Quasi-Static Axial Compression of Thi-walled Circular Aluminium Tubes

International Journal of Mechanical Sciences 43 (2001) 2103–2123

Quasi-static axial compression of thin-walledcircular aluminium tubes

S.R. Guillowa, G. Lua ; ∗, R.H. GrzebietabaSchool of Engineering and Science, Swinburne University of Technology, PO Box 218, Hawthorn,

Victoria 3122, AustraliabDepartment of Civil Engineering, Monash University, Clayton, Victoria 3168, Australia

Received 5 October 2000; received in revised form 26 March 2001

Abstract

This paper presents further experimental investigations into axial compression of thin-walled circulartubes, a classical problem studied for several decades. A total of 70 quasi-static tests were conductedon circular 6060 aluminium tubes in the T5, as-received condition. The range of D=t considered wasexpanded over previous studies to D=t=10–450. Collapse modes were observed for L=D6 10 and a modeclassi5cation chart developed. The average crush force, FAV, was non-dimensionalised and an empiricalformula established as FAV=MP = 72:3(D=t)0:32. It was found that test results for both axi-symmetricand non-symmetric modes lie on a single curve. Comprehensive comparisons have been made betweenexisting theories and our test results for FAV. This has revealed some shortcomings, suggesting that furthertheoretical work may be required. It was found that the ratio of FMAX=FAV increased substantially withan increase in the D=t ratio. The e;ect of 5lling aluminium tubes with di;erent density polyurethanefoam was also brie<y examined. ? 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Axial compression; Circular tube; Foam; Plastic collapse; Thin-walled tubes

0. Introduction

The behaviour of thin-walled metal tubes subjected to axial compression has been studied formany years. Such tubes are frequently used as impact energy absorbers and Reid [1] has pre-sented a general review of deformation mechanisms. Fig. 1 shows a typical force–displacementcurve for quasi-static loading. Generally speaking, the axial load rises until a 5rst buckle isformed at a characteristic maximum force value, FMAX. This initial buckling behaviour is well

∗ Corresponding author. Fax: +61-3-9214-8264.E-mail address: [email protected] (G. Lu).

0020-7403/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0020-7403(01)00031-5

2104 S.R. Guillow et al. / International Journal of Mechanical Sciences 43 (2001) 2103–2123

Nomenclature

D average diameterFAV average axial forceFMAX maximum axial force for 5rst peakg acceleration due to gravityH half-wavelength of foldL lengthMP full plastic bending moment of tube wall per unit lengthm geometric eccentricity factor—i.e. ratio of outwards fold length to total fold lengthN 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 distance�f density of foam�0 <ow stress�0:2 0.2% proof stress�ult ultimate tensile stress

Fig. 1. Typical load–de<ection curve for an axially loaded thin-walled metal tube which collapsed by progressivefolding.

S.R. Guillow et al. / International Journal of Mechanical Sciences 43 (2001) 2103–2123 2105

Fig. 2. Examples of various collapse modes for thin-walled circular 6060-T5 aluminium tubes under axial load-ing (more examples shown in Fig. 11): (a) axi-symmetric mode (D = 97:9 mm; t = 1:9 mm; L = 196 mm);(b) non-symmetric mode (D = 96:5 mm; t = 0:54 mm; L = 386 mm); (c) mixed mode (D = 97:5 mm;t = 1:5 mm; L= 350 mm).

known and will not be studied in depth here. Thereafter, depending on geometrical parameterssuch as the ratios of D=t (diameter=thickness) and L=D (length=diameter) and also on materialproperties, there are a variety of possible modes of collapse. Generally, collapse involves plastic


Fig. 3. Schematic axial view of non-symmetric or diamond collapse mode. Two cases are shown, N = 3 and 4circumferential lobes.

buckling and the formation of progressive folds (whether axi-symmetric or non-symmetric). Theformation 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. 2 shows sometypical examples : (i) axi-symmetric concertina bellowing, (ii) non-symmetric buckling (alsoknown as diamond or Yoshimura mode), with a variable number of circumferential lobes orcorners (refer to Fig. 3), (iii) mixed mode (combination of the two previous modes), (iv) Euleror global buckling; and (v) other (simple compression, single folds, etc.).

