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doi:10.1016/j.tw

CorrespondE-mail addrThin-Walled Structures 44 (2006) 11851191

www.elsevier.com/locate/twsTheoretical prediction and numerical simulation of multi-cell squarethin-walled structures

Xiong Zhang, Gengdong Cheng, Hui Zhang

State Key Laboratory of Structural Analysis for Industrial Equipments, Department of Engineering Mechanics,

Dalian University of Technology, Dalian 116023, China

Received 5 June 2006; accepted 26 September 2006

Available online 19 December 2006Abstract

The axial crushing of square multi-cell columns were studied analytically and numerically. Based on the Super Folding Element

theory, a theoretical solution for the mean crushing force of multi-cell sections were derived by dividing the profile into 3 parts: corner,

crisscross, and T-shape. Numerical simulations of square multi-cell sections subjected to dynamic axial crushing were conducted and an

enhancement coefficient was introduced to account for the inertia effects for aluminum alloy AA6060 T4. The analytical solutions show

an excellent agreement with the numerical results. It was found that the crisscross part was the most efficient component for energy

absorption and the energy absorption efficiency of a single-cell column can be increased by 50% when the section was divided into 3 3cells. Finally, the proposed method was extended to analyze the plateau stress of square cell honeycomb subjected to out-plane axial

crushing and to some extent validate the mechanical insensitivity of honeycomb to cell size.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Multi-cell column; Energy absorption; Axial crushing; Honeycomb1. Introduction

Thin-walled metal tubes, particularly those of square orcircular cross-section, are widely used as energy absorberssince they are relatively cheap and efficient for absorbingenergy. The behaviors of structural collapse in axialcrushing of them have been extensively studied over thepast decades. The axial crushing of circular tube was firstlyanalyzed by Alexander [1] and found to be an excellentmechanism for energy absorption. Also, an approximatetheoretical expression is developed to derive the averagecrushing force. Then experimental research and theoreticalpredictions on axial crushing of circular and square tubeseither statically or dynamically are detailed by Wierzbickiand Abramowicz [2,3], Abramowicz and Jones [4,5],Andrews et al.[6] and others. The general characteristicsof the forcedisplacement curves of circular tube andsquare tube are similar: the axial force first reaches aninitial peak, followed by a drop and then fluctuates.e front matter r 2006 Elsevier Ltd. All rights reserved.

s.2006.09.002

ing author. Tel./fax: +86 0411 84706599.

ess: [email protected] (X. Zhang).However, their collapse modes are very different. For acircular tube, it can collapse with the concertina mode,diamond mode, mixed mode or global buckling, whichdepends primarily on the ratio of dimensions of the tube,namely length, diameter and thickness. For a square tube,symmetric mode, asymmetric mode, extensional mode orglobal buckling may appear during axial compression.While hollow tubes were still addressed by various

authors (e.g. Guillow et al.[7], Langseth et al.[8,9], Huangand Lu [10], Tarigopula et al.[11]), recently tubes filled bycellular metal material are becoming the focus of manyresearchers and a number of authors have contributed tothis topic. Because of low density and nearly constant stressuntil compressed to a high densified strain, cellularmaterials such as aluminum honeycomb or foam, has thepotential for increasing energy absorption of thin-walledstructures. The main mechanisms providing enhancements(compared to an empty tube) are the compression of thefiller material itself and interaction effects between the fillerand tube. Seitzberger et al. [12] have shown that the energyabsorption characteristics of thin-walled columns can beconsiderably improved by aluminum foam filling. Hanssen

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Fig. 1. Representation of a multi-cell column. (a) Top view, and (b)

general view.

X. Zhang et al. / Thin-Walled Structures 44 (2006) 118511911186et al. [13,14] also studied the influence of aluminum foamto square and circular aluminum extrusions; Simpleexpressions on the mean crushing force and maximumcrushing force were presented. Efficiency improvement isalso shown when numerical method was adopted bySantosa and Wierzbicki [15] to study axially compressedsquare aluminum tubes, which were filled with honeycombor foam.

