Int. J. Mech. Sci. Vol. 35, No. 7, pp. 597 613, 1993 0020 7403/93 $6.00 + .00 Printed in Great Britain. ('~ 1993 Pergamon Press Ltd

E F F E C T O F A N N E A L I N G , S I Z E A N D C U T - O U T S O N A X I A L C O L L A P S E B E H A V I O U R O F C I R C U L A R T U B E S

N. K. GUPXA* and S. K. GVPTA* *Department of Applied Mechanics; and tCentre for Materials Science and Technology, Indian Institute of

Technology, Hauz Khas, New Delhi 110016, India

(Received 9 August 1991; and in revised form 15 June 1992)

Abstract--Axial compression tests were performed on round tubes of different sizes and made of a luminium and mild steel, both in as-received and annealed conditions. Length to diameter and diameter to thickness ratios of these tubes were varied in different tests, and cut-outs in the form of circular holes, varying in diameter, number and position, were laterally drilled in them. Typical histories of their deformation and load-compression curves are presented and the influence thereon of the annealing process, the tube size, or the cut-outs is discussed. Relations are presented to describe these influences and to express the first peak and mean collapse loads in terms of the Vickers hardness number. It is seen that the presence of holes in the tubes alters their mode of collapse and, as a consequence, affords the possibility of avoiding Euler buckling even when relatively much longer tubes are employed.

I N T R O D U C T I O N

Collapse of tubes under axial compression provides an efficient way of absorbing kinetic energy--except when they buckle in the Euler mode [1]--and study of the plastomechanics of their post-collapse deformation has attracted considerable attention [2-7] in recent years.

A tube of mean diameter D and thickness t may deform either in axisymmetric concertina mode or in a nonaxisymmetric diamond pattern. Studies I-2] have shown that thicker tubes with D/t less than 80-90 deform axisymmetrically, while those with larger values of D/t buckle in diamond mode. Experiments have also shown [8] that some tubes switch during a test from axisymmetric to diamond mode. Modes of deformation and load-compression curves of round aluminium tubes of t/D = 0.016-0.25; LID = 0.17-8.75, and without any cut-outs, were studied experimentally by Andrews et al. [7]. They presented a classification chart, which reveals that the concertina mode of deformation occurs in tubes of t/D varying from 0.016 to 0.1 (D/t = 10-62.5); the values of L/D, however, vary for different t/D values, the maximum being about 5 for t/D = 0.056. Theoretical formulations to compute the mean collapse load for both round- and square-section tubes without holes have been reviewed by Jones and Abramowicz [2]. They have discussed the improvements made in respective analyses wherein considerations of strain rate, interaction of bending and in-plane stret- ching and rolling of plastic hinges have been considered.

We present here some experimental results of mainly quasi-static axial compression tests on round tubes of aluminium and mild steel in both as-received and annealed conditions. The tubes were of different diameters and thicknesses and their length was varied in different experiments. The range of D/t for these tubes was varied from 10 to 33, to be able to get the range within which Andrews et al. [7] have shown that the concertina mode is expected for certain LID ratios in tubes without holes. It is seen in the present experiments that all these tubes did not deform in concertina mode and apart from the size, the extent of work hardening during manufacture and subsequent annealing has a great influence on the mode of deformation.

Typical modes of the deformation of the tubes and the corresponding load-compression curves are presented. The manner in which the tube collapse response is influenced by the annealing process and the size of the tube is discussed. Further, relations are presented to express the first peak and the mean collapse loads in terms of the Vickers hardness number (see Ref. I-9]) for tubes which deform in non-Euler mode. In the tubes of aluminium and mild

597

598 N.K. GUPTA and S. K. GUPTA

steel, cut-outs in the form of circular holes were drilled laterally; diameter, number and position of the holes were varied in different tests. Modes of deformation of these tubes with holes are presented and some main features of their collapse behaviour and load-compression curves are discussed. It is seen that the tubes with holes deform in Euler mode for much greater LID ratios than those without holes.

