Quasi-static axial compression of thin-walled circular aluminium tubes

International Journal of Mechanical Sciences 43(2001)2103–2123Quasi-static axial compression of thin-walled circular aluminium tubesS.R.Guillow a ,G.Lu a ;∗,R.H.Grzebieta ba School of Engineering and Science,Swinburne University of Technology,PO Box 218,Hawthorn,Victoria 3122,Australiab Department of Civil Engineering,Monash University,Clayton,Victoria 3168,AustraliaReceived5October 2000;receivedin revisedform 26March 2001AbstractThis paper presents further experimental investigations into axial compression of thin-walledcircular tubes,a classical problem studied for several decades.A total of 70quasi-static tests were conducted on circular 6060aluminium tubes in the T5,as-receivedcond ition.The range of D=t considered was expanded over previous studies to D=t =10–450.Collapse modes were observed for L=D 610anda mod e classiÿcation chart developed.The average crush force,F AV ,was non-d imensionalisedandan empirical formula establishedas F AV =M P =72:3(D=t )0:32.It was foundthat test results for both axi-symmetric and non-symmetric modes lie on a single prehensive comparisons have been made between existing theories andour test results for F AV .This has revealedsome shortcomings,suggesting that further theoretical work may be required.It was found that the ratio of F MAX =F AV increasedsubstantially with an increase in the D=t ratio.The e ect of ÿlling aluminium tubes with di erent density polyurethane foam was also brie y examined.?2001Elsevier Science Ltd.All rights reserved.Keywords:Axial compression;Circular tube;Foam;Plastic collapse;Thin-walledtubes0.IntroductionThe behaviour of thin-walledmetal tubes subjectedto axial compression has been stud iedfor many years.Such tubes are frequently usedas impact energy absorbers andReid[1]has pre-sented a general review of deformation mechanisms.Fig.1shows a typical force–displacement curve for quasi-static loading.Generally speaking,the axial load rises until a ÿrst buckle is formedat a characteristic maximum force value,F MAX .This initial buckling behaviour is well ∗Corresponding author.Fax:+61-3-9214-8264.E-mail address:glu@.au (G.Lu).0020-7403/01/$-see front matter ?2001Elsevier Science Ltd.All rights reserved.PII:S 0020-7403(01)00031-52104S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123NomenclatureD average diameterF AV average axial forceF MAX maximum axial force forÿrst peakg acceleration due to gravityH half-wavelength of foldL lengthM P full plastic bending moment of tube wall per unit lengthm geometric eccentricity factor—i.e.ratio of outward s foldlength to total foldlength N number of circumferential lobes(or corners)in non-symmetrical bucklingR average radiust wall thickness of tubeV Vickers hardness number(kg=mm2)e e ective crushing distancef density of foam0 ow stress0:20.2%proof stressult ultimate tensile stressFig.1.Typical load–de ection curve for an axially loaded thin-walled metal tube which collapsed by progressive folding.S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–21232105Fig.2.Examples of various collapse modes for thin-walled circular6060-T5aluminium tubes under axial load-ing(more examples shown in Fig.11):(a)axi-symmetric mode(D=97:9mm;t=1:9mm;L=196mm);(b)non-symmetric mode(D=96:5mm;t=0:54mm;L=386mm);(c)mixedmod e(D=97:5mm; t=1:5mm;L=350mm).known and will not be studied in depth here.Thereafter,depending on geometrical parameters such as the ratios of D=t(diameter=thickness)and L=D(length=diameter)and also on material properties,there are a variety of possible modes of collapse.Generally,collapse involves plastic2106S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123Fig.3.Schematic axial view of non-symmetric or diamond collapse mode.Two cases are shown,N=3and4 circumferential lobes.buckling andthe formation of progressive fold s(whether axi-symmetric or non-symmetric).The formation of these folds causes the characteristic uctuation in the axial force shown in Fig.1. This plastic collapse behaviour is of primary interest in this paper.Experimentally the following modes of collapse have been observed and Fig.2shows some typical examples:(i)axi-symmetric concertina bellowing,(ii)non-symmetric buckling(also known as diamond or Yoshimura mode),with a variable number of circumferential lobes or corners(refer to Fig.3),(iii)mixed mode(combination of the two previous modes),(iv)Euler or global buckling;and(v)other(simple compression,single fold s,etc.).Research on circular tubes in the past has generally concentratedon annealedaluminium or steel tubes with D=t ratios