Research on circular tubes in the past has generally concentrated on annealed aluminiumor steel tubes with D=t ratios between 10 and 150. It is common industrial practice to usealuminium alloys in the heat treated as-received condition, but little research appears to haveaddressed this particular case. Moreover, Gupta and Gupta [2] have identi5ed metal temper asone of the signi5cant factors in determining behaviour. Hence, it was decided to undertake anexperimental program to extend the range of research up to approximately D=t = 450 and totest aluminium alloy tubes which were in the heat treated as-received condition. This work isof potential application in civil, mechanical, marine and aeronautical engineering 5elds.

1. Review of previous studies

The following section summarises the available literature on the plastic collapse behaviour ofthin-walled circular metal tubes subject to quasi-static axial loading. It is arranged broadly in achronological order.

The 5rst signi5cant work to address the mechanics underlying the observed behaviour ofaxially loaded thin-walled tubes was by Alexander [3]. He proposed a simple model for theaxi-symmetric fold pattern (refer to Fig. 4) based on experiments with metal tubes of D=t=29–89. At a global level, external work done was equated with internal work from bending atthree stationary plastic hinges and circumferential stretching of the metal between the hinges.


Fig. 4. Axi-symmetric collapse mechanism assumed by Alexander [3].

Thus the following theoretical equation was obtained for average crush force (axi-symmetricfolds):

FAV = K�ot1:5√D; (1)

where K is a constant and �o is the <ow stress. Also, plastic half-wavelength, H (refer toFig. 4) was determined as follows:

H = C√Dt; (2)

where C is a constant. The experimental results observed by Alexander were generally inagreement with the above two equations. Although simple, this model seems to re<ect theunderlying physical processes involved and many subsequent researchers have used it as astarting point.

Pugsley and Macaulay [4] were among the 5rst researchers to consider the non-symmetricfolding mode, their study being largely empirical. Johnson et al. [5] attempted to develop atheory for the non-symmetric mode based on the actual geometry of folding, with the tubematerial at the mid-surface being considered inextensional. Hence they were able to developequations to predict average axial crush force, FAV. However, agreement between their modeland test results for P.V.C. tubes was not particularly good.

In 1978 Magee and Thornton [6] conducted a review of previous work by researchers whohad conducted axial crushing tests on circular metal tubes. By considering these collected datathey developed a number of empirical equations which involved the speci5c ultimate tensilestrength of the metal.

Andrews et al. [7] conducted a comprehensive series of tests on annealed aluminium alloytubes covering a wide range of D=t (4–60) and L=D (0:2–8:8). Consequently, they developed acollapse mode classi5cation chart which predicted the mode of collapse for any given D=t andL=D combination.


Fig. 5. Axi-symmetric model used by Abramowicz and Jones [8,9]. H is the half-wavelength of the fold.

Abramowicz and Jones [8,9] conducted axial compression tests on a range of thin-walled cir-cular and square steel tubes. They analytically considered both axi-symmetric and non-symmetricmodes. Abramowicz introduced the important concept of e;ective crushing distance, �e (referto Fig. 5), where a fold consisted of two equal radii segments of length H , curved in oppositedirections and the material had 5nite thickness.

For axi-symmetric folds, Abramowicz and Jones [9] developed the following equation in 1986(and a similar one in 1984 [8]):

FAVMP

=[25:23

√D=t + 15:09]

[0:86− 0:568√t=D]

; (3)

where MP = �o(t2=4):For non-symmetric folds, in 1984 [8] and 1986 [9] Abramowicz and Jones commenced with

two di;erent starting relationships. Taking into account e;ective crushing distance, materialstrain rate, etc., resulted in two di;erent equations for average crush force. The simple relation-ship developed in 1984 [8] applied regardless of the number of lobes:

FAVMP

= 86:14(Dt

)0:33: (4)

However, this 5nding appears to have developed from work carried out by Wierzbicki andAbramowicz [10] on rectangular rather than circular tubes. On the other hand, the relationship


Fig. 6. Collapse mechanism assumed by Grzebieta [12] for axi-symmetric mode.

