at

MiUniv008istr

Multi-cell tubeAxial crushingCrashworthinessEnergy absorptionTheoretical prediction

xprelyin

mization were also performed for the three tubes. A Pareto sets were obtained by the linear weighted

ve be

four modes of deformation that are concertina, diamond, mixed,and global buckling. What appears in the process of deformationmainly depends on the ratio of thickness-diameter-length of circu-lar tubes. Nonetheless, the general characteristics of crushingforcedisplacement curve of square tubes are similar to those of

rushing. Wierzb-ierzbicki [4he cross-srption lea

the thin-walled multi-cell tubes study.Chen and Wierzbicki [6] simplied the SFE theory to stu

axial crushing performance of single-cell, double-cell, triphollow square tubes and foam-lled tubes under quasi-static axialloading. By dividing the cross-sectional tube into distinct angesections, and assuming that each ange contributed to the similarrole in structure and that the ange was completely attened afterdeformation of the wavelength, the average folding wavelengthand the theoretical expression for the mean crushing force were

Corresponding authors at: College of Mechanical and Vehicle Engineering,Hunan University, Changsha, Hunan 410082, PR China.

E-mail addresses: shujuanhou@hnu.edu.cn (S. Hou), hanxu@hnu.edu.cn (X. Han).

Composite Structures 119 (2015) 422435

Contents lists availab

Composite S

sevetc.The collapse mode of square tube is different from that of circu-

lar tube. The square tubes collapse with the asymmetric mode,symmetric mode, or global buckling while the circular tubes have

cylindrical surfaces during the process of axial cicki and Abramowicz [29], Abramowicz and Wcluded that the number of angle elements on tof tubes decided the efciency of the energy absohttp://dx.doi.org/10.1016/j.compstruct.2014.09.0190263-8223/ 2014 Elsevier Ltd. All rights reserved.] con-ectionsding to

dy thele-cellenergy absorbers in the vehicle design for decades because of theirrelatively cheap price and better weight efciency. The earlyresearch aimed to investigate the mechanisms of structural col-lapse under axial crushing. Wierzbicki and Abramowicz [29], Abra-mowicz and Wierzbicki [4], Abramowicz and Jones [2,3] pioneeredin the experimental and theoretical solutions of the axial crushingforce of square and circular tubes under static and dynamic loadingwhich also inspired DiPaolo et al. (2004, 2009), Guillow et al. [10],

crushing force.According to the Super Folding Element (SFE) theory described

by Wierzbicki and Abramowicz [29], Abramowicz and Wierzbicki[4], deformed elements were described in several principal defor-mation folding mechanisms consisted of inextensional, quasi-inex-tensional and extensional mode. The key aspect of SFE was therecognition of the formation of moving hinge lines dening theboundaries of the component trapezoidal, toroidal, conical andMulti-objective optimization

1. Introduction

Thin-walled multi-cell tubes haaverage method. Deb and Gupta method was utilized to nd out knee points from the Pareto frontiersfor multi-cell tubes. The simulation results showed that the multi-cell tube type I with right corner, T-shape and criss-cross angle elements was the best one among the three tubes in the aspect of specicenergy absorption (SEA). For all the tubes, the stable and progressive folding deformation patterns weredeveloped. Finally, the theoretical predictions well coincided with the numerical results, and also vali-dated the efciency of the numerical optimization design method.

2014 Elsevier Ltd. All rights reserved.

en widely used to be

circular tubes [1]. The crushing curves of forcedisplacement ofall the proles show that the crushing force rst reaches an initialpeak, then declines and then uctuates around a value of the meanKeywords:square multi-cell tubes were divided into several basic angle elements: right corner, T-shape, 3-panel,criss-cross, and 4-panel angle element. Numerical simulations and multi-objective crashworthiness opti-Crushing analysis and numerical optimizstructures under axial impact loading

TrongNhan Tran a,b,c, Shujuan Hou a,b,, Xu Han a,b,,a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, HunanbCollege of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan 41c Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, Go Vap D

a r t i c l e i n f o

Article history:Available online 21 September 2014

a b s t r a c t

In this paper, theoretical etubes were derived by app

journal homepage: www.elion of angle element

nhQuang Chau c

ersity, Changsha, Hunan 410082, PR China2, PR Chinaict, HCM City, Viet Nam

ssions of the mean crushing force of the three different square multi-cellg the Simplied Super Folding Element (SSFE) theory. The proles of three

le at ScienceDirect

tructures

ier .com/locate /compstruct

derived. The work of Chen and Wierzbicki indicated that the addi-tion of interior walls made the specic energy absorption (SEA)increase by approximately 15% in comparison with the single-cellmodel.

