Thin-Walled Structures 48 (2010) 946–954

Contents lists available at ScienceDirect

Thin-Walled Structures

0263-82

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/tws

Comparative analysis of energy absorption and deformations of thin walledtubes with various section geometries

Ali Alavi Nia a,n, Jamal Haddad Hamedani b

a Mechanical Engineering department, Bu-Ali Sina University, Hamedan, Iranb Mechanical Engineering department, Bu-Ali Sina University, Hamedan, Iran

a r t i c l e i n f o

Article history:

Received 23 November 2009

Received in revised form

21 July 2010

Accepted 22 July 2010Available online 4 August 2010

Keywords:

Crushing

Thin-walled tube

Energy absorption

Folding

31/$ - see front matter & 2010 Elsevier Ltd. A

016/j.tws.2010.07.003

esponding author. Tel.: +98 811 8245704; fa

ail address: alavi1338@yahoo.com (A. Alavi N

a b s t r a c t

In this paper, deformations and energy absorption capacity of thin walled tubes with various section

shapes (circular, square, rectangular, hexagonal, triangular, pyramidal and conical) are investigated

both experimentally and numerically. The tubes have the same volume, height, average section area,

thickness and material and are subjected under axial quasi static loading. The results of simulations are

in good agreement with the experimental data and show that the section geometry has considerable

effect on the energy absorption. The circular tube has the most energy absorption capacity and the most

average force among all investigated sections. Since the maximum force is concerned in impact events,

pyramidal and conical tubes are recommended, due to their uniform load–displacement curves and

therefore, less difference between the maximum and the average forces.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, vehicles are used extensively and a large number ofhorrible accidents related to them occur widely. Increasing thesafety for passengers is a valuable aim and a lot of investigationsare carried out in this region. Using energy absorbers is anappropriate option for this purpose. These parts have differentshapes and are made from low density materials. In designingthese parts, investigation of their collapse behavior and energyabsorption capacity is necessary and a wide variety of studies isdone about these structures, especially about thin walled tubes.

Alexander [1] accomplished the first studies on the collapse ofcylindrical tubes under axial loads to access relations for designingnuclear fuel tanks. Inversion of tubes and inversion specificationswere studied by Al-Hassani et al. [2]. Mamalis et al. [3] presented anew theoretical model for collapse of steel conical tubes based onexperimental observations. Abramowicz and Wierzbicki [4] studiedcrushing of thin walled structures with polygon sections consideringfixed plastic hinges. Abramowicz and Jones [5] calculated theaverage crushing load of square tubes under axial static loads.Mamalis et al. [6] studied experimentally the effect of circulargrooves around outer surface of cylindrical tubes on the bucklingload, and showed that these grooves can control the maximum loadof collapse. Chirwa [7] investigated the inversion of thin walledtubes with varying thickness, both analytically and experimentally,and showed that energy absorption capacity of these tubes is about50% higher than those of tubes with constant thickness. Aljawi and

ll rights reserved.

x: +98 811 8257400.

ia).

Alghamdi [8] studied the inversion collapse of frusta using Abaqussoftware. They divided inversion of frusta into three types: externalflattening, internal flattening and folding mode. Alghamdi [9] madean overview about collapsible energy absorbers. Alavi Nia andLiaghat [10] investigated different deformation mechanisms ofhoneycomb panels and their crushing under axial quasi static loads.They [11] studied also crushing of short thin walled columns withsquare sections under the impact of cylindrical projectiles andcalculated the minimum velocity of impact needed for completefolding of such columns. Aljawi et al. [12] investigated energyabsorption of steel square tubes both experimentally and numeri-cally and observed that the maximum collapse load reduces about10% when these tubes are filled with foam. Tarigopula [13] studiedquasi static and dynamic loading of simple and top-hat tubesexperimentally and concluded that energy absorption of top-hattubes is bigger. Sedghi and Alavi Nia [14] investigated the effect ofouter grooves with different sections on the crushing and energyabsorption of cylindrical tubes, and showed that the crushingdistance and folding efficiency of these tubes are almost the sameand the maximum load in their load–displacement curve is reduceddue to grooves.