Abramowicz and Jones derived in 1986 [9] was of the form

FAVMP

= AN1

√Dt+ AN2; (5)

where AN1 and AN2 are constants which were a function of the number of lobes. For furtherdetails the reader is directed to this reference.

In Refs. [8,9], Abramowicz and Jones observed that reasonable agreement existed betweenpredictions of average crush loads based on the above noted equations and their experimentalresults for steel tubes with D=t = 9–65. In a subsequent work, Abramowicz and Jones [11]reported on further tests and summarised their 5ndings for both static and dynamic loadingcases in two failure mode maps, adding to the previous work by Andrews et al. [7].

Gupta and Gupta [2] performed a series of quasi-static axial compression tests on thin-walledaluminium and mild steel circular tubes in both the annealed and as-received conditions. Theycombined all results and developed empirical equations for the average crushing force in termsof the Vickers hardness and D=t.Grzebieta [12–14] used a strip method to analyse both axi-symmetric and non-symmetric

folding modes. He equated external work done with internal energy from horizontal, inclined andtravelling plastic hinges as well as stretching of the metal, to produce equations for determiningthe instantaneous forces involved. Unfortunately, these equations do not yield simple expressionsfor determining the average crush force.

Grzebieta’s collapse mechanism model for axi-symmetric mode (refer to Fig. 6) was a mod-i5cation of Alexander’s. A fold consisted of three equal lengths, two of which were curvesof equal radius and the third a straight line segment. For the non-symmetric mode Grzebietaanalysed the folds as a half-diamond mechanism. Grzebieta carried out static and dynamic testson steel tubes with D=t = 30–300.


Fig. 7. Axi-symmetric model used by Wierzbicki [15] and Singace et al. [16,17].

Wierzbicki et al. [15] introduced a new model for the axi-symmetric collapse mechanism(shown simpli5ed in Fig. 7) which allows for both inwards and outwards radial displacement.The geometry is governed by an arbitrary geometric eccentricity factor, m, which is de5nedas the ratio of outward fold length to total fold length. By considering energy rate equationsWierzbicki et al. [15] were able to develop equations for not only determining average crushload 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 singlefold.

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 form, which when solved gave a theoretical constant value of m = 0:65. In their second pa-per [17], Singace et al. reported good experimental agreement with the predicted value of0:65. Later, Singace et al. [18] re-considered the eccentricity factor, m, for the non-symmetricmode 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 = 0:65 for this modealso.

The equations developed by Singace et al. [17,18] for calculating average axial crush forceare as follows. For axi-symmetric mode:

FAVMP

= 22:27

√Dt+ 5:632: (6)

For non-symmetric mode:

FAVMP

=−�3N +

2�2

Ntan

( �2N

) Dt: (7)

One particular problem of interpretation arises with most theoretical equations developed fornon-symmetric mode collapse, for example Eq. (7). They require a knowledge of the number oflobes, N , at a given D=t ratio. We have not found any published equation entirely satisfactoryin determining N .


Fig. 8. Experimental set-up.

With regard to energy absorption, it has been suggested that 5lling metal tubes with low-density polyurethane foam (to provide wall stability) may be preferable to increasing the wallthickness. Early investigations into the e;ectiveness of this foam 5lling method were conductedby Thornton [19] and Lampinen and Jeryan [20]. Reid, Reddy and Gray [21] have conductedexperiments on the axial compression of thin-walled rectangular metal tubes which had been5lled with foam. Reddy and Wall [22] subsequently tested foam 5lled circular aluminium alloycans. Academic opinion appears to be divided about the relative bene5ts of foam 5lling versusincreasing the wall thickness.