between10and150.It is common ind ustrial practice to use aluminium alloys in the heat treatedas-receivedcond ition,but little research appears to have addressed this particular case.Moreover,Gupta and Gupta[2]have identiÿed metal temper as one of the signiÿcant factors in determining behaviour.Hence,it was decided to undertake an experimental program to extendthe range of research up to approximately D=t=450andto test aluminium alloy tubes which were in the heat treatedas-receivedcond ition.This work is of potential application in civil,mechanical,marine andaeronautical engineeringÿeld s.1.Review of previous studiesThe following section summarises the available literature on the plastic collapse behaviour of thin-walledcircular metal tubes subject to quasi-static axial load ing.It is arrangedbroad ly in a chronological order.Theÿrst signiÿcant work to address the mechanics underlying the observed behaviour of axially load edthin-walledtubes was by Alexand er[3].He proposeda simple mod el for the axi-symmetric foldpattern(refer to Fig.4)basedon experiments with metal tubes of D=t= 29–89.At a global level,external work done was equated with internal work from bending at three stationary plastic hinges andcircumferential stretching of the metal between the hinges.S.R.Guillow et al./International Journal of Mechanical Sciences 43(2001)2103–21232107Fig.4.Axi-symmetric collapse mechanism assumedby Alexand er [3].Thus the following theoretical equation was obtainedfor average crush force (axi-symmetric folds):F AV =K o t1:5√D;(1)where K is a constant and o is the ow stress.Also,plastic half-wavelength,H (refer to Fig.4)was determined as follows:H =C √Dt;(2)where C is a constant.The experimental results observedby Alexand er were generally in agreement with the above two equations.Although simple,this model seems to re ect the und erlying physical processes involvedandmany subsequent researchers have usedit as a starting point.Pugsley andMacaulay [4]were among the ÿrst researchers to consid er the non-symmetric folding mode,their study being largely empirical.Johnson et al.[5]attempted to develop a theory for the non-symmetric mode based on the actual geometry of folding,with the tube material at the mid-surface being considered inextensional.Hence they were able to develop equations to predict average axial crush force,F AV .However,agreement between their model andtest results for P.V.C.tubes was not particularly good .In 1978Magee andThornton [6]cond ucteda review of previous work by researchers who hadcond uctedaxial crushing tests on circular metal tubes.By consid ering these collectedd ata they d evelopeda number of empirical equations which involvedthe speciÿc ultimate tensile strength of the metal.Andrews et al.[7]conducted a comprehensive series of tests on annealed aluminium alloy tubes covering a wide range of D=t (4–60)and L=D (0:2–8:8).Consequently,they developed a collapse mode classiÿcation chart which predicted the mode of collapse for any given D=t and L=D combination.2108S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123Fig.5.Axi-symmetric mod el usedby Abramowicz andJones[8,9].H is the half-wavelength of the fold. Abramowicz andJones[8,9]cond uctedaxial compression tests on a range of thin-walledcir-cular andsquare steel tubes.They analytically consid eredboth axi-symmetric andnon-symmetric modes.Abramowicz introduced the important concept of e ective crushing distance, e(refer to Fig.5),where a foldconsistedof two equal rad ii segments of length H,curvedin opposite d irections andthe material hadÿnite thickness.For axi-symmetric folds,Abramowicz and Jones[9]developed the following equation in1986 (anda similar one in1984[8]):F AV M P =[25:23D=t+15:09][0:86−0:568t=D];(3)where M P= o(t2=4):For non-symmetric fold s,in1984[8]and1986[9]Abramowicz andJones commencedwith two di erent starting relationships.Taking into account e ective crushing distance,material strain rate,etc.,resultedin two d i erent equations for average crush force.The simple relation-ship d evelopedin1984[8]appliedregard less of the number of lobes:F AV M P =86:14Dt0:33:(4)However,thisÿnding appears to have developed from work carried out by Wierzbicki and Abramowicz[10]on rectangular rather than circular tubes.On the other hand,the relationshipS.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–21232109Fig.6.Collapse mechanism assumedby Grzebieta[12]for axi-symmetric mod e. Abramowicz andJones d erivedin1986[9]was of the formF AV M P =A N1Dt+A N2;(5)where A N1and A N2are constants which were a function of the number of lobes.For further details the reader is directed to this reference.In Refs.