Theoretical prediction and crashworthiness optimizationof multi-cell triangular tubes

TrongNhan Tran a,c, Shujuan Hou a,b, Xu Han a,b,n, Wei Tan a,b, NhatTan Nguyen a,d

a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, Hunan, PR Chinab College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, Hunan, PR Chinac Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, Go Vap District, HCM City, Vietnamd Center for Mechanical Engineering, Hanoi University of Industry, Tu Liem District, Ha Noi, Vietnam

a r t i c l e i n f o

Article history:Received 2 January 2014Received in revised form13 March 2014Accepted 26 March 2014Available online 17 May 2014

Keywords:CrashworthinessMultiobjective optimizationTriangular tubeMulti-cellEnergy absorptionImpact loading

a b s t r a c t

The triangular tubes with multi-cell were first studied on the aspects of theoretical prediction andcrashworthiness optimization design under the impact loading. The tubes' profiles were divided into2-, 3-, T-shapes, 4-, and 6-panel angle elements. The Simplified Super Folding Element theory wasutilized to estimate the energy dissipation of angle elements. Based on the estimation, theoreticalexpressions of the mean crushing force were developed for three types of tubes under dynamic loading.When taking the inertia effects into account, the dynamic enhancement coefficient was also considered.In the process of multiobjective crashworthiness optimization, Deb and Gupta method was utilized tofind out the knee points from the Pareto solutions space. Finally, the theoretical prediction showed anexcellent coincidence with the numerical optimal results, and also validated the efficiency of thecrashworthiness optimization design method based on surrogate models.

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1. Introduction

Thin-walled extrusions have been extensively applied in vehiclecrashworthiness components to absorb impact energy in the pastthree decades. The tests and the theoretical expressions of squareand circular tubes under axial quasi-static and dynamic loading caseswere first described by Wierzbicki and Abramowicz [1] and Abra-mowicz and Jones [2]. From then on, DiPaolo et al. [3,4], Guillow et al.[5], Ullah [6] Zhang and Zhang [7], Alavi Nia and Parsapour [8] alsodid many researches on these aspects. Beside square and circulartubes, several other profiles were also studied on their quasi-static ordynamic responses, such as triangular tubes [9–12], hexagonal tubes[13], etc. The structural collapse modes of triangular and square tubesare different from those of circular tubes. Nevertheless, the crushingcurves of force–displacement of triangular and square tubes aresimilar to those of circular tubes. The crushing curves of force–displacement of all the profiles show that the crushing force firstreaches an initial peak, then drops down and then fluctuates arounda value of the mean crushing force. The extensional deformation has

more dominant effect on the crushing responses while the quasi-inextensional mode occurs normally [14].

According to studies by Wierzbicki and Abramowicz [1], thenumber of “angle” elements on cross-section of tube decided,to a certain extent, the effectiveness of energy absorption. Asa matter of fact, it is necessary to design thin-walled multi-celltubes for weight-efficient energy absorption. Chen and Wierzbicki[15] examined the axial crushing resistance of single-cell, double-cell and triple-cell hollow tubes, and the respective foam-filledtubes under the quasi-static axial loading. The Simplified SuperFolding Element (SSFE) theory was applied to simplify SFE theory,and three extensional triangular elements and three stationaryhinge lines were comprised instead of the kinematically admissiblemodel of SFE [1]. The average folding wavelength and the theore-tical expression of the mean crushing force were deduced bydividing the cross-sectional tube into distinct panel section andangle element, assuming that the roles of each panel and of angleelement were at the same level. The work of Chen and Wierzbicki[15] showed that the multi-cell tube could increase the specificenergy absorption SEA by approximately 15%, compared to therespective hollow tube. Kim [16] used Chen and Wierzbicki's model[15] to study multi-cell tubes with four square elements at thecorner. The SEA of new multi-cell tube was reported to increase by190%, compared to conventional square tube. Zhang et al. [17] alsoapplied SSFE theory to derive a theoretical expression of the mean

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Thin-Walled Structures

http://dx.doi.org/10.1016/j.tws.2014.03.0190263-8231/Crown Copyright & 2014 Published by Elsevier Ltd. All rights reserved.

n Corresponding author 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).

