## Abstract

A variational model is formulated that accounts for the localization of deformation due to buckling under pure bending of thin-walled elastic tubes with circular cross-sections. Previous studies have successfully modelled the gradual process of ovalization of the cross-section with an accompanying progressive reduction in stiffness but these theories have had insufficient freedom to incorporate any longitudinal variation in the tube. Here, energy methods and small-strain nonlinear elastic theory are used to model the combined effects of cross-section deformation and localized longitudinal buckling. Results are compared with a number of case studies, including a nanotube, and it is found that the model gives rise to behaviours that correlate well with some published physical experiments and numerical studies.

## 1. Introduction

The buckling of structural components under various types of loading is a common cause of failure in structural engineering and other areas of applied mechanics. Elastic instability theory can be be used as a tool to investigate the transition from a safe structure to an unsafe one. It is known that a wide range of buckling phenomena is possible; contrasting cases are global Euler buckling seen in compressed struts and thoroughly localized buckling which is common in shell-type structures (Hunt *et al*. 1999). There are also instances when one type of buckling can herald the onset of another: one classical example is provided by a sandwich panel comprising two stiff face plates bounding a soft core material. Such a structural element, which is of great importance in the aeronautical industry, normally initially buckles in a global sense when subjected to an axial compression but a secondary instability can lead to localization in one of the face plates (Hunt & Wadee 1998).

A less well-studied problem from the perspective of classical elastic instability theory is that of localized buckling in tubes under pure bending. The initial bending behaviour of tubes is characterized by a uniform ovalization of the cross-section; this has been known for a long time (Brazier 1927). The transition from uniform ovalization to localized deformation in the form of kinking, though frequently observed in practice (Seide & Weingarten 1961; Kyriakides & Ju 1992; Wang & Cao 2001) has proved difficult to model mathematically. On the other hand, numerical simulations of macroscopic (Gellin 1980; Ju & Kyriakides 1992) and nanoscale (Pantano *et al*. 2003) tubes have successfully reproduced this effect. Our aim here, rather than rely on direct numerical simulations, is to formulate a phenomenological model to account for kinking as an elastic instability which can be applied to general isotropic circular tubes. It is worth contrasting this structural problem to that of the buckling of thin cylindrical shells under axial compression, where localization has been the subject of intense study in recent times (Champneys *et al*. 1999). The latter system is characterized by localization axially while a form of periodicity is maintained in the circumferential direction.

Previous work on this problem has concentrated largely on modelling the ovalization phenomenon taking place uniformly along the length of the tube (Calladine 1983; Harursampath & Dewey 1999), to the detriment of the localization. The gradual change in cross-section leads to a reduction in the bending stiffness of the tube with increasing applied moment. Many attempts, for example the work of Molyneaux & Li (1996), have been made to model this destiffening without addressing the fact that tubes can undergo kinking when the curvatures become sufficiently large. This may occur either wholly within the elastic range of the material or after plasticity has set in. This usually is manifested as a secondary bifurcation of the equilibrium path leading to kinking. A result of this further instability is that deformations may become so large as to cause the material to become plastic. This advanced phenomenon is beyond the scope of the current model.

Several modelling and approximation techniques have been applied to the study of tube bending (Fabian 1977; Emmerling 1984; Tatting *et al*. 1996). If the tube is supposed to be thin, series solutions can be developed which describe small perturbations from linear elasticity. An alternative methodology stems from the use semi-membrane theory (Libai & Bert 1994). One deficiency common to these various studies is that they are only able to account for a homogeneous deformation of the tube (cf. the experimental findings described earlier). In this work we present an analysis of tube bending that has the capability of incorporating both the ovalization and localization facets of the observed behaviour. Our model is developed as a combination of a so-called Reissner model with Timoshenko beam theory.

Reissner (1961) used a small-strain bending theory to derive equations for a closed tube of arbitrary cross-section and variable thickness. The novelty in his formulation was that it relaxed usual engineering beam bending assumptions and permitted the section to deform as load is applied. Consideration of the appropriate potential energy functional enabled Reissner to find minimizing series solutions which reproduced Brazier's softening moment–curvature response at the lowest order of approximation (see appendix A for an outline of this analysis for a circular tube). However, in order to develop a model capable of exhibiting localization as well as Brazier ovalization, appropriate nonlinearities must be incorporated from the mechanics of the system; softening properties usually provide such effects (Hunt & Wadee 1991).