EXPERIMENTAL

Round tubes of aluminium and mild steel of different diameters and thicknesses of commercial quality were obtained and later cut, to obtain specimens of the required length to diameter (L/D) ratios; for most of the tests, these ratios were 2 or 3. In some cases, however, these ratios were varied from 0.5 to 12. In some of the tubes, two circular holes varying in diameter from 3 mm to 12 mm were drilled opposite to each other at the tube mid-height. In a few other tubes, two, three or four holes, each of diameter 9 mm, were drilled in different configurations in one, two or three planes spaced symmetrically along the tube height. These tubes were tested under quasi-static compression in an Instron machine in both as-received and annealed conditions. Annealing was done by soaking aluminium tubes for 60 min at 300°C and mild steel tubes for 10 min at 910°C, and allowing these to cool in the furnace. The Vickers hardness number was determined for each specimen material as an average of 8 to 10 readings in both as-received and annealed conditions.

Round tubes employed in the test were of five different sizes of aluminium and mild steel each. The mean diameters of the aluminium tubes varied from 23.1 to 99.2 mm (nominally, 23-99 mm) and their thicknesses from 1.29 to 2.99 mm. The mild steel tubes were of mean diameters varying from 23.2 to 52.6 mm (nominally, 23-53 mm) and their thicknesses varied from 1 to 2.44 mm. The D/t ratio of these tubes varied from 17.9 to 33.3 in the case of aluminium and from 10.6 to 31.2 in the case of mild steel. Details of the dimensions of the tubes and their respective Vicker hardness values in both as-received and annealed conditions are given in Table 1.

TUBES WITHOUT HOLES

The t/D ratios of aluminium and mild steel tubes employed for these tests varied from 0.033 to 0.06 and from 0.034 to 0.096, respectively. Their LID ratios were 2 and 3 for aluminium, and 3 for mild steel tubes. In this range, Andrews et al. [7] showed that for LID = 2, all tubes should deform in ring mode, and tubes of t/D = 0.04 -0.08 should deform in

ring mode for LID = 3 also. These tubes were axially compressed in the Instron machine at a cross-head speed of 2 mm min- 1. Their load-compression curves were obtained on the machine chart recorder, and their typical deformation histories were observed and photo- graphs were taken.

The present tests, however, reveal that the initial state of the tube material and subsequent annealing process, as discussed below, have a great influence on the mode of deformation of the tubes of both aluminium and mild steel.

Aluminium tubes From the tests on several tubes for LID = 2 and 3, it was observed that in as-received

condition:

(i) Tubes of mean diameters 99 and 78 mm deformed in diamond mode. Their measured Vicker hardness values were 87 and 75, respectively. (ii) Tubes of diameters 49 and 23 mm initially deformed in concertina mode and after two rings were formed, the collapse mode switched to diamond mode. The hardness values of these tubes are 70 and 71, respectively. (iii) The tube of diameter 36 mm deformed in ring mode. Its Vicker hardness number is 60.

On annealing at 300°C for 60 min, the value of hardness of all these tubes dropped to 40-48 (see Table 1) and the tubes of diameter 99, 78, 49 and 36 mm deform in ring mode; 23 mm diameter tubes, however, form two rings and then switch to the diamond mode of deformation.

Axial collapse behaviour of circular tubes

TABLE 1. TUBE DIMENSIONS, MEAN COLLAPSE LOAD AND MODE OF DEFORMATION

599

Pm (kN)

Mean

diameter Thickness D t VHN Mode of

Material (mm) (mm) D/t H Experimental Calculated deformation

Aluminium 99.1 2.98 33.3 87.4 111.0 104.4 (3) Diamond as received 78.2 2.61 30.0 75.3 62.5 67.3 (3) Diamond

49.3 1.50 33.0 70.3 25.0 19.9 (2) 2 rings + 21.2 (3) Diamond

36.2 1.64 22.0 60.4 19.3 18.8 (2) Concertina 23.1 1.29 17.9 71.5 7.6 13.2 (2) 2 rings +

14.2 (3) Diamond

Aluminium 99.2 2.99 33.2 47.8 52.6 54.0 (2) Concertina annealed 78.4 2.52 31.1 48.0 37.5 37.9 (2) Concertina