2. Test procedure and material properties

A series of approximately 70 axial compression tests were conducted under quasi-static condi-tions. Tests were carried out on a SHIMADZU universal testing machine which applied the axialload through <at end platens (refer to Fig. 8). Cross-head speed was approximately 5 mm=min.A LABTECH data-logger recorded the data digitally for later analysis.

The tubes tested were made from commercial quality extruded 6060 aluminium alloy inthe as-received, heat treated T5 condition. Mechanical properties were determined from tensiletesting of coupons cut from several tubes. Fig. 9 shows a stress–strain curve for a typicaltensile test specimen which had a 0.2% proof stress, �0:2, of 180 MPa, an ultimate stress, �ult,of 212 MPa and a Vickers hardness, V , of 73 kg=mm2. By averaging results from several tensiletests we were able to determine an empirical relation between the 0.2% proof stress and Vickershardness for this particular type of alloy as follows: Vg=�0:2 = 3:92, where g= 9:81 m=s2 is thegravitational acceleration.


Fig. 9. Typical tensile stress–strain curve for 6060-T5 aluminium.

Fig. 10. Stress–strain curves for polyurethane foams of three di;erent densities.

The variety of commercial tubes available was insuScient to achieve the full range of D=tvalues we required. To produce tubes with very large D=t ratios, the outside surface of stocktubes was machined to produce the wall thickness desired. The 5nal thickness, mean diameterand Vickers hardness were measured for each compression testpiece. Vickers hardness read-ings were used to quantify the mechanical properties of each tube testpiece through the aboveequation. A representative sample of these tube properties and the results of our testing may befound in the appendix.

Most tests involved empty aluminium alloy tubes. However, some tests were carried out onaluminium alloy tubes which had been 5lled with polyurethane foam. This polyurethane foamusually comes as a two part mix (base and accelerator) and we used three di;erent densities(35, 60 and 140 kg=m3) during testing. Fig. 10 shows typical compressive stress–strain curves


for the three di;erent density foams, which were obtained from axial compression tests on96 mm diameter cylindrical foam blanks unrestrained laterally.

3. Experimental results and discussion

Our experimental results are summarised on the following pages in terms of the collapsemode, average force FAV, force ratio FMAX=FAV, eccentricity factor m and the e;ect of foam5lling.

3.1. Collapse mode

Further 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 multiplecorners (or lobes). We observed that for tubes with an increasing D=t ratio, the number of

Fig. 11. Further examples of collapse modes for axially loaded thin-walled 6060-T5 aluminium tubes:(a) mixed mode (D = 57:1 mm; t = 1:15 mm; L = 628 mm); (b) three sided non-symmetric folding(D= 57:1 mm; t = 1:15 mm; L= 628 mm); (c) Euler buckling (D= 58 mm; t = 2:0 mm; L= 566 mm).


Fig. 12. Schematic axial view of spiralling non-symmetric folding with N = 312 lobes, from Grzebieta [13].

Fig. 13. Mode classi5cation chart for circular 6060-T5 aluminium tubes.

circumferential lobes also increased from 2 up to 5 or 6. At high values of D=t (¿ 200), thenumber of lobes often varied during testing (in one case erratically between 3, 4 and 5 lobes).The number of lobes, N , was not always an integer—for example, in some cases we observeda relatively stable pattern with 31

2 lobes in a spiralling arrangement (refer to Fig. 12). In othercases the lobes were simply incompletely formed.

From our test results for as-received 6060-T5 aluminium tubes a mode classi5cation chartwas 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 producedby Andrews et al. [7], who tested annealed aluminium tubes. However, there are noticeabledi;erences in the location of the lines delineating the various areas. For example, consider analuminium tube with D=t = 50 and L=D= 10. From our chart we would expect a mixed mode


Fig. 14. Plot of non-dimensional experimental average force FAV=MP versus D=t.

collapse but from Andrews et al. chart an Euler collapse is indicated. Note that a logarithmicscale is used for D=t on our chart in order to cover the wider range of D=t values considered. Itmay be observed broadly from our chart that non-symmetric mode is present when D=t¿ 100,while axi-symmetric mode occurs when D=t¡ 50 and L=D¡ 2.

