Elastic Stability of Tubes in Pure BendingM. Khurram Wadee

AbstractWork at Exeter on structural buckling

has been concentrated on localizationphenomena where large deformations ofa structure are limited to a small regionrather than being distributed along it.Localized buckling is important instructures such as pipelines; structuralelements used in civil, mechanical andaeronautical engineering; railway tracksand in the bending of tubes which is alsoapplicable to aeronautical and biologicalsystems (e.g. the kinking of bloodvessels) and nanotechnology.

1 Introduction

Buckling is a fundamental way in whichstructural components can fail. It isassociated with the loss of stability of anequilibrium of a structure and can lead todramatic dynamic responses which arehighly nonlinear in nature. This means thatlinear analysis fails to account forpost-buckling behaviour and so wenecessarily turn to approximate methodswhich can be asymptotic or numerical or evena hybrid of these two types.

Localization of buckle patterns can occurin long, uniform structures where actuallarge deformation is limited to a small regionof the structure. Also the precise location ofthis region may be in any place with equalprobability.

Our work has concentrated on studyingthe phenomenology of localization usingheuristic and other mathematical models toinvestigate the nature of localized bucklingwhich is to be found in many physicalsituations. The outcome of the work hasresulted from a strong collaboration on theinterface between engineering and appliedmathematics and has led to pioneeringresults which have uncovered the nature oflocalization in the context of elasticstructures.

2 Tube bending: the Braziereffect and kink formation

When circular tubes are put into pure flexurethey are known to destiffen and ovalize.Brazier [1] modelled uniform ovalization andthis was later extended by Reissner [2].

Figure 1 The bending of an elastic transparentplastic tube depicting ovalization and kinking (aform of localization) followed by completeunloading.

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2

Reissner

Increasing t

α

m

Figure 2 Comparison of bending moment, m,vs curvature, α, for various tube wallthicknesses t.

0 100 200–10

010

–15–10

–505

101520

0 100 200–10

010

–15–10

–505

101520

0 100 200–10

010

–15–10

–505

101520

Figure 3 Modelling of steel tubes in pureflexure before, at and beyond the limit point.Dimensions in mm.

These fundamental analyses developedquantitative theories and accounted foruniform ovalization. However, we know thattubes do not just deflect equally everywherebut usually form a kink and fail before thepredicted moment (see Figure 1).

The new formulation, which is based ontotal potential energy, is able to model thevariation of deflection along the axis. Themodel uses an approach borrowed fromstudies of the buckling of sandwich panels(common in aeronautical structures) [3].

3 Conclusions and futurework

Work on tube bending has been driven fromthe need to understand the phenomenologyand aided through the development ofsophisticated mathematical techniques.Extensions of the model including study oforthotropic behaviour show great promise.

0 2000 4000 6000 8000 –1000 0 1000

–500

0

500

1000

1500

0 2000 4000 6000 8000 –1000 0 1000

–500

0

500

1000

1500

Figure 4 Buckling results for a nanotube takenfrom the literature (c.f. Pantano et al., J. Mech.Phys. Solids, 2004). Dimensions in pm.

External fundingMajor grant (GR/N05666/01) £138K2000–03: Elastic localization andrestabilization in the post-buckling ofmodel structures funded by:

PersonnelAcademic collaborators

• Professor Andrew P. Bassom(University of Western Australia)

• Dr M. Ahmer Wadee(Imperial College)

Post-doctoral research assistants

• Dr Ciprian D. Coman (2000–02)• Dr Andreas A. Aigner (2002–03)

References[1] L. G. Brazier. On the flexure of thin cylindrical

shells and other “thin” structures. Proc. R. Soc.Lond., A 116:104–114, 1927.

[2] E. Reissner. On finite pure bending of cylindricaltubes. Osterr. Ing. Arch., 15:165–172, 1961.

[3] M. K. Wadee, M. A. Wadee, A. P. Bassom, and A. A.Aigner. Longitudinally inhomogeneous deflectionpatterns in isotropic tubes under pure bending.Proc. R. Soc. Lond., A 462:817–838, 2006.