Thin-Walled Structures 82 (2014) 183–195

crush force of multi-cell square tubes under the dynamic impactloading. In Zhang's work, the cross-section of tube was divided intothree basic angle elements, and the study also came to thecontribution that plastic energy of each element type was dis-sipated through membrane action. It was assumed from thetheoretical expression that the average wavelength for the dissim-ilar folds developed at corners. Thereafter, the SSFE theory was alsoadopted by Zhang et al. [18] to predict the mean crushing force of 3-panel angle element. At the same time, the SFE theory wasextended by Najafi and Rais-Rohani [19] to explore the crushingcharacteristics of multi-cell tubes with two different types of three-panel elements. A closed form expression of mean crushing forcewas also put forward by Najafi and Rais-Rohani [19].

Dynamic progressive buckling of thin-walled multi-cell tubesunder axial impact loadings was studied by Jensen et al. [20] andKaragiozova and Jone [21]. Then, the structural dynamic progres-sive buckling under the axial loading was summarized by Kar-agiozova and Alves [22] from a phenomenological point of view.Consequently, the desirable energy-dissipating mechanism wasa stable and progressive folding deformation pattern for thestructural deformation. On the other hand, the global bending ona structure was an undesirable energy-dissipating mechanismmode. At the beginning, multi-cell tubes were mostly employedfrom the aspects of theoretical researches, such as by Kim [16] andNajafi and Rais-Rohani [19], Nowadays, either FE solutions [23] orsurrogate models [13,24–26] developed appeared in the searchfield of multi-cell tubes under the impact loading. However, thereis seldom a combination study of theory, numeric and optimalmethod for thin-walled multi-cell tubes.

Above all, the axial crushing of tube types I, II, III was studiedon both theoretical prediction and numerical optimization designin this paper. Based on the SSFE theory, theoretical expressions ofthe mean crushing force for the three types were derived. All theprofiles studies in this paper were divided into 2-, 3-, T-shape,4 and 6-panel angle elements. In order to obtain the optimalprofiles under the crashworthiness criterion, dynamic finiteelement analysis code ANSYS/LS-DYNA was executed to simulatetubes and to obtain the numerical results at the design samplingpoints. The multiobjective optimization design was utilized toobtain the optimal configurations. Finally, the theoretical expres-sions are employed to validate the numerical optimal solutions.

2. Theoretics

2.1. Theoretical prediction of multi-cell triangular tube

The SSFE theory was applied to solve the axial collapse oftriangular multi-cell thin-walled tubes. In the SSFE theory, the wallthickness was assumed to be constant and the variation of

wavelength 2H for different lobes was ignored. To analyze energydissipation over the collapse of a fold, the triangular multi-cellthin-walled tubes were divided into several basic elements: the 2-,3-, 4- and 6-panel angle element as shown in Fig. 1.

Based on the principle of global equilibrium for shells, theinternal and external energy dissipations are of equal rate ð _Eext ¼_EintÞ. The external energy work for a complete single fold is equal tothe sum of dissipated bending and membrane energy. That is

Pm2H ¼ 1ηðEbþEmÞ ð1Þ

where Pm, 2H, Eb and Em respectively denote the mean crushingforce, the length of the fold, the bending energy and the membraneenergy, and η is the effective crushing distance coefficient. The panelof folding element after deformation is not completely flattened asshown in Fig. 2. Hence, the available crushing displacement issmaller than 2H. In this study, the value of ηwas taken as 0.7 since itwas found between 0.7 and 0.8 [1].

2.1.1. The bending energy of tubeThe SSFE theory was applied to calculate the dissipated energy

in bending of each panel. Only three extensional and compres-sional triangular elements and three stationary hinge lines wereused in SSFE theory, which was different from SFE theory. In SFEtheory, a model was built with trapezoidal, toroidal, conical andcylindrical surfaces of moving hinge lines.

In the work of Chen and Wierzbicki [15], the energy dissipationfor bending of each panel Eb was estimated by summing up theenergy dissipation at three stationary hinge lines. Then

Efb ¼ ∑4

i ¼ 1M0αib ð2Þ

where M0 ¼ s0t2=4 is the fully plastic bending moment of thepanel, b is the sectional breadth and α denotes the rotation angleat the stationary hinge lines.

In this case, the panel was supposed to completely flatten afterthe deformation of the wavelength 2H. Consequently, the fourrotation angles α at three stationary hinge lines were π/2 one byone (as shown in Fig. 2). By applying Eq. (2), the bending energy at

Fig. 1. Cross-sectional geometry of triangular multi-cell tube and typical angle element. (a) Tube type I (b) Tube type II and (c) Tube type III.