Kim [19] applied the model of [6] to multi-cell tubes with foursquare elements at the corner under the dynamic loading. The SEAof the multi-cell structure was reported to increase by 190% thanthe square single-cell tube. Zhang et al. [30] also adopted themodel of [6] to derive a theoretical solution for calculating themean crushing force of multi-cell square tubes under the dynamicloading. In Zhang et als work, the cross-section of tube was dividedinto three basic elements, and their study also measured the con-tribution of each element type to the plastic energy dissipationthrough membrane action. The resulted theoretical solutionassumed an average wavelength for the dissimilar folds developedat the corners. Then, the model of [6] was also utilized by Zhang

paper. Based on the Simplied Super Folding Element (SSFE) the-ory, theoretical expressions of the mean crushing force for thethree multi-cell square tubes were derived. All of the studied pro-les were divided into several angle elements which were right-corner, 3-panel, T-shape, criss-cross and 4-panel angle element.To obtain the optimal proles under the crashworthiness criterion,Dynamic nite element analysis code ANSYS/LS-DYNA was exe-cuted to simulate tubes and to obtain the numerical results atthe design sampling points. The multi-objective optimizationdesign was utilized to obtain the optimal congurations. Finally,the theoretical expressions are employed to validate the numericaloptimal solutions.

2. Theoretics

rner

T. Tran et al. / Composite Structures 119 (2015) 422435 423et al. [3035] to predict the mean crushing force of 3-panel angleelement and X-shape elements with different angles. Additionally,the model of [6] was also applied by Hanssen et al. [11] in order topredict the mean crushing force of complex aluminum extrusion.

Naja and Rais-Rohani [20,21] extended SFE theory to investi-gate the crushing characteristics of multi-cell tubes with two dif-ferent types of three-ange elements. An equation of closed formfor prediction of the mean crushing force was also proposed by[20,21]. In addition, dynamic bucklings of thin-walled square andcircular tubes under axial impact were studied by Jensen et al.[16], and Karagiozova and Jones [18]. From a phenomenologicalviewpoint, a review of dynamic buckling of axially loaded tubeswas summarized by Karagiozova and Alves [17]. Therefore, the glo-bal bending of tubes was an undesirable energy-dissipating mech-anism. In other words, the desirable energy-dissipating mechanismwas a stable and progressive folding deformation pattern.

In order to obtain a simplication to replace the kinematicaladmissible model of SFE theory, the Simplied Super Folding Ele-ment (SSFE) theory was proposed by comprising the extensionaltriangular elements and the bending hinge line. This theory wasalso used to investigate the crushing response of multi-cell tubesunder oblique impact loading [27].

Thin-walled multi-cell tubes were also theoretically consideredby Kim [19], Naja and Rais-Rohani [20,21]. Then, with the devel-opment of nite element methods, FE solutions [22,25] or a surro-gate model technology were widely used in the crashworthinessdesign of the tubes [1315]. However, there is seldom a combina-tion study of theory, numeric and optimization design for multi-cell square tubes.

Above all, the crushing of multi-cell square tubes was studiedon both theoretical prediction and optimization design in this

(a) (b)

Criss-cross T-shape Right-co(d)Fig. 1. Cross-sectional geometry of tubes and typical angle elements:2.1. Theoretical prediction of multi-cell square tube

The SSFE theory was utilized to deal with the multi-cell squaretube type I, II and III in Fig. 1. This theory assumed that the wallthickness is a constant over the cross-section, and that the varia-tions of the wavelength 2H for different lobes were ignored inthe analysis. In order to analyze the energy dissipation over thecollapse of a fold, the proles of tubes were divided into the basicelements: the right corner, 3-panel, T-shape, criss-cross and 4-panel angle element (as shown in Fig. 1).

The instantaneous crushing force P is dened in accordancewith the balance rates between internal and external energy dissi-pations in shell: _Eext _Eint. Therefore, the external work from com-pression to form a complete collapse of a single fold was equal tothe sum of dissipated bending and membrane energy. That is

Pm2H 1g Eb Em 1

where Pm, Eb and Em denote the mean crushing force, the bendingenergy and the membrane energy. 2H and g are, respectively, thelength of the fold and the effective crushing distance coefcient.In reality, the ange of folding element after deformation is notcompletely attened as described in Fig. 2. Thus, the availablecrushing distance is smaller than 2H. Wierzbicki and Abramowicz[29], Abramowicz and Wierzbicki [4] showed that the effectivecrushing distance coefcient varied in the range 0.70.75. In thiscase, the value of g is taken as 0.7 for simplicity.

2.1.1. The bending energyIn order to apply the SFE theory to the multi-cell square tube,

the SSFE theory was proposed. In this approach, instead of buildinga model consisting of trapezoidal, toroidal, conical and cylindrical

(c)

3-panel 4-panel(a) type I; (b) type II; (c) type III and (d) typical angle elements.

surfaces with moving hinge lines as in SFE theory, the basic foldingelement in the SSFE theory contains triangular elements and onebending hinge line. The bending energy Eb of each ange can bedetermined by summing up the energy dissipation at the bending

Esymm rc 2Esymm f 8M0H2

t6

The energy dissipation in membrane of right-corner element inthe case of asymmetric mode has been analysed by Chen and Wie-rzbicki [6]. After the deformation, the triangular elements includ-ing one extensional and two compressional parts were developedfor each ange. The membrane energy Em of one ange, duringone wavelength crushing, was evaluated by integrating the exten-sional and compressional area (shaded areas in Fig. 3(b)):

Easymm f Zsr0tds 12r0tH

2 2M0 H2

t7

the dissipated membrane energy of T-shape element can be calcu-

additional panels. Due to the similarity in deformation mode, it

2 =

Bending hinge line

Fig. 2. Bending hinge line and rotation angle on basic folding.