[8,9],Abramowicz andJones observedthat reasonable agreement existedbetween pred ictions of average crush load s basedon the above notedequations andtheir experimental results for steel tubes with D=t=9–65.In a subsequent work,Abramowicz andJones[11] reportedon further tests andsummarisedtheirÿnd ings for both static andd ynamic load ing cases in two failure mode maps,adding to the previous work by Andrews et al.[7].Gupta andGupta[2]performeda series of quasi-static axial compression tests on thin-walled aluminium andmildsteel circular tubes in both the annealedandas-receivedcond itions.They combinedall results andd evelopedempirical equations for the average crushing force in terms of the Vickers hardness and D=t.Grzebieta[12–14]useda strip methodto analyse both axi-symmetric andnon-symmetric folding modes.He equated external work done with internal energy from horizontal,inclined and travelling plastic hinges as well as stretching of the metal,to produce equations for determining the instantaneous forces involved.Unfortunately,these equations do not yield simple expressions for determining the average crush force.Grzebieta’s collapse mechanism model for axi-symmetric mode(refer to Fig.6)was a mod-iÿcation of Alexand er’s.A foldconsistedof three equal lengths,two of which were curves of equal rad ius andthe thirda straight line segment.For the non-symmetric mod e Grzebieta analysedthe fold s as a half-d iamondmechanism.Grzebieta carriedout static andd ynamic tests on steel tubes with D=t=30–300.2110S.R.Guillow et al./International Journal of Mechanical Sciences 43(2001)2103–2123Fig.7.Axi-symmetric mod el usedby Wierzbicki [15]andSingace et al.[16,17].Wierzbicki et al.[15]introduced a new model for the axi-symmetric collapse mechanism (shown simpliÿed in Fig.7)which allows for both inwards and outwards radial displacement.The geometry is governedby an arbitrary geometric eccentricity factor,m ,which is deÿned as the ratio of outwardfoldlength to total foldlength.By consid ering energy rate equations Wierzbicki et al.[15]were able to develop equations for not only determining average crush load but also a representative load–de ection history.The latter helped explain the experi-mental observation that sometimes there are two force peaks during the formation of a single fold.Singace et al.[16,17]extended upon the previous work by Wierzbicki et al..For the axi-symmetric mode,they considered a global energy balance leading to an implicit equation for m ,which when solvedgave a theoretical constant value of m =0:65.In their secondpa-per [17],Singace et al.reportedgoodexperimental agreement with the pred ictedvalue of 0:ter,Singace et al.[18]re-considered the eccentricity factor,m ,for the non-symmetric mode case.From the results of experiments on a small range of circular metal tubes they de-duced that the factor m was (surprisingly)approximately constant at m =0:65for this mode also.The equations developed by Singace et al.[17,18]for calculating average axial crush force are as follows.For axi-symmetric mode:F AV M P =22:27 D t+5:632:(6)For non-symmetric mode:F AV M P =− 3N +2 2N tan 2N D t :(7)One particular problem of interpretation arises with most theoretical equations developed for non-symmetric mode collapse,for example Eq.(7).They require a knowledge of the number of lobes,N ,at a given D=t ratio.We have not foundany publishedequation entirely satisfactory in determining N .S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–21232111Fig.8.Experimental set-up.With regardto energy absorption,it has been suggestedthatÿlling metal tubes with low-density polyurethane foam(to provide wall stability)may be preferable to increasing the wall thickness.Early investigations into the e ectiveness of this foamÿlling methodwere cond ucted by Thornton[19]and Lampinen and Jeryan[20].Reid,Reddy and Gray[21]have conducted experiments on the axial compression of thin-walledrectangular metal tubes which hadbeen ÿlledwith foam.Red d y andWall[22]subsequently testedfoamÿlledcircular aluminium alloy cans.Academic opinion appears to be divided about the relative beneÿts of foamÿlling versus increasing the wall thickness.2.Test procedure and material propertiesA series of approximately70axial compression tests were conducted under quasi-static condi-tions.Tests were carriedout on a SHIMAD ZU universal testing machine which appliedthe axial loadthrough at endplatens(refer to Fig.8).Cross-headspeedwas approximately5mm=min.A LABTECH data-logger recorded the data digitally for later analysis.The tubes tested were