Fig. 2. Bending hinge lines and rotation angles on basic folding.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195184

stationary hinge lines of panel is obtained as

Efb ¼ 2πM0b ð3ÞSince the role of structural each panel is similar and the multi-

cell tube is constituted bym panels (as shown in Fig. 1), the energydissipation for bending of multi-cell tube is inferred as

Etubeb ¼ 2πM0mb¼ 2πM0B ð4Þwhere B is the sum of side and internal web lengths.

2.1.2. The membrane energy of angle element2.1.2.1. The membrane energy of 2-panel, 3-panel and T-shape angleelement. The basic folding element (BFE) was created by using thetriangular elements and the stationary hinge lines (Fig. 3) so as tocalculate the membrane energy of right-corner through asymmetricor symmetric deformation mode in the SSFE theory. Two possiblecollapse modes of the asymmetric and symmetric deformation weresupposed in the establishment of BFE. The symmetric and asymmetricmodes came from the extensional and quasi-inextensional modesrespectively. As for the asymmetric mode, the three triangularelements were developed for each web after the deformation.However, the symmetric mode had two triangular elements for eachpanel after the deformation. Thus for the asymmetric mode duringone wavelength crushing, the energy dissipation in membrane Em ofeach panel was evaluated by integrating the area of triangularelements (shaded areas in Fig. 3a). Then

Easymm� f ¼Zss0tds¼

12s0tH

2 ¼ 2M0H2

tð5Þ

Each panel was assumed to have the similar contribution. For theasymmetric mode, the dissipated membrane energy of the rightcorner element is double of that in one single panel. Then

Easymm_r� c ¼ 2Efm ¼ 4M0H2

tð6Þ

For the symmetric mode during one wavelength crushing, thedissipated energy in membrane of each panel was estimated byintegrating the area of triangular elements (shaded areas in Fig. 3b).Then

Esymm� f ¼Zss0tds¼ s0tH

2 ¼ 4M0H2

tð7Þ

The energy dissipation in membrane of right corner elementfor the symmetric mode is determined as

Esymm_r� c ¼ 2Esymm_f ¼ 8M0H2

tð8Þ

Concerning hollow tubes such as triangular, square, pentagonaland hexagonal, the cross-section profiles were formed by 2-panelangle elements. The crushing forces of hollow tubes were strength-ened in turn from triangular tube to hexagonal tube. When thecentral angle φ varies from 301 to 1201, the crushing force of 2-panelangle elements has an appropriate increase [27]. Eq. (1) shows thatthe mean crushing force is a function of bending and membraneenergy. Because the role of each structural panel is assumed to besimilar, the bending energy of angle elements with the samenumbers of panel will not change. Then, the mean crushing forcewill vary with respect to the membrane energy. Thus, the membraneenergy of right corner element is bigger than that of 2-panel angleelement with central angle of 601. From Fig. 4a, the membraneenergy of a 2-panel angle, during one wavelength crushing, iscalculated as

Eind:2�panelm ¼ Easymm_r� c cos γ ¼ 4M0

H2

tcos γ ð9Þ

The energy dissipation in membrane of 3-panel angle elementwas calculated by Zhang and Zhang [18]. Then, the membrane

Fig. 3. Basic folding element: (a) asymmetric mode [15] and (b) symmetric (extensional) mode.

Fig. 4. (a) Relationship between the membrane energy of right corner and of 2-panel angle element and (b) 3-panel angle element.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195 185

energy of 3-panel angle element, during one wavelength crushing,is

E3�panelm ¼ 4M0

H2

tð1þ2 tan ðϕ=2ÞÞ ð10Þ

T-shape element was created by one right-corner element andone additional panel as shown in Fig. 5. For this reason, themembrane energy of T-shape element was calculated by a sumbetween membrane energy of right-corner element for the sym-metric mode and membrane energy of one additional panel.Because the contribution of each panel is assumed to be similar,the energy dissipated in membrane of T-shape element, duringone wavelength crushing, is more than triple each panel's mem-brane energy. Then