424 T. Tran et al. / Composite Structures 119 (2015) 422435hinge line. Then

Efb Xmi1

M0abi 2

where b is the sectional width, M0 = r0t2/4 is the fully plastic bend-ing moment of the ange and a denotes the rotation angle at thebending hinge line.

In this case, the ange was assumed to be completely attenedafter the deformation of the wavelength 2H. As a consequence, therotation angle a at the bending hinge line is 2p (as shown in Fig. 2).On employing Eq. (2), the dissipated bending energy at bendinghinge line of ange can be determined as

Efb Xmi1

2pM0bi 3

Since each ange shares the same role and multi-cell tube iscreated by m panels (Fig. 1), the bending energy of multi-cellsquare tube can be estimated, as follows

Etubeb 2pM0mb 2pM0B 4where B is the sum of side and internal ange lengths.

2.1.2. The membrane energy2.1.2.1. The membrane energy of right corner element. To determinethe membrane energy of right-corner deformed through symmet-ric mode in the SSFE theory, the basic folding element is formed bythe triangular elements and the bending hinge line (Fig. 3(a)). Thedissipated energy in membrane of one ange, during one wave-length crushing, can be estimated by integrating the triangulararea (shaded areas in Fig. 3(a))

Esymm f Zsr0tds r0tH2 4M0 H

2

t5

It was assumed that the each ange had a similar role in struc-ture; the membrane energy of right corner element in the case ofsymmetric mode is double of that in one single ange, then

b2H

2H

045

(a)Fig. 3. Illustration of the deformation mode: (a) sywas assumed that the independent right-corner element wasequivalent to the corresponding right corner element in a 4-panelangle element (Fig. 5) [1]. Simultaneously, the deformation modeof right-corner element is a symmetric mode. The independentright-corner element has a similar geometric parameter as 4-panelangle element with b = 45 excepting the latter had two additional

Corner line

b

045

(a)lated by the sum of right-corner elements membrane energy in thecase of symmetric mode and one additional panels membraneenergy. In fact, each panel had a similar role in structure. To sim-plify calculation, the energy dissipated in membrane during onewavelength crushing of T-shape angle element was more than tri-ple each anges membrane energy

ETshapem 3Esymm f 12M0H2

t9

The membrane energy of 3-panel angle element has been deter-mined by Zhang and Zhang [32]. In consequence of their work, thedissipated energy in membrane of 3-panel angle element, duringone wavelength crushing, was

E3panelm 4M0H2

t1 2 tan/=2 10

2.1.2.3. The membrane energy of 4-panel and criss-cross angle ele-ment. The 4-panel angle element was a symmetric structure andcreated by a combination of one right corner element and twoAccording to above mentioned literature, the energy dissipationin membrane of the right corner element in the case of asymmetricmode was estimated as follow

Easymm rc 2Easymm f 4M0H2

t8

2.1.2.2. The membrane energy of 3-panel angle and T-shape ele-ment. The structure of T-shape element was formed by oneright-corner element and one additional panel (Fig. 4). Accordingly,(b)mmetric mode and (b) asymmetric mode [6].

M0

H

t48 15

2H

b

Additional panel

(b)(a)

R

X

W

J

U

Q, V

R

X

W

J

U

Q

V

2

(c)al el

T. Tran et al. / Composite Structures 119 (2015) 422435 425panels at top of right corner element. Therefore, the dissipatedmembrane energy of 4-panel angle element was calculated bythe sum of right corner elements membrane energy in the caseof symmetric mode and two additional panels membrane energy.

As calculated above, the membrane energy Em of right cornerelement in the case of symmetric mode is Esymm rc 8M0H2=t. Itwas not easy or quite impossible to give a precise calculation ofthe membrane energy of additional panel. In this case, the SFE the-ory was too complicated to apply. In consequence, a simplieddeformation model of the additional panels was suggested andthe SSFE theory was used to deal with this problem [27]. Repre-sented in Fig. 6(b), the areas of ABC and of ABF are dened asextensional elements of two additional panels. Thus, the mem-brane energy of one additional panel, during one wavelengthcrushing, was evaluated by integrating the triangular areas

Eapanelm Zsr0tds r0t H

2

cosb 4M0 H

2

t cos b11

Then, the dissipated membrane energy of 4-panel angle ele-ment is

E4panelm Ercm 2Eapanelm 8M0H2

t1 1

cosb

12

Being a symmetric structure and formed by four panels, theenergy dissipation in membrane of a criss-cross element wasdetermined by the sum of membrane energy absorbed by all fourpanels (Fig. 7). It is assumed that the angle elements contributedsimilar roles in structure, the four panels create two right-corner

