Energy absorption of an axially crushed square tube with a buckling initiator

International Journal of Impact Engineering 36 (2009) 402–417

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Energy absorption of an axially crushed square tube with a buckling initiator X.W. Zhang, H. Su, 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 i n f o

a b s t r a c t

Article history: Received 23 November 2007 Received in revised form 26 February 2008 Accepted 26 February 2008 Available online 10 May 2008

Axially crushed thin-walled square tubes have been widely used as energy absorbers because of their high specific energy absorption capacity and long stroke. However, they render extremely high initial peak forces which may cause serious injury or damage to the people or structures being protected. This paper proposes a novel idea that by installing a buckling initiator near the impact end which is composed of a pre-hit column and pulling strips, the initial peak force of the square tube could be greatly reduced while its deformation mode and excellent energy absorption are retained. Both experimental and numerical investigations are conducted on aluminum alloy square tubes. The peak force, mean force and half-length of a fold of the tested specimens are examined. The results show that the mean crushing force and deformation mode are not affected by the buckling initiator, while the reduction of the peak force strongly depends on the pre-hit height. It is also found that the buckling initiator can ensure the deformation more stable and uniform. Finally, a simplified analytical model is developed to study the relationship between the reduction of the peak force and the geometric imperfections; and the model can successfully predict the effectiveness of the buckling initiator. Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved.

Keywords: Square tube Buckling initiator Progressive buckling Peak force Energy absorption

1. Introduction Thin-walled tubes are extensively applied as structural members in engineering, because of their low cost, high stiffness and strength combined with a relatively low density. During impact events, thin-walled tubes can effectively absorb the kinetic energy of the structures by performing plastic deformation so as to protect the people, structures and/or equipments being involved. As energy absorbers, they can be used in many different ways, such as lateral compression, axial crushing, splitting and curling, expanding and so on [1,2]. In particular, the axial progressive buckling of thinwalled tubes notably display large specific energy absorption and a relatively long stroke. With the increasing demands on lighter and more efficient energy absorbing structural components in transportation systems, axial crushing behaviors of aluminum alloy and high-strength steel square tubes attract great attentions of researchers [3,4]. A great number of studies have been carried out on the axial crushing behaviors of thin-walled tubes since the pioneering work of Alexander [5] and Pugsley [6] in 1960s. Wierzbicki and Abramowicz [7] developed a simplified model based on rigid plastic assumptions and obtained the mean crushing force for square tubes. Abramowicz and Jones [8,9] conducted a series of experiments, and modified the collapse models by taking account of

* Corresponding author. Tel.: þ86 852 2358 8652. E-mail address: [email protected] (T.X. Yu).

strain-rate sensitivity and effective crushing distance. Later, Karagiozova and Jones [10,11] studied the dynamic effects of the axial crushing of square tubes and pointed out that for high-velocity impact, the inertia effect, wave propagation and material characteristics play very important roles in the initial stage of dynamic response. Moreover, the transition conditions between global buckling and progressive buckling were investigated by Abramowicz and Jones [12], and Jensen et al. [13]. Though the axially crushed thin-walled tube offers an excellent energy absorbing efficiency, it usually renders an extremely high initial buckling force, which may cause severe injury or damage to the people or structures being protected, so is highly undesirable in practical applications. In order to reduce the initial buckling force and make the deformation easier, researchers introduce imperfections to the tubes, such as corrugations [14] or grooves [15]. Marsolek and Reimerdes [16] employed a special device to induce non-axisymmetric folding patterns with different wave number and minimize the initial peak force of circular tubes. More recently, Sastranegara et al. [17] reported that although the introduction of imperfections can help to reduce the initial peak force, it can also cause a decrease of the structural stiffness which is not desirable for most applications. Therefore, they proposed that by introducing transverse impact to an axially impacted column just before the impact happens, its energy absorbing characteristics can be improved without any sacrifice of the structural stiffness. Also, Chung and Nurick [18] used explosive to generate holes on the walls of a steel square tube so that its axial peak force was reduced. However, both the pre-impact and explosive schemes seem

0734-743X/$ – see front matter Crown Copyright Ó 2008 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2008.02.002

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

impractical because both require complex additional devices and may cause safety concerns. Since in real applications, for example, in automobiles, the square tubes are not only energy absorbing members, but also serve as load-carrying structures. Therefore, their stiffness and vibration performance are quite important. However, conventional initial dents which could achieve reduction of peak force would also cause a significant reduction in the bending stiffness of local cross-sections (refer to Appendix 1). In this paper, a novel idea is proposed; that is, by installing a buckling initiator attached at the impact end, the initial peak force of a square tube could be effectively reduced, whilst its deformation mode, excellent stiffness and energy absorption capacity could be retained. Also, this buckling initiator starts to function just before the axial impact on the square tube happens and it is mechanically simple. With some prototypes being designed and manufactured, a series of quasi-static and dynamic experiments were conducted to investigate the effectiveness of the buckling initiators. Various design parameters as well as impact velocities and masses are analyzed. Moreover, numerical simulations are performed to verify the experiments, and special attentions are paid to the initial impact stage and the dynamic peak force. Finally, based on the experimental and numerical results, the effects of the design parameters of the buckling initiator on the reduction of peak force and the induced deformation are comprehensively discussed.

2. Experimental details 2.1. Specimen and deformation process of the tube The square tubes in the experiments were made of aluminum alloy 6063-T5. The total length of the tubes was 250 mm with the lower portion of 30 mm being clamped by a fixture as shown in Fig. 1(a). In order to generate a progressive buckling, the width and wall-thickness of the tubes’ cross-section were chosen as b ¼ 44.3  0.2 mm and t ¼ 1.25  0.03 mm, respectively, as shown in Fig. 1(b). The buckling initiator was composed of an aluminum pre-hit column and a steel pulling strip, as shown in Fig. 1(c). The column was vertically fixed at the middle of the pulling strip with its top surface slightly higher than the top edge of the tube, while the ends of the strip were connected to two opposite tube walls. Under an axial impact by a rigid mass, the entire deformation process of the square tube specimen can be divided into two distinct stages as follows: (1) Pulling stage: a striker of mass M and initial velocity V0 impinges the pre-hit column which forces the steel pulling strip to pull the two opposite tube walls inwards, creating some imperfections on those walls;