ET� shapem ¼ 3Esymm� f ¼ 12M0

H2

tð11Þ

2.1.2.2. The membrane energy of 4-panel angle element. The 4-panelangle element is a symmetric structure and is constituted by one2-panel angle element and two additional panels. Therefore, theenergy absorption in membrane of a 4-panel angle element wascalculated by a sum of membrane energy of 2-panel angle elementand membrane energy of two additional panels. Due to similardeformation mode shown in Fig. 6, it is assumed that 2-panelangle element and the corresponding 2-panel angle elementin the 4-panel angle element are of equal crushing resistance.Simultaneously, the deformation mode of 2-panel angle element isa symmetric mode in this case. For this reason, the dissipated

membrane energy of 4-panel angle element was estimated bysumming up the membrane energy of 2-panel angle element forsymmetric mode and the membrane energy of two additionalpanels.

It is not easy to give a precise calculation for membrane energyof two additional panels. In this case, it is too complicated for theSFE theory to be applied here. So, a simplified deformation modelof two additional panels was proposed and the SSFE theory wasutilized to solve this problem. Fig. 7b shows the membranetriangular elements developed at the corner line. Accordingly,the membrane energy Em of 4-panel angle element, during onewavelength crushing, is evaluated by integrating the triangularareas as

E4 panelm ¼ 8M0

H2

t1þ 1

cos β

� �ð12Þ

2.1.2.3. The membrane energy of 6-panel angle element. Asa symmetric structure and being composed of six panels, thedissipated energy in membrane of a 6-panel angle element wascalculated by the sum of membrane energy absorbed by all6 panels. Because of the symmetric structure and the similarcontribution of each angle element, the 6-panel angle element wasformed by two 4-panel angle elements (Fig. 8a). As a result, themembrane energy of 6-panel angle element, during onewavelength crushing, was estimated by the sum of membraneenergy absorbed by two 4-panel angle elements. This is

E6�panelm ¼ 2E4panelm ¼ 16M0

H2

t1þ 1

cos β

� �ð13Þ

Fig. 6. (a) 2-panel angle element and (b) 4-panel angle element.

Fig. 5. (a) Collapse mode of T-shape element and (b) extensional elements.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195186

2.1.3. The mean crushing force in quasi-static caseTo build a quantitative prediction of crushing resistance of

triangular multi-cell tubes and to find out how angle elementaffects the structure of tube, the theoretical solution of the meancrushing force of multi-cell tube was introduced. Fig. 1a shows thethree independent 2-panel angle elements and three 4-panelangle elements form profile of tube type I. Substituting the termsof Eqs. (4), (9) and (12) into Eq. (1), the general theoreticalequation of the mean crushing force of tube type I was obtained.This is

Pm� I2Hη¼ Etubeb þ3ðEind:2�panelm þE4�panel

m Þ

¼ 2πM0Bþ2M0H2

t6 cos γþ12 1þ 1

cos β

� �� �: ð14Þ

Transforming Eq. (14), we obtain

Pm� IηM0

¼ πBH

þHt

6 cos γþ12 1þ 1cos β

� �� �¼ πB

HþH

tFðγ;βÞ ð15Þ

The half-wavelength was obtained under the stationary condi-tion of the mean crushing force ∂Pm=∂H ¼ 0, then

0¼ �πBH2þ

Fðγ;βÞt

) H ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiπBt

Fðγ;βÞ

sð16Þ

Substituting Eq. (16) back into Eq. (15), the mean crushing force inquasi-static loading of tube type I is obtained as

Pm� I ¼πM0BηH

þM0Hηt

Fðγ;βÞ ¼ π0:5s0t1:5B0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiFðγ;βÞ

p2η

ð17Þ

where

Fðγ;βÞ ¼ 6 cos γþ12 1þ 1cos β

� �:

As presented in Fig. 1b, the profile of tube type II wasconstituted by three 3-panel angle elements, three T-shape angle

Fig. 7. (a) Collapse mode of 4-panel angle element and (b) extensional elements.

Fig. 8. (a) Collapse mode of 6-panel angle element and (b) extensional elements.