In this paper, deformation modes and energy absorption capacityof thin walled tubes with various section geometries are investi-gated and compared both experimentally and numerically.

2. Test specimens

Since the tubes with desirable sections were not available, wemade them in workshop from sheets by welding. Due to Argon

Table 1Mechanical properties of samples based on tension test results.

Thickness ofspecimen (mm)

Ultimatestrength (MPa)

Elongation atbreak (mm)

Stress at0.2% yield

1.5 131.9 6.5 129.5

1.5 134.0 7.3 127.2

1.5 147.8 6.8 131.0

1 140.0 6.5 132.0

1 135.0 6.4 129.0

1 138.0 6.6 130.5

Fig. 1. Stress–strain curve of one of the samples.

Table 2Material composition of samples.

Composition Percentage (1.5 mm thicknessplate)

Percentage (1 mm thicknessplate)

Al 97.82 97.81

Si 0.41 0.38

Zn 0.05 0.05

Mn 1.03 1.06

Sn 0.00015 0.00017

Fe 0.52 0.55

Cu 0.17 0.15

Sum 100 100

Table 3Specifications of the samples (dimensions in millimeters).

Specimen shape Dimensions )mm)

Cylindrical Length Diameter

100 60

Hexagonal prism Length Rib

100 31.4

Square prism Length Rib

100 47.1

Rectangular prism Length Cross section

100 31.4�62.8

Triangular prism Length Rib

100 62.8

Frusta Length Minimum diamete

100 43.32

Pyramidal Length Minimum cross se

100 40�41.37

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954 947

welding, this process can affect on the results, but we did our bestto reduce these unwanted effects as possible.

2.1. Material selection

Energy absorbers are made mainly from aluminum alloys, dueto their light weights. In this research, we used 1 and 1.5 mmthicknesses Al 3003 H12 plates.

2.2. Tension test

Mechanical properties of plates are determined based on ASTME 8M-98 standard using an Instron 8305 apparatus. Results ofthese tests are listed in Table 1 and stress–strain curve of one ofthe samples is shown in Fig. 1. Average values of yield andultimate stresses are 130 and 137.8 MPa, respectively.

2.3. Material composition

The composition of plates materials are determined and listedin Table 2.

2.4. Samples specifications

The samples are made from the same materials with 1 and1.5 mm thicknesses and have the same length, average sectionarea and volume. Specifications of all samples are listed in Table 3.Some samples are tested in both values of thicknesses.

2.5. Coding of the specimens

In order to indentify the samples, they are coded. Each codeincludes two parts. The first part constitutes two letters which

Thickness

1 and 1.5

Thickness

1 and 1.5

Thickness

1 and 1.5

Thickness

1 and 1.5

Thickness

1 and 1.5

r Maximum diameter Thickness

76.67 1 and 1.5

ction Maximum cross section Thickness

40�66.91 1 and 1.5

Table 4Abbreviations for samples and their number.

Specimen shape Code Number of tested specimens

1.5 mm thickness 1 mm Thickness

Cylindrical Cr 5 3

Hexagonal prism He 4 -

Square prism Sq 5 3

Rectangular prism Re 5 -

Triangular prism Tr 5 -

Frusta Fr 5 3

Pyramidal Pr 5 3

Fig. 2. Instron 8305 apparatus.

Table 5Results of experiments for all of the specimens.

