Elastic Stability of Tubes in Pure...
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Elastic Stability of Tubes in Pure Bending M. Khurram Wadee Abstract Work at Exeter on structural buckling has been concentrated on localization phenomena where large deformations of a structure are limited to a small region rather than being distributed along it. Localized buckling is important in structures such as pipelines; structural elements used in civil, mechanical and aeronautical engineering; railway tracks and in the bending of tubes which is also applicable to aeronautical and biological systems (e.g. the kinking of blood vessels) and nanotechnology. 1 Introduction Buckling is a fundamental way in which structural components can fail. It is associated with the loss of stability of an equilibrium of a structure and can lead to dramatic dynamic responses which are highly nonlinear in nature. This means that linear analysis fails to account for post-buckling behaviour and so we necessarily turn to approximate methods which can be asymptotic or numerical or even a hybrid of these two types. Localization of buckle patterns can occur in long, uniform structures where actual large deformation is limited to a small region of the structure. Also the precise location of this region may be in any place with equal probability. Our work has concentrated on studying the phenomenology of localization using heuristic and other mathematical models to investigate the nature of localized buckling which is to be found in many physical situations. The outcome of the work has resulted from a strong collaboration on the interface between engineering and applied mathematics and has led to pioneering results which have uncovered the nature of localization in the context of elastic structures. 2 Tube bending: the Brazier effect and kink formation When circular tubes are put into pure flexure they are known to destiffen and ovalize. Brazier [1] modelled uniform ovalization and this was later extended by Reissner [2]. Figure 1 The bending of an elastic transparent plastic tube depicting ovalization and kinking (a form of localization) followed by complete unloading. 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 wall thicknesses t. 0 100 200 –10 0 10 –15 –10 –5 0 5 10 15 20 0 100 200 –10 0 10 –15 –10 –5 0 5 10 15 20 0 100 200 –10 0 10 –15 –10 –5 0 5 10 15 20 Figure 3 Modelling of steel tubes in pure flexure before, at and beyond the limit point. Dimensions in mm. These fundamental analyses developed quantitative theories and accounted for uniform ovalization. However, we know that tubes do not just deflect equally everywhere but usually form a kink and fail before the predicted moment (see Figure 1). The new formulation, which is based on total potential energy, is able to model the variation of deflection along the axis. The model uses an approach borrowed from studies of the buckling of sandwich panels (common in aeronautical structures) [3]. 3 Conclusions and future work Work on tube bending has been driven from the need to understand the phenomenology and aided through the development of sophisticated mathematical techniques. Extensions of the model including study of orthotropic 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 taken from the literature (c.f. Pantano et al., J. Mech. Phys. Solids, 2004). Dimensions in pm. External funding Major grant (GR/N05666/01) £138K 2000–03: Elastic localization and restabilization in the post-buckling of model structures funded by: Personnel Academic 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 cylindrical tubes. ¨ Osterr. Ing. Arch., 15:165–172, 1961. [3] M. K. Wadee, M. A. Wadee, A. P. Bassom, and A. A. Aigner. Longitudinally inhomogeneous deflection patterns in isotropic tubes under pure bending. Proc. R. Soc. Lond., A 462:817–838, 2006.
Transcript of Elastic Stability of Tubes in Pure...
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.