made from commercial quality extruded6060aluminium alloy in the as-received,heat treated T5condition.Mechanical properties were determined from tensile testing of coupons cut from several tubes.Fig.9shows a stress–strain curve for a typical tensile test specimen which hada0.2%proof stress, 0:2,of180MPa,an ultimate stress, ult, of212MPa anda Vickers hard ness,V,of73kg=mm2.By averaging results from several tensile tests we were able to determine an empirical relation between the0.2%proof stress and Vickers hardness for this particular type of alloy as follows:Vg= 0:2=3:92,where g=9:81m=s2is the gravitational acceleration.2112S.R.Guillow et al./International Journal of Mechanical Sciences43(2001)2103–2123Fig.9.Typical tensile stress–strain curve for6060-T5aluminium.Fig.10.Stress–strain curves for polyurethane foams of three di erent densities.The variety of commercial tubes available was insu cient to achieve the full range of D=t values we required.To produce tubes with very large D=t ratios,the outside surface of stock tubes was machined to produce the wall thickness desired.Theÿnal thickness,mean diameter and Vickers hardness were measured for each compression testpiece.Vickers hardness read-ings were usedto quantify the mechanical properties of each tube testpiece through the above equation.A representative sample of these tube properties andthe results of our testing may be foundin the append ix.Most tests involvedempty aluminium alloy tubes.However,some tests were carriedout on aluminium alloy tubes which hadbeenÿlledwith polyurethane foam.This polyurethane foam usually comes as a two part mix(base andaccelerator)andwe usedthree d i erent d ensities (35,60and140kg=m3)during testing.Fig.10shows typical compressive stress–strain curvesfor the three di erent density foams,which were obtained from axial compression tests on 96mm diameter cylindrical foam blanks unrestrained laterally.3.Experimental results and discussionOur experimental results are summarisedon the following pages in terms of the collapse mode,average force F AV,force ratio F MAX=F AV,eccentricity factor m andthe e ect of foam ÿlling.3.1.Collapse modeFurther examples of collapse modes are shown in Fig.11,in addition to those shown in Fig.2.Of particular interest was the non-symmetric mode(refer to Fig.2b)which has multiple corners(or lobes).We observedthat for tubes with an increasing D=t ratio,the number ofFig.11.Further examples of collapse modes for axially loaded thin-walled6060-T5aluminium tubes: (a)mixedmod e(D=57:1mm;t=1:15mm;L=628mm);(b)three sided non-symmetric folding (D=57:1mm;t=1:15mm;L=628mm);(c)Euler buckling(D=58mm;t=2:0mm;L=566mm).Fig.12.Schematic axial view of spiralling non-symmetric folding with N=312lobes,from Grzebieta[13].Fig.13.Mode classiÿcation chart for circular6060-T5aluminium tubes. circumferential lobes also increasedfrom2up to5or6.At high values of D=t(¿200),the number of lobes often variedd uring testing(in one case erratically between3,4and5lobes). The number of lobes,N,was not always an integer—for example,in some cases we observed a relatively stable pattern with312lobes in a spiralling arrangement(refer to Fig.12).In other cases the lobes were simply incompletely formed.From our test results for as-received6060-T5aluminium tubes a mod e classiÿcation chart was produced,see Fig.13.This chart is divided up into areas which correspond,approximately, to the di erent modes of collapse.The general shape of our chart is similar to that produced by And rews et al.[7],who testedannealedaluminium tubes.However,there are noticeable di erences in the location of the lines delineating the various areas.For example,consider analuminium tube with D=t=50and L=D=10.From our chart we wouldexpect a mixedmod eFig.14.Plot of non-dimensional experimental average force F AV=M P versus D=t.collapse but from Andrews et al.chart an Euler collapse is indicated.Note that a logarithmic scale is usedfor D=t on our chart in order to cover the wider range of D=t values considered.It may be observed broadly from our chart that non-symmetric mode is present when D=t¿100, while axi-symmetric mode occurs when D=t¡50and L=D¡2.3.2.Average crush forceOne of the most signiÿcant parameters for quantifying the behaviour of axially compressed tubes is the average crush force F AV.This is usually expressednon-d imensionally as a ratio F AV=M P.When calculating the plastic moment,M P,di erent researchers have used various di erent measures for the ow stress, 0.Since our tests involvedonly aluminium we chose to take the value of0.2%proof stress, 0:2,as the ow stress.Thus e ectively:M P= 0:2(t2=4):Fig.14shows our test results for non-dimensionalised average axial force,F AV=M P,plotted logarithmically versus D=t.There is only a relatively small amount of experimental scatter(some points shown represent more than one test result).Note that when calculating the average axial force,F AV,results for the initial peak have been ignored.From Fig.14it can be seen that when plottedlogarithmically,all the results(whether axi-symmetric,non-symmetric or mixed modes)approximately form a straight line.Hence,we obtained the following empirical relation for6060-T5aluminium alloy tubes:F AV M P =72:3Dt0:32:(8)This equation is of similar form to Eq.