Deformation and Energy Absorption of Aluminum Square Tubes

with Dynamic Axial Compressive Load*1

Makoto Miyazaki*2 and Hideaki Negishi*3

Department of Mechanical Engineering and Intelligent Systems, The University of Electro-Communications, Chofu 182-8585, Japan

Buckling, impact resistance and energy absorption of dynamic axial compressed aluminum square tube are discussed. Numerical analysisof the deformation of the square tube is made by a finite element method. The result shows that ripples of buckling are produced in the surfaces ofthe tube wall when the striking mass reaches to a certain value. The wave pattern on the deformed tube wall of 1mm thick is concave-convexpattern in adjoining surface, while that of 2mm thick is convex pattern in all surfaces. Absorbed energy of the deformed tube increases inproportion to axial displacement of the tube. Experimental results agree approximately with those of the finite element method analysis.

(Received January 15, 2003; Accepted June 3, 2003)

Keywords: square tube, impact test, numerical analysis, deformation-load property, absorbed energy

1. Introduction

Square tube has been used for a framework and reinforce-ment member of the structures. The behavior of dynamicdeformation must be studied in order to use square tube as animpact absorption member. The study of the dynamicdeformation is mentioned in the following. Meng et al.1)

carried out an experiment and a plastic hinge analysis onlarge deformation mechanism of the square tube. Okamoto etal.2) carried out the experiment on axially compressedcircular tube and steel square tube, and compared the staticdeformation and the dynamic deformation by the plastichinge analysis. Sawairi et al.3) examined the influence of thecorner radius on deforming behavior of axially compressedsquare tube by the finite element method. About the staticdeformation of the square tube, the examination of bucklingmode of the aluminum square tube was carried out by Utsumiand Sakaki.4) Authors5) examined the deforming behavior ofaxially compressed thin square tube. However, the deformingbehavior and the deformation energy have not been clarified.

This paper deals with the deformation and energyabsorption of axially compressed aluminum square tube.

2. Experiment

2.1 ProcedureA drop hammer testing machine, shown in Fig. 1, is used

for dynamic axial compression of the square tube specimen.The drop hammer weight is 7.92 kg and its impact velocitydepends on the falling height of the hammer. The impactvelocities v are changed from 1.4 to 5.4m/s. The ends of thespecimen are fixed into the groove on the steel plate (2.5mmwidth, 5mm depth). The specimen, shown in Fig. 2, is analuminum square tube (JIS A6063–T5, 40mm width,100mm or 150mm length, 1mm or 2mm thickness) and itis annealed for 1 hour at 673K. Material properties are shownin Table 1. Corner radius R of the aluminum square tube is

Striking mass

Support frame

Base

Grooved plate

1800

mm

Specimen

Guide shaft

Fig. 1 Drop-hammer testing machine.

w = 40

w

t

l =100,150

t =1.0, 2.0

Fig. 2 Shapes and dimensions of specimens.

Table 1 Material Properties.

Young’s modulus E/GPa 69

Poisson’s ratio � 0.33

Density �/kg/m3 2:71� 103

F value F/MPa 222

n value n 0.25

*1This Paper was Originally Published in Japanese in Journal of Japan

Institute of Light Metals, 52-7 (2002) 308–312.*2Researcher, The University of Electro-Communications.*3Emeritus Professor, The University of Electro-Communications.

Materials Transactions, Vol. 44, No. 8 (2003) pp. 1566 to 1570#2003 The Japan Institute of Light Metals

R � 0:2mm. The axial strain of the specimen was measuredby the lattice of 4mm interval described on the surface of thespecimen.

2.2 ResultsFigures 3 and 4 show examples of the dynamically

deformed tube. Deformation shape of 1mm thickness tube isconcave-convex in adjoining surfaces at an impact velocityof 1.4m/s. The tendency appeared markedly at impactvelocities of 2.8 and 3.7m/s. The buckling deformation of

2mm thickness tube was observed at an impact velocity of2.8m/s. On the impact velocity of 3.7m/s, the convexshapes appeared in all surfaces of the tube. The tendency wasobserved markedly at the impact velocity of 5.4m/s.