V0

M

w

(2) Buckling stage: when the striker contacts the top edges of the tube itself, the square tube is axially compressed, resulting in a progressive buckling. It should be noted that the major role of the buckling initiator is to introduce geometric imperfections to the tube, while most of the kinetic energy of the striker is absorbed by the tube in the second stage. As illustrated in Fig. 1(c), three geometric parameters should be chosen in the design of buckling initiators, i.e., h (the pre-hit height of the column), d (distance between the pulling point and the top edge) and q (the initial inclination angle of the strip). It was observed in preliminary tests that in the pulling stage two typical deformation modes may appear, depending on the magnitude of parameter d. If d is small, Mode 1 appears, in which the middle plane OP1P2 is as shown in Fig. 2(a); but if d is large enough, Mode 2 appears, as shown in Fig. 2(b). However, when the tube deforms by Mode 2, the pulling strip is relatively weak compared with the lateral stiffness of the tube, so it is difficult to generate large imperfections. Under either Mode 1 or Mode 2, the deformation of the lower part of the tube below the pulling points can be simplified as shown in Fig. 2(c), where point A (and its new position A0 ) is the same as point P1 in Fig. 2(b). The length of the influenced region on the tube by the pulling from the bar is supposed to be l. In the xyz coordinate system shown in Fig. 2(c), when point A is pulled inward to A0 , line AE rotates an angle a about the y-axis. On the neighboring side wall, point B deforms outwards and line BF rotates an angle b about the x-axis, while the inclination angle of the corner CD is denoted by g. A rigid, perfectly-plastic analysis (see the Appendix pffiffiffi2 for details) confirms that the influenced length is about lzb= 2, and the deviation angle of the corner g is much smaller than that of a and b. By taking a as an indicator of the imperfection induced by the buckling initiator, the relationship between a and the displacement of the pre-hit column, d can be expressed as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2  u 2 b0 b0 u b0 t 2 d ¼ lð1  cos aÞ þ   l sin a  2 2 sin q tan q

θ

hzd < dmax ¼ 19:2 mm

t=1.25

A P2

P1

a

O

b=44.3

b

c

Fig. 1. A square tube specimen and its cross-section.

P1

P2

λ

O b 30

(2)

d P2

(1)

where b0 ¼ b  w  t is the effective width of the modified tube. In our tests, the width of the compressive column was chosen as w ¼ 10 mm, so that b0 ¼ 33 mm. The preliminary tests showed that for the case of q < 60 , the steel strip was easily broken due to a large pulling force. Hence, in the subsequent experiments the inclination of the pulling strip was chosen as qz84 . To avoid excessive compression of the tube, the pulling points (such as point A0 ) should not contact the pre-hit column before the striker contacts the top edges of the tube, which requires that a < sin1 ðb0 =2lÞz33 . In this case, g < 10 and the vertical displacement of the top edge of the tube can be ignored. Thus,

h

P1 L=250

403

λ O

Mode 1

Mode 2

a

b

z A’ C

B B’

y α

x

E D γ

F

β

c

Fig. 2. The deformation of the tube in the pre-hit stage.

404

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

neglected, the axial force applied to the loadcell can be calculated by

Table 1 Dynamic test results Test no

M (kg)

V (m/s)

D (mm)

Ein (J)

Fm (kN)

Fp (kN)

Folds

Tube-A Tube-B Tube-C Tube-D Tube-E D-05-10-A D-05-10-B D-05-10-C D-05-10-D D-05-10-E D-05-15-A D-05-20-A D-05-20-B D-05-20-C D-05-25-A D-30-10-A D-50-10-A D-30-15-A D-50-15-A

68 68 68 54 40 68 68 68 54 40 68 68 68 68 68 68 68 68 68

7.00 6.20 5.22 6.74 6.85 6.85 6.22 5.33 6.94 6.95 7.16 7.05 6.32 5.48 7.0 7.02 7.17 7.03 6.94

119 93 69.5 83 54 130 108 76 103 75 140 130 113.5 76 127 124 134 134 131

1745.3 1368.9 972.8 1270.5 959.6 1682.0 1387.4 1016.5 1354.9 995.5 1836.3 1776.5 1433.7 1071.7 1750.6 1758.2 1837.2 1769.6 1724.9

14.7 14.7 14.0 15.3 17.8 12.9 12.8 13.4 13.2 13.3 13.2 13.7 12.6 14.1 13.8 14.2 13.7 13.2 13.2

52.8 54.0 42.3 49.5 54.4 51.1 50.6 42.9 52.1 50.0 44.0 36.1 31.0 29.0 31.7 52.8 54.8 46.7 41.2

7 6 4 5 4 7 6 4.5 5.5 4 7 7 6 4 7 7 7 7 7

Note: In D-X-Y, X means the value of d, and Y means the value of h.

In our experiments, four different values of h were taken in the range of 10–25 mm and three values of d were examined. The parameters of the specimens are listed in Table 1. 2.2. Dynamic test program The dynamic experiments were conducted on a drop tower located at HKUST Impact Lab. As shown in Fig. 3(a), the drop tower is 7.2 m in height, and capable of carrying a drop weight of up to 68 kg. By considering the energy absorbing capacity of the specimens, three different masses were used, i.e., 40 kg, 54 kg and 68 kg, and the largest height used in the experiments was 2.5 m. At the bottom of the drop tower, there was a concrete block. For the measurement of the impact force, a loadcell was placed between the specimen and the concrete block. As shown in Fig. 3(b), this loadcell was composed of a hollow high-strength steel cylinder and two plates. The inner and outer diameters of the hollow cylinder were d1 ¼ 50 mm and d2 ¼ 56 mm, respectively, and its total length was 70 mm. On the outer surface of the loadcell, four strain gages were placed to record the axial strain signals during the impact. Besides, a laser sensor was adopted to measure the impact velocity and trigger the oscilloscope before the hammer impinges onto the specimen. During the impact process, the deformation of the loadcell should be kept in its elastic range and its material’s Young’s modulus is supposed to be E. If the effect of wave propagation is

Drop Weight Hammer Guideway Laser sensor

FðtÞ ¼ EeðtÞ

p 4

d22  d21



(3)

Then, the dynamic force applied to the hammer is Fd zFðtÞ, so the acceleration, velocity and displacement of the hammer can be calculated by

mx_ ¼ mg  Fd

x_ ¼ v0 þ gt 

(4a)

Z

t 0

1 x ¼ v0 t þ gt 2  2

Fd dt m ZZ

Fd dtdt m

(4b)

(4c)