Fig. 4. (a) Collapse mode of T-shape element, (b) Extensionelements, and that the deformation mode of right-corner elementis a symmetric mode. Consequently, the membrane energy of

Fig. 5. (a) Right corner element anThe half-length of the fold can be determined on the stationarycondition, that is @Pm

@H 0: Then,

H pBt48

r16criss-cross element, during one wavelength crushing, was calcu-lated by the sum of membrane energy absorbed by two right-cor-ner angle element in the case of symmetric mode as follow,

Eccm 2Esymm rc 16M0H2

t13

2.1.3. The mean crushing force of multi-cell tubeFor the prediction of mean crushing force of multi-cell tube, the

theoretical expressions of mean crushing force of tube type I, II andIII were introduced in below formulas of section. The prole of tubetype I is formed by a combination among of the four right-corner,four T-shape and one criss-cross angle elements (Fig. 1(a)). Substi-tuting the terms in Eqs. (4), (6), (9), (13) into Eq. (1), the proposedexpression of mean crushing force of tube type I as follows:

PmI2Hg Etubeb 4Esymm rc 4ETshapem Eccm

2pM0B 2M0 H2

t16 24 8 14

Transforming from Eq. (14), a new alternative form was

PmIg pB H

ements of T-shape element and (c) 3-panel angle element.Substituting Eq. (16) into Eq. (15), the nal expression of themean crushing force for tube type I under quasi-static loading was

d (b) 4-panel angle element.

Additional panel

A

ele

ructures 119 (2015) 4224352HI

G C

D

A

B

EF

(a)

Fig. 6. (a) Collapse mode of 4-panel angle

426 T. Tran et al. / Composite StPmI pM0BgH M0Hgt

48 p0:5r0t1:5B0:548

p

2g17

The prole of tube type II was created by a combination of four3-panel angle element and one criss-cross angle elements(Fig. 1(b)). Substituting Eqs. (4), (10) and (13) into Eq. (1), the the-oretical expression of mean crushing force for tube type II wasobtained as

PmII2Hg Etubeb 4E3panelm E6panelm

2pM0B 2M0 H2

t16 16 tan/=2 18

An alternative form of Eq. (16) was

PmIIgM0

pBH H

t16 16 tan/=2 pB

H H

tG/ 19

By using the stationary condition of the mean crushing force,the half-wavelength was obtained as @Pm

@H 0. Then,

0 pBH2

G/t

) H pBtG/

s20

Substituting Eq. (20) back into Eq. (19), the mean crushing forcefor tube type II under quasi-static loading was

F

P045

2H

K

L

M

NO

Fig. 7. Collapse mode and extensional elements of criss-cross angle element.PmII pM0BgH M0Hgt

G/ p0:5r0t1:5B0:5G/p2g

21

where G(/) = 16 + 16tan(//2).The prole of tube type III included four right-corner and four

4-panel angle elements. According to the equilibrium energy ofsystem, the external work on the tube and the sum of dissipatedenergy in bending and in membrane of all angle elements mustbe an equal value. Substituting Eqs. (4), (6) and (12) into Eq. (1),obtain

PmIII2Hg Etubeb 4 Esymm rc E4panelm

2pM0B 2M0 H2

t32 16

cosb

22

From Eq. (22), we obtained

PmIIIgM0

pBH H

t32 16

cosb

pB

H H

tQb 23

The half-wave length was obtained by the stationary conditionof the mean crushing force (oPm/oH = 0). Then

0 pBH2

Qbt

) H pBtQb

s24

B

C

D E

FI, G

cosADI ACIS S =(b)

b

ment and (b) extensional elements [28].Substituting Eq. (24) into Eq. (23), the nal expression of meancrushing force for tube type III under quasi-static loading was

PmIII pM0BgH M0Hgt

Qb p0:5r0t1:5B0:5Qbp2g

25

where Qc; b 32 16cosb .

2.2. Optimization design methodology

In the process of crashworthiness optimization, the analyticalobjectives must be dened. Among all the crashworthiness indica-tors, the energy absorption (EA) was the most important one. It was

Rigid wall

v = 10 m/s Lumped Mass

L = 250 mm

a

Fig. 8. Schema of the computational model.

tructT. Tran et al. / Composite Sdetermined by the total strain energy absorbed during the plasticdeformation. Hence, the EA was calculated by using the curve ofcrushing forcedisplacement as

EA Z d0

Pxdx 26

where P(x) was instantaneous crushing force.Simultaneously, it was expected that the optimized multi-cell

structure was able to absorb as much strain energy as possible ina unit structural weight. Consequently, this crashworthiness indi-cator was dened as the specic energy absorption (SEA) [9] inEq. (27)

SEA EAm

27

where m was the total mass of the considered structure. The higherthe SEA was, the better the capability of energy absorption was.