In the square tube of 1mm thickness and 100mm length,the partial buckling was generated in the center of the axialdirection of the tube. The buckling of 2mm thickness tubewas generated on the edges. In this experiment, the wholetube crushing was not observed in both of the thicknesses.

(1) 2.8 m/s (2) 3.7 m/s (3) 5.4 m/sFig. 4 Dynamically deformed specimens (t ¼ 2mm l ¼ 100mm).

(1) 1.4 m/s (2) 2.8 m/s (3) 3.7 m/sFig. 3 Dynamically deformed specimens (t ¼ 1mm, l ¼ 100mm).

Deformation and Energy Absorption of Aluminum Square Tubes with Dynamic Axial Compressive Load 1567

3. Numerical Analysis

3.1 MethodThe numerical analysis is carried out by non-linear

structure analysis program (Marc 2000) and pre-post pro-cessor (Mentat 2000). The nodes in the edge of the squaretube are fixed except for an axial direction of the impact edge.The boundary condition is similar to one of the experiments.The tube (40mm width, 100mm length, 1mm or 2mmthickness) is discretized into 1344 bilinear four-node shellelements. The tube (40mm width, 150mm length, 1mm or2mm thickness) is discretized into 2304 bilinear four-nodeshell elements. The hammer (120mm� 120mm� 40mm)is an un-discretized three-dimensional, eight-node, first-order, isoparametric element. The deformed tube is regardedas an isotropic material following to von-Mises yieldcondition and the flow stress-strain relationship is eq. (1)because the effect of the strain rate of the aluminum issmaller than other materials like iron, etc. The strain rate inthis experiment was 50 s�1.

� ¼ F"n ð1Þ

The time step width is 17.5 ms in this analysis. The Newton-Raphson method and updated Lagrangian formulation areused as the solution methods of the non-linear equation, andthe Newmark � of implicit solution time-integration methodis used for the analysis of dynamic deformation. The impactvelocities v are changed from 1.4 to 7.5m/s.

3.2 ResultsFinal deformations of the square tube for various impact

velocities are shown in Figs. 5–8. Strain distributions inbuckling region are shown in Figs. 9 and 10. Strain-timecurve in buckling region is shown in Figs. 11 and 12. In thecase of 1mm thickness tube, shown in Figs. 5 and 6, theirregular pattern on the surface of the tube is a concave-convex pattern on the adjoining surface. When the impactvelocity is high, the second buckling is generated in theadjacency part in the first buckling. As seen in Fig. 9,compressive strain near the corner of the tube is largecompared to that of the other part of the tube. In the case of1mm thickness tube, Face A has a convex surface and Face Bhas a concave surface. As seen in Fig. 11, strain near thecorner of the tube in the beginning of the deformation is small

X

Z

Y

Fig. 5 Dynamically deformed shapes obtained by FEM (t ¼ 1mm,

v ¼ 2:8m/s, l ¼ 100mm).

X

Z

Y

Fig. 6 Dynamically deformed shapes obtained by FEM (t ¼ 1mm,

v ¼ 7:5m/s, l ¼ 100mm).

X

Z

Y

Fig. 7 Dynamically deformed shapes obtained by FEM (t ¼ 2mm,

v ¼ 5:4m/s, l ¼ 100mm).

X

Z

Y

Fig. 8 Dynamically deformed shapes obtained by FEM (t ¼ 2mm,

v ¼ 7:5m/s, l ¼ 100mm).

