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Axial crushing of circular tubes with buckling initiators
ARTICLE IN PRESS Thin-Walled Structures 47 (2009) 788–797
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Axial crushing of circular tubes with buckling initiators X.W. Zhang, Q.D. Tian, T.X. Yu Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 2 October 2008 Received in revised form 20 January 2009 Accepted 20 January 2009 Available online 4 March 2009
This paper presents a study of the effectiveness of adding a buckling initiator which is used to reduce the initial peak force of a thin-walled circular tube under axial impact loadings. The buckling initiator is installed near the impact end of the circular tube and is composed of a pre-hit column along the axis of the tube and several pulling strips uniformly distributed around the top edge of the tube. This device functions just before the impact happens and does not affect the structural stiffness under its normal working conditions. By using two kinds of aluminum-alloy circular tubes, a series of quasi-static compression tests were conducted. The deformation mode, the initial peak force and the mean crushing force of the tubes with different number of pulling strips N, pre-hit height h and inclined angle of the pulling strips y0 were studied in the experiments. The results reveal that by using this buckling initiator, the large progressive deformation of the axially crushed circular tube switches from ring mode or mixed mode to diamond mode. Although specimens with N ¼ 2, 3 and 4 were tested, the stable deformation tended to diamond mode with lobe number N ¼ 3. With suitable selection of pre-hit height h, the initial peak force could be reduced by more than 30%. In addition, a simplified theoretical analysis is conducted to illustrate the reduction of the initial force as well as the energy dissipation mechanisms, leading to good agreements with the experimental results. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Circular tube Axial crushing Peak force Energy absorption Buckling initiator
1. Introduction Thin-walled circular tubes are widely employed in engineering as structural elements, because of their low cost, high strength and stiffness, excellent loading–carrying efficiency and energy absorption capacity. In particular, when subjected to axial impact loadings, the thin-walled circular tubes could deform in a progressive crushing mode and serve as excellent energy absorbers with a long stroke, stable loading and high specific energy absorption capacity. With the rapidly increasing requirements of safety and energy-saving in automobile, high-speed railway and aerospace industries, the axial crushed thin-walled tubes are thought to be very promising in the future applications. The earliest study on the axial crushing of thin-walled circular tubes started around 1960s, when Alexander [1] and Pugsley and Macaulay [2] developed theoretical models to predict the energy absorption of axially crushed circular tubes deformed in the axisymmetric mode (concertina or ring mode) and nonaxisymmetric mode (diamond mode), respectively. The energy absorption of non-axisymmetric mode was further studied by Pugsley [3] and Johnson and Soden [4]. Andrews et al. [5] classified the deformation modes of the axial collapse of
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cylindrical tubes and found that the deformation mode greatly depends on the length/diameter and diameter/thickness ratios. Later on, Jones and Abramowicz [6,7] conducted numerous experiments and improved the previous theoretical models on the ring mode by introducing the effective stroke and dynamic effects. In addition, many other researchers made their efforts to further improve the theoretical models for the axial crushing of circular tubes, such as Wierzbicki and Bhat [8], Grzebieta [9], Gupta et al. [10] and Huang and Lu [11], etc. In the last decade, the dynamic buckling under high-velocity impact as well as the transition between the global buckling and the progressive buckling modes were investigated by Abramowicz and Jones [12], and Jensen et al. [13]. Although, the axially loaded circular tubes have so many advantages as energy absorbers, they have a big shortcoming that when subjected to axial impact, they sustain an extremely high initial peak force. According to the injury criteria (e.g. HIC), this high peak force may cause serious damage or injury to the occupants or cargo. Therefore, in order to reduce the initial peak force, many methods have been proposed, such as introducing grooves or dents [14], using corrugated tubes [15] or patterns [16], and so on. It is true that these methods could reduce the peak force; but they also reduce the stiffness of the structure under the normal working conditions [17]. Therefore, a device, which does not affect the performance of the structure in normal use, whilst it could improve its energy absorption characteristics, is highly desired.
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In a previous study [18], the authors have studied the energy absorption of aluminum-alloy square tubes with buckling initiators subjected to axial impact loadings, and found that the buckling initiator could reduce the initial peak force as much as 30% without changing the other excellent characteristics of the square tubes; besides, the buckling initiator also makes the deformation more stable and uniform. As a further study, this paper investigated the effectiveness of a similar buckling initiator attached to an aluminum-alloy circular tube. Different from that of square tubes, the buckling initiators could possess N (N ¼ 2, 3, 4) pulling strips uniformly distributed around the top edge of the circular tubes. A series of quasi-static experiments was conducted and the effects of the pre-hit height, the number of pulling strips and the inclined angle of the strips were examined. Based on the experimental results, a simplified model is developed which could explain the deformation mechanism and predict the reductions in the peak force and mean force of the circular tubes with buckling initiators.
2. Experimental details
Material-TA Material-TB
Stress (MPa)
250 200 150 100 50 0 0.00
0.01
0.02
2r0 h 0
t
L0 D0
N=3
Fig. 2. The design of the circular tube with buckling initiator.
During the compression tests of tubes, the deformation of the column and strips can be ignored. The initial angle between the strips and the horizontal level is denoted by y0. Thus, the length of the strips is ls ¼ ðR0 r 0 Þ= sin y0
To investigate the effectiveness of the buckling initiators, two series of aluminum-alloy circular tubes were employed, which had the same dimensions, i.e. their outer radius and thickness were, D0 ¼ 50.570.5 mm and t ¼ 1.370.05 mm, respectively. However, their material properties are different because of the heat treatment. These two kinds of circular tubes were denoted by TA and TB, respectively. Fig. 1 plotted the quasi-static tensile test results for the two materials, showing that they have the same Young’s modulus E ¼ 65 GPa. It is noted that the material of TA is much stronger than that of TB. The yield and ultimate stresses of two materials are s0.2A ¼ 245 MPa and s0.2B ¼ 181 MPa, suA ¼ 270 MPa and suB ¼ 210 MPa, respectively. Since, aluminum alloys are conventionally strain-rate insensitive, the effect of strain-rate is ignored in this study. As illustrated in Fig. 2(a), a buckling initiator is installed at the top end of the circular tube, and is composed of a cylindrical aluminum-alloy column with a radius r0 along the axis of the circular tube and N pulling strips which are uniformly distributed around the top edge of the tube. The top surface of middle column is higher than the edge of the tube by h. The N pulling strips are made of steel and connect to the bottom of the column and the top edge of the tube by screws. The width and thickness of the pulling strips are 10 and 0.5 mm, respectively, with their yield stress and ultimate strain being 350 MPa and 0.2, respectively.
300
789
0.03 0.04 0.05 Strain (mm/mm)
0.06
Fig. 1. Quasi-static tensile tests for the materials of the tubes.
0.07
(1)
where R0 ¼ D0/2 is the radius of the tube. A typical specimen is shown in Fig. 2(b). When the specimen is subjected to axial compression (or axial impact), the crushing process has two distinct stages: in the first stage, the crosshead (or the impinger) contacts the top of the middle column and makes the strips to pull the tube wall inward; in the second stage, the tube is compressed directly by the crosshead and a progressive crushing in diamond mode occurs. Because of the imperfections introduced by the buckling initiator in the first stage, the initial buckling force in the second stage will be greatly reduced and the deformation mode would be changed from ring mode or mixed mode to diamond mode. The quasi-static compression tests of the specimens were conducted on a UTM machine at HKUST, whose upper limit of load is 200 kN. The loading speed was set as 5 mm/min. In the tests, the effects of the pre-hit height h, the number of pulling points N and the inclined angle y were studied.
3. Experimental results 3.1. Compression of the original tubes First, the original tubes of TA and TB were compressed between the crosshead and the base plate on a UTM machine. The tested original tubes are shown in Fig. 3. It can be seen that the specimen of TA deformed in mixed mode, which means that initially it deformed in ring mode, but after one or two layers of ring folds, the deformation mode switched to diamond mode with lobes N ¼ 3. By considering the friction between the crosshead and the top layer of fold, the initial ring mode is resulted from this friction constraint and the real natural crushing mode should be the diamond mode. On the other hand, as shown in Fig. 3(c), the deformation of TB was always in the ring mode. This result reveals that although these two circular tubes have the same geometric characteristics, their crushing modes could still be different because of different yield stresses, and the stronger tubes tend to deform in diamond mode while the weaker ones tend to be ring mode. Fig. 4 depicts a comparison of the loading curves of the two kinds of circular tubes, where the mean crushing force Fm is calculated by the total work done divided by the total displacement. It is shown that initial peaks of the two circular tubes are Fpeak ¼ 51.5 and 37.8 kN, while the mean forces at the
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Fig. 3. The deformation mode of the original circular tubes: (a) TA-side view; (b) TA-top view; (c) TB-side view.
60 Tube-A Mean force T-A Tube-B Mean force T-B
Force (kN)
40
20
0 0
20
40 Displacement (mm)
60
Fig. 4. The compression loading curves for the original tubes.
displacement 55 mm are Fm ¼ 23.8 and 17.7 kN, respectively. According to the results from Ref. [18] it is found that the initial crushing force for the relatively thick tubes can be predicted by the yield force of the cross-section. By using the yield stress measured above, the initial yield force for the two circular tubes are FY ¼ 2pR0sYt ¼ 50.5 and 37.3 kN which show very good agreements with the real measured peak forces.