Rigid wall

v = 10 m/sLumped Mass

L0 = 250 mm

a

Fig. 9. Schematic of the computational model.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195 187

elements and one 6-panel angle element. Substituting the termsof Eqs. (4), (10), (11) and (13) into Eq. (1), the general theoreticalexpression of the mean crushing force of tube type II is obtainedas

Pm� II2Hη¼ Etubeb þ3ðE3�panelm þET� shape

m ÞþE6�panelm

¼ 2πM0Bþ2M0H2

t32þ12 tan ðϕ=2Þþ 8

cos β

� �ð18Þ

An alternative form of Eq. (18) is

Pm� IIηM0

¼ πBH

þHt

32þ12 tan ðϕ=2Þþ 8cos β

� �¼ πB

HþH

tGðϕ;βÞ ð19Þ

By using the stationary condition of the mean crushing force,the half-wavelength can be obtained as ∂Pm=∂H ¼ 0. Then

0¼ �πBH2þ

Gðϕ;βÞt

) H¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπBt

Gðϕ;βÞ

sð20Þ

Fig. 10. Deformation process of three tubes.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195188

Substituting Eq. (20) back into Eq. (19), the equation of the meancrushing force for tube type II under the quasi-static loading isobtained as

Pm� II ¼πM0BηH

þM0Hηt

Gðϕ;βÞ ¼ π0:5s0t1:5B0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGðϕ;βÞ

p2η

ð21Þ

where

Gðϕ;βÞ ¼ 32þ12 tan ðϕ=2Þþ 8cos β

:

The structure of the tube type III (Fig. 1c) was formed by threeindependent 2-panel angle elements, six 4-panel angle elements,

and one 6-panel angle element. Substituting the terms of Eqs. (4),(9), (12) and (13) into Eq. (1), the general theoretical solution tothe mean crushing force of tube type III is obtained as

Pm� III2Hη¼ Etubeb þ3E2�panelm þ6E4�panel

m þE6�panelm

¼ 2πM0Bþ2M0H2

t32þ6 cos γþ 32

cos β

� �ð22Þ

Transforming Eq. (22), we obtain

Pm� IIIηM0

¼ πBH

þHt

32þ6 cos γþ 32cos β

� �¼ πB

HþH

tQ ðγ;βÞ ð23Þ

By using the stationary condition of the mean crushing force,the half-wavelength is expressed as ∂Pm=∂H ¼ 0. Then

0¼ �πBH2þ

Q ðγ;βÞt

) H ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπBt

Q ðγ;βÞ

sð24Þ

Substituting Eq. (24) into Eq. (23), the mean crushing force of tubetype III under quasi-static loading is

Pm� III ¼πM0BηH

þM0Hηt

Q ðγ;βÞ ¼ π0:5s0t1:5B0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ ðγ;βÞ

p2η

ð25Þ

where

Q ðγ;βÞ ¼ 32þ6 cos γþ 32cos β

:

2.2. Optimization design methodology

Among all the indicators of crashworthiness optimizationdesign, the vital analytical objective was the energy-absorption.Hence, in order to estimate the energy absorption of structuralunit mass m, specific energy absorption (SEA) was formulated as

SEA¼ EAm

ð26Þ

In fact, a higher SEA indicates a better capability of energyabsorption. In Eq. (26), the total strain energy during crushing is

Fig. 11. The crushing fore–displacement curve of (a) tube I, (b) tube II and(c) tube III.

Table 1Design matrix of three types of tube for crashworthiness.

n t (mm) a (mm) Tube type I Tube type II Tube type III

SEA(kJ/kg)

PCF (kN) SEA(kJ/kg)

PCF (kN) SEA(kJ/kg)

PCF (kN)

1 1 80 18.262 39.362 17.356 49.572 23.339 52.5922 1.3 80 20.226 52.274 19.876 65.965 25.227 69.5993 1.6 80 22.774 65.287 21.86 82.467 28.199 85.8014 1.9 80 23.854 77.82 22.003 98.125 29.092 101.1355 2.2 80 24.732 88.936 23.107 111.611 29.644 114.5416 1 85 17.635 42.151 16.842 53.133 22.753 56.0387 1.3 85 19.491 55.941 19.272 70.384 24.273 74.2028 1.6 85 21.079 69.78 20.924 88.192 26.388 92.4629 1.9 85 22.231 82.904 21.602 104.633 27.564 109.979