In theory aspect, after a classical plastic foldinganalysis of a cylindrical shell under axial crush loadingdone by Alexander in 1960 [1], plastic deformation ofprismatic columns was analyzed by the Super FoldingElement theory proposed by Wierzbicki and Abramowicz[2,3]. In the Super Folding Element theory, by adoptinga rigid-plastic material and using the condition ofkinematic continuity on the boundaries between rigid anddeformable zones, a model consisting of trapezoidal,toroidal, conical and cylindrical surfaces with movinghinge lines was analyzed. Main plastic energy dissipationmechanisms of typical crumpled thin-walled metal struc-tures were included in the model. Experimental validationof the theory was then performed by Abramowicz andJones [4].

The number of angle elements on a tubes cross-section decides, to a large extent, on the efficiency ofthe energy absorption [2,3]. It is therefore desirable todesign thin-walled extrusion with multiple cells for weight-efficient energy absorption. For multi-cell columns, due tothe complicacy of the problem, few literatures areavailable. Chen and Wierzbicki [16] presented closed-formsolutions to calculate the mean crushing strength of single-cell, double-cell and triple-cell hollow aluminum profilesand corresponding foam-filled extrusion. The analyticalsolution for calculating the mean crushing force of multi-cell profiles with four square elements at the corner isderived by Kim [17]. Both solutions were shown tocompare very well with the numerical results. Triggerswere introduced to the numerical models of multi-cellextrusions in order to obtain stable and regular crushingmodes.

The axial crushing of multi-cell aluminum extrusion asshown in Fig. 1 was studied analytically and numerically inthis paper. Based on the proposed Super Folding Elementtheory [2,3], an analytical solution for mean crushing forceof multi-cell square sections was developed, and it was veryconvenient to apply.

Nonlinear explicit finite element (FE) codes such asPAM-CRASH, ABAQUS Explicit and LS-DYNA arepowerful tools in numerical simulations of the largedeformation dynamic response of structures. A largeamount of numerical work [1117] has been carriedout on thin-walled structures by these commercialcodes apart from the above analytical and experimentalinvestigations. LS-DYNA [18] was used to conduct thenumerical simulations in this paper. The analytical solu-tions derived show an excellent agreement with thenumerical results.2. Theoretical analysis of multi-cell square columns

As mentioned above, Super Folding Element methodwas developed by Wierzbicki and Abramowicz [2] topredict the mean crushing force of thin-walled structures,and the theory was applied to the problem of progressivefolding of a thin-walled rectangular column. For rigidperfectly plastic material, the mean crushing force Pm canbe calculated by

Pm 9:56s0b1=3t5=3, (1)

where s0 denotes the flow stress of the material; b thesectional width and t the wall thickness. Abramowicz andJones [5] later changed the constant 9.56 to 13.06.To take strain hardening effects into account, the energy

equivalent flow stress for material with power law hard-ening can be calculated by using [19]

s0 ffiffiffiffiffiffiffiffiffiffiffisysu1 n

r, (2)

where sy and su denote the yield strength and the ultimatestrength, respectively; and n is the strain hardeningexponent.Chen and Wierzbicki [16] adopted a simplified approach

to derive the analytical solution of mean crushing force ofmulti-cell sections. Rather than building the model con-sisting of trapezoidal, toroidal, conical and cylindricalsurfaces with moving hinge lines, they proposed a basicfolding element consisting of 3 extensional triangularelements and 3 stationary hinge lines. Expressions of meancrushing force of double cell and triple cell were given.They are Pm 9:89s0b1=2t3=2, and Pm 12:94s0b1=2t3=2,respectively. In their work, the membrane energy of themulti-cell structure was determined by the number offlange, the contribution of each flange was considered to bethe same.For a complete collapse of a single fold of a tube,

considering the energy equilibrium of the system, the















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2

2

a b

Fig. 3. Illustration of dissipation in bending (Chen and Wierzbicki [16]).

(a) Stationary hinge line, and (b) rotation angles.