3.2. Deformation mode of the tubes with buckling initiators For the tube specimens with buckling initiators, considering the radius of the original tubes was R0 ¼ 25 mm, the pre-hit height h was chosen between 5 and 25 mm. For N ¼ 2 and 4, the angle y0 remained to be 901; while for N ¼ 3, y0 ¼ 601 and 901 were studied. In the compression tests, although tubes TA and TB had different material properties, their deformation modes were quite similar. The deformation of a typical specimen in the first stage of crushing is shown in Fig. 5(a). Initially, due to the pulling of the strips, the tube wall near the pulling points moved inwards while the parts far from the pulling points kept their original shape. As a result, the top edge of the circular tube deformed from a circle into a flower with N petals as shown in Fig. 5(b). However, in the second stage of crushing, the deformation mode switched to diamond mode, in which the tube wall progressively folded into equilateral polygons with N corners and the angle between two neighboring layers was p/N. Also, in the axial direction, the corners at every layer of fold had a finite height as shown in Fig. 5(c).
Figs. 6–8 show the top, bottom and side views of the tested specimens for TB tubes with N ¼ 2, 3 and 4, respectively. It is shown from Fig. 6(a) that the top layer of all the specimens deformed in equilateral flowers with N ¼ 2 petals. With the increase of pre-hit height h, the inward movements of the pulling points were more serious and the corner at every petal became sharper. This phenomenon is more clear for N ¼ 3 and 4 as shown in Figs. 7(a) and 8(a). It was also observed in the experiments that when h was smaller than 20 mm, the extension of the pulling strips was very little; however, when h was larger than 25 mm, the extension of the strips became significant due to the yielding of the strips. For the specimens with h ¼ 30, the pulling strips broke, which means that further increasing h after this value would be more difficult. From the bottom view in Figs. 6(b)–8(b), it is found that although for these specimens N ¼ 2 and 4, the subsequent deformations of most specimens had transferred to diamond mode with N ¼ 3, i.e., the diamond mode with N ¼ 3 seemed to be the most stable mode. The side views of the specimens show that although most of the specimens could crush progressively, the cases for N ¼ 2 and 4 had an unstable tendency to switch to Euler buckling mode. The folding details will be analyzed in the next sections. The measured geometric data of the tested specimens are listed in Table 1, in which l and de are the half fold-length and effective stroke coefficient, respectively. The half fold-length l is defined as the arc-length in the axial direction between the center of a horizontal plastic hinge and the neighboring mid-point of a corner (e.g., the arc-length between A and D in Fig. 13(b)). It is clear that for the specimens with N ¼ 3, l and de are both independent of h and y0, which have their average values l ¼ 19 mm and de ¼ 0.74. For N ¼ 2 and 4, because their deformations are not so stable and uniform, most of the geometric data were not obtained.
3.3. The mean force and peak force A typical loading–displacement curves of the specimen is depicted in Fig. 9, showing that in the first stage, the load was quite low and did not have a peak, because the compressive loading was only resisted by the pulling strips. The total displacement in the first stage was about h. When the crosshead contacted the top edge of the tube wall in the beginning of the second stage, the load increased sharply until a critical value Fpeak. Then, the tube buckled and a progressive crushing took place. Due to the imperfections generated in the first stage, Fpeak was significantly lower than the original peak as shown in Fig. 4. The mean forces and peak forces for all the specimens are listed in Table 1, while Fig. 10(a) and (b) plot the comparison of the peak force and mean force for the specimens TB with respect to
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791
Fig. 5. The deformation of the specimens: (a) side view in the first stage; (b) top view in the first stage; (b) side view in the second stage.
h=10mm
h=15mm
h=20mm
h =25mm
h=30mm
Fig. 6. Tested specimens for TB with N ¼ 2 and y ¼ 901: (a) top view; (b) bottom view and (c) side view.
the pre-hit height h. It is shown from Fig. 10(a) that with the same number of pulling strips, the increase of the pre-hit height h will reduce the initial peak force. After h was larger than 25 mm, the reduction of the peak force became insignificant, because the initial peak force was lower than the subsequent peaks. With the same pre-hit height, the initial peak for N ¼ 2 is slightly higher than those of N ¼ 3 and 4. As shown in Fig. 10(b), it is evident that with the same number N, although the inclined angles of the pulling strips are different, given suitable h, the same reduction of the peak force can be achieved. On the other hand, it can be seen that the mean force only depends on N, and the mean forces for specimens of N ¼ 2 and 4 are higher than that of specimens of N ¼ 3. Nevertheless, the change in the mean force is not so significant.
4. Theoretical considerations and discussions Based on the experimental results, the deformation and loading details of the specimens with buckling initiators are analyzed theoretically in this section. It should be noted that all the analysis are based on rigid and perfectly plastic material idealization.
4.1. Geometric analysis In the first stage of crushing, the middle column is firstly pushed downwards so that the N strips pull the top edge of the tube walls at the connected points. According to the observations in Fig. 5, the deformation of axial cross-section through a pulling point in this stage can be simplified as sketched in Fig. 11(a), in which A is an original pulling point, O is the center of the column, and D is the boundary point of the influenced area on the tube wall. At the end of this stage, the column moves a distance h and reaches the same horizontal level as the top edge of the tube. As a result, point A moves and rotates to position A0 . Assume the influenced length of the tube wall AD is l0, and the length of the pulling strips is ls, while the final rotation angles of the inclined tube wall and pulling strips are b and y, respectively. Then, at the end of the first crushing stage, we have,
l0 sin b þ ls sin y ¼ R0 r 0
(2)
l0 ð1 cos bÞ þ ls cos y ¼ ls cos y0 þ h
(3)
On the other hand, from the top view, the deformation of the tube wall in the first crushing stage can be simplified as shown in Fig. 11(b) (e.g., for N ¼ 3), in which the pulled parts deform
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h=10mm
h=15mm
h=20mm
h=25mm
h=30mm
Fig. 7. Tested specimens for TB with N ¼ 3 and y ¼ 901: (a) top view; (b) bottom view and (c) side view.
h=10mm
h=15mm
h=20mm
h=25mm
h=30mm
Fig. 8. Tested specimens for TB with N ¼ 4 and y ¼ 901: (a) top view; (b) bottom view and (c) side view.
inwards and the parts far from the pulling points remain with their original shape. The inward part A0 B curves inwards and has the similar curvature as its original sharp AB, which can be seen in Fig. 5(b). Therefore, the relationship between the horizontal distance AA0 and the width of the undeformed part of tube edge can be approximated by h
l0 sin b ¼ 2R0 1 cos
p N
i
a
(4)
where a denotes the half central angle of an undeformed section as shown in Fig. 11(b). In the above equations, the influenced
length l0 is independent of N and y0, and from the experimental measurements, it can be estimated by the half fold-length associated with elastic buckling as follows: pffiffiffiffiffiffiffi
l0 ¼ 1:72 R0 t
(5)
By combining Eqs. (2)–(5), the relationship between h/R and a for different N and different initial inclined angle of strips y0 can be obtained numerically. The results for N ¼ 3 are plotted in Fig. 12. It is shown that with the increase of h/R0, the half central angle a decreases. However, when h reaches a critical value, b becomes p/2 and a renders a minimum. It is revealed that even
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Table 1 Experimental result for circular tubes with buckling initiators. No.
h (mm)
l (mm)
TA TB TB-N3-60
0 0 5 10 15 20 25 30
8 19.25 19.25 19 19.5 19.25 19
TB-N3-90
10 15 20 25 30
TB-N2-90
Fm (kN)
Fp (kN)
0.74 0.75 0.75 0.74 0.74 0.74 0.73 0.70
23.8 17.7 16.9 16.1 16.3 16.1 15.8 16.3
51.5 37.8 32.8 30.0 28.2 20.2(23.4) 16.7(23.7) 17.2(23.3)
19 19.25 18.5 18.5 18.25
0.74 0.74 0.73 0.73 0.71
17.2 15.9 16.2 16.4 16.8
34.3 30.7 26.9 21.4(24.3) 19.2(24.2)
10 15 20 25 30
16.4 18.75 15.2
0.70 0.76 0.74
18.3 19.7 18.0 17.1 18.5
37 34.3 29 27.6 27.7
TB-N4-90
10 15 20 25 30
18.25 16.9 19 18.25 18.75
0.75 0.70 0.74 0.74 0.71
18.2 19.2 20.0 18.4 17.9
31.1 32.0 27.9 18.9(23.3) 15.4(23.4)
TA-N3-90
10 15 20 25
18.5 19.25 17.75 18.75
0.77 0.74 0.76 0.75
24.5 22.6 22.4 22.1
47.0 40.7 36.4 33.9
de
Notes: TA-N4-90 means specimen TA with N ¼ 4 and y0 ¼ 901; ‘ ’ means the value is not measured.