10 2.2 85 23.891 94.376 22.693 118.899 28.107 125.57211 1 90 16.689 48.183 16.559 60.979 21.338 64.54412 1.3 90 18.63 63.406 18.9 69.095 24.05 85.47813 1.6 90 19.883 80.385 20.273 99.84 25.536 107.64114 1.9 90 21.425 97.058 20.679 120.552 26.272 128.51215 2.2 90 22.916 111.899 21.524 138.568 26.42 146.4816 1 95 16.092 50.984 15.253 65.978 20.932 68.19617 1.3 95 17.793 67.207 16.954 83.194 22.371 90.66518 1.6 95 19.279 85.237 18.229 105.95 24.579 114.11219 1.9 95 20.865 102.754 19.146 128.907 25.306 135.85620 2.2 95 22.276 117.81 19.71 148.295 25.948 153.82521 1 100 15.569 50.155 14.398 63.148 19.337 66.64922 1.3 100 17.083 66.443 16.139 83.857 20.011 87.85223 1.6 100 19.26 83.029 17.748 104.85 22.979 110.01624 1.9 100 20.628 98.954 18.55 124.628 23.92 130.95125 2.2 100 21.053 113.035 18.715 142.093 24.152 157.552

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195 189

estimated as

EA ¼Z d

0PðxÞdx ð27Þ

where P(x) is the instantaneous crushing force. In addition, theinitial peak crushing force (PCF) of multi-cell thin-walled tube wasused for estimating the impact characteristics. Another crash-worthiness indicator is the mean crushing force Pm which iscomputed by

Pm ¼ EAd¼ 1d

Z d

0PðxÞdx ð28Þ

where d is the crushing displacement at a specific time.

2.2.1. Response Surface Method (RSM)A typical surrogate modelling technique was considered appro-

priate in the multivariate optimization process involving material,geometrical nonlinearities and contact-impact loading nonlinea-rities. The primary concept of RSM was applied to the constructionof regression functions for crashworthiness indicators by using thefunction values at the design sampling points. The mathematical

expression of RSM is expressed as

yðxÞ � ~yðxÞ ¼ ∑m

i ¼ 1βiψ iðxÞ ð29Þ

where ~yðxÞ and y(x) are respectively the surrogate surface approx-imation and the numerical solution denoting y(x). m representsthe total number of basic functions ψi(x), and βi is the unknown

Fig. 12. The response surface of (a) peak crushing force and (b) SEA.

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

Fig. 14. Pareto spaces for multi-objective optimization: (a) tube type I, (b) tubetype II and (c) tube type III.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195190

coefficient. Taking n dimensional problem for example, the fulllinear polynomial basis function is

1; x1; x2;…; xn ð30Þand the full quartic polynomial basis function is expressed 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

ð31Þ

The full quartic polynomial basis function was proved to bea better choice for the regression analysis [13,24–26]. The quarticresponse surface models were consequently adopted in this study.

2.2.2. Multi-objective optimizationWith two objectives of SEA and PCF, the multiobjective opti-

mization problem for minimizing PCF and maximizing SEA wasdefined by the linear weighted average methods (LWAM) [24].Then, the mathematical definition for the crashworthiness opti-mization in terms of the LWAM is given as

Minimize Fðt; aÞ ¼wPCFðt;aÞPCFn þð1�wÞ SEAn

SEAðt;aÞs:t wA ½0;1�1rtr2:2 mm80rar100 mm

8>>>><>>>>:

ð32Þ

where SEAn and PCFn are the given normalizing values for eachcross-sectional profile.

2.2.3. Knee pointIn some certain cases, the designer must choose the most

preferred solution (termed as “knee point”) from optimal solutionsto meet their requirement. Several methods were proposed todetermine a “knee point” from Pareto set such as Turevsky andSuresh [28] and Sun et al. [29]. However, if there is a great deviationamong the orders of magnitude of different objectives, thesemethods [29] seem to be less effective. A modified multi-objectiveevolutionary algorithm suggested by Branke et al. [30] was utilized toseek the knee regions. Deb and Gupta [31] have recently suggested asolution to find a “knee point” with maximum bend-angle, which is

mathematically given as

Maximize θðx; xL; xRÞ ¼ θL�θR ð33Þwhere θL ¼ arctanf 2ðxLÞ� f 2ðxÞ=f 1ðxÞ� f 1ðxLÞ and θR ¼ arctanf 2ðxÞ�f 2ðxRÞ=f 1ðxRÞ� f 1ðxÞ are the left and right bend-angles of x.