Since this analysis is based on neglecting the effect of wave propagation, it is more accurate in the late stage of the impact process when the stress wave has been greatly dissipated. For the initial impact stage, the peak force measured by this loadcell is greatly influenced by the wave propagation and may not represent the true value of the dynamic peak force. This issue will be discussed later in this paper. The dynamic force was recorded by the oscilloscope at a rate of 500,000 points/s. The original data for a representative specimen are shown in Fig. 4(a). It is seen that this loadcell can record the impact force history very clearly. To filter the noise of the equipment and remain the characteristics of the peak force (including the wave propagation), a 30-point FFT smoothing was conducted. As shown in Fig. 4(b), this smoothing filter can eliminate the system noise and keep the key features of the peak force and wave propagation effect. Moreover, before the impact experiments, both static and dynamic calibrations of the loadcell were conducted, which showed that the static force was in good agreement with MTS readings and the displacement obtained by integration of the force history was very close to that measured from the length change of the tube specimen after tests. 3. Experimental results 3.1. Material properties Quasi-static uni-axial tensile tests for material AA6063-T5 were conducted on MTS SINTECH 10/D whose upper limit of load is 50 kN, and the loading speed was 0.5 mm/min. As depicted in Fig. 5(a), the samples were cut from tube walls along the axial direction of a tube. Fig. 5(b) plotted the typical true stress–strain curves, showing that its Young’s modulus is E ¼ 65 GPa, the yield stress is sY ¼ s0:2 ¼ 180 MPa, ultimate stress su ¼ 220 MPa and the maximum elongation before break is about 10%. As aluminum alloys are strain-rate insensitive, and the impact velocities in our dynamic experiments were lower than 10 m/s, the effect of strainrate is ignored.

Specimen Loadcell Strain gages

a

3.2. Quasi-static tests

b Fig. 3. (a) The experimental set-up; (b) the loadcell.

56

50

50

Concrete block

The axial compression experiments of square tubes were previously conducted and described by many researchers, such as Abramowicz and Jones [8], Langseth and Hopperstad [3] and others. The phenomena observed are similar. When the length-towidth ratio of the tube is relatively large, the global Euler buckling

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

a

a

405

120

60

Force(kN)

5

15

Original data for Tube-A

50

40

30 30

50

20

b

250 strain-stress curve

10

0.01

0.02

0.03

0.04

0.05

0.06

Time(s)

b

60 Measured Peak

Original data After 30-point FFT

50

True stress (MPa)

200 0 0.00

150

100

50

Force(kN)

40 0 0.00

30

0.02

0.04

0.06

0.08

0.10

True strain (mm/mm) 20

Fig. 5. (a) The sample for tensile test; (b) a typical stress–strain curve from the material tests.

10

0 0.0080 0.0082 0.0084 0.0086 0.0088 0.0090 0.0092 0.0094

Time(s) Fig. 4. (a) The original force history obtained from a dynamic test; (b) the force history in the initial impact stage.

mode may be generated. For short tubes, one of four different progressive deformation modes may appear, depending on the width-to-thickness ratio; and they are the symmetric mode, nonsymmetric mode, non-compact progressive mode and the extensional mode [1,8].

In our experiments, some square tube specimens with and without buckling initiators (described as original and monitored tubes, respectively, in the following text) were compressed on MTS with a loading speed of 5 mm/min. During these tests, progressive buckling modes were observed. The entire compression process of the original tube has two deformation stages, i.e., an axial compression stage, followed by a progressive buckling stage. First, the compressive stress along the tube was uniformly distributed, and the applied compressive force F increased almost linearly with the displacement. When F reached a critical value Fcr, a local buckling occurred; after which the force gradually decreased and subsequently a progressive buckling took place. The initial buckling position of the original tubes was quite sensitive to the

Fig. 6. Deformed configurations of tube specimens after quasi-static test: (a) original tube; (b) monitored tube S-05-20.

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X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

imperfections and varied from test to test. For the monitored tubes, on the other hand, due to the pulling of the steel strip, certain geometric imperfections had appeared on the tube before the compression of the tube itself. As a result, the axial compression force was significantly smaller than that for the original tube, and the progressive buckling always started from the pulling positions. Typical specimens after quasi-static compression are shown in Fig. 6. It is seen that in the original tube, the folds are quite uniform and no facture is observed, while the first fold of the monitored tube is notably larger than others and fractures can be observed in some regions along the corners. The force–displacement curves of these two tests are plotted in Fig. 7. The initial buckling force of the original tube is Fcr ¼ 40:0 kN. By installing the buckling initiator of d ¼ 5 mm and h ¼ 20 mm, the initial buckling force is reduced to 27.7 kN. After the initial peak force at buckling, many subsequent peak forces exist, each of which is associated with a fold on the deformed specimen. The average force Fm associated with Fig. 7 is calculated from

Rx 0

Fm ðxÞ ¼

FðxÞdx x

(5)

When the displacement is small, the average force is quite high, and with the increase of the displacement, the average force approaches stable. The mean force of the original tube is Fm ¼ 11.5 kN, while the monitored tube is Fm ¼ 12.1 kN. Besides, the fluctuation of the monitored tube is larger than the original tube.

a

Original Tube Average force

35

Force(kN)

30 25 20 15

5 0 0

30

60

90

120

150

Displacement(mm)

b S-05-20 Average force

40

20

0 0

60

120

"

(6)

where Es ¼ s=e and Et ¼ vs=ve which are the secant and tangent modulus of elasticity, respectively. By using the stress–strain curve shown in Fig. 5(b), the plastic buckling stress is found to be scr zsY ¼ 180 MPa, and consequently the initial buckling force is Fcr ¼ 39:6 kN which is very close to the measured force (40 kN). As for the average compression force, according to the rigid, perfect-plastic analysis of square tubes given by Abramowicz and Jones [7], the average force should be

 13 s t2 Fm b ¼ 52:22 where M0 ¼ Y M0 4 t

(7)

By taking b ¼ 44.3 mm, t ¼ 1.25 mm and sY ¼ 180 MPa, the theoretical average force is found as Fm ¼ 12.0 kN. Hence, the theoretical prediction agrees well with the experimental results. 3.3. Dynamic tests

Ein ¼

10

Force(kN)

scr

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#  Et t 2 ¼ Es 2 þ 1 þ 3 9 Es b

p2

The dynamic test results are summarized in Table 1. Limited by the experimental facility, the impact velocity was not higher than 10 m/s. First, the sensitivity of the original and monitored tubes to impact velocity and mass was studied on Tube-X and D-05-10-X series. Then, tubes with different buckling initiators were tested and three different combinations of impact mass and velocity were used. The impact conditions are labeled by A–E, as shown in Table 1. The impact velocity V0 was measured by the laser sensor, while the ultimate deformation D was determined from the change of the tube length before and after the test. Therefore, the input energy Ein d can be obtained as follows: and the mean impact force Fm

45 40

To predict the initial buckling force of a square tube, the buckling theory of simply-supported plate can be used. The experiment indicated that when the buckling occurred, the tube materials had entered the plastic range, so the following formula [3] can be used:

180

Displacement(mm) Fig. 7. Force–displacement curves obtained from quasi-static tests of an original tube and a monitored tube S-05-20.