Standard engineering bending theory is based on the Euler–Bernoulli assumptions in which beam bending is assumed to occur in the absence of shear strain. In contrast, Timoshenko beam theory allows shear deformations to develop with the result that deformation at one cross-section can have an effect on its neighbours thereby leading to a longitudinally inhomogeneous variation of deformation of the section under constant applied moment. This type of analysis has been used for modelling bending deformation in sandwich panels and is incorporated within the current problem.

Here, we shall model the phenomenon of tube bending with the approach outlined above. Enough of the physics of the problem is incorporated so as to permit axially varying section profiles which can capture the essence of kinking phenomena. The mathematical model is introduced in §2, where the complementary features of Reissner's ovalization model and Timoshenko beam theory are described. An expression for the total potential energy in the system is obtained and variational techniques are employed to derive the governing ordinary differential equations (ODEs) for the problem. This system of equations is derived in §3 and some numerical solutions are discussed in §4. This computational work was executed using the Auto97 package (Doedel *et al*. 1997) and the results are analysed in the light of those obtained by Reissner. It is also an important test of a new model that it is of some relevance to practical problems and, to this end, we benchmark it against two sets of results taken from the literature. One of these concerns experiments using sections of aluminium alloy while the second relates to numerical simulations of bending nanotubes. Our work concludes in §5 with a brief discussion of the results and suggestions of avenues for further study.

## 2. Model formulation

Consider an initially straight tube of length *L* with radius *r* and wall thickness *t* made from a homogeneous and isotropic linear elastic material with Young's modulus *E* and Poisson's ratio *ν*. Suppose that a coordinate system is aligned so that the centreline of the tube is horizontal and coincides with the *z*-axis. Moreover, the remaining axes are rotated so that the *x*-direction is also horizontal and the tube is acted upon by a bending moment *M*—the configuration being summarized in figure 1*a*. Note that under such loading, at least to first order, the longitudinal strain in the *z*-direction *ϵ*_{z} is proportional to the distance from the *x*-axis.

The remaining sub-parts of figure 1 illustrate the plausible behaviour of a typical cross-section of the tube as it is subjected to increasing loading. In the unloaded state the tube is straight and the section is perfectly circular (figure 1*b*,*e*) but if a small moment is applied it attains a constant curvature and ovalizes (figure 1*c*,*f *). Notice that at this stage the deformation continues to have both vertical and horizontal axes of symmetry. However, once the load is sufficiently high, the tube becomes unstable and a breaking of symmetry (about the horizontal axis) occurs, see figure 1*d*,*g*. The cross-section is now characterized by the presence of a kinking phenomenon but we would not expect this to occur simultaneously at all sections along the tube. Rather, the kink forms over a relatively short length giving rise to the desired localization of buckle pattern along the tube.

We start our modelling by borrowing ideas from the analysis of sandwich panels (Hunt & Wadee 1998) in which the structural element is assumed to behave according to the bending theory of Timoshenko. Implementation of this approach for a tube under constant end moment shows that the vertical deflection of the neutral axis, *W*(*z*), is a circular arc approximated by a parabola. Moreover, if *θ*(*z*) is the angle made with the normal of the deformed longitudinal neutral axis by a plane section which was initially perpendicular to the undeformed axis of the tube, then(2.1)Here *W* is measured downwards and the quantities *q*_{s} and *q*_{t} denote the dimensionless amplitudes of the shear and tilt of initially planar vertical faces. The magnitude of tilt is readily identified with the radius of curvature (*R*) of the bent tube via(2.2)For Euler–Bernoulli beams *q*_{s} and *q*_{t} are always equal; this is the upshot of requiring that the shear strain be zero (i.e. d*W*/d*z*=*θ*). However, in Timoshenko's more general beam theory, this constraint is relaxed and *q*_{s} and *q*_{t} can be different; the adoption of this beam theory is necessary in order to allow for the potential variation in the shape of the cross-section along the tube.

By itself Timoshenko's theory is incapable of capturing ovalization but this is addressed by incorporating Reissner's model which includes the Brazier effect in a systematic manner. We begin by summarizing the uniform phenomenon before embarking on modelling of the localization.

