving a wider dimple DBD, joined to the parent circle by tangents DP. When we impose equality of lengths PDBDP and PACAP, we find that BD/BA = 2/√3, and ψ_D/ψ_A = √3 – at least, when the entire circular arc is, reasonably, regarded as ‘shallow’. (The diagram is not strictly to scale.) This in turn implies that the strain energy per unit length of boundary layer at D is about three times that previously calculated for A; and hence, that the total strain energy of the dimple is approximately double the previous estimate.

In this connection we might add that the concentration of elastic strain energy in the boundary layer at the ends of the dimple’s horizontal diameter is consistent with the tendency of the initially circular dimple to become diamond-shaped as it grows – as was seen in the computations of both Guggenberger (2006) and Zhu et al. (2002). We shall return to this point later.

A model for self-weight buckling

We are now in a position to make a simple first-order theoretical analysis of the post-buckling mode for the open-topped shell. Our basic idea (see Figure 4(a)) is to equate the self-weight of the portion of the shell ABBA above (say) the centre-line of the dimple to the constant force F necessary to hold the dimple in place. Since the weight of the portion ABBA is proportional to the area multiplied by the thickness t, it is immediately clear that the post-buckling behaviour will involve thickness as t^1.5, just as in the experimental result (2).

Putting F = C E t^{2.5} / R^{0.5}

(12)

where C is a constant whose value is to be determined, and substituting for F the weight of the shell moiety ABBA directly above the centre of the dimple – at various stages of the ABAQUS results – we find that the value of C rises from ∼1.5 to ∼2 as the inward deflection at the centre of the dimple increases from 0.01R to 0.03R. Since the distribution of membrane stress within the buckled shell is actually rather complicated (see Mandal and Calladine, 2000), this result is encouraging; for C = 1.4 (in equation (5)) corresponds fairly well to our re-computation of Guggenberger’s case (Figure 4(b)) with appropriate boundary conditions.

Experimental ‘scatter’ of buckling loads for ‘ordinary’ shells

As mentioned above (see Figure 2(b)), the self-weight-buckling of our open-topped cylindrical shells showed little experimental ‘scatter’. How then can we explain the large scatter that is shown consistently – as in Figures 1 and 2(a)– in the testing of what we might call ‘ordinary’ shells ? That constitutes the second main task of this article.

It is clear that the post-buckling mode sketched in Figure 4(a) would not be able to occur if the top of the shell were to be held circular by a closing diaphragm: the inextensional ‘storage-tank-like’ mode of the upper moiety of the shell obviously requires the top edge to be free. That free upper edge also makes the shell statically determinate as a membrane (e.g. Calladine, 1983: §6.5.1). Thus, our crude estimate of F in Figure 4(a) as the weight of the portion of the shell directly above the dimple would cease to be reasonable if the top were to be held circular. Hence we may argue that the ‘scatter’ observed in the testing of ‘ordinary’ shells is somehow directly related to the statical indeterminacy of those structures.

In this connection we may mention that von Kármán et al. (1940) describes buckling tests on a shell with R/t = 1820, loaded in a specially constructed displacement-controlled rig, so that the development of buckles as the shell was slowly shortened could be recorded photographically. He states that ‘the initial wave form is elliptical in shape and scattered randomly through the specimen’. And the accompanying photograph (Figure 17) shows an isolated elliptical dimple at mid-height, which is the first sign of impending collapse. In the next frame this dimple has grown somewhat and is accompanied by a second, partner dimple nearby; thereafter, more-or-less diamond-shaped dimples progressively cover the shell, as the shortening of the shell proceeds.

What kind of imperfection?

In Koiter’s studies, the ‘imperfections’, which play a crucial role in reducing the failure load of the shell below the classical prediction (1), are stress-free geometric perturbations in the form of the eigenmodes of classical buckling, or combinations thereof. Whether such hypothetical ‘imperfections’ are representative of observable imperfections in physical shells is, of course, open to question; but it seems likely that manufacturing processes involving thermal welding of curved plates will leave behind imperfections which are neither in the form of classical eigenmodes, nor stress-free (see for example Rotter and Teng (1989), Holst et al. (2000) or Rotter (2004)). And the same would be true, presumably, for shells subject to accidental damage, for example, by inelastic impact.