793
crushing stage can be simplified as sketched in Fig. 13. Section ACGD is 1/2N of an entire circle along the circumference and has a half fold-length l in the axial direction. When the tube is compressed compactly, the cross-section along AD will be as shown in Fig. 13(b). The tube wall between A and E has been folded, while ED is always vertical during the deformation process. Finally, E and F reach the same horizontal level. It should be noted that the vertical space between D and E is taken by the folded tube wall at FG. Therefore, the vertical distance of DE is equal to that of AE. Supposed the arc-length between A and E is lh, the horizontal distance of AE after folding is H, and the curvature radius at point A is rh. Then, if the influence of the thickness is ignored, the fold-length and effective energy absorbing stroke coefficient de can be obtained by
l ¼ lh þ r h
and
de ¼ 1
2r h
l
¼
lh r h lh þ r h
(6)
For the deformation shape in Fig. 13(b), if de is given, the length lh and radius rh can be obtained as functions of H, with the details being given in the Appendix A. Fig. 14 shows the top cross-sectional view of a circular tube with N ¼ 3 at point A after compression reaches compaction, in which the solid line is the plastic hinges at this horizontal level. The cross-section at point A changes from initially a circle shown by the dashed curve to an equilateral triangle. With the assumption that the tube wall experiences no extension along the circumference, and the curvature radius at the corners is ignored, we have jACj ¼ Lx ¼
pR0 N
(7)
Considering the symmetry along the axis in Fig. 14, the angle +ACE ¼ pR0/2N and the horizontal distance between A and E can be found as p pR0 tan (8) H¼ N 2N Also, the inclined plastic hinge EF in Fig. 13(a) can be obtained
30
as
N=3, θ=90°, h=25mm F Fm
25
Lcp ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlh r h Þ2 þ L2x
(9)
Force (kN)
20 4.2. Estimate of the peak forces
15 10 5 0 0
20
40 60 Displacement (mm)
80
Fig. 9. Loading curve for the specimen TB-N3-90 with h ¼ 25 mm.
the initial inclined angle y0 is different, a always takes the same minimum value a ¼ 0.4, which depends on l0 only. Besides, the influence of middle column is also considered. The results for two values r0/R0 ¼ 0.1 and 0.2 are obtained, indicating that when r0/R0o0.2, its influence can be ignored. Based on the observations in Fig. 5(c), the deformation of a representative section on the circular tube wall in the stable
To estimate the initial peak force of the circular tubes with buckling initiators, the deformation details at a typical section near the top edge are analyzed. As shown in Fig. 15(a), at the end of the first crushing stage, the tube wall near the pulling point A has been pulled inwards to A0 , while section BC is not influenced by the pulling strips and maintains its original shape. It should be noted that after moving inwards, A0 B is lower than the undeformed edge BC. As a result, in the second stage when the crosshead compresses the tube directly, the contact area will be reduced to N corner sections like BC. Because, the tube wall A0 BD is curved but BCGD remains unchanged, all the compressive loads will be carried by BCGD. Consider the compression of a typical section BCGD in Fig. 15(a). If the compressive stress is large enough, section BCGD will buckle and a plastic hinge will occur along the dashed curve through point T as shown in Fig. 15(b), where point T is located near the middle of CG. Since, the original tube deforms in ring mode or mixed mode whose initial peak force can be approximated by the product of the yielding force and the cross-sectional area (see Section 4.1), as a part of the tube, the buckling force of section BCGD can be estimated by the yield force along the midline KT.
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1.2
45 Fp N=2 Fm N=2 Fp N=3 Fm N=3 Fp N=4 Fm N=4
40 35 Force (kN)
30
1.0 0.8
25 0.6 20 0.4
15 10
0.2 5 0.0
0 0
5
10
0
5
10
15 20 Prehit height (mm)
15
20
25
30
25
30
1.2
45
Thus, the subsequent deformation of this fold at section A0 CGD can be considered to consist of bending of A0 JKD and buckling of JCKT. Since, KT located at the middle of CG and CD is approximately straight, the buckling force of KT is estimated as F p1 ¼
F-p θ=60
40
Fm θ=60 Fp θ=90
35
1.0
Fm θ=90
0.8
30 Force (kN)
As the relation between a and h/R0 has been obtained in the previous section, by using Eq. (11), the initial buckling force of the tube in the second stage can be estimated. However, it should be noted that if the initial peak force is lower than the subsequent peaks, further reduction of the initial peak will be meaningless. Therefore, the magnitude of the subsequent peaks still needs to be determined and compared with the initial one. The formation of every fold in the stable crushing stage starts from a shape as shown in Fig. 16(a). Different from the top fold of the tube as shown in Fig. 15(a), in this stage, points B and C have merged together and A0 C is a straight horizontal plastic hinge. Also, the shape along CD is much closer to a straight line. The distance AA0 is h p i p (12) jAA0 j ¼ R0 1 cot N N
25 0.6 20 0.4
15 10
0.2 5 0.0
0 0
5
10 15 20 Prehit height: h (mm)
25
30
Fig. 10. Experimental results for peak forces and mean forces: (a) TB specimens with y ¼ 901 and different N; (b) TB specimens with N ¼ 3 and different y.
pR0 2N
A
A’
D ’
0 R0
ls
O A’
A B
C
Fig. 11. Geometric relationships of the specimen in the first crushing stage: (a) side view; (b) top view.
Suppose the central angle of BC is a and the hinge along BKD can be approximated by a quarter of sine curve with the point T located in the middle of CG, then the length of KT is 2 p jKTj ¼ R0 (10) aþ 3 3N Therefore, the initial buckling force of the whole circular tube in the second crushing stage can be approximated by 2 p F p 2N a 1 (11) and F p ¼ sY t2NR0 aþ ¼ þ 3 3 3N FY 3p
(13)
The compressive stress along A0 J comes from bending of section A0 JKD, and the total force is the integration of the stress on every strip of width dx as shown in Fig. 16(b). If the effect of the transition section is ignored, the horizontal and vertical distances between A0 and D are lx ¼ l sin c and ly ¼ l cos c
(14)
In the above equations, l can be obtained from Eq. (6) and c can be calculated by lx ¼ |AA0 |. After a finite bending, the part below DK changes very little, the compression force on a strip of width dx can be obtained as dF ¼
2M p
l sin c
dx
(15)
pffiffiffi where M p ¼ ð1=2 3ÞsY t 2 is the maximum plastic bending moment per unit width of tube wall. For the strip at x position, the inclined angle is j, but l is a function of x as follows:
lðxÞ ¼ l
sY t
Nx
pR0
(16)
Therefore, the total buckling force for a typical fold can be approximated by ! Z 1 F ps t 1 1 dx þ ¼ pffiffiffi (17) 2 FY 3R0 ½1 ðp=NÞ ð1= tanðp=NÞÞ 0:5 x According to Eq. (17), if t/R0 is smaller, Fps/FY will be smaller. It should be pointed out that the method of estimating the peak forces in this section can only be applied to the case of relatively thick tubes which deform in ring mode or mixed mode. 4.3. Mean crushing force Although numerous papers published have discussed the modeling of axial crushing of circular tubes in ring mode, very few studies have been made about the energy absorption of circular tubes deformed in diamond mode. Pugsley and Macaulay [2] proposed the first theoretical analysis in 1960s, followed by Pugsley [3] and Johnson and Soden [4], who made further improvements. However, these models only provided some basic ideas in analyzing this problem and were not accurate enough. Recently, Marsolek and Reimerdes [16] used some patterns to generate diamond mode and developed a model to predict the
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mean crushing force. However, some of their assumptions do not agree with the observations in our experiments. To estimate the mean crushing force, an energy method is applied to a typical tube section of length l. During the entire compression process, the energy is dissipated by four distinct mechanisms as follows:
Half central angle of a corner α
1.1 1.0
θ=90°, r0 /R 0=0.2
0.9
θ=72°, r0 /R 0=0.2 θ=60°, r0 /R 0=0.2
0.8
795
θ=60°, r0 /R 0=0.1
(a) First, the original cylindrical tube wall is compressed to become flat and the energy dissipated is
0.7
E1 ¼ 2plM p
0.6 0.5 0.4 0.0
0.2
0.4 0.6 0.8 Pre-hit height h/R0
1.0
(18)
where Mp is the maximum plastic bending moment per unit width of tube wall. (b) A portion of energy is dissipated by the rotation of the horizontal plastic hinges. According to the analysis given in the Appendix A, the rotation angle along every horizontal hinge is kcp, where kc is 1.2 for de ¼ 0.74. Therefore, the energy dissipated is
1.2
Fig. 12. Relations between the pre-hit height h/R0 and half central angle of a corner.
E2 ¼ 2pR0 M p kc p
(19)
(c) The energy dissipated at the inclined plastic hinges is E3 ¼ 2NMp pLcp
A 2rh A
C
D
F T S
E D
E
λ Δ
G
H Fig. 13. Diagram for the deformation of the tubes in a typical region: (a) side-view; (b) cross-sectional view.
A
O
H
C (F)
W ¼ F m lde
Fig. 14. The cross-sectional view in the stable crushing stage of the tube.
B D
2Mp NLx l (21) r According the measurements in our experiments, the curvature radius r is about 1.5 times of the tube’s wall thickness. To sum up the above four portions, the total energy dissipated by a half fold is X R0 Ei ¼ 2pMp l þ kc pR0 þ 2NLcp þ l (22) r i¼1;4 E4 ¼
On the other hand, if the mean crushing force during this compression process is Fm, then the work done by the external force is
E
A’
(20)
(d) Besides the horizontal and inclined plastic hinges, a great portion of energy is dissipated by the traveling plastic hinges. As shown in Fig. 13(a), the buckling of section EFGD starts from an inclined plastic hinge ET. However, with the progress of compression, this plastic hinge will gradually move from EC to EF and then to ES. In the area where the plastic hinge passed by, the tube wall experienced bending and revisedbending processes. According to the deformation mechanism, the shadow area in Fig. 13(a) is the region where the plastic hinge passed through. Suppose the curvature radius of plastic hinge is r, then the energy dissipated by the traveling hinges is
O
(23) P
In view of energy balance, we have, W ¼ iEi. By using Eqs. (18)–(23), we can obtain the mean crushing force of the circular tubes deforming in diamond mode as follows, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 ðlh r h Þ2 þ L2x R0 Fm 2p 4 R0 (24) ¼ 1 þ kc p þ N þ 5 Mp de l l r
C B
C 4.4. Discussions
G K T D
The initial peak forces estimated by Eq. (11) for N ¼ 3 and
y0 ¼ 901 with different h are plotted and compared with G
Fig. 15. The initial buckling of the specimen in the second stage: (a) three dimensional diagram; (b) details at a typical corner.
experimental results in Fig. 17. It is shown that this analysis can predict very well the decreasing tendency of the initial peak force with the increase of h, and the maximum reduction of the peak force could be more than 30%. However, the predicted curve is
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dF
O A’
A’
x
J
A
K
T
D
5. Summary and remarks
ly
C
G
D
lx R0
Fig. 16. Buckling of a fold in the stable crushing stage: (a) three dimensionaldiagram; (b) cross-sectional view.