Specimen code Absorbed energy (Nm) Mean force (kN) Maximum force (kN)

Cr1.5-1 1170.0 14.63 34.50

Cr1.5-2 1190.0 15.25 36.00

Cr1.5-3 1220.0 15.38 34.78

Cr1.5-4 1150.0 14.37 35.55

Cr1.5-5 1110.0 14.05 35.10

Cr1-1 629.0 8.00 25.62

Cr1-2 622.0 7.90 25.28

Cr1-3 640.0 8.10 25.35

Sq1.5-1 830.0 10.50 34.40

Sq1.5-2 820.0 10.60 33.90

Sq1.5-3 823.0 10.40 34.10

Sq1.5-4 840.0 10.56 34.00

Sq1.5-5 825.0 10.27 33.80

Sq1-1 511.0 6.46 23.50

Sq1-2 482.0 6.02 23.20

Sq1-3 491.0 6.33 23.40

Fr1.5-1 1075.0 13.27 25.59

Fr1.5-2 1087.0 13.20 25.61

Fr1.5-3 1081.0 13.34 25.26

Fr1.5-4 1112.0 13.64 25.91

Fr1.5-5 1098.0 13.47 25.32

Fr1-1 561.0 6.68 15.99

Fr1-2 556.0 6.66 15.89

Fr1-3 560.0 6.55 15.91

Tr1.5-1 658.0 7.74 32.40

Tr1.5-2 610.0 7.30 30.15

Tr1.5-3 677.0 8.01 30.98

Tr1.5-4 663.0 8.08 31.70

Tr1.5-5 647.0 7.73 31.50

Re1.5-1 710.0 9.08 32.57

Re1.5-2 692.0 8.76 32.72

Re1.5-3 688.0 8.76 32.80

Re1.5-4 730.0 9.01 33.00

Re1.5-5 690.0 8.57 32.80

He1.5-1 961.0 12.64 37.00

He1.5-2 947.0 12.62 36.80

He1.5-3 939.0 12.49 36.30

He1.5-4 956.0 12.50 37.40

Pr1.5-1 728.0 8.18 26.40

Pr1.5-2 752.0 8.10 26.90

Pr1.5-3 776.0 8.20 26.60

Pr1.5-4 763.0 8.30 27.10

Pr1.5-5 780.0 8.22 26.70

Pr1-1 461.0 5.55 12.99

Pr1-2 422.0 5.17 12.16

Pr1-3 427.0 5.11 12.20

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954948

refer to the section geometric shape and one number,which shows its thickness. The second part which is separatedfrom the first part by a hyphen refers to the number of the samplebetween the samples with the same section geometry. Forexample, Tr 1.5-1 refers to the first sample of thin walled tubeswith triangle section and a thickness of 1.5 mm. The abbreviationsused for different samples and the number of tested samples arelisted in Table 4.

3. Experiment

Axial quasi static loading of samples is carried out usingInstron 8503 apparatus (Fig. 2). This apparatus has two jaws: theupper one is stationary and the lower one is moveable. Thesample is set between two jaws vertically and is compressedaxially. Since the upper and the lower jaws of the apparatus arestationary and moveable, respectively, the upper and the lowerends of the specimen are named ‘‘fixed end’’ and ‘‘moving end,’’respectively. The rate of loading is 100 mm/s and the stroke isconsidered equal to 90 mm. This stroke is given so that all of thesamples can absorb the maximum energy. During the test, theload–displacement curve is drawn for the sample.

In Table 5 values of the maximum and the average forces intests are listed, the samples crushing length at the end of loading,