X. Zhang et al. / Thin-Walled Structures 44 (2006) 11851191 1187external work done by compression has to be dissipated byplastic deformation in bending and membrane. Namely

2HPmk Wbending Wmembrane, (3)

where H is the half-length of the fold, and Pm denotes theaverage crushing force over the collapse of a fold. Wbendingand Wmembrane are, respectively, the energy dissipation inbending and membrane deformation. In a real structure, anactual folding element can never be completely flattened.The crush distance is actually less than 2H. Wierzbicki andAbramowicz [2,3] found that the effective crush distance isabout 7075% of the wavelength. This value kis taken as71% in the following derivation.

The simplification adopted by Chen and Wierzbicki [16]was also employed here to deal with the multi-cell squaresections. The wall thickness is assumed constant over thecross-section. Extensional triangular elements and station-ary hinge lines constitute the basic folding element. Toanalyze the energy dissipation over the collapse of a fold,the multi-cell square section was divided into three basiccomponents: the corner part, the crisscross part and the T-shape part. Fig. 2 shows the illustration.

2.1. The energy dissipated in bending

The bending energy Wbending can be calculated bysumming up the energy dissipation at stationary hingelines. For each flange, 3 horizontal stationary hinge linesare developed. See Fig. 3(a):

Wbending X3i1

M0yiLc, (4)

where M0 1=4s0t2is the fully plastic bending momentof the flange, y the rotation angle at each hinge lineandLcdenotes the total length of all flanges. For simplicity,it is assumed that the flanges are completely flattened(Fig. 3(b)) after the axial compression of 2H [16], whichmeans that the rotation angles at 3 hinge lines are p/2, pFig. 2. Illustration of the division of multi-cell section.and p/2, respectively. Consequently

Wbending 2pM0Lc. (5)

2.2. The energy dissipated in membrane deformation

The membrane energy Wmembrane dissipated during onewavelength crushing can be calculated by integrating theextensional and compressional areas. The membraneenergy of the three basic parts are discussed in thefollowing.

2.2.1. The corner part

The corner part has been analyzed by Chen andWierzbicki [16]. After deformation, three membraneelements (one in extension and two in compression) weredeveloped for each flange. By integrating the extensionaland compressional area, the membrane energy dissipatedby the corner part during one wavelength crushing was

Mcorner 21

2s0tH2 4M0H2=t. (6)

2.2.2. The crisscross part

The deformation mode of the crisscross part is muchmore complicated than the corner part. Extracted from thenumerical results of multi-cell columns (see next section), atypical collapse mode of the crisscross part is shown inFig. 4. The centerline of the crisscross part was observed tofold in the 451 direction between two flanges as shown bythe dash line. Based on the collapse mode, a simplifiedfolding mode is shown in Fig. 5(a). The bold line denotesthe section before deformation. Two flanges compressedeach other and another two flanges were elongated. Theillustration of membrane elements is shown in Fig. 6. Forall four flanges, two flanges were in extension and two incompression. Again integrating the extensional and com-pressional area, the membrane energy dissipated by the








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Fig. 4. Typical collapse mode of the crisscross part by FEM. (a) Top view,

and (b) side view.

45 H

45

H

a b

Fig. 5. Simplified folding modes. (a) The crisscross part, and (b) the T-

shape part.

Fig. 6. Illustration of the membrane elements (a) in extension, and (b) in

compression.

X. Zhang et al. / Thin-Walled Structures 44 (2006) 118511911188crisscross part during one wavelength crushing was

Mcrisscross 4s0tH2 16M0H2=t. (7)

2.2.3. The T-shape part

The collapse mode of T-shape part is relatively simple asshown in Fig. 7. The centerline of the T-shape wasobserved to fold in the plane of the protruded flanges asshown by the dashed line. Based on the collapse mode, asimplified folding mode is shown in Fig. 5(b). The bold linedenotes the section before deformation. Two flangescompressed each other and another flange was unaffected.The shape of the membrane element of T-shape part is thesame as crisscross part as shown in Fig. 6. For all threeflanges, two flanges were in compression and another onewas unaffected. Again integrating the extensional andcompressional area, the membrane energy dissipated by theT-shape part during one wavelength crushing was

MTshape 2s0tH2 8M0H2=t. (8)

Therefore the whole energy dissipated by membranedeformation is

Wmembrane NcMcorner NoMcrisscross NTMTshape 4Nc 16No 8NTM0H2=t, 9

where Nc, No, and NT denote the number of corner,crisscross and T-shape, respectively.By comparing the three parts, one can see that the

crisscross part is the most efficient energy absorbingcomponent of the multi-cell square column since themembrane energy dissipation per weight of the crisscrosspart is two times that of the corner part.