40 1.0 35
force (kN)
30
0.8
25 0.6 20 15
0.4 Theoretical Exp Fpeak Exp Fmean Fs Fm
10 5 0
0.2 0.0
0
5
10 15 20 pre-hit height h (mm)
25
strips in our device greatly reduce the inertia, because in the initial pulling stage, the involved mass is only the small part at the end of the tube instead of the entire tube. In addition, in this theoretical analysis, the extension of the pulling strips is not considered, because it is only significant when h is quite large.
30
Fig. 17. Comparison between theoretical analysis and the experiments for N ¼ 3.
slightly lower than the experimental results, because in the analysis the radius of the corners and inclined plastic hinges are neglected. The value of peak force in the stable crushing stage is also estimated and plotted as the dotted line. When the initial peak force is lower than the subsequent peaks, the further reduction will be meaningless. Therefore, the level of subsequent peaks can be regarded as the lower limit of the peak force. Considering the stroke coefficient of the tubes is about de ¼ 0.74 and using Eq. (6), we have rh/lh ¼ 0.15. Therefore, from Fig. A2 in the Appendix A, j ¼ 0.3, so that kc ¼ 1.20 and lh ¼ 1.25 H. For N ¼ 3 and H ¼ 0.6R0, the mean crushing forces predicted by Eq. (24) for TA and TB are Fm ¼ 22.7 and 16.8 kN, respectively. These estimations agree very well with the experiments (Fm ¼ 23.6 and 17.7 kN, for TA and TB, respectively). It should be noted that under impact scenarios two kinds of dynamic effects, namely, strain-rate and inertia effects, will make the structure and material stronger. The strain-rate effect depends on the material itself; for the strain-rate insensitive materials, this effect can be ignored. On the other hand, the inertia effect depends on the impact velocity V as well as the mass ratio M/m (where M and m denote the masses of the impinger and the tube, respectively); it will become much weaker for larger mass ratio and lower velocity. According to the study in [18], in the axial crushing of aluminum-alloy tubes, the dynamic effects can be ignored for M/m4100 and Vo10 m/s. Also, different from the conventional study of tubes under axial crushing, the pulling
In this paper, a buckling initiator designed for reducing the initial peak force of axially crushed circular tubes is investigated. This buckling initiator is installed near the impact end of a circular tube and is composed of a pre-hit column along the tube’s axis and several pulling strips uniformly distributed around the top edge of the tube. It functions just before the impact happens, and can effectively reduce the initial buckling force without affecting the structural stiffness at normal working conditions. By using aluminum-alloy thin-walled circular tubes with R/t ¼ 19, the influences of the number of pulling strips N, pre-hit height h and the inclined angle of the pulling strips have been all studied. The results reveal that this buckling initiator could effectively reduce the initial peak force and retain the excellent energy absorption of the circular tube. By using this buckling initiator, the deformation mode of the axial crushed circular tube switches from ring mode or mixed mode to diamond mode. For the relatively thick tubes which could deform in ring mode or mixed mode, although specimens with N ¼ 2, 3 and 4 were tested, the stable deformation tended to diamond mode with lobe number N ¼ 3. With suitable selection of pre-hit h, the initial peak force could be reduced by more than 30%. Based on the experimental results, the deformation of the thin-walled tubes in diamond mode is analyzed in detail and a simplified theoretical analysis is conducted to reveal the reduction of the initial force as well as the energy dissipation mechanisms, showing good agreement with the experiments. For the application of this kind of devices, they can be used as energy absorbers at the roadside or as part of the landing gears of small flying vehicles; for the automobile structures, in which the proximal end of the tube is fixed, allocating the pulling points to the middle of the tubes may be considered.
Acknowledgements This study is funded by the Hong Kong Research Grant Council (RGC) under CERG Grant no. 621505 and the National Natural Science Foundation of China under Key Project no. 10532020. Their supports are gratefully acknowledged.
Appendix A. Geometric analysis for the folds in the axial direction Consider a fold of the tube wall shown in Fig. 13(b). The configuration of the folded part can be simplified as shown in
r
l
H Fig. A1. The cross-sectional view of a typical half fold.
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Thus,
0.25
residual height of a half-fold Δ/λ
797
l ð2 cos j 1Þ r 1 ¼ and ¼ ¼ 1 de r sin j l ðp þ 2jÞ þ ð2 cos j1Þ
0.20
(A4)
sin j
The horizontal fold-length is cos jð2 cos j 1Þ H ¼ 2rð1 þ sin jÞ þ l cos j ¼ r 2ð1 þ sin jÞ þ sin j
0.15
(A5)
0.10 and
0.05
H¼l
0.00 0.0
0.2
0.4
0.6
0.8
1.0
1.2
inclined angle of the straight segment ϕ Fig. A2. The relationship between D/l and j.
½2ð1 þ sin jÞ þ ðcos jð2 cos j 1Þ= sin jÞ ðp þ 2jÞ þ ðð2 cos j 1Þ= sin jÞ
(A6)
After folding, the residual height of a half fold is D ¼ r. Hence, Eqs. (A4)–(A6) give the relationships among D/l, l/H and j, which are depicted in Figs. A2 and A3. For a given j, kc ¼ (1+2j/p), and the parameters de ¼ D/l, kl ¼ l/H. Therefore, the relation between de and kl could be obtained. References
1.45 1.40 1.35
λ/H
1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.0
0.2 0.4 0.6 0.8 1.0 Inclined Angle of the straight section ϕ
1.2
Fig. A3. The relationship between l/H and j.
Fig. A1. Suppose the curved sections have the same curvature radius r, the length of the straight section between two curved regions is l, and the angle between the inclined straight section and the horizontal level is j. Then, if the influence of the wall thickness is ignored, the arc-length of this half fold is p (A1) l ¼ 2r þ j þ l 2 The vertical height of this half fold is r ¼ lð1 de Þ
(A2)
Also, from geometric relations in the vertical direction, r cos j ¼ l sin j þ rð1 cos jÞ
(A3)
[1] Alexander JM. An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q J Mech Appl Math 1960;XIII:10–5. [2] Pugsley SA, Macaulay M. The large-scale crumpling of thin cylindrical columns. Q J Appl Math 1960;XIII:1–9. [3] Pugsley AG. On the crumpling of thin tubular struts. Q J Mech Appl Math 1979;XXXII:1–7. [4] Johnson W, Soden PD. Inextensional collapse of thin-walled tubes under axial compression. J. Strain Anal 1977;12(4):317–30. [5] Andrews KRF, England GL, Ghani E. Classification of the axial collapse of cylindrical tubes under quasi-static loading. Int J Mech Sci 1983;25(9): 687–96. [6] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. Int J Impact Eng 1984;2(3):263–81. [7] Abramowicz W, Jones N. Dynamic axial crushing of circular and square tubes. Int J Impact Eng 1986;4(4):243–70. [8] Wierzbicki T, Bhat SU. A moving hinge solution for axi-symmetric crushing of tubes. Int J Mech Sci 1986;28(3):135–51. [9] Grzebieta RH. An alternative method for determining the behavior of round stocky tubes subjected to an axial crush load. Thin-walled struct 1990(1–4): 61–89. [10] Gupta NK, Sekhon GS, Gupta PK. A study of fold formation in axi-symmetric axial collapse of round tubes. Int J Impact Eng 2002;27(1):87–117. [11] Huang X, Lu G. Axisymmetric progressive crushing of circular tubes. Int J Crashworthiness 2003;8(1):87–95. [12] Abramowicz W, Jones N. Transition from initial global bending to progressive buckling of tubes loaded statically and dynamically. Int J Impact Eng 1997;19(5–6):415–37. [13] Jensen Q, Langseth M, Hopperstad OS. Experimental investigations on the behavior of short to long square aluminum tubes subjected to axial loading. Int J Impact Eng 2004;30(8–9):973–1003. [14] Hosseinipour SJ, Daneshi GH. Energy absorption and mean crushing load of thin-walled grooved tubes under axial compression. Thin-walled struct 2003;41(1):31–46. [15] Singace AA, El-Sobky H. Behavior of axially crushed corrugated tubes. Int J Mech Sci 1997;39(3):249–68. [16] Marsolek J, Reimerdes HG. Energy absorption of metallic cylindrical shells with induced non-axisymmetric folding patterns. Int J Impact Eng 2004; 30(8–9):1209–23. [17] Sastranegara A, Adachi T, Yamaji A. Improvement of energy absorption of impacted column due to transverse impact. Int J Impact Eng 2005; 31(4):483–96. [18] Zhang XW, Su H, Yu TX. Energy absorption of an axially crushed square tube with a buckling initiator. Int J Impact Eng 2009;36(3):402–17.