1 E d ¼ in MV0 þ Mg D and Fm D 2

(8)

Table 1 shows that for both the original and monitored tubes, the ultimate deformation and the number of folds vary linearly with the input energy. The average force of the original tubes d will be large greatly depends on the ultimate deformation D. Fm when D is small. As D increases, it becomes stable. However, for the monitored tubes, the average force is quite stable for different input energies. Compared with the corresponding original tubes under the same impact conditions, the ultimate deformations of monitored tubes are slightly larger than their counterparts, as a side effect of the reduction in the initial peak force. For the same tubes, the peak force is more sensitive to the impact velocity than to the mass of the drop hammer; while larger pre-hit height h leads to larger reduction in the peak force. The peak forces shown in Table 1 are taken from the experimental results, which contained the influence of the wave propagation; however, they still can be used as references, and the details about the dynamic peak force will be discussed in the later section. Fig. 8 presents the photographs of the original tubes after impact. It is seen that symmetric progressive buckling modes were generated, but the initial buckling position was randomly located. For Tubes-A, -B and -D, the first folder occurred near the distal end of the tube, while in Tubes-C and -E, the initial buckling started at the impact end. Besides, the folds were non-uniform and the compression of the first two layers of folds was much more serious than the subsequent ones.

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

407

Fig. 8. The original tubes after impact tests.

The deformed configurations of monitored tubes are illustrated in Fig. 9. For the group D-5-10-X, their buckling was initiated near the pulling points and the deformation was rather uniform. As the pre-hit height increases, when h > 15 mm, the buckling became a little unstable and the inclined deformation was observed. Fig. 9(c) shows that the specimen D-05-20-A had a little inclination, while the inclination in specimen D-05-25 was more serious. On the other hand, as shown in Fig. 9(d), when d is larger, part of the tube walls above the pulling points made no contribution to the energy absorption. In particular, for large d, the pre-hit column has to be longer, which will waste the energy absorbing stroke of the tube and may induce Euler buckling mode. The dynamic loading curves of some representative specimens are plotted in Fig. 10, in which the average force and energy absorption with respect to the displacement are obtained according to Eqs. (3)–(5). It is seen that all the specimens have a high initial peak force, and each subsequent peak on the dynamic loading curves is associated with the formation of a fold as shown in Figs. 8 and 9. Besides, the ultimate deformation and the total energy absorption obtained from the dynamic loading curves show good agreements with those calculated by the final length change and impact velocity. It is also revealed that the initial impact stage of original tubes displays two peaks, but the monitored tubes have only one peak, after which the force decreases quickly. In relation to the wave propagation, the implication is that the duration of the initial peak of an original tube must be longer than that of a monitored tube. 4. Numerical simulation 4.1. Numerical results To verify the experimental results and investigate the peak forces, a numerical simulation of the axial impact event is performed by means of FEM Code ABAQUS/explicit. Shell elements S4R (1.95 mm  2.2 mm) are employed to model the square tube (Fig. 11(a)), and the striker is regarded as a point mass attached to a rigid surface. As the lower part of the tube is clamped, the dimensions of the tube are taken as L ¼ 220 mm, b ¼ 43 mm and the wall-thickness t ¼ 1.25 mm. The material is strain-rate insensitive AA6063-T5 with Young’s modulus and density being E ¼ 65 GPa and r ¼ 2700 kg=m3 , respectively; and its stress–strain curve follows Fig. 5(b). A monitored tube is modeled as shown in Fig. 11(b), in which the pulling strip is a steel bar of thickness 0.8 mm and width 15 mm;

while its Young’s modulus and yield stress are 210 GPa and 350 MPa, respectively; the flow stress is s ¼ 400 MPa at ep ¼ 0:2. Similar to the one used in experiments, the pre-hit column is made of aluminum with cross-section 10 mm  10 mm, so that its deformation is negligible and no buckling would take place. Initially the striker has velocity V0 along z-direction, and the bottom of the tube is clamped in a rigid fixture which is stationary. Also, the degrees of freedom of the lower edges of the tube in the xand y-directions are constrained. The contact between the tube and the striker is defined as surface–surface interaction with a friction coefficient 0.25. Besides, self-contact is defined on the tube walls and gravitational acceleration is applied to the whole model. For the original tubes, in order to initiate a symmetric buckling mode, some initial geometric imperfections are introduced to the model. The imperfection is composed of the first 10 buckling modes with the maximum amplitude of 0.02 mm. It should be noted that the initial imperfections at this order do not affect the initial buckling force. The final shapes of some representative specimens are shown in Fig. 11(c)–(e). It can be seen that for Tube-A, the buckling is initiated at the top of the tube; while in dynamic experiments, some specimens began to buckle from the bottom. The deformation modes of the monitored tubes obtained by numerical simulation are very close to the specimens as shown in Fig. 9. Also, the numbers of folds in the deformed tubes are the same as observed in the experiments. Fig. 12 shows the comparison of the force–displacement curves between numerical simulations and experiments for Tube-A and D05-25-A. For the original tube Tube-A in Fig. 12(a), the configurations from simulations and experiments are similar but do not match very well. This is because, in the experiments the initial buckling position is randomly located, while in simulations it depends on the imperfections imposed on the model. For the monitored tubes, not only the ultimate deformation and average force matched well, but also the configurations agree with the experiments. More detailed results are given in Table 2. It can be seen that except the exact initial peak force, the other parameters predicted by numerical simulations, such as the ultimate displacement and average force, are all in good agreement with the experimental results. 4.2. Investigation of the dynamic peak force From the above comparison, it has been found that the numerical simulation can predict the deformation mode, the ultimate displacement and the mean force very well. However, the

408

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

a

D-05-10-A

D-05-10-B

D-05-10-C

D-05-10-D

D-05-10-E

b

D-05-20-A

D-05-20-B

D-05-20-C

c

D-05-10-A

D-05-15-A

D-05-20-A

D-05-25-A

d

D-30-10-A

D-30-15-A

D-50-10-A

Fig. 9. The monitored tubes after impact tests.