49.4 1.55 31.9 40.0 14.4 12.0 (2) Concertina 36.3 1.65 22.0 40.2 12.4 12.6 (2) Concertina 23.3 1.29 18.1 46.0 5.8 8.5 (2) 2 rings +

9.1 (3) Diamond

Mild steel 52.6 2.44 21.6 177.5 115.3 121.6 (2) Concertina as received 46.2 1.57 29.4 185.6 59.4 56.2 (2) Concertina

37.4 1.58 23.7 154.2 47.4 45.2 (2) Concertina 26.8 1.02 26.3 195.0 23.8 24.3 (2) Concertina 23.2 2.18 10.6 111.0 52.5 59.0 (3) Diamond

Mild steel 52.6 2.22 23.7 90.3 60.5 55.6 (3) Diamond annealed 46.2 1.48 31.2 88.8 27.6 25.8 (3) Diamond

37.4 1.51 24.7 82.6 24.5 23.7 (3) Diamond 26.6 1.00 26.6 95.0 9.8 12.1 (3) Diamond 23.2 2.02 11.5 83.7 43.6 38.5 (3) Diamond

(2), (3): values calculated from Eqn (2) or (3).

Typical load-compression curves for aluminium specimens of mean diameter 78 mm are shown in Fig.la, wherein curves for as-received and annealed specimens are compared.

The tensile stress-strain curves obtained from tests performed on specimens machined from the aluminium tube of 78 mm diameter tube in the as-received and annealed conditions are shown in Fig. lb. Though the yield point of as-received material is much higher than the annealed material, its plastic curve is relatively fiat. The annealed material curve on the other hand rises more gradually. At the onset of plastic deformation, the slopes of the engineering stress vs engineering strain curves in as-received and annealed specimens are 350 and 775 MPa, respectively, and the latter deformed in concertina mode.

In the as-received condition, aluminium is heavily cold worked. On the other hand, annealing of aluminium tubes at 300°C causes recrystallization and grain growth. Normal grain growth is observed in 36 and 23 mm diameter tubes. There is abnormal grain growth in the tubes of 99, 78 and 49 mm diameters, resulting in nonuniform grain size distribution, see Fig.lc, and reduced ductility of these tubes (Fig. lb). These deformed in diamond mode in as-received condition and in concertina mode when annealed.

Mild steel tubes The response of mild steel tubes is quite opposite to that of the aluminium tubes; they

deformed in ring or concertina mode in as-received condition. The only exception was the tube of diameter 23 mm, t/D = 0.09 and LID = 2 and 3, which deformed in diamond mode:

(i) Tubes of diameters 53, 46, 37 and 27 mm deformed in concertina mode, and their hardness values (see Table 1) are 178, 186, 154 and 195, respectively. (ii) Tube of diameter 23 mm deformed in diamond mode and its hardness value is 111.

600 N.K. GUPTA and S. K. GUPTA

After annealing, all these tubes deformed in diamond mode and their hardness reduced to 83-95 (see Table 1 for individual values). Typical steel specimens of diameter 53 mm, t/D = 0.04 and L/D = 3 in annealed and as-received conditions and deformed up to 50% compression, are shown in Fig. 2a and b respectively. Load-compression curves for these specimens are shown in Fig. 3a.

In as-received condition, all steel tubes except the one with 23 mm diameter are severely strain hardened. This can be seen from the tensile tests performed in the as-received and annealed conditions on specimens machined from the steel tubes (Fig. 3b). The 23 mm diameter steel tube in the as-received condition shows yielding prior to first peak load during the compression test, which is similar to the behaviour of all annealed steel tubes. Due to the yielding phenomenon, there is near-zero strain hardening at the onset of plastic deformation. On the other hand, the slope of the engineering stress vs engineering strain curve is much higher in the as-received condition; for example, it is 2120 MPa in 53 mm diameter steel tube.

Besides eliminating work hardening, the annealing of steel tubes coarsens the grain size, as can be seen in the microstructure (Fig. 3c). However, the grain size is not observed to be nonuniform as in some aluminium tubes.