Fig. 9. Deformation processures 119 (2015) 422435 427Continuously, the mean crushing force was another very crucialindicator of crashworthiness, which was dened as

Pm EAd 1d

Z d0

Pxdx 28

where dwas the crushing distance at a specic time. In addition, theinitial peak crushing force (PCF) was also utilized to estimate thestructural crashworthiness.

2.2.1. Response surface method (RSM)TheRSMwas considered as an efcient approximationmethod in

the multivariate optimization problems involving complex nonlin-ear mechanics such as contact-impact. The basic idea of RSM wasto express a complex function f(x) in terms of a series of simple basisfunctionui(x). The mathematical equations of RSM was written as

f x ~fx Xmi1

diuix 29

of three types of tube.

Fig. 10. The crushing forcedisplacement curve: (a) tube type I, (b) tube type II and (c) tube type III.

428 T. Tran et al. / Composite Structures 119 (2015) 422435

)tructTable 1Design matrix of three types of tubes for crashworthiness.

n t (mm) a (mm) Tube I

SEA (kJ/kg) PCF (kN

1 1 50 22.930 40.5542 1.4 50 27.286 64.9993 1.8 50 30.281 90.5344 2.2 50 31.677 115.1465 2.6 50 30.897 138.9586 1 55 18.948 42.3677 1.4 55 22.796 64.1388 1.8 55 26.795 92.7689 2.2 55 28.497 122.527

10 2.6 55 29.352 151.19611 1 60 19.770 45.25612 1.4 60 23.609 71.61213 1.8 60 26.222 101.61114 2.2 60 27.136 131.91215 2.6 60 26.322 159.937

T. Tran et al. / Composite Swhere f(x) and ~fx were the response surface approximation andthe numerical solution denoting for f(x), respectively. m representsthe total number of basic functions ui(x), and di was the unknowncoefcient. A typical class of basic functions was the polynomialswhose full linear form was given as

1; x1; x2; . . . ; xn 30and full quartic form was given as

1;x1;x2; . . . ;xn;x21;x1x2; . . . ;x1xn; . . . ;x2n;x

31;x

21x2; . . . ;x

21xn;x1x

22; . . . ;

x1x2n; . . . ;x3n;x

41;x

31x2; . . . ;x

31xn;x

21x

22; . . . ;x

21x

2n; . . . ;x1x

32; . . . ;x1x

3n; . . . ;x

4n 31

In the literature, the full quartic polynomial function provided abest approximation [1315]. Hence, the quartic response surfacemodel was therefore adopted in this study.

2.2.2. Multi-objective optimizationFrom the point of view of practicality, it was indicated that

these two objectives of SEA and peak crushing force (PCF) competeagainst each other strongly. It was apparent that no improvement

16 1 65 17.361 48.24817 1.4 65 20.847 72.72518 1.8 65 23.616 101.73119 2.2 65 24.786 135.76920 2.6 65 25.565 171.91121 1 70 17.530 50.75622 1.4 70 20.980 78.23223 1.8 70 23.040 110.53724 2.2 70 23.770 145.34625 2.6 70 23.467 178.408

Fig. 11. The response surface of (a) PTube II Tube III

SEA (kJ/kg) PCF (kN) SEA (kJ/kg) PCF (kN)

16.950 39.510 20.402 43.43219.595 60.837 24.142 66.31923.632 85.308 26.320 92.94025.480 111.988 27.570 121.10224.640 137.414 27.800 148.71314.489 43.188 18.410 47.07017.300 64.487 22.660 70.43418.809 88.003 24.605 98.55521.295 115.374 25.917 129.16322.950 146.232 25.903 160.11414.400 47.106 17.379 50.58416.364 69.463 21.019 75.49619.599 95.941 22.910 104.19321.052 125.110 23.636 136.50022.692 156.953 23.880 170.162

ures 119 (2015) 422435 429in one function could be achieved without deteriorating the otherfunction. In this paper, the multi-objective optimization design ofminimizing PCF and maximizing SEA was dened by using the lin-ear weighted average method (LWAM) [15].

The multi-objective optimization was expressed in terms of theLWAM as

Minimize Ft; a wPCFt;aPCF 1w SEA

SEAt;as:t w 2 0;1

1 6 t 6 2:6 mm50 6 a 6 70 mm

8>>>>>>>:

32

where SEA and PCF were the given normalizing values for eachcross-sectional prole.

2.2.3. Knee pointIn most cases, the designers can chose suitable schemes based

on what they need. However, the most preferred solution (termedas Knee point) must be subjected to certain designers in their

14.254 51.208 16.310 53.82616.085 74.753 20.290 80.72118.048 100.618 21.465 109.98020.952 131.965 22.475 143.90021.500 164.165 22.330 179.39714.056 55.270 15.770 56.95814.776 80.148 18.288 85.94016.590 107.093 20.073 116.18018.141 136.140 21.003 150.49819.423 169.304 21.158 188.223

eak crushing force and (b) SEA.