1568 M. Miyazaki and H. Negishi

compared to that of the other part of the tube. With thedeforming, Face A changes to tensile strain. It is consideredthat the buckling is produced in the side. After the side isbuckled, the axial compressive strain increases greatly nearthe corner. It is considered that the buckling is produced inthe corner. The buckling happens first in the side, and thenanother buckling happens near the corner. In the 2mmthickness tube, shown in Figs. 7 and 8, the convex shapeappeared in all surfaces at the end of the tube. In the case of2mm thickness, one side in adjoining surface is Face A, andthe other side is Face B. As seen in Fig. 10, compressivestrain near the corner of the tube is small compared to that ofthe other part of the tube, because buckling shape is convex inall surfaces. As seen in Fig. 12, after the compressive strainreaches to the peak, it changes to the tensile strain in the side.It is considered that the buckling is produced in the side.Unlike the case of the 1mm thickness tube, the deformationof the 2mm thickness tube is not wavily generated. The firstbuckling happened in the edge, and a little later the secondbuckling is produced in the opposite edge. Even under otherconditions, the strain distributions that are obtained with theexperiment and analysis are almost the same. The size andkind of element seems to be appropriate because experimen-tal results agree approximately with those of the finiteelement method analysis.

In the case of the 1mm thickness tube and 2mm thicknesstube, there is a difference in the deformation shape. Thedeformation near the corner is simplified as shown in Fig. 13.The continuous line of the figure shows before the deforma-tion of the tube, and the broken line of the figure shows afterthe deformation of the tube. The deformation of Fig. 13(a) isobtained by greatly twisting of the corner. It is thedeformation of 1mm thickness of the tube. The deformationof Fig. 13(b) is the deformation obtained by the extension of

the side without twisting in the corner. The torsional rigidityin the corner is given in eq. (2).

� ¼ GIp ¼ G1

6t4 ð2Þ

G is modulus of transverse elasticity and Ip is polarmoment of inertia of the area. In this case, Ip is t

4=6. From eq.(2), the polar moment of inertia of the area in the cornerbecomes large, when the thickness is large. The cornertorsional rigidity increases. The torsional rigidity in thecorner of 2mm thickness tube is larger than 1mm thicknesstube 16 times. The torsional rigidity in the corner of the 1mmthickness tube becomes smaller than torsional rigidity in thecorner of the 2mm thickness tube, and the corner of the 1mmthickness tube becomes easy to twist. Therefore, 1mmthickness tube becomes a concave-convex pattern on theadjoining surface. The torsional rigidity in the corner seemsto contribute greatly to the decision of the deformation shape.

4. Energy Absorption

The relationship between the axial displacement andabsorbed energy are shown in Figs. 14 and 15. Q is an

−40 −20 0 20 40

−0.2

−0.1

0

0.1

Distance from the corner, d / mm

Axi

al s

trai

n ε x

ExperimentAnalysis

Fig. 9 Axial strain distribution along the hill and valley of a concave-

convex surface (t ¼ 1mm, v ¼ 2:8m/s, l ¼ 100mm).

−40 −20 0 20 40−0.08

−0.06

−0.04

−0.02

0

Distance from the corner, d / mm

Axi

al s

trai

n ε x

ExperimentAnalysis

Fig. 10 Axial strain distribution along the hill of a convex surface

(t ¼ 2mm, v ¼ 5:4m/s, l ¼ 100mm).

0 1 2 3

−0.2

−0.1

0

Time, T / ms

Axi

al s

trai

n ε x

Face ACornerFace B

Fig. 11 Axial strain-time curve at the center and corner of Face A and Face

B (t ¼ 1mm, v ¼ 2:8m/s, l ¼ 100mm).

0 0.5 1 1.5−0.05

−0.04

−0.03

−0.02

−0.01

0

Face ACorner

(Face B)

Time, T / ms

Axi

al s

trai

n ε x

Fig. 12 Axial strain-time curve at the center and corner of Face A and Face

B (t ¼ 2mm, v ¼ 5:4m/s, l ¼ 100mm).

(a) (b)

Before deformationAfrer deformation

Fig. 13 Deformation patterns in corner part of square tube.