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Axial crushing of circular tubes with buckling initiators X.W. Zhang, Q.D. Tian, T.X. Yu Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 2 October 2008 Received in revised form 20 January 2009 Accepted 20 January 2009 Available online 4 March 2009
This paper presents a study of the effectiveness of adding a buckling initiator which is used to reduce the initial peak force of a thin-walled circular tube under axial impact loadings. The buckling initiator is installed near the impact end of the circular tube and is composed of a pre-hit column along the axis of the tube and several pulling strips uniformly distributed around the top edge of the tube. This device functions just before the impact happens and does not affect the structural stiffness under its normal working conditions. By using two kinds of aluminum-alloy circular tubes, a series of quasi-static compression tests were conducted. The deformation mode, the initial peak force and the mean crushing force of the tubes with different number of pulling strips N, pre-hit height h and inclined angle of the pulling strips y0 were studied in the experiments. The results reveal that by using this buckling initiator, the large progressive deformation of the axially crushed circular tube switches from ring mode or mixed mode to diamond mode. Although specimens with N ¼ 2, 3 and 4 were tested, the stable deformation tended to diamond mode with lobe number N ¼ 3. With suitable selection of pre-hit height h, the initial peak force could be reduced by more than 30%. In addition, a simplified theoretical analysis is conducted to illustrate the reduction of the initial force as well as the energy dissipation mechanisms, leading to good agreements with the experimental results. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Circular tube Axial crushing Peak force Energy absorption Buckling initiator
1. Introduction Thin-walled circular tubes are widely employed in engineering as structural elements, because of their low cost, high strength and stiffness, excellent loading–carrying efficiency and energy absorption capacity. In particular, when subjected to axial impact loadings, the thin-walled circular tubes could deform in a progressive crushing mode and serve as excellent energy absorbers with a long stroke, stable loading and high specific energy absorption capacity. With the rapidly increasing requirements of safety and energy-saving in automobile, high-speed railway and aerospace industries, the axial crushed thin-walled tubes are thought to be very promising in the future applications. The earliest study on the axial crushing of thin-walled circular tubes started around 1960s, when Alexander [1] and Pugsley and Macaulay [2] developed theoretical models to predict the energy absorption of axially crushed circular tubes deformed in the axisymmetric mode (concertina or ring mode) and nonaxisymmetric mode (diamond mode), respectively. The energy absorption of non-axisymmetric mode was further studied by Pugsley [3] and Johnson and Soden [4]. Andrews et al. [5] classified the deformation modes of the axial collapse of
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E-mail address: [email protected] (T.X. Yu). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.01.002
cylindrical tubes and found that the deformation mode greatly depends on the length/diameter and diameter/thickness ratios. Later on, Jones and Abramowicz [6,7] conducted numerous experiments and improved the previous theoretical models on the ring mode by introducing the effective stroke and dynamic effects. In addition, many other researchers made their efforts to further improve the theoretical models for the axial crushing of circular tubes, such as Wierzbicki and Bhat [8], Grzebieta [9], Gupta et al. [10] and Huang and Lu [11], etc. In the last decade, the dynamic buckling under high-velocity impact as well as the transition between the global buckling and the progressive buckling modes were investigated by Abramowicz and Jones [12], and Jensen et al. [13]. Although, the axially loaded circular tubes have so many advantages as energy absorbers, they have a big shortcoming that when subjected to axial impact, they sustain an extremely high initial peak force. According to the injury criteria (e.g. HIC), this high peak force may cause serious damage or injury to the occupants or cargo. Therefore, in order to reduce the initial peak force, many methods have been proposed, such as introducing grooves or dents [14], using corrugated tubes [15] or patterns [16], and so on. It is true that these methods could reduce the peak force; but they also reduce the stiffness of the structure under the normal working conditions [17]. Therefore, a device, which does not affect the performance of the structure in normal use, whilst it could improve its energy absorption characteristics, is highly desired.
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In a previous study [18], the authors have studied the energy absorption of aluminum-alloy square tubes with buckling initiators subjected to axial impact loadings, and found that the buckling initiator could reduce the initial peak force as much as 30% without changing the other excellent characteristics of the square tubes; besides, the buckling initiator also makes the deformation more stable and uniform. As a further study, this paper investigated the effectiveness of a similar buckling initiator attached to an aluminum-alloy circular tube. Different from that of square tubes, the buckling initiators could possess N (N ¼ 2, 3, 4) pulling strips uniformly distributed around the top edge of the circular tubes. A series of quasi-static experiments was conducted and the effects of the pre-hit height, the number of pulling strips and the inclined angle of the strips were examined. Based on the experimental results, a simplified model is developed which could explain the deformation mechanism and predict the reductions in the peak force and mean force of the circular tubes with buckling initiators.
2. Experimental details
Material-TA Material-TB
Stress (MPa)
250 200 150 100 50 0 0.00
0.01
0.02
2r0 h 0
t
L0 D0
N=3
Fig. 2. The design of the circular tube with buckling initiator.
During the compression tests of tubes, the deformation of the column and strips can be ignored. The initial angle between the strips and the horizontal level is denoted by y0. Thus, the length of the strips is ls ¼ ðR0 r 0 Þ= sin y0
To investigate the effectiveness of the buckling initiators, two series of aluminum-alloy circular tubes were employed, which had the same dimensions, i.e. their outer radius and thickness were, D0 ¼ 50.570.5 mm and t ¼ 1.370.05 mm, respectively. However, their material properties are different because of the heat treatment. These two kinds of circular tubes were denoted by TA and TB, respectively. Fig. 1 plotted the quasi-static tensile test results for the two materials, showing that they have the same Young’s modulus E ¼ 65 GPa. It is noted that the material of TA is much stronger than that of TB. The yield and ultimate stresses of two materials are s0.2A ¼ 245 MPa and s0.2B ¼ 181 MPa, suA ¼ 270 MPa and suB ¼ 210 MPa, respectively. Since, aluminum alloys are conventionally strain-rate insensitive, the effect of strain-rate is ignored in this study. As illustrated in Fig. 2(a), a buckling initiator is installed at the top end of the circular tube, and is composed of a cylindrical aluminum-alloy column with a radius r0 along the axis of the circular tube and N pulling strips which are uniformly distributed around the top edge of the tube. The top surface of middle column is higher than the edge of the tube by h. The N pulling strips are made of steel and connect to the bottom of the column and the top edge of the tube by screws. The width and thickness of the pulling strips are 10 and 0.5 mm, respectively, with their yield stress and ultimate strain being 350 MPa and 0.2, respectively.
300
789
0.03 0.04 0.05 Strain (mm/mm)
0.06
Fig. 1. Quasi-static tensile tests for the materials of the tubes.
0.07
(1)
where R0 ¼ D0/2 is the radius of the tube. A typical specimen is shown in Fig. 2(b). When the specimen is subjected to axial compression (or axial impact), the crushing process has two distinct stages: in the first stage, the crosshead (or the impinger) contacts the top of the middle column and makes the strips to pull the tube wall inward; in the second stage, the tube is compressed directly by the crosshead and a progressive crushing in diamond mode occurs. Because of the imperfections introduced by the buckling initiator in the first stage, the initial buckling force in the second stage will be greatly reduced and the deformation mode would be changed from ring mode or mixed mode to diamond mode. The quasi-static compression tests of the specimens were conducted on a UTM machine at HKUST, whose upper limit of load is 200 kN. The loading speed was set as 5 mm/min. In the tests, the effects of the pre-hit height h, the number of pulling points N and the inclined angle y were studied.
3. Experimental results 3.1. Compression of the original tubes First, the original tubes of TA and TB were compressed between the crosshead and the base plate on a UTM machine. The tested original tubes are shown in Fig. 3. It can be seen that the specimen of TA deformed in mixed mode, which means that initially it deformed in ring mode, but after one or two layers of ring folds, the deformation mode switched to diamond mode with lobes N ¼ 3. By considering the friction between the crosshead and the top layer of fold, the initial ring mode is resulted from this friction constraint and the real natural crushing mode should be the diamond mode. On the other hand, as shown in Fig. 3(c), the deformation of TB was always in the ring mode. This result reveals that although these two circular tubes have the same geometric characteristics, their crushing modes could still be different because of different yield stresses, and the stronger tubes tend to deform in diamond mode while the weaker ones tend to be ring mode. Fig. 4 depicts a comparison of the loading curves of the two kinds of circular tubes, where the mean crushing force Fm is calculated by the total work done divided by the total displacement. It is shown that initial peaks of the two circular tubes are Fpeak ¼ 51.5 and 37.8 kN, while the mean forces at the
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Fig. 3. The deformation mode of the original circular tubes: (a) TA-side view; (b) TA-top view; (c) TB-side view.
60 Tube-A Mean force T-A Tube-B Mean force T-B
Force (kN)
40
20
0 0
20
40 Displacement (mm)
60
Fig. 4. The compression loading curves for the original tubes.
displacement 55 mm are Fm ¼ 23.8 and 17.7 kN, respectively. According to the results from Ref. [18] it is found that the initial crushing force for the relatively thick tubes can be predicted by the yield force of the cross-section. By using the yield stress measured above, the initial yield force for the two circular tubes are FY ¼ 2pR0sYt ¼ 50.5 and 37.3 kN which show very good agreements with the real measured peak forces.