### (a) Homogeneous ovalization of the tube

Here, we will recount briefly a few key results from Reissner's analysis (1961). It is convenient to introduce the following elastic constants(2.3)Additionally we denote by *α* a dimensionless measure of curvature(2.4)Figure 2 summarizes some of the salient notation that will be referred to in the course of the development. Suppose that a point (*x*, *y*) lies on the original undeformed circular section; it will be helpful to make reference to the angle *φ* denoting the angle made by the radius from the origin to (*x*, *y*) with the negative *y*-axis. When the section is ovalized we suppose that the point originally at (*x*, *y*) is now at and that the deviation of the tangent to the oval from its direction when the tube is undeformed is *β*; in all that follows *β* is assumed to be small. Reissner (1961) derived expressions for the deflection of the tube section under the action of a constant moment *M* and proved that the point (*x*, *y*) on the undeformed circle moves to , where(2.5)These approximations come from considering the elastic deflection of an elementary piece of the tube under uniform applied moment under a modified bending theory, where incremental deformation of the section under load is allowed (see appendix A). Formally, expressions (2.5) are truncations of infinite series in *α* but they converge rapidly and so only the terms up to *α*^{4} have been shown. Notice that these forms ensure that the deflection of the centreline of the tube is symmetric about both the *x*- and *y*-axes (figure 1*c*,*e*).

The corresponding moment–curvature relationship can be written in dimensionless terms as(2.6)and is noted for future reference in the discussion of our results.

### (b) Deviation of the cross-section from Reissner's formulation

Reissner's approach accounts for the ovalization of the cross-section under increasing moment. As the section becomes flattened so the moment–curvature graph destiffens. Among the assumptions made in the analysis is that the uniform deformation of the tube under load is such that no stretching of the material occurs in the circumferential direction—hence an element of the section of length d*s* remains unchanged during deformation. We shall not enforce this requirement and instead allow the tube to undergo an additional small radial displacement *w*(*x*, *φ*) which in turn accounts for any *z*-dependent variation in the profile. It is this that enables the tube to exhibit axially varying deformation after an initial global ovalization phase.

According to linear bending theory the first-order axial strain in the *z*-direction is(2.7)In order to incorporate nonlinear strain effects the von Kármán expression for strain is used(2.8)This quantity includes the influence of in-plane deflection, , which we assume to vary linearly with vertical distance from the neutral axis of the section, i.e. . This term is necessary to include the effect of deflection in the plane of the surface of the tube on strain in the *z*-direction. Note that if the section were to remain undeformed during loading then these terms would reduce to the standard expression for bending energy.

### (c) Energy components

The strategy underpinning the derivation of the salient governing equations relies on variational techniques applied to an appropriate potential energy functional. So far we have seen that with the ovalization followed by localization the point originally at (*x*, *y*) moves to with *w*=*w*(*z*, *φ*). In order to proceed we need to deduce the appropriate form for the potential energy in terms of these quantities and remark that contributions arise from the following five sources.

#### (i) Bending energy

Under conditions of constant moment, the tube initially behaves like a beam with a general cross-section and deflects into a circular arc. For small deflections, this can be approximated by a parabola, see equation (2.1). As we intend to account for ovalization of the tube during this deformation, it is necessary to incorporate the incremental change in shape as it occurs. Hence, the conventional term for second moment of area of an elemental piece of the section, *y*^{2}*t* d*s*, is replaced by the term . The strain energy of this component plus that due to additional *z*-dependent local deflection of the section is then(2.9)where a dot denotes differentiation with respect to *z*. Note that we have retained only the terms that will make non-zero contributions to the final forms of the governing equations. In particular, the cross-term proportional to arising from the strain energy is omitted since *W* is assumed to be quadratic in *z* and so the cross-term is linear in and has no effect on our final model.

Depending upon the precise form of *w* this term can account for asymmetry of deflection about the *x*-axis meaning that the neutral axis may no longer coincide with that axis (see equation (3.1)). However, since *w* is assumed to be sufficiently small throughout we have neglected any secondary complications such as progressive increase in local bending moments.

#### (ii) Membrane energy

Using the von Kármán strain expression the membrane energy can be cast as(2.10)

#### (iii) Shear strain energy

Shear strain is incorporated within our formulation and, including the components arising from the incremental deformation *w*, we have (Hunt & Wadee 1998)(2.11)The contribution to the potential energy from shear strains is given by(2.12)where is the shear modulus of the material.

#### (iv) Circumferential bending energy

A small but important contribution to the energy arises due to the additional deformation *w*. The additional curvature is leading to the term(2.13)In passing notice that this contribution is analogous to the role played by the elastic foundation in the strut model—the focus of much study as an archetypal localized buckling problem. Its effect is to cause the response to become independent of the length of the structure as long as the boundaries are sufficiently far away from the position of significant deformation (Hunt & Wadee 1991).