(4)proposedby Abramowicz andJones in1984[8] for non-symmetric mode but quite di erent from the corresponding equation proposed by Gupta andGupta[2].parison of present experimental results for average crush force with empirical equations of Gupta and Gupta[2].parison of experiment and theory for average forceThe following paragraphs are a subjective comparison between our experimental results for average crush force,F AV,andvarious theories andempirical relationships.Fig.15shows our test results for F AV comparedwith empirical equations by Gupta andGupta [2].They usedVickers hard ness,V,to characterise material properties.These equations were determined from tests on metal tubes with a relatively small range of dimensions(D=t=10–33, L=D=2–3).In view of this,agreement for both axi-symmetric andnon-symmetric mod es is quite goodin the range D=t=10–100.For D=t¿100,their curve for axi-symmetric mode is closer to our experimental test points than their non-symmetric one,even though the actual collapse mode exhibited was non-symmetric.Fig.16shows our test results for average force,comparedwith equations d evelopedby Abramowicz andJones[8].From thisÿgure it may be seen that agreement for both axi-symmetric and non-symmetric modes is fair.Their axi-symmetric equation predicts average forces which are rather low comparedwith our test points.On the other hand,their equation for non-symmetric mode,Eq.(4),predicts average forces which are rather high compared with our test points. Nevertheless,it may be notedfrom Fig.16that the slope of the line representing Eq.(4) (non-symmetric mode)is almost the same as our test points.This is also evident from a comparison of Eqs.(4)and(8).Fig.17shows our test results for average force,F AV,comparedwith the theoretical equations d evelopedby Abramowicz andJones[9].Their axi-symmetric Eq.(3)estimates an average force which is still low comparedwith our test results,but closer to our test points than their 1984prediction.In the case of non-symmetric collapse,Abramowicz and Jones[9]developed Eq.(5),which produces a family of lines,one for each value of N.Thus,we needto know the number of lobes,N,in order to interpret Fig.17.From the appendix it will be noted that formost of the tubes we tested,N falls in the range N=3–4.Agreement between their theory andparison of present experimental results for average force with theory by Abramowicz andJones[8].parison of present experimental results for average force with theory by Abramowicz andJones[9]. our test points is goodin this range.For cases with low D=t values(¡50),where N¡3,their prediction for F AV is rather low.For high values of D=t(¿300),where N¿4,their predicted value for F AV is rather high.Nevertheless,overall it appears that this methodof pred icting F AV is satisfactory.Fig.18shows our test results for average force comparedwith the equations d evelopedby Singace et al.[17,18].Their equation for axi-symmetric mode,Eq.(6),gives values for F AV which are much too low comparedwith our test points.In the case of non-symmetric mod e, their Eq.(7),when plottedon logarithmic axes prod uces a series of very steep lines,one for each number of lobes,N.This makes the process of interpretation even more di cult. Determining the precise number of corners or lobes for each test specimen presents somepractical di culties.As has previously been noted,if D=t¿200we sometimes observedthatparison of present experimental results for average force with theory by Singace et al.papers [16–18].the number of lobes variedd uring the one test.Nevertheless,on Fig.18test points are shown for which we felt conÿdent of the lobe number.It will be noted that the lines representing the Singace et al.Eq.(7)are not inconsistent with our test points although agreement is not close.However,we observedthat when the number of lobes variedd uring testing there was not a corresponding variation in the instantaneous crush force.This observation casts doubt on the validity of Eq.