3.2. Deformation mode of the tubes with buckling initiators For the tube specimens with buckling initiators, considering the radius of the original tubes was R0 ¼ 25 mm, the pre-hit height h was chosen between 5 and 25 mm. For N ¼ 2 and 4, the angle y0 remained to be 901; while for N ¼ 3, y0 ¼ 601 and 901 were studied. In the compression tests, although tubes TA and TB had different material properties, their deformation modes were quite similar. The deformation of a typical specimen in the first stage of crushing is shown in Fig. 5(a). Initially, due to the pulling of the strips, the tube wall near the pulling points moved inwards while the parts far from the pulling points kept their original shape. As a result, the top edge of the circular tube deformed from a circle into a flower with N petals as shown in Fig. 5(b). However, in the second stage of crushing, the deformation mode switched to diamond mode, in which the tube wall progressively folded into equilateral polygons with N corners and the angle between two neighboring layers was p/N. Also, in the axial direction, the corners at every layer of fold had a finite height as shown in Fig. 5(c).
Figs. 6–8 show the top, bottom and side views of the tested specimens for TB tubes with N ¼ 2, 3 and 4, respectively. It is shown from Fig. 6(a) that the top layer of all the specimens deformed in equilateral flowers with N ¼ 2 petals. With the increase of pre-hit height h, the inward movements of the pulling points were more serious and the corner at every petal became sharper. This phenomenon is more clear for N ¼ 3 and 4 as shown in Figs. 7(a) and 8(a). It was also observed in the experiments that when h was smaller than 20 mm, the extension of the pulling strips was very little; however, when h was larger than 25 mm, the extension of the strips became significant due to the yielding of the strips. For the specimens with h ¼ 30, the pulling strips broke, which means that further increasing h after this value would be more difficult. From the bottom view in Figs. 6(b)–8(b), it is found that although for these specimens N ¼ 2 and 4, the subsequent deformations of most specimens had transferred to diamond mode with N ¼ 3, i.e., the diamond mode with N ¼ 3 seemed to be the most stable mode. The side views of the specimens show that although most of the specimens could crush progressively, the cases for N ¼ 2 and 4 had an unstable tendency to switch to Euler buckling mode. The folding details will be analyzed in the next sections. The measured geometric data of the tested specimens are listed in Table 1, in which l and de are the half fold-length and effective stroke coefficient, respectively. The half fold-length l is defined as the arc-length in the axial direction between the center of a horizontal plastic hinge and the neighboring mid-point of a corner (e.g., the arc-length between A and D in Fig. 13(b)). It is clear that for the specimens with N ¼ 3, l and de are both independent of h and y0, which have their average values l ¼ 19 mm and de ¼ 0.74. For N ¼ 2 and 4, because their deformations are not so stable and uniform, most of the geometric data were not obtained.
3.3. The mean force and peak force A typical loading–displacement curves of the specimen is depicted in Fig. 9, showing that in the first stage, the load was quite low and did not have a peak, because the compressive loading was only resisted by the pulling strips. The total displacement in the first stage was about h. When the crosshead contacted the top edge of the tube wall in the beginning of the second stage, the load increased sharply until a critical value Fpeak. Then, the tube buckled and a progressive crushing took place. Due to the imperfections generated in the first stage, Fpeak was significantly lower than the original peak as shown in Fig. 4. The mean forces and peak forces for all the specimens are listed in Table 1, while Fig. 10(a) and (b) plot the comparison of the peak force and mean force for the specimens TB with respect to
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791
Fig. 5. The deformation of the specimens: (a) side view in the first stage; (b) top view in the first stage; (b) side view in the second stage.
h=10mm
h=15mm
h=20mm
h =25mm
h=30mm
Fig. 6. Tested specimens for TB with N ¼ 2 and y ¼ 901: (a) top view; (b) bottom view and (c) side view.
the pre-hit height h. It is shown from Fig. 10(a) that with the same number of pulling strips, the increase of the pre-hit height h will reduce the initial peak force. After h was larger than 25 mm, the reduction of the peak force became insignificant, because the initial peak force was lower than the subsequent peaks. With the same pre-hit height, the initial peak for N ¼ 2 is slightly higher than those of N ¼ 3 and 4. As shown in Fig. 10(b), it is evident that with the same number N, although the inclined angles of the pulling strips are different, given suitable h, the same reduction of the peak force can be achieved. On the other hand, it can be seen that the mean force only depends on N, and the mean forces for specimens of N ¼ 2 and 4 are higher than that of specimens of N ¼ 3. Nevertheless, the change in the mean force is not so significant.
4. Theoretical considerations and discussions Based on the experimental results, the deformation and loading details of the specimens with buckling initiators are analyzed theoretically in this section. It should be noted that all the analysis are based on rigid and perfectly plastic material idealization.
4.1. Geometric analysis In the first stage of crushing, the middle column is firstly pushed downwards so that the N strips pull the top edge of the tube walls at the connected points. According to the observations in Fig. 5, the deformation of axial cross-section through a pulling point in this stage can be simplified as sketched in Fig. 11(a), in which A is an original pulling point, O is the center of the column, and D is the boundary point of the influenced area on the tube wall. At the end of this stage, the column moves a distance h and reaches the same horizontal level as the top edge of the tube. As a result, point A moves and rotates to position A0 . Assume the influenced length of the tube wall AD is l0, and the length of the pulling strips is ls, while the final rotation angles of the inclined tube wall and pulling strips are b and y, respectively. Then, at the end of the first crushing stage, we have,
l0 sin b þ ls sin y ¼ R0 r 0
(2)
l0 ð1 cos bÞ þ ls cos y ¼ ls cos y0 þ h
(3)
On the other hand, from the top view, the deformation of the tube wall in the first crushing stage can be simplified as shown in Fig. 11(b) (e.g., for N ¼ 3), in which the pulled parts deform
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h=10mm
h=15mm
h=20mm
h=25mm
h=30mm
Fig. 7. Tested specimens for TB with N ¼ 3 and y ¼ 901: (a) top view; (b) bottom view and (c) side view.
h=10mm
h=15mm
h=20mm
h=25mm
h=30mm
Fig. 8. Tested specimens for TB with N ¼ 4 and y ¼ 901: (a) top view; (b) bottom view and (c) side view.
inwards and the parts far from the pulling points remain with their original shape. The inward part A0 B curves inwards and has the similar curvature as its original sharp AB, which can be seen in Fig. 5(b). Therefore, the relationship between the horizontal distance AA0 and the width of the undeformed part of tube edge can be approximated by h
l0 sin b ¼ 2R0 1 cos
p N
i
a
(4)
where a denotes the half central angle of an undeformed section as shown in Fig. 11(b). In the above equations, the influenced
length l0 is independent of N and y0, and from the experimental measurements, it can be estimated by the half fold-length associated with elastic buckling as follows: pffiffiffiffiffiffiffi
l0 ¼ 1:72 R0 t
(5)
By combining Eqs. (2)–(5), the relationship between h/R and a for different N and different initial inclined angle of strips y0 can be obtained numerically. The results for N ¼ 3 are plotted in Fig. 12. It is shown that with the increase of h/R0, the half central angle a decreases. However, when h reaches a critical value, b becomes p/2 and a renders a minimum. It is revealed that even
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Table 1 Experimental result for circular tubes with buckling initiators. No.
h (mm)
l (mm)
TA TB TB-N3-60
0 0 5 10 15 20 25 30
8 19.25 19.25 19 19.5 19.25 19
TB-N3-90
10 15 20 25 30
TB-N2-90
Fm (kN)
Fp (kN)
0.74 0.75 0.75 0.74 0.74 0.74 0.73 0.70
23.8 17.7 16.9 16.1 16.3 16.1 15.8 16.3
51.5 37.8 32.8 30.0 28.2 20.2(23.4) 16.7(23.7) 17.2(23.3)
19 19.25 18.5 18.5 18.25
0.74 0.74 0.73 0.73 0.71
17.2 15.9 16.2 16.4 16.8
34.3 30.7 26.9 21.4(24.3) 19.2(24.2)
10 15 20 25 30
16.4 18.75 15.2
0.70 0.76 0.74
18.3 19.7 18.0 17.1 18.5
37 34.3 29 27.6 27.7
TB-N4-90
10 15 20 25 30
18.25 16.9 19 18.25 18.75
0.75 0.70 0.74 0.74 0.71
18.2 19.2 20.0 18.4 17.9
31.1 32.0 27.9 18.9(23.3) 15.4(23.4)
TA-N3-90
10 15 20 25
18.5 19.25 17.75 18.75
0.77 0.74 0.76 0.75
24.5 22.6 22.4 22.1
47.0 40.7 36.4 33.9
de
Notes: TA-N4-90 means specimen TA with N ¼ 4 and y0 ¼ 901; ‘ ’ means the value is not measured.