From the above observations, it is seen that for the tubes of D/t = 11-33 considered in the present study, besides the geometry of the tube, its hardness, the microstructural parameters and the nature of its material constitutive behaviour influence the mode of deformation of the round tubes. The present results reveal that at the onset of plastic deformation, a higher value of the slope of the stress-strain curve leads to the concertina mode of deformation, while a relatively fiat curve would initiate a diamond mode.

From the load-compression curves of all the tubes of both the materials, the first peak load P~ and mean collapse load Pm were determined. The ratio of P~ to Pm is plotted against D/t for all the specimens in Fig. 4. This shows that P~/Pm can be related to D/t by the equation:

P,/Pm = KI(D/t) + K2, (1)

where K 1 = 0.026 and K2 = 1.04. The ratio PI/Pm is higher for the tubes of higher D/t and may need to be reduced for their

use in energy absorption devices. The results presented later show that this ratio is lower in the case of the tubes which have laterally drilled holes.

To express the mean collapse load for the tubes of both materials which deform in ring mode, an expression similar in form to that of Abramowicz and Jones [10] is written in terms of Vickers hardness number as:

Pm/(Ht 2) = K3(D/t) 1/2 + K 4. (2)

For the diamond mode of deformation, the expression is similar in form to that of Pugsley and Macauley [3]:

Pm/(Ht 2) = Ks(D/t ) + K 6. (3)

Values of the constants determined from the experimental results are:

K 3 = 10.1, K 4 = 68.1, K s = 1.0 and K 6 = 101.2.

Both P~ and Pm are in newtons. It is clear from the experimental results (see Figs la and 3a) that the annealing process

reduces the first peak load, as well as the mean collapse load. The values of Pm computed from Eqns (2) or (3) are given in Table 1 for each tube of aluminium and mild steel. Experimental values are also given for comparison. Here, we have considered only tubes of D/t in the "region of concertina" of Ref. [7]. The thicker and thinner tubes will be discussed in a future paper. The computed values from the above equations compare well with experiments (see Table 1). For example, the experimental values of Pm for a typical tube of steel (D = 53 mm, t = 2.44 mm) in as-received condition and a tube of aluminium (D = 36mm, t = 1.65mm) in annealed condition, are 115.3 and 12.4kN, whereas the computed values from Eqn (2) are 121.6 and 12.6 kN. These values, when computed as per

Axial collapse behaviour of circular tubes 601

(a)

12C

z 8(~

0 o

,_1

40

I t I I 20 40 60 80 100

C o m pressions rn m

(b) 2 0 0 t AS RECEIVED ALUMINIUM

I 0 1 , I , l , i , i , I 0 0.04 0.08 0 0.04 008 0.12

STRAIN

I 120

FIG. 1. (a) Load-compression curves for 78.2 mm diameter aluminium tube in the as-received and annealed conditions. (b) Engineering stress vs engineering strain curves for tensile test performed on specimens machined from 78 mm diameter aluminium tube in as-received and annealed states. (c)

Microstructure of annealed 49.4 mm diameter aluminium tube showing nonuniform grain size.

602 N . K . GUPTA and S. K. GUPTA

+

FIG. 2. Deformed shape of the 52.6 mm diameter steel tube (a) in annealed state and (b) in as-received state after 50% compression.

Axial collapse behaviour of circular tubes 603

Refs I-2] and [4] and taking the ultimate stress for computing the plastic moment, come out to be 117.5 and 111.1 kN in the case of steel, and 9.3 and 8.8 kN in the case of aluminium.

TUBES WITH HOLES

Two, three or four holes of 9 mm diameter were laterally drilled in tubes of all dimensions of aluminium in one, two or three planes spaced symmetrically along the height of the tubes. When holes were drilled in two or three planes, they were either parallel or in cross position. These tubes were compressed by about 20-30%, by which time the mode of deformation became clear.

Figure 5a shows load-compression curves for a specimen of diameter 49 mm and LID = 3, for tubes in which two holes are drilled in one, two or three planes, either in parallel or cross positions. Typical modes of deformation for tubes 1, 3 and 5 are shown in Fig. 5b; 3 and 5 are the tubes with two holes in two and three planes, respectively, but in cross positions. For such holes in cross positions, the modes of deformation of the tubes are distinctly different from those of the tubes with parallel holes, as illustrated in Fig. 5b. The holes in cross position lead to the rotation of the axis of symmetry of deformations by 90 ° . In the case of two layers, thus, one portion begins to fold into the other, see Fig. 5b. In the case of three layers, the lobes formed are symmetrical to the line joining the opposite holes in the first and third layers.