Deformation and Energy Absorption of Aluminum Square Tubes with Dynamic Axial Compressive Load 1569

energy absorbed by a whole square tube. The energyabsorption quantity was calculated from the plastic workingquantity to the end of deformation. In the case of 1mm and2mm thickness, absorbed energy of the impact load by thetube deformation increases in proportion to the axialdisplacement of the tube. There is no change on thistendency, even if the axial compression deformation in-creases. If the thickness and the axial maximum displacementare same, absorbed energy becomes the same in the case ofdifferent axial length. It is shown that the influence of thedifference of the axial length on the energy absorptioncharacteristic is small when the square tube normallybuckles. Energy absorption quantities at each impact velocityis shown in Tables 2 and 3. Qb is energy absorbed by thebuckling deformation region of the square tube. When weightcollides with the tube, the value of energy agrees approx-imately with the value of Q shown in Tables 2 and 3. In theinitial stage of deformation, the energy is absorbed by thewhole square tube. The energy absorption quantity in thebuckling deformation region increases, as the buckling isgenerated, and as the deformation increases. The effective-ness as impact absorption member of the aluminum squaretube is shown, because the energy is greatly absorbed in thebuckling deformation region and the deformation has beenpartially carried out.

5. Conclusions

From the experiment and numerical simulation, thefollowing conclusions were obtained.(1) In the case of the tube of thin wall, the deformation

shape is concave-convex in adjoining surfaces at the

center of the axial direction of the tube, and thedeformation is wavily generated.

(2) In the case of the tube of thick wall, the convex shapeappeared in all surfaces at the end of the tube. The firstbuckling happened in the edge, and a little later thesecond buckling is produced in the opposite edge.

(3) The buckling happens first in the side, and then anotherbuckling happens near the corner.

(4) The absorbed energy of the impact load by the tubedeformation increases in proportion to the axialdisplacement of the tube.

(5) The square tube of thin wall is effective for the impactabsorption member because the impact is absorbed bythe partial buckling and the deformation is wavilygenerated.

REFERENCES

1) Q. Meng, S. T. S. Al-Hassani and P. D. Soden: Int. J. Mech. Sci. 25

(1983) 747–773.

2) S. Okamoto, T. Tsuta, J. Bai and M. Doi: JSME 71st Fall Annual

Meeting B930-63 (1993) 568–570.

3) Y. Sawairi, M. Gotoh and M. Yamashita: Proc. of 1999 Japanese Spring

Conf. For the Technology of Plasticity, (1999) 157–158.

4) N. Utsumi and S. Sakaki: Proc. of 1999 Japanese Spring Conf. For the

Technology of Plasticity, (1999) 247–248.

5) M. Miyazaki, H. Endo and H. Negishi: J. Mater. Process. Technol. 85 1–

3 (1999) 213–216.

0 10 20 30 400

100

200

100 mm150 mm

Axial length

Abs

orbe

d en

ergy

, Q /

J

Axial displacement, x / mm

Fig. 14 Relationship between absorbed energy Q and axial displacement x

(t ¼ 1mm).

0 2 4 6 8 100

100

200

100 mm150 mm

Axial length

Abs

orbe

d en

ergy

, Q /

J

Axial displacement, x / mm

Fig. 15 Relationship between absorbed energy Q and axial displacement x

(t ¼ 2mm).

Table 2 Energy Absorption (t ¼ 1mm).

v/m�s�1 l/mm Q/J Qb/J

1.4100 7.39 5.50

150 6.96 4.86

2.8100 31.2 27.7

150 30.1 22.4

3.7100 54.2 48.6

150 51.9 39.8

5.4100 114.5 111.0

150 111.8 91.3

7.5100 220.3 218.5

150 212.4 178.0

Q: Absorbed energy

Qb: Absorbed energy at buckling region

Table 3 Energy Absorption (t ¼ 2mm).

v/m�s�1 l/mm Q/J Qb/J

2.8100 28.9 —

150 28.5 —

3.7100 51.2 —

150 50.6 —

5.4100 111.9 87.8

150 110.5 50.1

7.5100 217.7 200.1

150 211.6 138.2

1570 M. Miyazaki and H. Negishi