#### (v) Work done by load

The work done is the sum of the contributions from the internal and external moments. Each term is basically the moment multiplied by the local rotation, which is, therefore,(2.14)

## 3. Potential energy functional and governing equations

The total potential energy for the structural system is given by the sum of the contributions §§2*c*(i)–2*c*(iv) of strain energy formulated above in equations (2.9)–(2.13) minus the work done by the load (2.14). It is also helpful to note that the requisite extremal problem would be much simplified if the double integrals could be reduced to single ones and this can be done with relatively little difficulty. There are added bonuses as well: when performing the *s* (or equivalently, *φ*) integrals in equations (2.9)–(2.14) we can ensure that *w*(*z*, *φ*) is necessarily periodic in *φ*. To do this we assume a Fourier cosine series for *w*(*z*, *φ*) so(3.1)of course we must ensure that *w*(*z*, *φ*) is even-valued in *φ* for the deflection of the tube to be symmetric about the *y*-axis in the context of figure 2.

At this stage we now have the potential energy in terms of integrals of the functions *w*_{n}(*z*); application of variational principles would lead to an infinite system of ODEs for these quantities. In practice one needs to truncate this system at some point and here an admittedly very severe form was adopted—for all the computations described below we set *w*_{n}≡0 for *n*≥3. In practice, experiments report that tubes buckle via non-axisymmetric modes and the retention of *w*_{0}(*z*), *w*_{1}(*z*) and *w*_{2}(*z*) allows our model to capture non-axisymmetric forms of solution. In particular *w*_{0} is an axisymmetric dilation, *w*_{2} is a non-axisymmetric mode which maintains symmetry about the *x*- and *y*-axes and *w*_{1} is a mode with symmetry only about the *y*-axis. To illustrate the usefulness and power of the proposed model, our desire was to minimize the complexity of the underlying problem and some experimentation revealed that ignoring everything except for *w*_{0}, *w*_{1} and *w*_{2} was sufficient both to capture the ovalization and localization phenomena sought and provides good comparison with previous results (see §4). Of course a more refined picture could be constructed by inclusion of more terms (with a correspondingly more involved numerical task). Note that, for completeness, the form of the potential energy *V* when expressed in terms of *w*_{0}, *w*_{1} and *w*_{2} is recorded in appendix B. It is convenient to replace *α* with the expression (2.4) so that it can be manipulated more easily in subsequent expressions, where the identification of *q*_{t} becomes important.

The functional form of the potential energy is now typified by , where *Q* is any of *w*_{0}(*z*), *w*_{1}(*z*), *w*_{2}(*z*) or *u*(*z*). Such an integral is extremized if(3.2)With the assumption of simple supports at the ends of the tube the physical boundary conditions(3.3)eliminate most of the boundary terms in equation (3.2) when *Q*≡*w*_{n}, where *n*=0, 1, 2. However, slightly more complicated boundary conditions arise for :(3.4)The second part of the minimizing process is to eliminate the integrand in equation (3.2). This leads to sets of ODEs for each of the variables *w*_{0}(*z*), *w*_{1}(*z*), *w*_{2}(*z*) and *u*(*z*). Additionally, we minimize *V* with respect to the two parameters *q*_{s} and *q*_{t} and these requirements, that and , give rise to two integral constraints. In total the four ODEs and two integral conditions are(3.5)(3.6)(3.7)(3.8)(3.9)(3.10)It is worth noting that if the assumption of isotropy is maintained then the material Young's modulus *E* cancels from the ODEs and its effect is confined to the absolute level of the external moment *M*. We now face the task of solving system (3.5)–(3.8) subject to boundary conditions (3.3) and (3.4) and the integral constraints (3.9) and (3.10).

## 4. Numerical solutions and comparisons

The system (3.3)–(3.10) was solved using the numerical continuation package Auto97. To conserve the inherent symmetry of the system the equations were solved over half the length (*z*=[0, *L*/2]) and the symmetry conditions at *z*=*L*/2(4.1)which also minimize *V*, are imposed. Solutions were obtained from the initially undeformed and unloaded state () and *q*_{t} was increased parametrically with Auto97 solving the ODEs subject to the boundary conditions and integral constraints, giving numerical values of *q*_{s} and *M* that satisfied the equilibrium equations. The ability of the software to evaluate the location of bifurcation points is well known and was employed to determine the limit points and the post-buckling behaviour discussed below. Also it is important to realize that no knowledge of any initial imperfections is necessary in order to initiate buckling using this technique. Key features such as the wavelength of buckling on the compressed side develop gradually during evolution of the system from the undeformed state rather than in a more abrupt way that might occur with a pitchfork bifurcation. Comparison of this wavelength with independently obtained values is a good way to compare the current model with the literature; further comments on this are postponed until §4*d*.