In other tubes in which the holes are located in parallel position, the deformation begins at the location of the holes at one of the planes, see Fig. 5c for a tube which has two holes each in three planes. Figure 6a shows the load--compression curves for two, three and four holes in one, two or three planes. Figure 6b shows typical modes of deformation for the tubes having two, three or four holes in one plane. Up to about 20% of compression, there seems to be little change in the load-compression behaviour for this increase in the number of holes in any plane.

It is seen from the load-compression curves that the deformation in the tubes with holes is initiated at the location of the holes, and the corresponding first peak load required for this collapse mechanism to begin is much reduced in comparison with the load required in the case of tubes without holes for progressive collapse or even Euler buckling to occur. The presence of holes alters the mode of deformation and thus Euler buckling does not occur until the tube length is increased significantly such that the Euler load becomes lower than the load required to initiate progressive collapse at the location of holes. The extent of the variations in first peak load and mean collapse load as a function of the hole diameter and the influence of the changes in mode of deformation on the tube length for Euler buckling to occur are studied by considering tubes which have two opposite holes at their mid-height.

TUBES WITH TWO OPPOSITE HOLES AT MID-HEIGHT

Two diametrically opposite holes were drilled in the tubes of all dimensions of aluminium and mild steel. The diameters of these holes in different tests were 3, 6, 9 and 12 mm.

In tubes with holes, collapse begins at the location of the holes. It was seen in the experiments that these tubes did not buckle in the Euler mode, even for lengths which were much larger than those of tubes without holes. Figure 7 shows a typical load-deformation curve for aluminium tubes of D = 36 mm. It was seen that even for LID = 9 or 10, tubes with holes collapsed in progressive mode. The tubes without holes collapsed in the Euler mode for LID = 5, while the tubes with holes collapsed in the Euler mode for LID = 11. In comparison, the minimum LID ratio required for the Euler mode of buckling to occur in the above tube (without holes), as per the classification chart of Ref. [7], is 6.3.

Tubes with two opposite holes, considered below for studying their mode of deformation and load-compression curves, were those for which a tube without holes deformed in ring mode, i.e. annealed tubes of aluminium and as-received tubes of steel.

Aluminium tubes Deformed shapes of a typical aluminium specimen of D = 36 mm, L/D = 3 are shown in

Fig. 8 at 12 different stages of the test. The hole diameter is 9 mm in this case. It is seen that for 3 mm diameter holes, the deformations of the specimens are similar to the ones without

(a) 20¢

holes. For 6, 9 or 12 m m diameter holes, the deformation is initiated at the location of holes and it progresses with the formation of lobes there.

To be able to visualize the nature of their deformation, a PVC tube of D = 62.5 mm and t = 1 mm having two opposite holes of 12 m m diameter was compressed by about 10%. Its mode of deformation was seen to be similar to that of aluminium tube. Some of this deformation was recovered by putting the tube in warm water. The recovered tube showed the pattern of hinge formation which is given in Fig. 9. It is seen that the folds are formed below the centre line on one side of a holc, while they form symmetrically above the centre line on the opposite side of the hole.

Load-compression curves for a tube o fD = 49 mm and LID = 4 are shown in Fig. 10, for hole diameters of 3, 6, 9 and 12 mm, along with a curve for a tube without holes. F rom these

15C

Z J¢

.~10C 0 0

, .J

5 0

0 0

1 1 I I I 1 I I 10 20 30 40 50 60 70 80

Compress ion jmm

604 N . K . GuP'rA and S. K. GUPTA

I 9O

I 100

(b) I- AS RECEIVED STEEL

500

400

~E ANNEALED U1

300

20C

100

O ~ 0 0.08 0.16 0.24 0.32 0.40 O.Z~

STRAIN

FIG. 3. (a) Load-compression curves for 52.6 mm diameter mild steel tube in as-received and annealed conditions. (b) Engineering stress vs engineering strain curves for tensile test performed on specimens machined from 53 mm diameter mild steel tube in as-received and annealed states.