3. Numerical simulation and crashworthiness optimization

3.1. Numerical simulation

Dynamic nite element analyses were performed by usingANSYS/LS-DYNA to simulate three types of tube under axial crush-ing. The tubes were modeled with the BelytschkoTsay 4-nodeshell elements with three integration points through the thicknessand with one integration point in the element plane. The materialAA6060 T4 was modeled with material model #24 (Mat_Piece-wise_Linear_Plasticity) in ANSYS/LS-DYNA. Regarding the contactof surface, nodes to surface contact between the thin-walled tubeand rigid-wall was dened to simulate the real contact. Otherwise,

430 T. Tran et al. / Composite Structures 119 (2015) 422435work. For a big distance among the orders of magnitude of differ-ent objectives, an introduction for a modied multi-objective evo-lutionary algorithm was presented by Branke et al. [5] to nd outthe knee regions. Then, the approach was proposed by Deb andGupta [8] to identify knee point with the maximum bend-angleDeb, and Guptas method [8] was mathematically given as

Maximize hx; xL; xR hL hR 33

where hL arctan f 2xLf 2xf 1xf 1xL and hR arctanf 2xf 2xRf 1xRf 1x were the left

and right bend-angle of x.

Fig. 12. Pareto spaces for multi-objective optimization: (a) tube type I; (b) tubetype II and (c) tube type III.

Table 2Optimal results by using method of Deb and Gupta (knee points).

Type of cross-section Terms Optimal

Type I Approximate value t = 1.56,FE numerical valueRE

Type II Approximate value t = 1.64,FE numerical valueRE

Type III Approximate value t = 1.48,FE numerical valueREa single surface contact algorithm provided by ANSYS/LS-DYNAwas also utilized to consider the self-contact among the shell ele-ments. A coulomb friction coefcient of 0.3 among all surfaces incontact was used. A lumped mass of 400 kg was attached to oneend of the tube to impact on a rigid wall with an initial velocityof 10 m/s. Fig. 8 showed the schema of the computational model.

The tube was made of aluminum alloy AA6060 T4 withmechanical properties: Youngs modulus E = 68200 MPa, initialyield stress ry = 80 MPa, ultimate stress ru = 173 MPa, Poissonsration t = 0.3 and power law exponent n = 0.23. The engineeringstressstrain curve was also presented in literature [23]. Sincethe aluminum is insensitive to the strain rate effect, this effectwas neglected in the nite element modeling. For the tube struc-ture, the side-length (a) of the cross-sections and the thickness(t) were chosen for design variables.

The structures of tubes show that all of them were symmetricstructures. Thus, the tube type II and III have the same mass withthe values of thickness and side-length, while the mass of tube typeI was a smallest one. Fig. 9 shows deformation process of threetypes of tubes at different time points. The corresponding curvesof the crushing forcedisplacement for three types of tubes wereshown in Fig. 10. It also showed that the exact value of the effectivecrushing distance on the crushing forcedisplacement curve wassomehow not unique. Additionally, the effective crushing distancesof multi-cell tubes were equal to about 70% of the initial length.

3.2. Crashworthiness optimization

Regarding to response functions of SEA and PCF, a series of 25sampling points (based on a and t) were selected in the designspace to provide sampling designs for FEA. From the results inTable 1, the response surface of SEA and PCF were, respectively,established and described in Fig. 11(a) and (b). It showed thatthe SEAs and PCFs RS of tube type I, II and III cases behaved mono-tonically over the design domains.

By changing the weight w in Eq. (32), Pareto sets for three pro-les of tubes were obtained and were plotted as in Fig. 12. In fact,

design variables (mm) SEA (kJ/kN) PCF (kN)

a = 50 28.685 51.18728.561 51.410.434 0.434

a = 50 22.375 62.32522.264 61.970.499 0.573a = 50 24.667 55.44224.759 55.0030.372 0.798

value and RS approximate value for three types of tubes. Accord-ingly, the FE simulation value and RS approximate value at theKnee points were quite exactly similar. In addition, the curves inFig 13 illustrated the variation of SEA and Pm with changes inweight. Moreover, the tube types I and III were the better than tubetype II on the aspect of the energy absorption.

4. Theoretical validation and discussions

The theoretical solutions (17), (21) and (25) of mean crushingforce were created for three types of tubes under quasi-staticimpact. However, these expressions did not take the effect ofdynamic crushing into account. For the dynamic cases, thedynamic amplication effects consisting of inertia and strain rateones must be considered in these theoretical solutions. In fact,the aluminum alloy with No. AA6 series is not sensitive to thestrain rate [7]. A dynamic enhancing coefcient k was thus pro-posed to take the inertia effect into account (Alghamdi, 2001;Hsu, 2004; Hou et al., 2012). It was not simple to determine anaccurate value for the dynamic enhancing coefcient, and thiscoefcient kwas a variable used for different geometric parametersas described by Langseth et al. (1996, 1998), and Hanssen et al.[12]. According to these studies, this coefcient was proposed ina ranging of 1.31.6 for AA6060 T4 extruded tubes under axial

Fig. 13. (a) SEA vs structural weight and (b) Pm vs structural weight for tube.