D-50-15-A

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

Tube-A Average Force

1400 1200

40

1000 30

800 600

20

400 10

20

40

60

80

100

120

140

1800 Tube-D Average Force

50

1000 30 800 20

600 400 200

0 160

0

0

20

40

1000 30

800 600

20

400

60

80

100

120

140

160

800 600

20

0

f

1600

200 0 0

20

40

60

80

100 120 140 160 180

30

800 600

20

400 10

200 60

80

100

120

140

0 160

Force(kN)

1000

D-50-15-A Average Force

1600

Ea

1400 1200

40

1000 30

800 600

20

400 10 0

Energy Absorbed(J)

1200

1800

60 50

Energy Absorbed(J)

1400

40

1000 30

400

0 180

1800

20

1200

D-5-20-A

Ea

1400

40

Displacement(mm)

D-30-15-A Average Force

0

1600 Ea

D-05-15-A

40

0

0 120

1800 D-05-20-A Average Force

Displacement(mm)

60 50

Force(kN)

40

100

10

200 20

80

60 50

Force(kN)

1400 1200

0

60

Energy Absorbed(J)

Ea

Energy Absorbed(J)

Force(kN)

1600

10

e

d

1800 D-05-15-A Average Force

40

0

1200

Displacement(mm) Tube-D

60 50

1400

Ea

40

Displacement(mm) Tube-A

c

1600

10

200 0

60

Energy Absorbed (J)

Force(kN)

1600

Ea

Energy Absorbed (J)

50

0

b

1800

60

Force(kN)

a

409

200 0

20

40

60

80

100

Displacement(mm)

Displacement(mm)

D-30-15-A

D-50-15-A

120

140

0 160

Fig. 10. The force–displacement relation and energy absorption of the representative specimens.

numerical peak force is somehow different from the experimental result. As mentioned in Section 2, the experimental impact forces were measured by the strain gages on the loadcell below the specimen. Also it is seen from Fig. 4 that the peak force recorded by the loadcell exhibits a large fluctuation in the initial impact stage. To obtain the dynamic force history, one often uses an FFT filter or a moving averaging method to conduct the data processing, which is unable to obtain the accurate peak force. In order to clarify the sources of the fluctuations and reveal the true dynamic peak force, the initial impact stage has to be investigated in details. Except for the model (Model 1) shown in Fig. 11(a), two other models (Models 2 and 3, as shown in Fig. 13(a)) are also built, in which loadcells are added under the specimens. The loadcell has the same dimensions as in the experiments, and two steel plates A and B are rectangles with thickness 5 mm. In Model 3, a steel

cylindrical shell with length 250 mm is added under the plate B, and its inner and outer diameters are 98 mm and 80 mm, respectively. The impact velocity and mass are taken as 7 m/s and 68 kg, respectively. The top impact force FT is the interaction force on the impact interface, while FB is that between the bottom edges and the top surface of plate A. Besides, the force FL is obtained from the stress applied on the loadcell. The results show that for the square tubes without buckling initiator, FT and FB obtained by the three models remain the same, but FL greatly depends on the structures under the specimen. The variations of FT , FB and different FL with time in the initial impact stage (within 1 ms) are shown in Fig. 13(b), where FL1 and FL2 are obtained from the loadcells of Model 2 and Model 3, respectively. It can be seen that the force at the loadcell, FL , could be much larger than FT and FB . By considering the reflection on several interfaces,

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X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

Fig. 11. Numerical simulation.

the history of FL could be very complicated. For FT and FB , during the period from 0.25 ms to 0.55 ms, no fluctuation appears because the tube deforms in plastic range. On the other hand, for the monitored tubes, FT is also independent of the lower structures, but FB has large fluctuations. Therefore, for these numerical models, only FT is the characteristic of the tube itself, and the peak value of FT will be used as the true peak force for the axial crushing of the tubes.

a

60

Dynamic test Static compression Numerical

Force(kN)

40

20

0

0

20

40

60

80

100

120

140

Fig. 13(c) and (d) compare the experimental and numerical results. It is seen from Fig. 13(c) that at t ¼ 0.2 ms, the striker impinges the top edge of the tube, and at t ¼ 0.25 ms, the stress wave reaches the bottom of the tube and is then almost doubled. Between t ¼ 0.25 ms and t ¼ 0.5 ms, the tube is compressed axially and the impact force has a plateau, in which the entire square tube deforms plastically. After t ¼ 0.5 ms, the plastic region will be localized in the regions close to the folds. For the monitored tubes, the buckling initiator makes the buckling much easier and the plastic deformation only takes place around the pulling points. In order to evaluate the real peak force in the experiments, a method of taking the average of the first peak and valley values on the experimental curve is proposed. The results from FFT filtering by means of a threshold frequency 2.5 kHz are used as a comparison, because after many trials, this frequency can give the best results for these specimens. Fig. 13(c) and (d) show that the FFT filtered results are good for some specimens, but bad for others. Compared with the FFT filter method, the average value of the first peak and valley provides a better prediction of the peak force on the top impact interface, and it is independent of the FFT filter frequency. Therefore, this average value will be taken as the experimental peak force in the subsequent discussion. 5. Discussions

Displacement(mm) In this section, based on the experimental and numerical results, the effectiveness of the buckling initiator will be discussed.

Tube-A

b

ExperimentD-05-25-A Numerical simulation

50

Some representative specimens were cut out along the symmetry plane, and the cross-sections are depicted in Fig. 14. It demonstrates that the folds in the quasi-statically compressed specimens were uniform, while for the dynamically compressed original tubes the first two folds were irregular, but from the third

40

Force(kN)

5.1. Deformation mode and average force

30

20

Table 2 Comparison between experimental and numerical results Experimental

10

0 0

20

40

60

80

100

120

140

160

Displacement(mm) D-05-25-A Fig. 12. Comparison of the force–displacement curves between numerical and experimental results.

Tube-A Tube-B Tube-C D-05-10-A D-05-15-A D-05-25-A D-30-15-A D-50-10-A

Numerical

D (mm)

Fp (kN)

Fm (kN)

D (mm)

Fp (kN)

Fm (kN)

119 93 70 130 140 130 134 134

52.8 54.0 42.3 51.1 52.1 31.7 46.7 54.8

14.7 14.7 14.0 12.9 13.2 13.8 13.2 13.7

120 93 70 133 134 135 132 134

41.6 40.7 41.8 41.7 28.9 26.3 32.6 38.8

13.6 14.3 13.4 12.7 13.1 12.7 12.6 13.2

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

a

b

411

FT (Top Interface) FB (Bottom Interface) FL-1 (Loadcell for Model-2) FL-2 (Lodacell for Model-3)

70 60

Force(kN)

50 Plate-A

40 30 20 10 0 0.0

0.6

0.8

1.0

Model 2 and Model 3

Numerical dynamic forces for tube-A. FT FB Experiment FFT2.5kHz

Fp

50

d

D-05-20-A FT Experimental FFT-2.5kHz

40 FP

35 30

(Fp+Fv)/2

30 20

Fv

Force(kN)