It was on account of such scepticism about the kinds of ‘imperfection’ assumed, apparently without question, in so many theoretical studies of shell buckling that Lancaster, Palmer and Calladine (2000) conducted tests on an ‘ordinary’ cylindrical shell of diameter of 0.9 m and height of 0.7 m, made from Melinex sheet (R/t = 1800), to which the end discs – covered with emery cloth grade 400 – were secured frictionally by circumferential belt-like clamps. The upper disc was heavy enough to buckle the shell; but it was supported by a central instrumented vertical rod, controlled at the base by a screw. That scheme limited the overall axial shortening of the buckled shell, thereby preventing serious damage to the specimen.

The point of this method of clamping the shell to the end-plates was twofold. First, it would enable us to impart local initial-stress imperfections to the shell by applying specific local ‘uplift’ at the lower boundary; and second, it would enable the same physical shell to be tested repeatedly.

The basic idea behind tests with local uplift was that such action would induce, locally, compressive stress in the shell. This could even be of similar magnitude to the classical buckling stress; thus, it might lead to premature buckling of the shell under applied axial load.

Some 30 tests were performed without the deliberate introduction of any imperfection. As can be seen in Figure 1, the mean buckling load for these tests lay at almost 2 standard deviations above the mean of the wide data-base of buckling test results. Furthermore, the standard deviation of these 30 buckling loads was about 20% of the standard deviation of the points of Figure 1 in the region of R/t = 1800. That was our first major surprise: an unexpected consequence of our ‘frictional’ end conditions was a striking improvement in the buckling performance of the shell.

Thirty more tests were conducted with local ‘uplift’, typically of the order of 0.1 mm, deliberately applied at the base. Here, the mean buckling load lay at about 1 standard deviation above the mean (see Figure 1). Thus, there was some reduction in buckling load on account of the local uplift; but still the performance was good in relation to the data-base of Figure 1.

In some of these tests, the initial local uplift produced a single small, diamond-shaped dimple near the base of the shell at zero applied load, about 90 mm wide, with maximum inward displacement of about 2.5 mm, or 10 wall thicknesses. (We were not able to produce circular dimples in these assays.) Then, as the load level increased, the dimple grew both in width and height; but the shell did not actually buckle until the centre of the dimple reached almost the mid-height of the shell. In one particular test, the specimen was left overnight with a single large, stable diamond-shaped dimple about 170 mm wide, and having a maximum inward displacement of about 8 mm, or 30 wall thicknesses. But by the following morning half of the shell was covered in diamond dimples of roughly the same size, as shown in Figure 8. (Quite plausibly, the ‘triggering’ event was slip at the upper, frictional boundary.)

Figure 8.

Photograph of the shell of Lancaster et al. (2000) in its post-buckled state, in which approximately half of the circumference is covered by a more-or-less regular pattern of diamond-shaped dimples. Immediately before this state was reached, the shell supported a single large dimple, roughly at mid-height – which had grown, stably, as the axial load on the shell had increased, from an initially small dimple, induced by a small local ‘uplift’ at the base: see the cited paper for details.

That performance was strikingly different from the observation of von Kármán et al. (1940) – on a shell with a similar R/t value, but having ‘fixed’ upper and lower boundaries – of a small elliptical dimple spawning other small dimples nearby as it grows, as mentioned above.

How can we understand the growth of a large, stable, isolated dimple in Lancaster’s shell? The simplest explanation of this novel feature – which is not observed when the ends of the shell are securely fixed – is that since the presence of such a dimple requires definite small axial displacements at the ends, it is a direct consequence of the special, frictional end conditions. Growing dimples of that kind are not permitted, geometrically, by ‘fixed’ end conditions, as in von Kármán’s experiment; this may also explain why in that case the dimples had a clear tendency to multiply in number, before becoming unstable.

Another aspect of Lancaster’s frictional end conditions is that the pattern of multiple well-developed dimples shown in Figure 8 extends fully towards the ends of the shell, whereas in von Kármán et al.’s (1940) shell the final state involves multiple dimples confined to a more central area of the shell. And indeed, that same general feature may be seen in the photographs of Timoshenko and Gere (1961, Figure 11.6).

It should be noted that the final dimples seen in Figure 8 are somewhat larger – as a fraction of the shell’s diameter – than those of von Kármán’s, and of these other, shells. An obvious explanation of such an effect is that the ‘mobility’ of well-developed dimples in Melinex shells is likely to be greater than in metal shells, on account of metals having a smaller elastic range of strain than that of Melinex; thus, the localised zones of very high curvature are less mobile on account of irreversible localised deformation there.