3.2. Average crush force

One of the most signi5cant parameters for quantifying the behaviour of axially compressedtubes is the average crush force FAV. This is usually expressed non-dimensionally as a ratioFAV=MP. When calculating the plastic moment, MP, di;erent researchers have used variousdi;erent measures for the <ow stress, �0. Since our tests involved only aluminium we chose totake the value of 0.2% proof stress, �0:2, as the <ow stress. Thus e;ectively:

MP = �0:2(t2=4):

Fig. 14 shows our test results for non-dimensionalised average axial force, FAV=MP, plottedlogarithmically versus D=t. There is only a relatively small amount of experimental scatter (somepoints shown represent more than one test result). Note that when calculating the average axialforce, FAV, results for the initial peak have been ignored. From Fig. 14 it can be seen thatwhen plotted logarithmically, all the results (whether axi-symmetric, non-symmetric or mixedmodes) approximately form a straight line. Hence, we obtained the following empirical relationfor 6060-T5 aluminium alloy tubes:

FAVMP

= 72:3(Dt

)0:32: (8)

This equation is of similar form to Eq. (4) proposed by Abramowicz and Jones in 1984 [8]for non-symmetric mode but quite di;erent from the corresponding equation proposed by Guptaand Gupta [2].


Fig. 15. Comparison of present experimental results for average crush force with empirical equations of Gupta andGupta [2].

3.3. Comparison of experiment and theory for average force

The following paragraphs are a subjective comparison between our experimental results foraverage crush force, FAV, and various theories and empirical relationships.

Fig. 15 shows our test results for FAV compared with empirical equations by Gupta and Gupta[2]. They used Vickers hardness, V , to characterise material properties. These equations weredetermined 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 and non-symmetric modes isquite good in the range D=t = 10–100. For D=t¿ 100, their curve for axi-symmetric mode iscloser to our experimental test points than their non-symmetric one, even though the actualcollapse mode exhibited was non-symmetric.

Fig. 16 shows our test results for average force, compared with equations developed byAbramowicz and Jones [8]. From this 5gure it may be seen that agreement for both axi-symmetricand non-symmetric modes is fair. Their axi-symmetric equation predicts average forces which arerather low compared with our test points. On the other hand, their equation for non-symmetricmode, Eq. (4), predicts average forces which are rather high compared with our test points.Nevertheless, it may be noted from Fig. 16 that the slope of the line representing Eq. (4)(non-symmetric mode) is almost the same as our test points. This is also evident from acomparison of Eqs. (4) and (8).

Fig. 17 shows our test results for average force, FAV, compared with the theoretical equationsdeveloped by Abramowicz and Jones [9]. Their axi-symmetric Eq. (3) estimates an averageforce which is still low compared with our test results, but closer to our test points than their1984 prediction. In the case of non-symmetric collapse, Abramowicz and Jones [9] developedEq. (5), which produces a family of lines, one for each value of N . Thus, we need to know thenumber 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 and


Fig. 16. Comparison of present experimental results for average force with theory by Abramowicz and Jones [8].

Fig. 17. Comparison of present experimental results for average force with theory by Abramowicz and Jones [9].

our test points is good in this range. For cases with low D=t values (¡ 50), where N ¡ 3, theirprediction for FAV is rather low. For high values of D=t (¿ 300), where N ¿ 4, their predictedvalue for FAV is rather high. Nevertheless, overall it appears that this method of predicting FAVis satisfactory.