40

Force(KN)

0.4

Time(ms)

c 60

0.2

Plate-B

25 20

(FP+FV)/2

15 FV

10

10 5 0

0

-10

-5 0.0

0.2

0.4

0.6

0.8

1.0

2.6

2.8

3.0

3.2

Time(ms)

Time(ms)

Numerical and experimental results for Tube-A

Modified tube D-05-20-A

3.4

Fig. 13. The initial impact stage.

one, the folds gradually became stable and regular. In contrast, a very uniform and stable progressive buckling mode was observed on the monitored tubes. To examine the deformation mode, two parameters are usually used, i.e., the half-wave-length of a fold H and the effective stroke de . As shown in Fig. 14, suppose the arc-length between points J and K is L0 , and the vertical distance between J and K is Lu . Because arc JK contains two basic layers of the fold, we have L0 ¼ 4H and de ¼ ðL0  Lu Þ=L0 . Also, the theoretical prediction for a basic

collapse element of a perfect square tube can be obtained by the following equation [2],

 2 H b 3 and de ¼ 0:73 ¼ 0:983 t t

(9)

Therefore, the theoretical prediction can be obtained as H ¼ 13.0 mm and L0 ¼ 52:0 mm. Table 3 lists the average values of

Fig. 14. The cross-section views of the tested specimens.

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X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

Table 3 Fold parameters of the deformed tubes L0 (mm)

Lu (mm)

de

52.0 48.0 46.9 51.4 50.5 51.8 48.9

14.2 12.6 12.5 13.7 13.7 13.3 12.5

0.73 0.738 0.733 0.733 0.728 0.743 0.744

40

measured L0 , Lu and de , showing that all these parameters are not affected by the buckling initiators. The average impact force of the dynamically tested tube is plotted in Fig. 15, showing that the average force is also independent of the design parameters of the buckling initiators. The study in this section proves that after the initial buckling the subsequent deformation mode and energy absorption capacity of the monitored tubes are not affected by the buckling initiators.

30

20

10

0 0

5

10

15

25

Numericalh =10mm Experimentalh =10mm Numericalh =15mm Experimentalh =15mm

50 45 40

Peak force(kN)

First, the initial peak forces of the specimens D-05-X are investigated. In addition to the case of t ¼ 1.25 mm, other cases with thicknesses t ¼ 1.0 mm and t ¼ 1.5 mm are also analyzed numerically. Both the experimental and numerical initial peak forces of the monitored tubes are shown in Fig. 16, for which the impact conditions are V0 ¼ 7 m/s and M ¼ 68 kg. It can be seen that in case of t ¼ 1.25 mm, the numerical simulations are in good agreement with the experiments. When h is small, the initial peak force only has little reduction, but when h > 10 mm, the reduction in the peak force becomes notable. However, at h ¼ 20 mm, the peak force is about 26.7 kN, i.e., almost the same as that at h ¼ 25 mm. According to the analysis in Section 2, when hz20 mm, the pulling points contact the pre-hit column so it becomes very difficult to increase a. In this case, the further increase in h will cause the pulling strip yield and does not help in reducing the peak force. The cross-sections along the axis of specimens D-05-X from numerical simulation are presented in Fig. 17, showing that when h > 15 mm, the geometric imperfections increase fast; while at h ¼ 20 mm and h ¼ 25 mm, the imperfections are almost the same, because the pulling strip is completely yielded if h > 20 mm. The effect of the pulling position d is demonstrated in Fig. 16(b). For h ¼ 10 mm, the peak force will decrease with the increase of d, but for h ¼ 15 mm, the decrease of the peak force is very small. It is

20

Pre-hit h /mm

b

5.2. Initial peak force

Numerical t=1.0mm Numerical t=1.5mm Numerical t=1.25mm Experimment t=1.25mm

50

Peak force (kN)

Theoretical Static Tube-A D-5-10-A D-5-20-A D-30-15-A D-50-15-A

a

35 30 25 20 15 10 5 0 0

10

20

30

40

50

Pulling position d (mm) Fig. 16. (a) The initial peak force via the pre-hit height h for D-05-h; (b) the peak force via different pulling position when t ¼ 1.25 mm.

also observed in the experiments, for D-30-15 and D-50-15, the pulling strips were broken after test. Though, for the same h, the peak force can be reduced by increasing d, large d must require stronger pulling strip and longer pre-hit column, which may generate the global buckling mode. Therefore, the dominant parameter for the design of this buckling initiator is the pre-hit height h.

14

5.3. Analytical model for monitored tubes

Mean force (kN)

12 10

Experiments --- Average

8 6 4 2 0 0

5

10

15

20

25

30

Pre-hit height (mm) Fig. 15. Average force of the specimens with different design parameters.

For the theoretical study of the loading history of axial crushing behavior of tubes, Wierzbicki [19] suggested a method to evaluate the instantaneous crushing force of a square tube, in which the deformation mode and energy dissipation mechanisms described in Ref. [7] were used, but the half-wave-length of folds H could be variable. Additionally, Wierzbicki et al. [20] revised Alexander’s model and obtained the loading history of the crushing behavior. In their model, each layer of fold contains several segments of arcs and the eccentricity factor of the folds is considered. However, the energy dissipation mechanism of axial crushed circular tubes is quite different from that of square tubes. In this section, to investigate the relationship between the geometric imperfection and the initial peak force, Wierzbicki and Abramowicz’s model [7] is employed. As shown in Fig. 18(a), A1B1C1–A2B2C2 represents a quarter of a layer of fold, and the

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

413

Fig. 17. The cross-sections of the monitored tubes when the strikers contact the top edges of the tubes.

half-wave-length of this structural element is H. During the compression process, AiCi and BiCi are stationary plastic hinges, while point C0 will move towards point C00 along the hinge line A0C0. Thus, the material will flow from A1C1C0A0 and A2C2C0A0 to B1C1C0B0 and B2C2C0B0, respectively, through the inclined traveling hinges C1C0 and C2C0. Therefore, the energy is dissipated by three different mechanisms:

necessarily constant. Therefore, according to Ref. [19], the instantaneous crushing force can be obtained by

F ¼

E_ 1 ¼ E_ 2 ¼ E_ 3 or E_ 0 ¼ 3E_ 2

From Ref. [7], the total energy dissipation E_ 0 has a form as follows 2

(10)

where M0 ¼ s0 t 2 =4, r is the radius of the toroidal area along C1C0C2, and H, A1, A2, A3 are functions of the inclination angle q. On the other hand, the external work done by the axial crushing force can be expressed by