In order to obtain solutions of the problem it is obviously necessary to specify realistic values for the various physical parameters involved. In this we were guided by three contrasting cases: one dealing with an idealized tube, one taken from an experimental study on aluminium sections and the last concerned with finite element simulations of the behaviour of nanotubes; the configurations are summarized in table 1. It is worth remarking that the cases presented in the numerical study cover a reasonably wide spectrum of geometries from the short in terms of the ratio *L*/*r*=10 to the slender *L*/*r*=48, and from a thin-walled tube (*r*/*t*=100) to one approaching thick-walled proportions (*r*/*t*≈6).

### (a) Localization in steel tubes

In the first study we explore solutions within steel tubes of varying parameters. When loaded, previous work has shown that the moment–curvature relationship initially has an approximately linear elastic response for small curvatures followed by a reduction in stiffness caused by increasing cross-section ovalization leading to a limit point (or maximum moment capacity) being exhibited. Beyond the limit point is a régime of negative stiffness that would cause structural failure in practical situations under conditions of load control. In the current study many geometric parameters were varied but it turned out that the behaviour of the tube was relatively insensitive to changes in length, as long as the ratio of *L*/*r* did not become too small (e.g. less than 10). The only property that seemed to be strongly dependent on *L* was the number of waves appearing along the length of the tube. In particular it was found that the applied moment at the limit point was only a weak function of *L* but this independence of buckling load on length is a trait of localization such as that seen in the strut-on-foundation model (Hunt & Wadee 1991). We, therefore, decided to focus attention on varying the ratio *r*/*t*. This strategy yielded more significant results and the configurations chosen are given in the first line of table 1. The wall thickness was varied parametrically from *t*=0.1 mm to *t*=1.0 mm. The equilibrium paths of the tubes and the profiles of the cross-sections are plotted in figure 3 and show the comparisons between the ovalization of the cross-section at the limit point and far into unstable region (at *m*=0.5, with *m* as defined in equation (2.6)).

The equilibrium diagram in figure 3*a* is plotted as dimensionless moment *m* versus dimensionless curvature *α* (equation (2.3)). The inhomogeneous deflection in the model means that the measure *m* only defines the overall applied bending moment. This is a function solely of the end rotations and we maintain this measure as a means of directly comparing with other results. As can be seen from figure 3*a* the model curve, although starting with the same stiffness as Reissner's model, quickly begins to soften as the inhomogeneous ovalization takes hold within the structure that leads to localization; a limit point is encountered far earlier than Reissner's curve and unloading takes place. It is also worth noting that the relative magnitude of the dimensionless moment at the limit point increases consistently with increasing wall thickness *t* as might be expected as a stockier section intuitively should increase the buckling load of the tube towards Reissner's limit-point load.

Detailed solutions of *w*(*z*, *φ*) for the profile of the tube before, at, and beyond the limit points are shown in figures 4 and 5 for *t*=0.3 and 1 mm, respectively (the axes are those defined in figure 1*a*). Note the significant thinning of the cross-section at midspan far into the post-buckling range. The rippling of the compression (lower) side in the post buckled tube can be discerned in figure 6, which shows the deflection of the upper and lower fibres of the tube in the post-buckling range when *m*=0.5. This variation of the profile around the circumference is to be expected as buckling in the compressive region of the bent tube spreads around the circumference and may perhaps even penetrate part of the tensile region but in the parts where tensile stresses are high the modulated buckled profile is absent; this has also been observed experimentally (Wang & Cao 2001).

### (b) Kyriakides's aluminium alloy tubes

Kyriakides & Ju (1992) describe detailed results for 11 aluminium alloy tubes: we shall examine two of these here. These two were selected as they have relatively small imperfections and are as listed in the second (aluminium tube 1) and third lines (aluminium tube 2) of table 1. The respective numerical results are shown in figures 7 and 8. It is worth noting here that these tubes exceeded the material elastic limit and suffered from substantial plastic deformation during the experiments due to the large deformations present during the localization of the buckle pattern. Therefore, the quantitative comparisons for the maximum moment is not so good. However, the curvature at the limit point correlates well as does the qualitative nature of the ovalization of the cross-sections.