Thus, the presence of frictional end-fittings may well explain the relatively high experimental collapse loads recorded on our shell. This also suggests clearly that attention to the detailed design of end-fittings of real, physical shells may significantly improve the buckling performance. The presence of frictional ends, or end conditions that are mechanically equivalent to them, introduces a dissipative element into an otherwise elastic, reversible system; which will doubtless complicate analysis, but may well improve structural performance, as we have seen.

Concluding remarks

In this article we have striven to resolve a long-standing paradox in shell buckling, whereby, notwithstanding the classical prediction (1), experiments conform to a different formula (3), with a different power law. We have made progress by building on unexpected results from our self-weight buckling experiments, and using Guggenberger’s analysis of a dimple. The answer to the question where does the empirical 1.5-power law come from? appears to be directly related to the detailed mechanics of the boundary layer that surrounds the dimple.

In his celebrated Thesis on shell buckling, Warner Koiter (1945) (cf. Calladine, 1983: Chapter 15) built on von Kármán’s exposition of the highly unstable buckling of shells at the classical load. (His research was conducted in Holland during World War II, and he managed to persuade the occupying powers to supply him with the latest American papers of von Kármán and others.) He found that the classical analysis of axially compressed long cylindrical shells produced multiple, competing eigenmodes; he discovered non-linear interactions between selected groups of these, which produced highly unstable post-buckling; and he showed that small-amplitude geometric imperfections (in the form of eigenmodes) as perturbations could greatly reduce the buckling load below the classical prediction.

That reduction of the classical load, in proportion to the square root of the imperfection amplitude, provides the well known ‘imperfection sensitivity’ of theoretical shell buckling; for example, an imperfection of amplitude 20% of wall thickness can lower the buckling load to 50% of the classical prediction.

But this theoretical ‘imperfection sensitivity’ is independent of the R/t value for the shell. Thus, in particular, it is only consistent with the empirical 1.5-power law of shell buckling, as seen clearly in the data-collection of Figure 1, if a hypothetical relationship is postulated between imperfection amplitude and wall thickness (John Hutchinson, private communication, 2017). Such a manoeuvre does not explain, of course, our primary experimental observation of a 1.5-power law in self-weight buckling, and the questions thereby raised; neither does it explain the small scatter in those data.

The only direct way that I can see to explain the empirical 1.5-power law (3) is to invoke the mechanics of dimples, surrounded by boundary layers. This branch of non-linear mechanics is not obviously within the reach of techniques that synthesise deformation-fields exclusively from eigenmodes of the linearised classical analysis.

The second advance recorded in this article concerns the beneficial effects, in terms of buckling performance, of using frictional end-fittings for cylindrical shells. Just as in the case of self-weight buckling, the striking results emerged as unexpected consequences of an experiment set up in order to investigate a somewhat vague hunch. I like to think of these two outcomes as examples of a feature commonly found in historical scientific investigations; namely the benefit of following up surprising and unexpected experimental observations.

Footnotes

Acknowledgements

I am grateful to students and colleagues who have helped with this work: Nicholas Barber, Roger Denston, Jim Greenwood, Werner Guggenberger, Mark Holst, John Hutchinson, Earl Lancaster, Partha Mandal, Stephen Palmer and Zhu Enchun. And I specially thank Michael Rotter for many helpful discussions over the years.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The author received no financial support for the research, authorship, and/or publication of this article except for the experimental work described in Lancaster et al. (2000).

References

Brush

Almroth

(1975) Buckling of Bars, Plates, and Shells. New York: McGraw-Hill Book Company.

Bushnell

(1967) Bifurcation phenomena in spherical shells under concentrated and ring loads. AIAA Journal 5: 2034–2040.

Calladine

(1983) Theory of Shell Structures. Cambridge: Cambridge University Press.

Calladine

(1988) The theory of shell structures, 1888–1988. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 202: 141–149.

Calladine

(2000) A shell-buckling paradox resolved. In: Durban

Givoli

Simmonds

(eds) Advances in the Mechanics of Plates and Shells (The Avinoam Libai Anniversary Volume). Dordrecht: Kluwer Academic Publishers, pp. 119–134.