_ ¼ Ff ðqÞHq_ W

(11)

It should be noted that, in Ref. [7], H and r were assumed to be constant so that approximated solutions for the mean crushing force Fm could be obtained by means of integration over an entire fold and the principle of minimum energy dissipation rate. Although, H and r are always changing during the deformation process, it should be noted that as long as the deformation mode and energy dissipation mechanisms described in Ref. [7] remain correct, Eqs. (10) and (11) can still be applied, in which H and r are not

a

b F

α0

B1 A1 B0

C1

A0 C0’

θ

A

F A1 H0

B2

A1

A0 θ

H1 B0

A2

A2 H

(13)

For a monitored tube, when the striker contacts the tube, it already has imperfections. The configuration of a typical tube is shown in Fig. 19(a), while Fig. 19(b) and (c) are its cross-sections at two axial symmetric planes. The imperfections can be characterized by angles a and b. Angle a is on the sides pulled by the strips and b is the angle on the other two sides. Fig. 20(a) and (b) plot some configurations of cross-section through symmetric planes A–A and B–B during the progressive crushing of the tube, respectively. It is seen that the structure higher than A1 and B1 deforms very little, while the lower part exhibits a progressive buckling mode similar to that of the original tube. Therefore, the deformation of A1A0A2 can be simplified to that shown in Fig. 18(b), which also applies to B1B0B2. From the numerical simulations, it is shown that the length of A1A0 and A0A2 is H0 ¼ 1.5 H and H1 ¼ 2.0 H, respectively, where H is the half-wavelength obtained from Eq. (9). The analysis (see Fig. A3) shows that a is always larger than b, which means that the top of B–B cross-section is a little higher than that of A–A. Thus, the work done by the crushing force F to the whole structure can be given as

!

q_

(12)

In the above equation, the real values of H and r should make F a minimum, which means that the three terms in the right bracket of the above equation should be equal to each other. Consequently, at any time increment, the energy dissipated by horizontal plastic hinges is one third of the total energy dissipation,

E_ 1 : dissipated by the bending of stationary plastic hinges AiCi, BiCi, i ¼ 0, 1, 2; E_ 2 : dissipated by the material plastic flow through the traveling hinges C1C00 , C2C00 ; E_ 3 : dissipated by the traveling of the corner at point B00 .

Hr H E_ 0 ¼ E_ 1 þ E_ 2 þ E_ 3 ¼ M0 A1 þ A2 b þ A3 r t

  M0 r b H A1 þ A2 þ A3 t H r f ðqÞ

C2

Fig. 18. (a) A basic structure element for the progressive buckling of square tubes; (b) simplified model for deformation of A1A2.

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X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

Fig. 19. Configuration of a modified tube in the pulling stage: (a) the whole configuration; (b) cross-section A–A; and (c) cross-section B–B.

_ ¼ FH ðsin q þ cos q tan bÞq_ W 0

(14)

According to Eq. (13) and the analysis in Appendix 4, and considering a whole layer of fold that consists of two walls like A–A and B–B, the total energy dissipation rate should be

   H cos q cos q _ q E_ 0 ¼ 12M0 b 2 þ 1 þ H0 cos a cos b Therefore, the crushing force is

(15)

  H1 cos q cos q þ H0 cos a cos b F ¼ 12M0 b H0 ðsin q þ cos q tan bÞ 2þ

(16)

As mentioned in the previous sections, the initial peak force for an original square tube can be predicted by Fmax ¼ 4sY bt, and initially qz0. Therefore, the non-dimensional initial buckling force for a monitored tube is found as

F ¼

F 3 ¼ Fmax 4

2 þ 1:33

  1 1 þ cos a cos b t 2H tan b

Fig. 20. The change of the cross-sections along symmetry planes: (a) A–A; (b) B–B.

(17)

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

415

6. Conclusions

Initial Peak Force F/Fmax

1.0 0.8 0.6 0.4

t=1.00mm t=1.25mm t=1.50mm Numerical t=1.00mm Numerical t=1.25mm Numerical t=1.50mm

0.2 0.0

0

2

4

6

8

10

12

14

16

18

20

22

Pre-height h (mm) Fig. 21. The peak force vs. the pre-hit height for b ¼ 43 mm.

In the above equation, when bz0, F will be infinite. But actually, F should be always smaller than Fcr zFmax . Thus, the following approximation can be employed to predict the initial peak force,

" F ¼ min 1;

3 4

2 þ 1:33

  1 1  2 # þ 1 t 3 cos a cos b 2  0:983 b tan b

(18)

The applicability and effectiveness of adopting buckling initiators for axially loaded square tubes are investigated both experimentally and numerically. This kind of buckling initiator is attached to the tube near the impact end, and it is composed of a pre-hit column and a strip. When subjected to an axial impact, the pre-hit column will be impinged by the striker first and it will lead the strips to pull two opposite walls of the tube and create some geometric imperfections. The results show that by installing the buckling initiator, the objective of reducing the initial buckling force can be achieved with very little effects on the subsequent progressive crushing mode and the excellent energy absorption capacity of the square tube. Because the imperfections are generated just before the impact, the stiffness of the intact square tube in its normal structural function can be remained. Our investigation reveals that by using a buckling initiator, 30% reduction in the initial buckling force is achievable. It is also found that this buckling initiator can not only reduce the initial peak force, but also ensure a stable and uniform crushing mode. In the design of this kind of buckling initiators for square tubes, the pre-hit height h is the critical parameter to achieve the reduction of peak force, while the pulling position should be close to the top edge of the tube so as to avoid using a long column, which may induce a global buckling or waste the effective stroke of the tube for energy absorption. Acknowledgments

Which means that F takes the minimum value of the two terms in the bracket. Based on the relationships of a  b and a  h given in Section 2 and the Appendices 2 and 3, the non-dimensionlized buckling force of the monitored tubes with respect to the pre-hit height h can be obtained. In order to compare with the numerical results, b ¼ 43 mm is taken, and three cases for t ¼ 1.0 mm, 1.25 mm and 1.5 mm are considered. The results are plotted in Fig. 21. It can be seen that the analytical model can successfully predict the reduction of the peak force. Before h increases to h ¼ 10 mm, the peak force almost does not show any reduction, but when h is around 19 mm, the peak force reaches its minimum. After this point, further increase of the pre-hit height will no longer help for the reduction of the peak force. For the tubes with the same width and different wall-thickness, if the same buckling initiator is used, the peak force reduction of the thinner tube will be more significant than the thicker one, which means that the peak forces of thicker tubes are more difficult to reduce. The analysis has demonstrated that by using this kind of buckling initiator, the peak force can be reduced by more than 30%.