Fig. 18 shows our test results for average force compared with the equations developed bySingace et al. [17,18]. Their equation for axi-symmetric mode, Eq. (6), gives values for FAVwhich are much too low compared with our test points. In the case of non-symmetric mode,their Eq. (7), when plotted on logarithmic axes produces a series of very steep lines, one foreach number of lobes, N . This makes the process of interpretation even more diScult.Determining the precise number of corners or lobes for each test specimen presents some

practical diSculties. As has previously been noted, if D=t¿ 200 we sometimes observed that


Fig. 18. Comparison of present experimental results for average force with theory by Singace et al. papers [16–18].

the number of lobes varied during the one test. Nevertheless, on Fig. 18 test points are shownfor which we felt con5dent of the lobe number. It will be noted that the lines representing theSingace et al. Eq. (7) are not inconsistent with our test points although agreement is not close.However, we observed that when the number of lobes varied during testing there was not acorresponding variation in the instantaneous crush force. This observation casts doubt on thevalidity of Eq. (7), as the large gaps between the lines in Fig. 18 suggest there should be alarge variation in crush force with a change in the number of lobes, N .

3.4. Discussion of average force

At this stage the following observations may be made. In general, the existing theories producenumerical predictions for average force which are reasonable only for a limited range of D=t.Comparison of our test results with these theories has revealed two fundamental features whichremain inexplicable at present. The 5rst feature is that all our test points, regardless of modeof collapse (axi-symmetric or non-symmetric), lie on one curve whereas the theories treat thesemodes quite separately. Further, most theories for non-symmetric mode predict average forceswhich are a function of the number of lobes but experimentally this does not appear to be thecase.

The second, more important, feature relates to the functional dependence of average force onD=t. Our experiments clearly show that FAV=MP is empirically dependent on (D=t)0:32. Existingtheories for axi-symmetric mode, however, suggest that FAV=MP should be dependent on

√D=t.

In the case of non-symmetric mode, a wide variety of theories have been suggested; typicallyFAV=MP is seen as being a linear function of D=t as for the Singace et al. Eq. (7). An exceptionto this is Eq. (4) developed by Abramowicz and Jones [9], where FAV=MP was proportionalto (D=t)0:33. However, as previously noted, this equation appears to have developed from workby Wierzbicki and Abramowicz [10] on rectangular rather than circular tubes. Thus it seems


Fig. 19. FMAX=FAV force ratio versus D=t.

that for circular tubes a rigorous theoretical explanation of the D=t exponent of 13 is still to be

developed.

3.5. Force ratio FMAX=FAV

In a previous paper, Guillow and Lu [23] identi5ed the force ratio FMAX=FAV as being ofsome interest. In that paper it was noted that FMAX=FAV varied as a function of D=t ratio (thishas also been noted by other researchers). The variation in force ratio highlights the fact thatthe mechanics of formation of the initial and subsequent folds is substantially di;erent. Fig. 19shows the results of our more recent tests at larger values of D=t. The FMAX=FAV ratio appearsto be monotonically increasing up to D=t = 450. Variability in the FMAX=FAV ratio increasedmarkedly for D=t¿ 100. This scatter could be due to signi5cant variation in the initial bucklingforce, FMAX, 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 alternativeexplanation based on post-buckling considerations.

3.6. Eccentricity factor m

When folds occur during progressive buckling, they form partly on the outside and partlyon the inside of the original tube pro5le. As previously noted, Singace et al. [16–18] haveinvestigated this phenomenon by considering the eccentricity factor, m, (refer to Fig. 7 for itsde5nition). We were surprised at their claim that the factor m was approximately constant at0.65. Therefore, we decided to examine our test pieces to see if the Singace et al. 5ndingsalso applied to 6060-T5 aluminium alloy tubes. Our test results for axi-symmetric mode foldingare shown in Fig. 20. They appear to con5rm that a constant value of approximately 0.65 alsoapplies in this case. (It is not clear why the m value should be so di;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 foam 5lling in 6060-T5 aluminium tubes. All tubes of length 196 mm, averagediameter 97 mm and thickness 1:0 mm. Refer to Fig. 10 for stress–strain curves of polyurethane foam.