### (c) Pantano's nanotube

The new and rapidly developing field of nanotechnology has given fresh impetus to the study of various areas of solid mechanics. Despite the fact that the structures under investigation are composed of individually identifiable atoms continuum mechanics has nonetheless been found to be useful in studying these problems (Pantano *et al*. 2003). Following this lead we have applied our model to an example from the literature.

In Pantano *et al*. (2004) detailed results are given for the single-walled carbon nanotube described in the final line of table 1. Figure 9 shows the displaced shape of the tube at the limit point (*q*_{t}=0.549) and at the point shown in Pantano's paper (total rotation: 1.64 radians which equates to *q*_{t}=0.820). Figure 10 shows the change in the cross-section from the limit point to this advanced state; note the asymmetry in the deformation about the horizontal axis. Table 2 shows a comparison between the numerical results from Pantano *et al*. and the current model, which shows excellent correlation for the onset of the limit point instability but a relatively stiffer result far into the post-buckling range. This discrepancy can be accounted for by the very large end rotation present at this stage in the loading; the shear strains at this level would be sizeable and the Timoshenko beam approximation is probably breaking down—higher order beam theories may well rectify this in the future (Wang *et al*. 2000).

### (d) Wavelength comparisons

The wavelength of the buckle pattern on the compressed side of the tube provides an important measure of the validity of the current model. Table 3 gives some results for the wavelength of ripples on the compressed side of the tube for the configurations presented in table 1. The wavelengths quoted were measured directly from figures in the appropriate papers and so there is inevitably some uncertainty in the estimates. Nevertheless, as can be seen, the comparisons are very encouraging indeed. (Note that Kyriakides & Ju (1992) also identify a second, shorter, kink wavelength but that is not the one with which our comparisons have been made as it is associated with local plastic deformation of the material.)

## 5. Concluding remarks

A variational model has been developed which allows for longitudinally inhomogeneous ovalization in the pure bending of tubes. Unlike earlier work in this area, the localization phenomenon that is accounted for arises from purely geometric considerations by assuming a higher order bending model that gives deflection profiles which may vary with the axial coordinate. This gives quantitative results that are less stiff than would arise from uniform ovalization modelled by Brazier and Reissner and also buckle significantly earlier; a series of physical examples of various scales and materials have been tested from the literature with good correlation. Plasticity effects, although particularly important in the quantitative response of metallic tubular components, are deemed to be of higher order in the current study; in that they would just exacerbate the elastic deformation into plastic hinges and permanent deformation.

There is plenty of scope to extend this work for more complicated material models and differing loading configurations. More specifically, the assumption of isotropy can be removed to look at the practically important case of orthotropic materials. In that case there is no link between the shear modulus and Young's modulus and there is much greater freedom to vary those parameters. For structures that exhibit a length scale nonlinear interaction, for instance sandwich panels (Wadee & Hunt 1998), the severity of the instability is considerably increased with the use of orthotropic core materials rather than isotropic ones.

In terms of different loading conditions there are several cases which may merit further investigation, e.g. tubes under axial compression or tubes in bending with internal or external pressure. Under axial compression, unlike conventional shell buckling for which there is a huge amount of literature, the length scale interaction would be almost certainly begin with the Euler mode of overall buckling before evolving to localization via a secondary bifurcation in a similar way as found in sandwich struts and stiffened plated structures. As tubular structures have recently found increased popularity within the construction industry, physical modelling of their behaviour has become more important. Tubes under internal or external pressure commonly arise not only in structural engineering but also in the biomedical field (Heil & Pedley 1996; Pedley & Luo 1998)—the sudden overstress of collapsible tubes such as blood vessels, veins, arteries and so on under flexure from, for instance, a large impact can easily lead to physical shock or trauma. Modelling for this type of structural situation may lead to better understanding of this phenomenon and improve therapeutic treatments for these severe kinds of injury.

## Acknowledgments

Funding for A.A.A. was provided by a UK Engineering and Physical Sciences Research Council research grant (ref.: GR/N05666/01). We are grateful to the referees for their valuable comments.

## Footnotes

- Received March 7, 2005.
- Accepted October 14, 2005.

- © 2006 The Royal Society