Crushing length D (mm) Collapse mode Collapse starting point

80.0 Concertina and diamond Fixed end

78.0 Concertina and diamond Fixed end

79.3 Concertina and diamond Fixed end

80.0 Concertina and diamond Fixed end

79.0 Concertina and diamond Fixed end

78.6 Diamond Fixed end

77.6 Diamond Fixed end

79.0 Diamond Fixed end

79.0 Concertina and diamond Fixed End

77.3 Concertina Fixed end

79.0 Concertina Moving end

79.5 Concertina and diamond Fixed end

80.3 Concertina Fixed end

79.0 Diamond Fixed end

80.0 Diamond Fixed end

77.5 Diamond Moving end

81.0 Diamond Fixed end

82.3 Diamond Fixed end

81.0 Diamond Fixed end

81.5 Diamond Fixed end

81.5 Diamond Fixed end

84.0 Diamond Fixed end

83.5 Diamond Fixed end

85.5 Diamond Fixed end

85.0 Diamond Mid of specimen

83.5 Diamond Mid of specimen

84.5 Diamond Mid of specimen

82.0 Diamond Mid of specimen

83.7 Diamond Mid of specimen

78.2 Concertina Fixed end

79.0 Concertina and diamond Moving end

78.5 Concertina and diamond Fixed end

81.0 Concertina Fixed end

80.5 Concertina and diamond Fixed end

76.0 Concertina Fixed end

75.0 Concertina Fixed end

75.2 Concertina Fixed end

76.5 Concertina Moving end

82.3 Concertina and diamond Fixed end

81.5 Concertina and diamond Fixed end

83.3 Concertina Fixed end

78.8 Concertina Fixed end

84.0 Concertina and diamond Fixed end

83.1 Concertina and diamond Fixed end

81.6 Concertina and diamond Fixed end

83.5 Diamond Fixed end

Fig. 3. Some specimens before (left), during (middle) and after loading (right).

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954 949

the collapse modes and starting location of folding in each test areregistered and these parameters with the amount of energy absorbedduring the test and calculated from the load–displacement curve ofthe specimen are listed in Table 5.

Fig. 3 shows some of the specimens before, during and afterloading.

3.1. Calculation of the average values of the important parameters

from test results

The main parameters which are concerned in energy absorp-tion process of thin walled structures include the maximum

displacement of the end of the sample, wmax, absorbed energy, E,the maximum force, Fmax and the average force, Fmean. Averageamount of these parameters for each section shape with the samethicknesses are shown in Tables 6 and 7.

4. Numerical simulation

Numerical simulations for axial compression of thin walledsamples are carried out, using LSDYNA 970 software. Threedimensional models are constructed, due to assurance aboutaccuracy of results. The model geometry includes thin-walledtube between two rigid parts at its ends. Belytscho–Tsay shell

Table 7Average values of the important results for 1 mm thickness specimens.

Specimen shape Absorbed energy (Nm) Mean force (kN) Maximum force (kN) Crushing length D (mm) Collapse Mode Collapse starting Point

Cylindrical 630.3 8.00 25.42 78.4 Diamond Fixed end

Frusta 559.0 6.63 15.93 84.3 Diamond Fixed end

Square prism 496.6 6.27 23.36 78.8 Diamond Fixed end

Pyramidal 436.7 5.28 12.45 82.7 Concertina and diamond Fixed end

Fig. 4. Rectangular section tube before (left) and during loading (right).

Table 6Average values of the important results for 1.5 mm thickness specimens.

Specimen shape Absorbed energy (Nm) Mean force (kN) Maximum force (kN) Crushing length D (mm) Collapse mode Collapse starting point

Cylindrical 1168.0 14.74 35.19 79.3 Concertina and diamond Fixed end

Hexagonal prism 1090.6 13.38 25.54 81.5 Diamond Fixed end

Square prism 950.8 12.56 36.88 75.7 Concertina Fixed end

Rectangular prism 827.6 10.46 34.04 79.0 Concertina Fixed end

Triangular prism 651.0 7.77 31.35 83.7 Diamond Fixed end

Frusta 702.0 8.84 32.78 79.4 Concertina and diamond Fixed end

Pyramidal 759.8 8.20 26.74 82.0 Concertina Fixed end

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954950

elements with 1 and 1.5 mm thicknesses are used for tubes. Theboundary conditions are the same as the experimental tests;therefore, the upper rigid part is constrained completely, whereasthe lower rigid part can move upward vertically with a speedequal to 100 mm/s. The material models for the tube and the rigidparts are Mat_picewise_ linear_placticity and Rigid, respectively.In order to supply appropriate conditions for deformations,‘‘contact automatic surface to surface title’’ and ‘‘contact auto-matic single surface title’’ are used for tube-rigid part elementsand tube elements with each other, respectively. The finiteelement model of the specimen with rectangular section beforeand during loading is shown in Fig. 4.