3.7. E;ect of foam <lling

Most of our tests involved empty aluminium alloy tubes. However, a few tests were carriedout on aluminium alloy tubes which had been 5lled with polyurethane foam. Fig. 21 showssome of our test results for foam-5lled aluminium tubes and Table 1 presents the data for aver-age axial crush force. Stress–strain curves for foam only were presented earlier in Fig. 10. Allof the aluminium alloy tubes used in this stage of testing were identical (D=97 mm; t=1:0 mmand L= 196 mm). Test results for an identical empty aluminium tube are shown in Fig. 1.

We expected to observe an increase in the average crushing force, FAV, for aluminium alloytubes which had been 5lled with foam, as compared with identical empty aluminium tubes. Infact, there is a complex interaction between the metal tubes and the foam 5lling. The foamprovides support for the thin walls of the aluminium tubes leading to an increase in the overall


Table 1Comparison of average crush force for foam only, empty aluminium tubes and foam 5lled aluminium tubesa

Foam density FAV for FAV for empty Sum of two FAV for foam(kg=m3) foam only aluminium tube columns at 5lled A1

(kN) (kN) left (kN) tubes (kN)

35 0.908 15.53 16.44 19.3260 2.685 15.53 18.22 21.10140 12.43 15.53 27.96 28.45

aNote that average force was calculated over 126 mm stroke under identical conditions for 6060-T5 aluminiumalloy tubes, L= 196 mm; D= 97 mm and t = 1:0 mm.

crushing force. Less obvious is the sti;ening e;ect the aluminium tube provides by preventinglateral expansion of the foam when compressed. Table 1 shows that the average crush forcefor a foam 5lled aluminium tube is about 10% greater than the sum of the average force foran identical aluminium tube and foam cylinder blank tested separately. This point was notedpreviously by Reddy and Wall [22].

Foam 5lling of thin-walled tubes has previously been observed to a;ect the mode of collapse.A change in mode was demonstrated by the test cases considered. The collapse mode of anempty aluminium tube was non-symmetric (refer to Fig. 2(b)) and the same applied to anidentical tube 5lled with foam of density 35 kg=m3. However, 5lling with polyurethane foamof 60 kg=m3 forced the 5lled aluminium tube to collapse in a mixed mode and foam of density140 kg=m3 produced axi-symmetric mode collapse.

4. Conclusion

This paper details the results of axial compressive tests on thin-walled circular 6060 alu-minium tubes which were in the as-received, heat treated T5 condition. The load was appliedquasi-statically and the range of tubes considered was D=t=10–450 and L=D6 10. A collapsemode classi5cation chart for 6060-T5 aluminium tubes is presented, see Fig. 13.

An empirical relationship between average force, FAV, and D=t ratio for this particular mate-rial was developed, see Eq. (8). It was found that FAV=MP was proportional to (D=t)0:32 and thatthis equation could be applied to axi-symmetric, non-symmetric and mixed modes of collapse.Also, it was observed that the force ratio FMAX=FAV increased over the range of D=t consid-ered. Measurement of the eccentricity factor, m, for axi-symmetric folds indicates that a valueof 0.65 (as previously assumed) is appropriate for 6060-T5 aluminium. The e;ect of 5llingaluminium tubes with polyurethane foam and the resulting change in collapse mode were alsoexamined.

Available literature on the axial compression behaviour of thin-walled tubes was reviewedand comparisons made, in terms of the average force, between the theories and the presentexperimental results. It was found that reasonable numerical agreement existed between our testresults and the work by Abramowicz and Jones (Eq. (3) for axi-symmetric mode and Eq. (4)for non-symmetric mode). However, for circular tubes, a rigorous theoretical explanation of thefunctional dependence of FAV=MP on (D=t)0:32 for all modes of collapse, as observed in ourexperiment, needs to be developed.


Acknowledgements

The authors wish to thank the Australian Research Council for their 5nancial support of thisresearch and ORICA Chemicals for their assistance with polyurethane foam.