T. Tran et al. / Composite Structures 119 (2015) 422435 431any point in the Pareto frontier could be an optimum, and a rangeof optimal solutions was supplied to the decision maker. That iswhy some methods were proposed to nd out the best solution(Knee point) which has a large trade-off value compared to otherPareto-optimal points. The results of expression (33) showed thatPareto solutions (Knee points) for tube type I, II and III were0.769, 0.791 and 0.783, respectively. These Knee points were alsoplotted in Fig. 12 for three types of tubes. Deriving from the resultsof Eq. (33), the optimal design variables of multi-cell square sec-tions for tube type I, II and III were simulated in axial impact load-

ing case. Table 2 presents the relative errors (REs) of FE simulation

Table 3Differences of numerical results and theoretical predictions for three types of tubes.

n Tube type I Tube type II

Num. Pm (kN) Theo. Pm (kN) Diff. (%) Num. Pm (kN) Th

1 25.067 25.594 2.10 21.294 22 40.864 42.281 3.47 35.336 33 63.876 61.473 3.76 55.525 54 86.419 82.836 4.15 75.737 75 106.922 106.132 0.74 96.616 96 26.122 26.859 2.82 22.580 27 43.620 44.383 1.75 36.797 38 64.977 64.545 0.67 57.525 59 87.702 86.998 0.80 80.348 7

10 112.871 111.494 1.22 99.356 911 27.167 28.068 3.32 23.353 212 44.724 46.390 3.73 38.767 413 67.374 67.478 0.15 58.695 514 92.557 90.970 1.72 83.826 715 116.763 116.609 0.13 106.129 1016 27.923 29.227 4.67 24.355 217 46.749 48.314 3.35 40.346 418 67.960 70.288 3.43 59.706 619 92.941 94.776 1.97 85.892 820 121.167 121.509 0.28 110.804 1021 29.289 30.341 3.59 25.292 222 48.114 50.164 4.26 41.781 423 70.588 72.990 3.40 61.934 624 96.001 98.435 2.53 87.952 825 123.456 126.220 2.24 114.382 10impact loading. These propositions were also consistent with theinvestigation of Tarigopula et al. [26]. For simplicity, these coef-cients were 1.6, 1.6, and 1.4 for tube type I, II and III, respectively.Accordingly, the theoretical solution for tube type I was applied as

Pdym:mI kIPmI kIp0:5r0t1:5B0:548

p

2g34

For tube type II, that was

Pdym:mII kIIPmII kIp0:5r0t1:5B0:5G/p2g

35

where G(/) = 16 + 16tan(//2).And for tube type III, that was

Tube type III

eo. Pm (kN) Diff. (%) Num. Pm (kN) Theo. Pm (kN) Diff. (%)

2.278 4.62 24.825 25.470 2.606.794 4.13 42.713 42.065 1.523.480 3.68 63.568 61.141 3.822.046 4.87 86.114 82.366 4.352.282 4.49 109.871 105.501 3.983.382 3.55 25.586 26.731 4.478.627 4.97 44.771 44.160 1.376.160 2.37 66.322 64.204 3.195.677 5.81 90.713 86.517 4.636.960 2.41 114.062 110.849 2.824.435 4.63 26.630 27.935 4.900.376 4.15 44.910 46.160 2.788.717 0.04 67.641 67.128 0.769.141 5.59 91.914 90.477 1.561.423 4.43 118.958 115.951 2.535.445 4.47 27.752 29.090 4.822.053 4.23 46.185 48.077 4.101.167 2.45 68.891 69.929 1.512.460 4.00 94.408 94.272 0.145.697 4.61 123.283 120.837 1.986.416 4.45 28.846 30.200 4.693.666 4.51 47.757 49.920 4.533.523 2.57 70.742 72.622 2.66

5.650 2.62 98.255 97.919 0.349.805 4.00 125.607 125.534 0.06

Fig. 14. Comparison among Num. predictions and Theo. predictions: (a) tube type I; (b) tube type II and (c) tube type III.

432 T. Tran et al. / Composite Structures 119 (2015) 422435

tructT. Tran et al. / Composite SPdym:mIII kIIIPmIII kIp0:5r0t1:5B0:5Qbp2g

36

where Qc;b 32 16cosb.In Eqs. (34)(36), r0 was the ow stress of material with power

law hardening which was approximated by an energy equivalentstress [24] as

Fig. 15. (a) Def. result and (b) crushing forced

Fig. 16. (a) Def. result and (b) crushing forced

Fig. 17. (a) Def. result and (b) crushing forcedures 119 (2015) 422435 433r0 ryru1 n

r37

where ry and ru denoted the yield strength and the ultimatestrength of the material, respectively; and n was the strain harden-ing exponent.

isplacement curve of optimal tube type I.

isplacement curve of optimal tube type II.

isplacement curve of optimal tube type III.

ructThe Eqs. (34)(36) were utilized to predict the mean crushingforces of tubes. Then, the values of mean crushing force for tubesat the 50% displacement were used to compare with the valuesobtained by using Eqs. (34)(36). Obviously, these mean crushingforces were dened as the equivalent constant force with a corre-sponding amount of displacement. The differences among numer-ical predictions and theoretical equations above for all cases werelisted in Table 3. In respect of tube type I, II, the differences of Eq.(34), (35) and FE results were, respectively, ranging from 4.15% to4.67% and from 4.61% to 4.62%. For tube type III, the deviationsbetween Eq. (36) and numeric results were a range from 4.63%to 4.82%. The results of those comparisons showed that these dif-ferences were in the available range. Accordingly, a very strongsupport was veried between the theoretical solutions and thenumerical results in these cases (as shown in Fig. 14).