(7),as the large gaps between the lines in Fig.18suggest there should be a large variation in crush force with a change in the number of lobes,N .3.4.Discussion of average forceAt this stage the following observations may be made.In general,the existing theories produce numerical predictions for average force which are reasonable only for a limited range of D=t .Comparison of our test results with these theories has revealedtwo fund amental features which remain inexplicable at present.The ÿrst feature is that all our test points,regardless of mode of collapse (axi-symmetric or non-symmetric),lie on one curve whereas the theories treat these modes quite separately.Further,most theories for non-symmetric mode predict average forces which are a function of the number of lobes but experimentally this does not appear to be the case.The second,more important,feature relates to the functional dependence of average force on D=t .Our experiments clearly show that F AV =M P is empirically dependent on (D=t )0:32.Existing theories for axi-symmetric mode,however,suggest that F AV =M P should be dependent on D=t .In the case of non-symmetric mode,a wide variety of theories have been suggested;typically F AV =M P is seen as being a linear function of D=t as for the Singace et al.Eq.(7).An exception to this is Eq.(4)d evelopedby Abramowicz andJones [9],where F AV =M P was proportional to (D=t )0:33.However,as previously noted,this equation appears to have developed from work by Wierzbicki andAbramowicz [10]on rectangular rather than circular tubes.Thus it seemsFig.19.F MAX=F AV force ratio versus D=t.that for circular tubes a rigorous theoretical explanation of the D=t exponent of13is still to be developed.3.5.Force ratio F MAX=F AVIn a previous paper,Guillow andLu[23]id entiÿedthe force ratio F MAX=F AV as being of some interest.In that paper it was notedthat F MAX=F AV variedas a function of D=t ratio(this has also been notedby other researchers).The variation in force ratio highlights the fact that the mechanics of formation of the initial and subsequent folds is substantially di erent.Fig.19 shows the results of our more recent tests at larger values of D=t.The F MAX=F AV ratio appears to be monotonically increasing up to D=t=450.Variability in the F MAX=F AV ratio increased markedly for D=t¿100.This scatter couldbe d ue to signiÿcant variation in the initial buckling force,F MAX,at large values of D=t.Incidently,the common wisdom attributes the scatter of initial buckling force to imperfection sensitivity of thin-walled shells.However,Calladine[24]has recently provided an alternative explanation basedon post-buckling consid erations.3.6.Eccentricity factor mWhen folds occur during progressive buckling,they form partly on the outside and partly on the inside of the original tube proÿle.As previously noted,Singace et al.[16–18]have investigatedthis phenomenon by consid ering the eccentricity factor,m,(refer to Fig.7for its deÿnition).We were surprised at their claim that the factor m was approximately constant at 0.65.Therefore,we decided to examine our test pieces to see if the Singace et al.ÿndings also applied to6060-T5aluminium alloy tubes.Our test results for axi-symmetric mode folding are shown in Fig.20.They appear to conÿrm that a constant value of approximately0.65alsoapplies in this case.(It is not clear why the m value shouldbe so d i erent at D=t=20.)Fig.20.Eccentricity,m,as a function of D=t,for axi-symmetric mode.Fig.21.E ect of varying density of foamÿlling in6060-T5aluminium tubes.All tubes of length196mm,average diameter97mm and thickness1:0mm.Refer to Fig.10for stress–strain curves of polyurethane foam.3.7.E ect of foamÿllingMost of our tests involvedempty aluminium alloy tubes.However,a few tests were carried out on aluminium alloy tubes which hadbeenÿlledwith polyurethane foam.Fig.21shows some of our test results for foam-ÿlledaluminium tubes andTable1presents the d ata for aver-age axial crush force.Stress–strain curves for foam only were presentedearlier in Fig.10.All of the aluminium alloy tubes usedin this stage of testing were id entical(D=97mm;t=1:0mm and L=196mm).Test results for an identical empty aluminium tube are shown in Fig.1. We expectedto observe an increase in the average crushing force,F AV,for aluminium alloy tubes which hadbeenÿlledwith foam,as comparedwith id entical empty aluminium tubes.In fact,there is a complex interaction between the metal tubes andthe foamÿlling.The foamprovides support for the thin walls of the aluminium tubes leading to an increase in the overall。