Appendix

Table A1Sample compression test results for empty aluminium tubes

TEST D=t L=D Dimensions Collapse No. of V (kg �0:2 FMAX FAV FAV= FMAX

No. D × t (mm) mode lobes =mm2 (MPa) (kN) (kN) MP =FAV

4 51.5 2.0 97:9× 1:9 Axi-sym. — 79 198 109 43.0 241 2.536 97 2.1 97× 1:0 Non-sym. 3.5 79 198 56.6 15.4 311 3.688 97 3.6 97× 1:0 Mixed 3 79 198 57.5 15.8 319 3.649 A 97 3.6 97× 1:0 Non-sym. 3.5 79 198 54.5 16.25 328 3.3510 A 51.5 2.1 97:9× 1:9 Mixed 4 79 198 117.5 44.0 246 2.6713 A 65 3.6 97:5× 1:5 Mixed 3 79 198 87.5 30.18 271 2.9016 A 172 2.0 96:6× 0:56 Non-sym. 4 79 198 22.1 5.75 370 3.8424 A 97 2.0 97× 1:0 Non-sym. 3.5 79 198 58.6 15.72 318 3.7333 179 4.0 96:5× 0:54 Non-sym. 3.5 79 198 25.35 5.06 351 5.044 370 2.1 96:3× 0:26 Non-sym. 5.5 79 198 5.55 1.66 496 3.3454 D 430 2.0 96:2× 0:22 Non-sym. 5 75 188 6.0 1.10 466 5.4655 D 308 4.0 96:3× 0:31 Non-sym. 4 68.3 171 10.64 1.84 439 5.7756 D 331 2.0 96:3× 0:29 Non-sym. 4.5 72.2 181 9.33 1.825 476 5.1158 D 307 6.0 96:3× 0:31 Non-sym. 4.5 76.6 192 10.63 2.00 423 5.3259 D 186 7.1 96:5× 0:52 Non-sym. 3.5 75 188 21.4 5.06 400 4.2360 D 97.7 7.2 97× 0:99 Other-non- Indet. 84.2 211 44.0 15.0 290 2.93

Sym +Euler61 X 35.8 6.2 48:4× 1:35 Mixed 2 71 178 37.7 18.7 231 2.0267 F 9.84 1.24 29× 2:95 Other-1 ax — 78.3 196 70.9 — — —

Fold +comp69 F 9.85 3.75 29× 2:95 Other-non- — 78.4 196 70.6 — — —

Sym +Euler70 X 35.9 0.62 48:4× 1:35 Axi-sym. — 71 178 37.85 17.5 216 2.1672 X 35.8 2.06 48:4× 1:35 Mixed 2.5 71 178 36.5 19.0 234 1.9273 X 35.8 4.13 48:4× 1:35 Mixed 2.5 70.2 176 37.6 18.5 231 2.0374 X 35.8 1.45 48:3× 1:35 Axi-sym. — 72 180 38.0 18.75 228 2.0375 G 19.1 2.49 30:4× 1:59 Axi-sym. — 79.5 199 35.9 24.0 191 1.5079 F 9.85 1.75 29× 2:95 Axi-sym. — 80.4 201 70.8 63.1 144 1.1286 H 30.2 2.0 57:9× 1:92 Axi-sym. — 71.5 179 66.5 36.9 224 1.8095 H 99 8.0 56:5× 0:57 Non-sym. 3 82.2 206 16.65 4.50 269 3.7096 H 49.6 11.0 57:1× 1:15 Mixed 3 75.4 189 37.6 15.33 245 2.4597 H 49.7 13.0 57:1× 1:15 Non-sym. 3 75.4 189 37.0 15.1 242 2.4598 H 182 8.0 56:1× 0:31 Non-sym. 3.5 76.6 192 6.81 1.64 358 4.1599 H 307 8.0 56:2× 0:18 Non-sym. 4 75.4 189 — 0.625 395 —105 D 439 4.0 96:2× 0:22 Non-sym. 5 70 176 4.0 1.07 507 3.74


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Documents

Quasi-Static Axial Compression of Thi-walled Circular Aluminium Tubes