Calladine

Barber

(1970) Simple experiments on self-weight buckling of open cylindrical shells. Journal of Applied Mechanics: Transactions of the ASME 37: 1150–1151.

Guggenberger

(2006) Elastic stability and imperfection sensitivity of axially loaded cylindrical shells on narrow supports. Computational Mechanics 37: 537–550.

Harris

Suer

Benjamin

(1957) The stability of thin-walled circular cylinders under axial compression, including the effects of internal pressure. Journal of the Aeronautical Sciences 24: 587–596.

Hibbitt, Karlsson & Sorensen, Inc. (1995) ABAQUS User’s Manual (version 5.5). Providence, RI: Hibbitt, Karlsson & Sorensen, Inc.

10.

Holst

JMFG

Calladine

(1994) Inversion problems in elastic thin shells. European Journal of Mechanics A: Solids 13(Suppl. 4): 3–18.

11.

Holst

JMFG

Rotter

Calladine

(2000) Imperfections and buckling in cylindrical shells with consistent residual stresses. Journal of Constructional Steel Research 54: 265–282.

12.

Kamyab

Palmer

(1989) Analysis of displacements and stresses in oil storage tanks caused by differential settlement. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 303: 61–70.

13.

Koiter

(1945) On the stability of elastic equilibrium (in Dutch, with English summary). PhD Thesis, H.J. Paris, Delft (English translation, Air Force Dynamics Laboratory Technical Report, AFFDL-TR-70-25, Ohio, February 1970).

14.

Lancaster

Palmer

Calladine

(2000) Paradoxical buckling behaviour of a thin cylindrical shell under axial compression. International Journal of Mechanical Sciences 42: 843–865.

15.

Mallock

(1908) Note on the instability of tubes subjected to end pressure, and on the folds in a flexible material. Proceedings of the Royal Society of London Series A 81: 388–393.

16.

Mandal

Calladine

(2000) Buckling of thin cylindrical shells under axial compression. International Journal of Solids and Structures 37: 4509–4525.

17.

Mescall

(1965) Large deflections of spherical shells under concentrated loads. Journal of Applied Mechanics: Transactions of the ASME 32: 936–938.

18.

Nowinka

Lukasiewicz

(1991) Symmetric elements in geometrical analysis of large deformations of spherical shells. International Journal of Non-Linear Mechanics 26: 151–168.

19.

Owen

(1967) Some aerodynamic problems of grouped cooling towers. In: Hannah

Owen

(eds) Natural Draught Cooling Towers – Ferry Bridge and After. London: Institution of Civil Engineers, pp. 73–78.

20.

Pogorelov

(1967) Geometrical methods in the non-linear theory of shells. In: Niordson

(ed.) IUTAM Symposium on the Theory of Thin Shells. Berlin: Springer-Verlag, pp. 294–300.

21.

Ranjan

Steele

(1977) Large deflection of deep spherical shells under concentrated load. In: Proceedings of 18th structures, structural dynamics and materials conference, San Diego, CA, 21–23 March, pp. 269–278. Reston, VA: American Institute of Aeronautics and Astronautics.

22.

Rotter

(2004) Chapter 2. Cylindrical shells under axial compression. In: Teng

Rotter

(eds) Buckling of Thin Metal Structures. London: Spon Press, pp. 42–87.

23.

Rotter

Teng

J-G

(1989) Elastic stability of cylindrical shells with weld depressions. Journal of Structural Engineering 115: 1244–1263.

24.

Timoshenko

Gere

(1961) Theory of Elastic Stability. 2nd ed.New York: McGraw-Hill Book Company.

25.

von Kármán

(1940) The engineer grapples with nonlinear problems. Bulletin of the American Mathematical Society 46: 615–683.

26.

von Kármán

Tsien

(1941) The buckling of thin cylindrical shells under axial compression. Journal of the Aeronautical Sciences 8: 303–312.

27.

von Kármán

Dunn

Tsien

(1940) The influence of curvature on the buckling characteristics of structures. Journal of the Aeronautical Sciences 7: 276–289.

28.

von Kármán

Sechler

Donnell

(1932) The strength of thin plates in compression. Transactions of the American Society of Mechanical Engineers 54: 53–57.

29.

Zhu

Mandal

Calladine

(2002) Buckling of thin cylindrical shells: an attempt to resolve a paradox. International Journal of Mechanical Sciences 44: 1583–1601.