2.3. The fold length and mean crushing force

Substitute Eqs. (5) and (9) to (3), the mean crushingforce can be obtained:

0:71Pm

M0 2Nc 4No 2NT

H

t pLc

H. (10)

Lc is the total length of the wall of the cross-section.The folding wavelength H can be determined by the

stationary condition of the mean crushing force:

qPmqH

0. (11)

Therefore

H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipLct

2Nc 4No 2NT

s. (12)

Substituting Eq. (12) into Eq. (11), the mean crushingforce for a multi-cell section would be

Pm 2

0:71M0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Nc 4No 2NTpLc

t

r s0t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNc 4No 2NTpLct

p. 13

Obviously Lct is the material area of the cross-section.One can see that this expression is very convenient toapply.

3. Numerical simulations

The explicit nonlinear FE code LS-DYNA was used topredict the response of the multi-cell thin-walled sectionssubjected to axial crushing. The FE model was created bythe program ANSYS/LS-DYNA [20]. The full sectionswere modeled using the quadrilateral BelytschkoTsayshell element with four nodes and three integration pointswere used throughout the thickness. Prescribed velocity




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Fig. 7. Collapse mode of the T-shape part. (a) Top view, (b) side view, and (c) front view.

200

160

120

80

40

00.00 0.04 0.08 0.12 0.16 0.20

Strain

Str

ess (

MP

a)

Fig. 8. Engineering stressstrain curves for AA6060 T4.

X. Zhang et al. / Thin-Walled Structures 44 (2006) 11851191 1189boundary conditions were applied on the rigid plane usedto crush the columns. Clamped boundary conditions wereapplied at the bottom of the column.

The material of columns used here is the aluminum alloyAA6060 T4 with mechanical properties: Youngs modulusE 68.2GPa, initial yield stress sy 80MPa, the ultimatestress su 173MPa, Poissons ratio n 0.3 and the powerlaw exponent n 0.23. The complete stressstrain relationfor this material is shown in Fig. 8.

In the quasi-static crushing, the speed of loading isusually in the range 0.011mm/s. This range of velocity istoo slow for the numerical simulation. This is due to thefact that the explicit time integration method is onlyconditionally stable, and therefore in general very smalltime increments have to be used. To overcome thisproblem, dynamic crushing with a velocity of 10m/s wasadopted in all the simulations in this paper. A dynamicenhancing coefficient was introduced to account for theinertia and strain rate effects. In fact, as aluminum isinsensitive to the strain rate, strain rate effects are of minorimportance. The coefficient was proposed by Langseth andHopperstad [7] to be in the range of 1.31.6 for AA6060 T4columns without trigger and a relatively lower value forcolumns with trigger. For simplicity, this coefficient was setto 1.3 here. Therefore, the expression applied here is

Pm 1:3s0tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNc 4No 2NTpLct

p. (14)

A square box column of 80 80mm cross-section with16 cells, 0.6mm thickness, and 150mm length wasconsidered in this analysis. The FE model of the squarecolumn is shown in Fig. 9(a). Trigger was introduced nearthe top of the column. The deformed result is also shown inFig. 9(b). Fig. 10 shows the crushing forcedisplacementcurve and mean loaddisplacement curve. The meancrushing force was obtained to be 27.4 kN.Obviously for this cross-section, Nc 4, No 9, and

NT 12. Substituting them into Eq. (14), theoreticalprediction of the mean crushing force is

Pm 1:3 106 0:6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 36 24800 0:6p

p 25:7 103 N:

Similar simulation was conducted for cross-section withsame material area (same weight) and with various wallthicknesses. The width and length of the columns were kept80 and 150mm, respectively. Theoretical solutions andnumerical results of the mean crushing forces are listed inTable 1. The crushing forcedisplacement curves are shownin Fig. 11.One can see that the theoretical solutions compare very

well with the numerical predictions, except that for a singlecell, the theoretical solution is a little higher than thenumerical prediction, and for 4 4 cells and 5 5 cells, thetheoretical solutions are somewhat lower. This is due to thefact that the effective crush distance is taken as 71% of thewavelength all the time. However actually this value isunderestimated for single cell and overestimated for 4 4and 5 5 cells. Adopting the effective crush distanceobtained by numerical simulation, modified theoreticalsolutions were also listed in Table 1. More closer agreementwas reached between theoretical solutions and numericalpredictions. In addition, for multi-cell columns, since no

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Fig. 9. (a) FE model of a square column with 16 cells, and (b) deformed result after axial crushing.

Fig. 10. The crushing forcedisplacement curve and mean loaddisplace-

ment curve.

Table 1

Theoretical solutions and numerical results of the mean crushing forces

Column section Single cell 2 2 3 3 4 4 5 5

Wall thickness (mm) 1.5 1.0 0.75 0.6 0.5

Total wall length (mm) 320 480 640 800 960

Theoretical solution (kN) 16.1 21.4 24.1 25.7 26.8

Numerical prediction (kN) 12.6 21.0 25.1 27.4 30.4

Modified theoretical result (kN) 13.8 20.4 25.6 26.8 27.9

Fig. 11. The crushing forcedisplacement curves of a group of sections

with the same weight.

X. Zhang et al. / Thin-Walled Structures 44 (2006) 118511911190imperfection was introduced to the internal web of multi-cell columns, a dynamic enhancing coefficient of 1.3 wasappropriate. However, for single-cell column, a lower valueshould be applied considering the trigger introduced.Comparing to the numerical prediction 12.6 kN, thetheoretical mean force for single cell would be 12.7 kNwith an enhancing coefficient of 1.2. Furthermore, one canalso see from the table that the energy absorption efficiencyof a single-cell column can be increased by 50% when thesection is divided into 3 3 cells, which validates the factthat the energy absorption efficiency can be greatlyimproved by introducing internal webs to the columns.4. Discussion

From the theoretical and numerical results in Table 1, itseems that the mean crushing force of the columns withprescribed material area tends to a certain limit with thenumber of cells increasing. Now consider a NN-cellcolumn(see Fig. 12) with material area Am and width a.The number of corner Nc 4The number of crisscross No N 12:The number of T-shape NT 4N 1:The wall thickness t Am

2N1a :Substituting into Eq. (13),

Pm ffiffiffip

ps0

NA3=2mN 1a

. (15)

With N increasing,

Pm !ffiffiffip

ps0

A3=2ma

.

When N is large enough, the column can be taken ashoneycomb material. The relative density of the honey-comb is defined as r Am=a2, and the nominal stress can

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N cells

N cells

a

Fig. 12. Sketch map of a NN-cell section.

X. Zhang et al. / Thin-Walled Structures 44 (2006) 11851191 1191be defined as

sP Pm

a2

ffiffiffip

ps0r3=2. (16)

It shows that the plateau stress of square cell honeycombcan be determined by this method. However, experimentsshould be conducted to validate this expression. One cansee from Eqs. (15) and (16) that the plateau stress of thehoneycomb is insensitive to the number of cell (cell size)when N is above a certain value. For instance, keeping thewidth of the section constant, the plateau stress willincrease less than 5% when the number of cell increasesfrom 100 to 400. In this point it shows agreement with thefact that mechanical properties of cellular solids dependonly weakly on cell size [21].