(Continued opposite.)

Axial collapse behaviour of circular tubes 605

F1G. 3. (Continued.) (c) Microstructures of(i) as-received and (ii) annealed 53 m m diameter mild steel tube.

2

o

~ o o - o

I I I f 10 20 30 40

0/t

FIG. 4. Ratio of the first peak load to the mean load (PJPm) vs the ratio of the diameter to the thickness of the tube (D/t) for all the a luminium and mild steel specimens.

606 N . K . GUPTA and S. K. GUPTA

Z

0 o

._I

(a)

2O

1 0

-3

I I I I L L 0 5 I0 15 20 25 30

Compression .,mm

~ ~ ~!~ ~ii I i!i~ ~ •

FIG. 5. (a) Load-compress ion curves of a luminium tube of 49.4 m m diameter with two opposite holes of 9 m m diameter (1) in one plane, (2) in two planes (parallel), (3) in two planes (cross), (4) in three planes (parallel) and (5) in three planes (cross). (b) Deformed shapes of a luminium tube of 49.4 m m diameter with two opposite holes in one, two and three planes. (c) Deformed shape of a luminium tube of 49.4 mm diameter with two opposite holes in three planes located in parallel

position.

Axial collapse behaviour of circular tubes 607

(a)

30

z 20

@ _J

10

2

i I t t I L 5 10 15 20 25 30

Compression~ mm

FIG. 6. (a) Load-compress ion curves of a luminium tube of 49.4 mm diameter with two, three and four holes of 9 m m diameter in one, two and three planes. (b) Deformed shapes of a luminium tube of

49.4 m m diameter with two, three and four holes of 9 m m diameter at mid-height.

608 N.K. GUPTA and S. K. GUPTA

FIG. 8. Typical history of deformations of aluminium tube with two opposite holes at mid-height.

Axial collapse behaviour of circular tubes 609

FIG. 11. Deformed shape, (a) front and (b) top views, of aluminium tube with two opposite holes at mid-height.

FIG. 12. (a) Front and (b) top views of the deformed steel tube with two opposite holes at mid-height.

Axial collapse behaviour of circular tubes 611

32 F 2~

Without hole

7 , , , , - 10 20 30 60 50 60

Compression, mm

FIG. 7. Load-compresszon curves for relatively longer aluminium tubes with and without holes.

FIG. 9. Developed surface of a PVC tube with two opposite holes at mid-height after 10% compression showing the pattern of hinge formation.

16

12

o . e

i r / d = 6 m m ] , , A

a

I i i i I I l I I i I i I I i I 20 40 60 80 100 120 14 0 160

Compression (mm)

FIG. 10. Load-compression curves for aluminium tube with two opposite holes at mid-height of varying hole diameters.

612 N.K. GUPTA and S. K. GUPTA

2 0 0 -

150

z

100 "o O o ..i

5 0

,u• d = 0 m m d = 3mm

= 6ram = 9 r a m = 12ram

0 I 0 10

z

Hole d i a j m m

I I I I I I I t I 20 30 40 50 60 ?0 80 90 100

Compression, mm

FIG. 13. Load-compression curves for steel tubes with two opposite holes at mid-height of varying hole diameters.

9 oLo ° ~ o ~ o o o ,Steel tubes

I o ° '

3 I I i I L I i I I L 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0-40 0.45 0-50 0.55

d / D

FIG. 14. Plot of Eqn (4), along with the experimental points for both aluminium and mild steel tubes.

curves it is seen that the first peak load, P~, decreases with the increase in the hole diameter. The nature of this variation is shown in the insert to Fig. 10. It is further seen from the tests on all the tubes that the mean collapse load, for tubes with hole diameter > 3 mm, reduces to about 80% of the mean collapse load for corresponding tubes without holes. The reduced mean collapse load, however, does not change much for a given change in hole diameter. Figure 11 shows the deformed shape of a tube of diameter 49 mm corresponding to point "A" marked on the load-compression curve in Fig. 10.