793
crushing stage can be simplified as sketched in Fig. 13. Section ACGD is 1/2N of an entire circle along the circumference and has a half fold-length l in the axial direction. When the tube is compressed compactly, the cross-section along AD will be as shown in Fig. 13(b). The tube wall between A and E has been folded, while ED is always vertical during the deformation process. Finally, E and F reach the same horizontal level. It should be noted that the vertical space between D and E is taken by the folded tube wall at FG. Therefore, the vertical distance of DE is equal to that of AE. Supposed the arc-length between A and E is lh, the horizontal distance of AE after folding is H, and the curvature radius at point A is rh. Then, if the influence of the thickness is ignored, the fold-length and effective energy absorbing stroke coefficient de can be obtained by
l ¼ lh þ r h
and
de ¼ 1
2r h
l
¼
lh r h lh þ r h
(6)
For the deformation shape in Fig. 13(b), if de is given, the length lh and radius rh can be obtained as functions of H, with the details being given in the Appendix A. Fig. 14 shows the top cross-sectional view of a circular tube with N ¼ 3 at point A after compression reaches compaction, in which the solid line is the plastic hinges at this horizontal level. The cross-section at point A changes from initially a circle shown by the dashed curve to an equilateral triangle. With the assumption that the tube wall experiences no extension along the circumference, and the curvature radius at the corners is ignored, we have jACj ¼ Lx ¼
pR0 N
(7)
Considering the symmetry along the axis in Fig. 14, the angle +ACE ¼ pR0/2N and the horizontal distance between A and E can be found as p pR0 tan (8) H¼ N 2N Also, the inclined plastic hinge EF in Fig. 13(a) can be obtained
30
as
N=3, θ=90°, h=25mm F Fm
25
Lcp ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlh r h Þ2 þ L2x
(9)
Force (kN)
20 4.2. Estimate of the peak forces
15 10 5 0 0
20
40 60 Displacement (mm)
80
Fig. 9. Loading curve for the specimen TB-N3-90 with h ¼ 25 mm.
the initial inclined angle y0 is different, a always takes the same minimum value a ¼ 0.4, which depends on l0 only. Besides, the influence of middle column is also considered. The results for two values r0/R0 ¼ 0.1 and 0.2 are obtained, indicating that when r0/R0o0.2, its influence can be ignored. Based on the observations in Fig. 5(c), the deformation of a representative section on the circular tube wall in the stable
To estimate the initial peak force of the circular tubes with buckling initiators, the deformation details at a typical section near the top edge are analyzed. As shown in Fig. 15(a), at the end of the first crushing stage, the tube wall near the pulling point A has been pulled inwards to A0 , while section BC is not influenced by the pulling strips and maintains its original shape. It should be noted that after moving inwards, A0 B is lower than the undeformed edge BC. As a result, in the second stage when the crosshead compresses the tube directly, the contact area will be reduced to N corner sections like BC. Because, the tube wall A0 BD is curved but BCGD remains unchanged, all the compressive loads will be carried by BCGD. Consider the compression of a typical section BCGD in Fig. 15(a). If the compressive stress is large enough, section BCGD will buckle and a plastic hinge will occur along the dashed curve through point T as shown in Fig. 15(b), where point T is located near the middle of CG. Since, the original tube deforms in ring mode or mixed mode whose initial peak force can be approximated by the product of the yielding force and the cross-sectional area (see Section 4.1), as a part of the tube, the buckling force of section BCGD can be estimated by the yield force along the midline KT.
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1.2
45 Fp N=2 Fm N=2 Fp N=3 Fm N=3 Fp N=4 Fm N=4
40 35 Force (kN)
30
1.0 0.8
25 0.6 20 0.4
15 10
0.2 5 0.0
0 0
5
10
0
5
10
15 20 Prehit height (mm)
15
20
25
30
25
30
1.2
45
Thus, the subsequent deformation of this fold at section A0 CGD can be considered to consist of bending of A0 JKD and buckling of JCKT. Since, KT located at the middle of CG and CD is approximately straight, the buckling force of KT is estimated as F p1 ¼
F-p θ=60
40
Fm θ=60 Fp θ=90
35
1.0
Fm θ=90
0.8
30 Force (kN)
As the relation between a and h/R0 has been obtained in the previous section, by using Eq. (11), the initial buckling force of the tube in the second stage can be estimated. However, it should be noted that if the initial peak force is lower than the subsequent peaks, further reduction of the initial peak will be meaningless. Therefore, the magnitude of the subsequent peaks still needs to be determined and compared with the initial one. The formation of every fold in the stable crushing stage starts from a shape as shown in Fig. 16(a). Different from the top fold of the tube as shown in Fig. 15(a), in this stage, points B and C have merged together and A0 C is a straight horizontal plastic hinge. Also, the shape along CD is much closer to a straight line. The distance AA0 is h p i p (12) jAA0 j ¼ R0 1 cot N N
25 0.6 20 0.4
15 10
0.2 5 0.0
0 0
5
10 15 20 Prehit height: h (mm)
25
30
Fig. 10. Experimental results for peak forces and mean forces: (a) TB specimens with y ¼ 901 and different N; (b) TB specimens with N ¼ 3 and different y.
pR0 2N
A
A’
D ’
0 R0
ls
O A’
A B
C
Fig. 11. Geometric relationships of the specimen in the first crushing stage: (a) side view; (b) top view.
Suppose the central angle of BC is a and the hinge along BKD can be approximated by a quarter of sine curve with the point T located in the middle of CG, then the length of KT is 2 p jKTj ¼ R0 (10) aþ 3 3N Therefore, the initial buckling force of the whole circular tube in the second crushing stage can be approximated by 2 p F p 2N a 1 (11) and F p ¼ sY t2NR0 aþ ¼ þ 3 3 3N FY 3p
(13)
The compressive stress along A0 J comes from bending of section A0 JKD, and the total force is the integration of the stress on every strip of width dx as shown in Fig. 16(b). If the effect of the transition section is ignored, the horizontal and vertical distances between A0 and D are lx ¼ l sin c and ly ¼ l cos c
(14)
In the above equations, l can be obtained from Eq. (6) and c can be calculated by lx ¼ |AA0 |. After a finite bending, the part below DK changes very little, the compression force on a strip of width dx can be obtained as dF ¼
2M p
l sin c
dx
(15)
pffiffiffi where M p ¼ ð1=2 3ÞsY t 2 is the maximum plastic bending moment per unit width of tube wall. For the strip at x position, the inclined angle is j, but l is a function of x as follows:
lðxÞ ¼ l
sY t
Nx
pR0
(16)
Therefore, the total buckling force for a typical fold can be approximated by ! Z 1 F ps t 1 1 dx þ ¼ pffiffiffi (17) 2 FY 3R0 ½1 ðp=NÞ ð1= tanðp=NÞÞ 0:5 x According to Eq. (17), if t/R0 is smaller, Fps/FY will be smaller. It should be pointed out that the method of estimating the peak forces in this section can only be applied to the case of relatively thick tubes which deform in ring mode or mixed mode. 4.3. Mean crushing force Although numerous papers published have discussed the modeling of axial crushing of circular tubes in ring mode, very few studies have been made about the energy absorption of circular tubes deformed in diamond mode. Pugsley and Macaulay [2] proposed the first theoretical analysis in 1960s, followed by Pugsley [3] and Johnson and Soden [4], who made further improvements. However, these models only provided some basic ideas in analyzing this problem and were not accurate enough. Recently, Marsolek and Reimerdes [16] used some patterns to generate diamond mode and developed a model to predict the
ARTICLE IN PRESS X.W. Zhang et al. / Thin-Walled Structures 47 (2009) 788–797
mean crushing force. However, some of their assumptions do not agree with the observations in our experiments. To estimate the mean crushing force, an energy method is applied to a typical tube section of length l. During the entire compression process, the energy is dissipated by four distinct mechanisms as follows:
Half central angle of a corner α
1.1 1.0
θ=90°, r0 /R 0=0.2
0.9
θ=72°, r0 /R 0=0.2 θ=60°, r0 /R 0=0.2
0.8
795
θ=60°, r0 /R 0=0.1
(a) First, the original cylindrical tube wall is compressed to become flat and the energy dissipated is
0.7
E1 ¼ 2plM p
0.6 0.5 0.4 0.0
0.2
0.4 0.6 0.8 Pre-hit height h/R0
1.0
(18)
where Mp is the maximum plastic bending moment per unit width of tube wall. (b) A portion of energy is dissipated by the rotation of the horizontal plastic hinges. According to the analysis given in the Appendix A, the rotation angle along every horizontal hinge is kcp, where kc is 1.2 for de ¼ 0.74. Therefore, the energy dissipated is
1.2
Fig. 12. Relations between the pre-hit height h/R0 and half central angle of a corner.
E2 ¼ 2pR0 M p kc p
(19)
(c) The energy dissipated at the inclined plastic hinges is E3 ¼ 2NMp pLcp
A 2rh A
C
D
F T S
E D
E
λ Δ
G
H Fig. 13. Diagram for the deformation of the tubes in a typical region: (a) side-view; (b) cross-sectional view.
A
O
H
C (F)
W ¼ F m lde
Fig. 14. The cross-sectional view in the stable crushing stage of the tube.
B D
2Mp NLx l (21) r According the measurements in our experiments, the curvature radius r is about 1.5 times of the tube’s wall thickness. To sum up the above four portions, the total energy dissipated by a half fold is X R0 Ei ¼ 2pMp l þ kc pR0 þ 2NLcp þ l (22) r i¼1;4 E4 ¼
On the other hand, if the mean crushing force during this compression process is Fm, then the work done by the external force is
E
A’
(20)
(d) Besides the horizontal and inclined plastic hinges, a great portion of energy is dissipated by the traveling plastic hinges. As shown in Fig. 13(a), the buckling of section EFGD starts from an inclined plastic hinge ET. However, with the progress of compression, this plastic hinge will gradually move from EC to EF and then to ES. In the area where the plastic hinge passed by, the tube wall experienced bending and revisedbending processes. According to the deformation mechanism, the shadow area in Fig. 13(a) is the region where the plastic hinge passed through. Suppose the curvature radius of plastic hinge is r, then the energy dissipated by the traveling hinges is
O
(23) P
In view of energy balance, we have, W ¼ iEi. By using Eqs. (18)–(23), we can obtain the mean crushing force of the circular tubes deforming in diamond mode as follows, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 ðlh r h Þ2 þ L2x R0 Fm 2p 4 R0 (24) ¼ 1 þ kc p þ N þ 5 Mp de l l r
C B
C 4.4. Discussions
G K T D
The initial peak forces estimated by Eq. (11) for N ¼ 3 and
y0 ¼ 901 with different h are plotted and compared with G
Fig. 15. The initial buckling of the specimen in the second stage: (a) three dimensional diagram; (b) details at a typical corner.
experimental results in Fig. 17. It is shown that this analysis can predict very well the decreasing tendency of the initial peak force with the increase of h, and the maximum reduction of the peak force could be more than 30%. However, the predicted curve is
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dF
O A’
A’
x
J
A
K
T
D
5. Summary and remarks
ly
C
G
D
lx R0
Fig. 16. Buckling of a fold in the stable crushing stage: (a) three dimensionaldiagram; (b) cross-sectional view.