This study is supported 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. The authors also would like to thank Mr. Alex Fok for his assistance in experimental set-up. Appendix. Reduction of bending stiffness of a square tube with initial dents Fig. A1(a) shows a square tube with a conventional initial dent at a local cross-section. Fig. A1(b) illustrates the shape of this crosssection, which has a width b and thickness t, while two opposite walls are compressed inward by a distance d. Therefore, the moment of inertia with respect to its x-axis is

2 2   1 2d 2d I1 z b3 t 1  zI0 1  2 b b

(A-1)

y x

δ

δ

B b

A

φ

x

a Conventional dent

b Cross section at A-B

c Dent due to pulling

Fig. A1. (a) A conventional dent of a square tube; (b) the cross-section of a conventional dent; and (c) dent due to pulling.

c

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X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

For a given small dent with d ¼ 0:1b, I1 z0:64I0 , which means that the reduction of bending stiffness is about 36% at this local cross-section. Another case is shown in Fig. A1(c), the middle points of two opposite walls are pulled inward by a distance d, then the moment of inertia is

O A’

A

λ

B’

C

K2

A’

δ

D A1 C1

"

3 #  c 3  b3 t 1 2d I2 z  1  2 3 sinð45  fÞ b b

A1

B

(A-2)

B1

E

A

θ2 C1

K1

θ1 x

a

y

C

b

Fig. A2. (a) Rigid plastic deformation mode; (b) local deformation details.

where

f ¼ sin1



 b  2d pffiffiffi and c ¼ b sinð45 þ fÞ 2b

(A-3)

Thus, for a given dent with d ¼ 0:1b, the bending stiffness will be reduced by 20%. Rigid plastic analysis for the deformation of laterally pulled square tube Suppose the thin-walled square tube is rigid, perfectly-plastic. When it is subjected to a pair of pulling forces at the middle of two opposite walls near one end, the deformation mode is as shown in Fig. 2(c). The pulling point A moves inwards, while point B on the traction-free wall moves outwards. The influenced length of the tube is l, and the displacement of point A is d. In one of the corner areas, ACBB1C1A1, four plastic hinge lines, i.e., AC1, BC1, A1C1 and B1C1, are generated. For a small displacement d of point A, the deviation of corner C can be ignored and the rotation angles of A and B about the corner are equal to each other. Then the total energy dissipated by the plastic hinge lines is



ET ¼ 4 EAC1 þ EBC1 þ EA1 C1 þ EB1 C1

(A-4)

where EA1 C1 ¼ EB1 C1 ¼ M0 ðb=2Þðd=lÞ and EAC1 ¼ EBC1 . For the energy dissipation EAC1 , as shown in Fig. A2(a), the rotation angle of hinge line AC1 is q ¼ q1 þ q2 . In Fig. A2(b), AC ¼ x, CC1 ¼ y and AA0 ¼ d. CK1 and CK2 are perpendicular to AC1 and A0 C1, respectively. Therefore,

dy2 xy K1 K2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; CK1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ y2 2 x 2 2 2 2 x þy x þy þd 1

0

q1 ¼ sin1

(A-5)

d K1 K2 yC B ¼ sin1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A CK1 2 x 2 2 x þy þd

(A-6)

W ¼ 2F d

(A-10)

Minimizing the pulling force F by vF=vl ¼ 0, l can be obtained as

b

l ¼ pffiffiffi

(A-11)

2

Analysis for geometric imperfections due to buckling initiator When the displacement of point A becomes large, the deviation of the corners cannot be ignored. However, the locations of plastic hinge lines will not change. The subsequent deformation can be analyzed by rigid body rotations. In the xyz coordinate system as shown in Fig. 2(c), the initial locations of A, C and B are as given by

    b b A 0; ; l ; B ; 0; l ; Cð0; 0; lÞ: 2 2 After the tetrahedron ABCE rotates along the x-, y- and z-axis, the new locations of A, B and C can be obtained as,

0 0 0

xA ; yA ; zA ¼ Tz Ty Tx ðxA ; yA ; zA Þ

(A-12a)

0 0 0

xB ; yB ; zB ¼ Tz Ty Tx ðxB ; yB ; zB Þ

(A-12b)

0 0 0

xC ; yC ; zC ¼ Tz Ty Tx ðxC ; yC ; zC Þ

(A-12c)

Considering the symmetric conditions that point A and point B should always locate within the mid-plane, we have that yA ¼ b=2 pffiffiffi and xB ¼ b=2. By noting l ¼ b= 2, the relationship between a, b and g can be obtained numerically as shown in Fig. A3.

Similarly,

0

1

d

xC q2 ¼ sin1 B @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A y 2 x2 þ y2 þ d If d 

(A-7)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 , then

q ¼ q1 þ q2 z

d A0 C1



 x y þ y x

(A-8)

d l

ET ¼ 4 M0 b þ 2M0 d

l b 2

b

!#

þ2

l

The external work done by the force F is:

Considering the model shown in Fig. 18(b), the bar has an initial imperfection characterized by a0, and the length of A1A0 is H0. When the bar A1A2 is compressed axially, point A1 can only move vertically, and the length of A0A2 is H1. Then, after a small rotation q, the vertical displacement of point A1 is

D ¼ H1 ð1  cos qÞ þ H0 ðcos a0  cos aÞ

In the square tube, x ¼ b=2, y ¼ l, hence,

"

Modified model for an axial crushing bar

(A-13)

Since point A1 has no horizontal displacement, we have

(A-9)

H1 sin q ¼ H0 ðsin a  sin a0 Þ Therefore,

(A-14)

X.W. Zhang et al. / International Journal of Impact Engineering 36 (2009) 402–417

References

40 α γ β

35

Angles (degree)

30 25 20 15 10 5 0 0

5

10

15

20

25

30

35

40

(degree) Fig. A3. Relationship between b, g and a.

a_ ¼

417

H1 cos q _ q and D_ ¼ H1 ðsin q þ cos q tan aÞq_ H0 cos a

(A-15)

The total energy dissipation is

    H cos q _ q E_ ¼ 2M0 q_ þ a_ ¼ 2M0 1 þ 1 H0 cos a

(A-16)

_ ¼ H ðsin q þ cos q tan aÞq_ _ ¼ PD W 1

(A-17)

As a result, the instantaneous crushing force should be

H1 cos q H0 cos a P ¼ 2M0 H1 ðsin q þ cos q tan aÞ 1þ

(A-18)

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