Mild steel tubes The deformed shape of a typical specimen of D = 53 mm, L/D = 3 and hole diameter

6 mm is shown in Fig. 12. Here, a ring is first formed at the location of the hole and thereafter two lobes begin to form on the upper or lower half of the tube. Figure 13 shows typical load-compression curves for this tube for hole diameters of 3, 6, 9 and 12 mm. Here, the mean collapse loads in specimens with holes are not greatly different from those of the tubes

Axial collapse behaviour of circular tubes 613

without holes. The peak load P~, however, decreases with increase in hole diameter in a manner as illustrated in the insert to Fig. 13.

From the experimental results, it is seen that the first peak load for tubes with holes can be written in terms of the Vickers hardness number, dimensions of the tube and the hole diameter as:

PI/(HDt) = K7 + Ksd/D, (4)

where K7 = 8.68 and Ks = - 8.49. A plot of this Eqn (4) along with experimental points is shown in Fig. 14. For tubes

without holes, particularly of higher D/t ratio, the value of the first peak load P~ is quite high compared with Pm and in some situations, its value is required to be reduced. The presence of holes reduces P~, and Eqn (4) is useful in determining the diameter of the holes, d, required for obtaining its desired value.

C O N C L U S I O N S

Tubes of D/t = 11-33, and made of aluminium and mild steel, both with and without holes were compressed in an Instron machine. Results reveal that the mode of deformation of the tubes without holes depends on the initial state of work hardening and the subsequent annealing process, as well as on the tube geometry. A highly cold worked as-received aluminium tube deforms in diamond mode, and when annealed it deforms in a ring mode. As-received strain-hardened steel tubes deform in concertina mode and on annealing, they deform in diamond mode. The corresponding stress-strain curves reveal that their slope at the onset of plastic deformation is much higher in the case of aluminium when annealed and in the case of steel as received. Thus, besides the tube geometry, its state of strain hardening would determine the mode of its deformation. The first peak and the mean collapse loads can be related to the Vickers hardness number and the dimensions of the tube, whereas the ratio of the first peak load to mean collapse load is seen to depend linearly on the D/t ratio. The presence of holes, on the one hand reduces the first peak load and, on the other hand, affords the possibility of employing, without Euler buckling, much longer tubes than those without holes.

R E F E R E N C E S

I. S. R. REID, Metal tubes as impact energy absorbers, in Metal Forming and Impact Mechanics (edited by S. R. REID), p. 249. Pergamon Press, Oxford (1985).

2. N. JONES and W. ABRAMOWICZ, Static and dynamic axial crushing of circular and square tubes, in Metal Forming and Impact Mechanics (edited by S. R. REID), p. 225. Pergamon Press, Oxford (1985).

3. A. PUGSLEY and M. MACAULAY, The large scale crumpling of thin cylindrical columns. Quart. J. Mech. Appl. Maths 13, 1 (1960).

4. J. M. ALEXANDER, An approximate analysis of the collapse of thin cylindrical shells under axial loading. Quart. J. Mech. Appl. Maths 13, 10 (1960).

5. T. WIERZBICKI and W. ABRAMOWICZ, On the crushing mechanics of thin walled structures. J. Appl. Mech. 50, 727 (1983).

6. P. H. THORTON, H. F. MAHMOOD and C. L. MAGEE, Energy absorption by structural collapse, in Structural Crashworthiness (edited by N. JONES and T. WIERZBICK0, p. 96. Butterworths, London (1983).

7. K. R. F. ANDREWS, G. L. ENGLAND and E. GHANI, Classification of axial collapse of cylindrical tubes under quasi-static loading. Int. J. Mech. Sci. 25, 687 (1983).

8. A. G. MAMALIS and W. JOHNSON, The Quasi-static crumpling of thin-walled circular cylinders and frustra under axial compression. Int. J. Mech. Sci. 25, 713 (1983).

9. W. JOHNSON and S. R. REID, Metallic energy dissipating systems. Appl. Mech. Rev. 31, 277 (1978). 10. W. ABRAMOWICZ and N. JONES, Dynamic axial crushing of circular tubes. Int. J. Impact Engng 2, 263 (1984).