40 1.0 35
force (kN)
30
0.8
25 0.6 20 15
0.4 Theoretical Exp Fpeak Exp Fmean Fs Fm
10 5 0
0.2 0.0
0
5
10 15 20 pre-hit height h (mm)
25
strips in our device greatly reduce the inertia, because in the initial pulling stage, the involved mass is only the small part at the end of the tube instead of the entire tube. In addition, in this theoretical analysis, the extension of the pulling strips is not considered, because it is only significant when h is quite large.
30
Fig. 17. Comparison between theoretical analysis and the experiments for N ¼ 3.
slightly lower than the experimental results, because in the analysis the radius of the corners and inclined plastic hinges are neglected. The value of peak force in the stable crushing stage is also estimated and plotted as the dotted line. When the initial peak force is lower than the subsequent peaks, the further reduction will be meaningless. Therefore, the level of subsequent peaks can be regarded as the lower limit of the peak force. Considering the stroke coefficient of the tubes is about de ¼ 0.74 and using Eq. (6), we have rh/lh ¼ 0.15. Therefore, from Fig. A2 in the Appendix A, j ¼ 0.3, so that kc ¼ 1.20 and lh ¼ 1.25 H. For N ¼ 3 and H ¼ 0.6R0, the mean crushing forces predicted by Eq. (24) for TA and TB are Fm ¼ 22.7 and 16.8 kN, respectively. These estimations agree very well with the experiments (Fm ¼ 23.6 and 17.7 kN, for TA and TB, respectively). It should be noted that under impact scenarios two kinds of dynamic effects, namely, strain-rate and inertia effects, will make the structure and material stronger. The strain-rate effect depends on the material itself; for the strain-rate insensitive materials, this effect can be ignored. On the other hand, the inertia effect depends on the impact velocity V as well as the mass ratio M/m (where M and m denote the masses of the impinger and the tube, respectively); it will become much weaker for larger mass ratio and lower velocity. According to the study in [18], in the axial crushing of aluminum-alloy tubes, the dynamic effects can be ignored for M/m4100 and Vo10 m/s. Also, different from the conventional study of tubes under axial crushing, the pulling
In this paper, a buckling initiator designed for reducing the initial peak force of axially crushed circular tubes is investigated. This buckling initiator is installed near the impact end of a circular tube and is composed of a pre-hit column along the tube’s axis and several pulling strips uniformly distributed around the top edge of the tube. It functions just before the impact happens, and can effectively reduce the initial buckling force without affecting the structural stiffness at normal working conditions. By using aluminum-alloy thin-walled circular tubes with R/t ¼ 19, the influences of the number of pulling strips N, pre-hit height h and the inclined angle of the pulling strips have been all studied. The results reveal that this buckling initiator could effectively reduce the initial peak force and retain the excellent energy absorption of the circular tube. By using this buckling initiator, the deformation mode of the axial crushed circular tube switches from ring mode or mixed mode to diamond mode. For the relatively thick tubes which could deform in ring mode or mixed mode, although specimens with N ¼ 2, 3 and 4 were tested, the stable deformation tended to diamond mode with lobe number N ¼ 3. With suitable selection of pre-hit h, the initial peak force could be reduced by more than 30%. Based on the experimental results, the deformation of the thin-walled tubes in diamond mode is analyzed in detail and a simplified theoretical analysis is conducted to reveal the reduction of the initial force as well as the energy dissipation mechanisms, showing good agreement with the experiments. For the application of this kind of devices, they can be used as energy absorbers at the roadside or as part of the landing gears of small flying vehicles; for the automobile structures, in which the proximal end of the tube is fixed, allocating the pulling points to the middle of the tubes may be considered.
Acknowledgements This study is funded by the Hong Kong Research Grant Council (RGC) under CERG Grant no. 621505 and the National Natural Science Foundation of China under Key Project no. 10532020. Their supports are gratefully acknowledged.
Appendix A. Geometric analysis for the folds in the axial direction Consider a fold of the tube wall shown in Fig. 13(b). The configuration of the folded part can be simplified as shown in
r
l
H Fig. A1. The cross-sectional view of a typical half fold.
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Thus,
0.25
residual height of a half-fold Δ/λ
797
l ð2 cos j 1Þ r 1 ¼ and ¼ ¼ 1 de r sin j l ðp þ 2jÞ þ ð2 cos j1Þ
0.20
(A4)
sin j
The horizontal fold-length is cos jð2 cos j 1Þ H ¼ 2rð1 þ sin jÞ þ l cos j ¼ r 2ð1 þ sin jÞ þ sin j
0.15
(A5)
0.10 and
0.05
H¼l
0.00 0.0
0.2
0.4
0.6
0.8
1.0
1.2
inclined angle of the straight segment ϕ Fig. A2. The relationship between D/l and j.
½2ð1 þ sin jÞ þ ðcos jð2 cos j 1Þ= sin jÞ ðp þ 2jÞ þ ðð2 cos j 1Þ= sin jÞ
(A6)
After folding, the residual height of a half fold is D ¼ r. Hence, Eqs. (A4)–(A6) give the relationships among D/l, l/H and j, which are depicted in Figs. A2 and A3. For a given j, kc ¼ (1+2j/p), and the parameters de ¼ D/l, kl ¼ l/H. Therefore, the relation between de and kl could be obtained. References
1.45 1.40 1.35
λ/H
1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.0
0.2 0.4 0.6 0.8 1.0 Inclined Angle of the straight section ϕ
1.2
Fig. A3. The relationship between l/H and j.
Fig. A1. Suppose the curved sections have the same curvature radius r, the length of the straight section between two curved regions is l, and the angle between the inclined straight section and the horizontal level is j. Then, if the influence of the wall thickness is ignored, the arc-length of this half fold is p (A1) l ¼ 2r þ j þ l 2 The vertical height of this half fold is r ¼ lð1 de Þ
(A2)
Also, from geometric relations in the vertical direction, r cos j ¼ l sin j þ rð1 cos jÞ
(A3)
[1] Alexander JM. An approximate analysis of the collapse of thin cylindrical shells under axial loading. Q J Mech Appl Math 1960;XIII:10–5. [2] Pugsley SA, Macaulay M. The large-scale crumpling of thin cylindrical columns. Q J Appl Math 1960;XIII:1–9. [3] Pugsley AG. On the crumpling of thin tubular struts. Q J Mech Appl Math 1979;XXXII:1–7. [4] Johnson W, Soden PD. Inextensional collapse of thin-walled tubes under axial compression. J. Strain Anal 1977;12(4):317–30. [5] Andrews KRF, England GL, Ghani E. Classification of the axial collapse of cylindrical tubes under quasi-static loading. Int J Mech Sci 1983;25(9): 687–96. [6] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. Int J Impact Eng 1984;2(3):263–81. [7] Abramowicz W, Jones N. Dynamic axial crushing of circular and square tubes. Int J Impact Eng 1986;4(4):243–70. [8] Wierzbicki T, Bhat SU. A moving hinge solution for axi-symmetric crushing of tubes. Int J Mech Sci 1986;28(3):135–51. [9] Grzebieta RH. An alternative method for determining the behavior of round stocky tubes subjected to an axial crush load. Thin-walled struct 1990(1–4): 61–89. [10] Gupta NK, Sekhon GS, Gupta PK. A study of fold formation in axi-symmetric axial collapse of round tubes. Int J Impact Eng 2002;27(1):87–117. [11] Huang X, Lu G. Axisymmetric progressive crushing of circular tubes. Int J Crashworthiness 2003;8(1):87–95. [12] Abramowicz W, Jones N. Transition from initial global bending to progressive buckling of tubes loaded statically and dynamically. Int J Impact Eng 1997;19(5–6):415–37. [13] Jensen Q, Langseth M, Hopperstad OS. Experimental investigations on the behavior of short to long square aluminum tubes subjected to axial loading. Int J Impact Eng 2004;30(8–9):973–1003. [14] Hosseinipour SJ, Daneshi GH. Energy absorption and mean crushing load of thin-walled grooved tubes under axial compression. Thin-walled struct 2003;41(1):31–46. [15] Singace AA, El-Sobky H. Behavior of axially crushed corrugated tubes. Int J Mech Sci 1997;39(3):249–68. [16] Marsolek J, Reimerdes HG. Energy absorption of metallic cylindrical shells with induced non-axisymmetric folding patterns. Int J Impact Eng 2004; 30(8–9):1209–23. [17] Sastranegara A, Adachi T, Yamaji A. Improvement of energy absorption of impacted column due to transverse impact. Int J Impact Eng 2005; 31(4):483–96. [18] Zhang XW, Su H, Yu TX. Energy absorption of an axially crushed square tube with a buckling initiator. Int J Impact Eng 2009;36(3):402–17.