The results of simulations including important parametersrelated to energy absorption capacity of samples are listed inTables 8 and 9.

5. Results and discussion

The results of simulations are compared with the experimentaldata in Tables 10 and 11. As it is shown from these tables,maximum difference between results is about 8.3%. Therefore,simulations can predict the behavior of tubes reasonably.Furthermore, the sequence of samples with various sectionsfrom viewpoint of energy absorption, the maximum force and theaverage force is the same in experiments and simulations.However, there are some mismatches about load–displacementcurves and at the starting location of folds, as it is seen in

Tables 10 and 11. These discrepancies may be due to weldingprocess effect on the test samples. The differences between thestarting locations of folds may be due to an inaccuracy onapproximating the friction coefficient in simulations. Since thereis no deterministic criterion for prediction of the starting place offolds [1], the results of experiments are more valid.

5.1. Deformation mode

Deformed sections of tested and simulated samples arecompared in Fig. 5. It is clear from this figure that in some casesthere are differences between these results, which could berelated to non-uniformities caused by welding.

The triangular cross-section does not fold progressively, due toan inherent incompatibility of folding modes of neighboringcorner elements. The desired in–out–in–out folding, whichguarantees diamond-like pattern in square, hexagonal or circularsections, cannot be developed as there is always an in–in or out–out element in a triangular folding lobe.

5.2. Absorbed energy

Energy absorption capacity of tested sections is compared inFigs. 6 and 7. It is clear from these figures that the circular andtriangular sections absorbed the most and the least amount ofenergy between the tested sections, respectively. The conical,pyramidal, rectangular and triangular tubes are set aftercylindrical tube. In axial quasi-static tests, the larger thenumber of section edges, the greater the energy absorptioncapacity. This is due to an increase of the number of folds andplastic hinges in sections with larger number of edges.

5.3. Maximum collapse load

The maximum force is a critical parameter during impact of bodiesand is the first peak in the load–displacement curve. Investigation ofthis parameter shows that pyramidal and conical tubes undergo alarge amount of reduction in maximum force, whereas for the tubeswith constant sections, this parameter is almost the same. However,this advantage can be easily counterbalanced in prismatic specimensby an introduction of triggering dents or other folding initiators, yetlarge stability of progressive folding of tapered tubes distinguishes

Table 8Results of simulations for 1.5 mm thickness specimens.

Specimen shape Absorbed energy (Nm) Mean force (kN) Maximum force (kN) Crushing length D (mm) Collapse mode Collapse starting point

Cylindrical 1080.0 13.67 35.80 79.0 Concertina and diamond Fixed end

Hexagonal prism 1030.0 13.20 24.80 78.0 Diamond Fixed end

Square prism 988.0 12.05 36.60 82.0 Concertina Moving end

Rectangular prism 894.0 11.10 35.30 80.6 Concertina Moving end

Triangular prism 662.0 8.34 32.50 79.0 Diamond Mid of specimen

Frusta 684.0 8.40 34.00 81.4 Concertina Moving end

Pyramidal 730.0 8.79 27.60 83.0 Concertina Fixed end

Table 9Results of simulations for 1 mm thickness specimens.

Specimen shape Absorbed energy (Nm) Mean force (kN) Maximum force (kN) Crushing length D (mm) Collapse mode Collapse starting point

Cylindrical 585.0 7.40 23.30 79.0 Concertina and diamond Fixed end

Hexagonal prism 533.0 6.58 15.90 81.0 Diamond Fixed end

Square prism 529.0 6.59 24.10 80.3 Concertina Moving end

Rectangular prism 512.0 6.32 23.60 81.6 Concertina Moving end

Triangular prism 383.0 4.73 15.50 81.0 Diamond Mid of specimen

Frusta 419.0 5.34 18.40 78.5 Concertina Fixed end

Pyramidal 465.0 5.60 14.99 83.0 Concertina and diamond Fixed end

Table 10Comparison between the results of simulations and experiments for 1.5 mm thickness samples.