From the optimal results in Table 2, the multi-cell square pro-les of three optimal tubes were considered in this analysis. Inregards to the optimal tube I, the deformation result and crushingforcedisplacement curves were shown in Fig. 15. The value ofmean crushing force obtained from FE analysis was 48.287 kN.The parameters of this prole was a = 50 mm, t = 1.56 mm andB = 293.76 mm. Substituting items into Eq. (34), the theoreticalprediction for optimal tube I was

Pdyn:mI 1:60:1061:561:5293:760:5481:4

49:678 kN 38

Fig. 16 represented the deformation result and the curves ofcrushing forcedisplacement for the optimal prole of tube II.Thus, the mean crushing force of optimal tube II was 44.868 kN.At the same time, the sum of side length and of internal web lengthB was of 333.221 mm. The side-length and the thickness were50 mm and 1.64 mm, respectively. Substituting items into Eq.(35), the theoretical prediction of mean crushing force was

Pdyn:mII 1:60:1061:641:5333:2210:5481:4

46:566 kN 39

The optimal prole of tube type III had 5 cells (as shown inFig. 17). The side-length and the thickness of this cross-sectionwere, respectively, 50 mm and 1.48 mm. In addition, the meancrushing force obtained from FE analysis was 45.233 kN. As a mat-ter of course, the parameter of prole of tube III wasB = 334.021 mm. Substituting items into Eq. (36), the theoreticalprediction of mean crushing force was

Pdyn:mI 1:40:1061:481:5334:2210:5481:4

45:694 kN 40

The differences between FE numerical value and Eqs. (38)(40)were, respectively, 2.88%; 3.78% and 1.019%. These differencesshowed a strong agreement between the proposed equations andthe numerical simulations. Additionally, the stable and progressivefolding deformation patterns developed for all three types of tubeswere the desirable energy-dissipating mechanism.

5. Conclusions

The cross-sections of tube type I, II and III were divided into sev-eral basic elements that were right corner, T-shape, 3-panel, criss-cross and 4-panel angle ones. Based on the SSFE theory, theoreticalexpressions of the mean crushing force for three types of tubeswere developed in this study. Numerical simulations of tube typeI, II and III under dynamic loading were also performed. Numericalresults showed that the stable and progressive collapses weredeveloped for tube type I, II and III. In addition, the tube type Iand II were the best and the worst structures in the aspect of

434 T. Tran et al. / Composite Stenergy absorption respectively. Meanwhile, tube III was the mostefcient structure in weight utilization.The specic energy absorption (SEA) and the peak crushing force(PCF) were dened to be the analytical objectives for the crashwor-thiness optimization design. The surrogated models of SEA and PCFwere constructed by using response surface method (RSM). Paretosets were obtained in the terms of the linear weighted averagemethod (LWAM). As the optimal solutions, Knee points were gotfrom the Pareto spaces of three tubes. The REs among RS approxi-mate values and FE numerical values at the Knee points wereacceptable. At the three knee points, the proposed equations wereused to validate the numerical solutions, and the theoretical solu-tions coincided very well with the numerical results.

Acknowledgments

The nancial supports from National Natural Science Founda-tion of China (Nos. 11232004, 11372106), New Century ExcellentTalents Program in University (NCET-12-0168) and Hunan Provin-cial Natural Science Foundation (12JJ7001) are gratefully acknowl-edged. Moreover, Joint Center for Intelligent New Energy Vehicle isalso gratefully acknowledged.

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T. Tran et al. / Composite Structures 119 (2015) 422435 435

Crushing analysis and numerical optimization of angle element structures under axial impact loading1 Introduction2 Theoretics2.1 Theoretical prediction of multi-cell square tube2.1.1 The bending energy2.1.2 The membrane energy2.1.2.1 The membrane energy of right corner element2.1.2.2 The membrane energy of 3-panel angle and T-shape element2.1.2.3 The membrane energy of 4-panel and criss-cross angle element

2.1.3 The mean crushing force of multi-cell tube

2.2 Optimization design methodology2.2.1 Response surface method (RSM)2.2.2 Multi-objective optimization2.2.3 Knee point

3 Numerical simulation and crashworthiness optimization3.1 Numerical simulation3.2 Crashworthiness optimization

4 Theoretical validation and discussions5 ConclusionsAcknowledgmentsReferences