5. Conclusions

A method for determining the mean crushing force ofsquare multi-cell columns subjected to axial loading isgiven. The profile was divided into three parts: corner,crisscross, and T-shape. Based on the Super FoldingElement theory, each part was analyzed independently.The crisscross part was found to be the most efficientcomponent for energy absorption. The mean crushingforce and the folding wavelength was then determined by aminimum principle that the collapse mechanism tends tominimize the mean crushing force. Numerical simulationsof square multi-cell sections subjected to dynamic axialcrushing were conducted and an enhancement coefficientwas introduced to account for the inertia effects foraluminum alloy AA6060 T4. The analytical solutions showan excellent agreement with the numerical results especiallywhen the effective crush distance was revised. By analyzinga group of sections with same weight, one can see that theenergy absorption efficiency can be greatly improved byintroducing internal webs to the columns. The energyabsorption efficiency of a single-cell column can beincreased by 50% when the section is divided into 3 3cells. The proposed method was also extended to analyzethe plateau stress of square cell honeycomb subjected toout-plane axial crushing and showed agreement with thefact that mechanical properties of cellular solids dependonly weakly on cell size.Acknowledgments

The present work was supported by the NationalNatural Science Foundation of China (no. 10332010),National Creative Research Team Program (no.10421002)and National Basic Research Program of China(2006CB601205).References

[1] Alexander JM. An approximate analysis of the collapse of thin

cylindrical shells under axial loading. Q J Mech Appl Math 1960;

13(1):105.

[2] Wierzbicki T, Abramowicz W. On the crushing mechanics of thin-

walled structures. J Appl Mech Trans ASME 1983;50(4 a):72734.

[3] Abramowicz W, Wierzbicki T. Axial crushing of multicorner sheet

metal columns. J Appl Mech Trans ASME 1989;56(1):11320.

[4] Abramowicz W, Jones N. Dynamic axial crushing of square tubes.

Int J Impact Eng 1984;2(2):179208.

[5] Abramowicz W, Jones N. Dynamic progressive buckling of circular

and square tubes. Int J Impact Eng 1986;4(4):24370.

[6] Andrews KRF, England GL, Ghani E. Classification of the axial

collapse of cylindrical tubes under quasi-static loading. Int J Mech Sci

1983;25(9-10):68796.

[7] Guillow SR, Lu G, Grzebieta RH. Quasi-static axial compression of

thin-walled circular aluminium tubes. Int J Mech Sci 2001;43(9):

210323.

[8] Langseth M, Hopperstad OS, Hanssen AG. Crash behaviour of thin-

walled aluminium members. Thin-Walled Struct 1998;32(1-3):12750.

[9] Langseth M, Hopperstad OS. Static and dynamic axial crushing of

square thin-walled aluminium extrusions. Int J Impact Eng 1996;

18(7-8):94968.

[10] Huang X, Lu G. Axisymmetric progressive crushing of circular tubes.

Int J Crashworthiness 2003;8(1):8795.

[11] Tarigopula V, et al. Axial crushing of thin-walled high-strength steel

sections. Int J Impact Eng 2006;32(5):84782.

[12] Seitzberger M, et al. Crushing of axially compressed steel tubes filled

with aluminium foam. Acta Mech 1997;125(1-4):93105.

[13] Hanssen AG, Langseth M, Hopperstad OS. Static and dynamic

crushing of square aluminium extrusions with aluminium foam filler.

Int J Impact Eng 2000;24(4):34783.

[14] Hanssen AC, Langseth M, Hopperstad OS. Static and dynamic

crushing of circular aluminium extrusions with aluminium foam filler.

Int J Impact Eng 2000;24(5):475507.

[15] Santosa S, Wierzbicki T. Crash behavior of box columns filled with

aluminum honeycomb or foam. Comput Struct 1998;68(4):34367.

[16] Chen W, Wierzbicki T. Relative merits of single-cell, multi-cell and

foam-filled thin-walled structures in energy absorption. Thin-Walled

Struct 2001;39(4):287306.

[17] Kim HS. New extruded multi-cell aluminum profile for maximum

crash energy absorption and weight efficiency. Thin-Walled Struct

2002;40(4):31127.

[18] Hallquist J. LS-DYNA users manual. Version: LS-DYNA 970 ed.

Livermore Software Technology Corporation, 2003.

[19] Santosa SP, et al. Experimental and numerical studies of foam-filled

sections. Int J Impact Eng 2000;24(5):50934.

[20] ANSYS Inc. Ansys documentation. Version 8.1, ANSYS LS-DYNA

users guide, 2004.

[21] Gibson LJ, Ashby MF. Cellular solids: structure and properties. 2nd

ed. Cambridge: Cambridge University Press; 1997.


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