Specimen shape Difference % Collapse mode Collapse starting point

Absorbed energy (Nm) Mean force (kN) Maximum force (kN) Crushing length D (mm)

Cylindrical 7.5 7.00 �1.70 0.3 Similar Similar

Hexagonal prism 5.5 1.10 2.80 4.0 Similar Similar

Square prism �3.9 4.00 0.75 �8.3 Similar Different

Rectangular prism �8.0 �6.00 �3.70 5.6 Similar Different

Triangular prism �1.7 �7.30 �3.70 5.6 Similar Similar

Frusta 2.5 4.90 �3.73 �2.5 Almost similar Different

Pyramidal 3.9 �7.20 �3.20 �1.2 Similar Similar

Table 11Comparison between the results of simulations and experiments for 1 mm thickness samples.

Specimen shape Difference % Collapse mode Collapse starting point

Absorbed energy (Nm) Mean force (kN) Maximum force (kN) Crushing length D (mm)

Cylindrical 7.2 7.50 8.30 �0.7 Almost similar Similar

Frusta 4.6 0.60 0.10 6.3 Similar Similar

Square prism �3.1 �0.78 �1.00 3.5 Almost similar Similar

Pyramidal 4.0 6.20 1.40 0.3 Almost similar Similar

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954 951

them from prismatic tubes. Reductions of the maximum force inpyramidal and conical tubes are due to small area of tube at theimpact location, which in turn decreases the required force foryielding and formation of plastic hinge.

5.4. Difference between the maximum and the average force

Reducing difference between the maximum and the averageforce is of concern in energy absorber systems, and there are somemethods for attaining this purpose (for example: foam filling,making internal or external grooves and continuous variation ofthe net area of the section). Comparison of the results of theexperiments and the numerical simulations show that differencebetween the maximum and the average force in conical

and pyramidal tubes is smaller. This is clearer in conical tubes,which have uniform load–displacement curves. This characteristicis related to the small net area under load at the impact end,which in turn is due to continuous growth of section in suchtubes.

In Figs. 8–11, the maximum and the average forces fordifferent sections are compared.

6. Conclusions

Based on experiments and simulations of this research, theimportant results related to thin walled tubes, which are used as

Fig. 5. Comparison between deformation modes of samples in experiments (left) and simulations (right).

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954952

energy absorbers are as follows:

�

Absorbed energy per unit mass is maximum for cylindrical tubes. � For samples with uniform polygon sections, the lower the

number of edges, the lower the energy absorption capacity, sothat this property has the least value for the triangular section.

� Gradual increase in section area of samples affects their energy

absorption capacity, the maximum and the average force;reducing the section area at the impact end reduces themaximum load considerably.

�

The maximum force for circular and hexagonal sections isgreater than the other one and is minimum for conicalsamples. � Difference between the maximum and the average force in

conical and pyramidal samples is considerable and it has theleast value for the conical samples.

� Changing the thickness of tubes from 1 to 1.5 mm does not

affect the sequence of their energy absorbance, the maximumand the average load.

� There is good agreement between test data and simulation results.

0

200

400

600

800

1000

1200

Abs

orbe

d en

ergy

(N.m

)

CrSpecimen shape

Experimental Numerical

Fr He Sq Tr Re Pr

Fig. 6. Comparison of energy absorption capacity of different sections in

experiments and numerical simulations (1.5 mm thickness).

0

100

200

300

400

500

600

700

Abs

orbe

d en

ergy

(N.m

)

Cr

Specimen shape

Experimental Numerical

Fr He Sq Tr Re Pr

Fig. 7. Comparison of energy absorption capacity of different sections in

experiments and numerical simulations (1 mm thickness).

0

5

10

15

20

25

30

35

40

Forc

e (k

N)

Cr

Specimen shape

Mean forceMax force

Fr He Sq Tr Re Pr

Fig. 8. Comparison of the maximum and the average force of various sections with

1.5 mm thickness (experiments).

0

5

10

15

20

25

30

35

40

Forc

e (k

N)

CrSpecimen shape

Mean forceMax force

Fr He Sq Tr Re Pr

Fig. 9. Comparison of the maximum and the average force of various sections with

1.5 mm thickness (simulations).

0

5

10

15

20

25

30Fo

rce

(kN

)

CrSpecimen shape

Mean forceMax force

Fr Sq Pr

Fig. 10. Comparison of the maximum and the average force of various sections

with 1 mm thickness (experiments).

0

5

10

15

20

25

Forc

e (k

N)

CrSpecimen shape

Mean forceMax force

Fr He Sq Tr Re Pr

Fig. 11. Comparison of the maximum and the average force of various sections

with 1 mm thickness (simulations).

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954 953

A. Alavi Nia, J. Haddad Hamedani / Thin-Walled Structures 48 (2010) 946–954954

References

[1] Alexander JM. An approximate analysis of the collapse of thin cylindricalshells under axial loading. Q J Mech Appl Math 1960;13(1):10–5.

[2] Al-Hassani STS, Johnson W, Lowe WT. Characteristics of inversion tube underaxial loading. J Mech Eng Sci 1972;14:370–81.

[3] Mamalis AG, Johnson W, Viegelahn GL. The crumpling of steel thin-walledtubes and frusta under axial compression at elevated strain-rate: someexperimental results. Int J Mech Sci 1984;26:537–47.

[4] Abramowicz W, Wierzbicki T. Axial crushing of multicorner sheet metalcolumns. J Appl Mech 1989;56(1):113–20.

[5] Abramowicz W, Jones N. Dynamic axial crushing of square tubes. Int J ImpEng 1984;2(2):179–208.

[6] Mamalis AG, Manolakos DE, Saigal S, Viegelahn G, Johnson W. Extensibleplastic collapse of thin-wall frusta as energy absorbers. Int J Mech Sci1986;28(4):219–29.

[7] Chirwa EC. Theoretical analysis of tapered thin-walled metal inverbucktube.Int J Mech Sci 1993;35(3/4):325–51.

[8] Aljawi AAN, Alghamdi AAA. Inversion of frusta as impact energy absorber. In:Hassan MF, Megahed SM, editors. Current advances in mechanical design andproduction VII. New York: Pergamon; 2000. p. 234–43.

[9] Alghamdi A. Collapsible impact energy absorber: an overview. Thin-WalledStruct 2001;39:189–213.

[10] Alavi Nia A, Liaghat GH, Investigation of characteristics and quasi staticanalysis of honeycombs, In: Proc 12th Annu. 8th Int Conf Mech Eng (2004)Tarbiat Modarres University, Tehran, Iran.

[11] Alavi Nia A, Liaghat GH, Dynamic crushing of thin walled columns underimpact of projectiles, In: Proc 12th Annu. 8th Int Conf Mech Eng (2004)Tarbiat Modarres University, Tehran, Iran.

[12] Aljawi. AAN, Abd-Rabou M, Asiri S. Finite element and experimental analysisof square tubes under dynamic axial crushing. ECCOMAS 2004.

[13] Tarigopula V, Langseth M, Hopperstad OS, Clusen AH. Axial crushing of thin-walled high-strength steel sections. Int J Imp Eng 2006;32:847–82.

[14] Sedghi M, Alavi Nia A, Labbafi H, Attari P, Effect of circumferential groovesgeometries on crashworthiness of axially loaded cylindrical tubes, In: Proc16th Annu 12th Int Conf Mech Eng (2008) Bahonar University, Kerman, Iran.