3. Numerical simulation and crashworthiness optimization

3.1. Numeric simulation

In this section, the FE model was carried out by ANSYS/LS-DYNAto simulate the triangular multi-cell thin-walled tubes subjected toaxial dynamic loading with 4, 6 and 9 cells (as shown in Fig. 1). Theside-length a of the cross-sections and the thickness t were chosento be design variables, and the design interval is given in Eq. (32).The total length L0 of all the tubes is 250 mm.

In this study, the thin-walled tubes were modelled with theBelytschko–Tsay four-node shell element with the optimal meshdensity of 2.5�2.5 mm. The material AA6060 T4 was modelledwith material model #24 (Mat_Piecewise_Linear_Plasticity) withmechanical properties: Young's modulus E¼68,200 MPa, initialyield stress sy¼80 MPa, ultimate stress su¼173 MPa, Poisson'sration υ¼0.3, and power law exponent n¼0.23 [32]. Since thealuminum was insensitive to the strain rate effect, this effect wasneglected in the finite element analysis. An automatic node tosurface contact between thin-walled tube and rigid wall wasdefined to simulate the real contact. Alternatively, an automaticsingle surface contact algorithm was utilized for the self-contactamong the shell elements to avoid interpenetration of foldinggenerated during the axial collapse. In the contact definition,a friction coefficient of 0.3 among all surfaces was employed. Togenerate enough kinetic energy, one end of tube was attachedwith a lumped mass of 500 kg whereas another end impacted ontoa rigid wall with an initial velocity of 10 m/s. The schematic of thecomputational model is shown in Fig. 9.

All of the tubes were axial symmetric structures. Despite thesame length, same side-length and same thickness, the three tubes

Table 2Difference of numeric result and theoretical prediction for three tubes.

n Tube type I Tube type II Tube type III

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

1 22.23 23.016 3.54 25.894 27.148 4.84 39.973 41.221 3.122 34.107 34.028 �0.23 41.1 40.143 �2.33 60.35 60.954 1.003 46.497 46.344 �0.33 55.331 54.680 �1.18 85.345 83.028 �2.724 61.379 59.817 �2.54 72.1 70.589 �2.10 111.642 107.183 �3.995 77.295 74.337 �3.83 91.64 87.738 �4.26 139.519 133.223 �4.516 23.173 23.736 2.43 26.75 27.996 4.66 41.129 42.510 3.367 34.142 35.098 2.80 42.39 41.404 �2.33 61.05 62.868 2.988 47.275 47.809 1.13 58.3 56.406 �3.25 87.9 85.648 �2.569 62.465 61.717 �1.20 74.137 72.827 �1.77 113.899 110.581 �2.91

10 79.196 76.710 �3.14 94.125 90.533 �3.82 144.17 137.467 �4.6511 23.787 24.435 2.73 27.482 28.820 4.87 42.716 43.760 2.4412 35.095 36.137 2.97 43.184 42.627 �1.29 62.715 64.726 3.2113 48.408 49.230 1.70 59.954 58.080 �3.13 89.541 88.190 �1.5114 64.341 63.561 �1.21 76.129 74.998 �1.49 117.161 113.878 �2.8015 81.044 79.013 �2.51 95.039 93.244 �1.89 148.092 141.583 �4.3916 24.151 25.115 3.99 28.264 29.620 4.80 43.862 44.976 2.5417 35.645 37.146 4.21 43.859 43.816 �0.10 63.768 66.531 4.3318 49.676 50.611 1.88 60.278 59.707 �0.95 91.482 90.660 �0.9019 66.14 65.352 �1.19 77.219 77.108 �0.14 120.81 117.082 �3.0920 82.19 81.250 �1.14 95.897 95.879 �0.02 151.03 145.584 �3.6121 24.819 25.776 3.86 29.109 30.400 4.44 45.268 46.160 1.9722 36.812 38.129 3.58 44.984 44.974 �0.02 65.507 68.290 4.2523 51.437 51.956 1.01 60.994 61.291 0.49 93.192 93.066 �0.1424 67.423 67.096 �0.49 79.107 79.162 0.07 122.495 120.201 �1.8725 83.731 83.427 �0.36 96.597 98.443 1.91 153.514 149.477 �2.63

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195 191

were different in weight. Tube I is the lightest one while tube III isthe heaviest. The axial crushing of multi-cell tubes was presentedwith a displacement equal to about 70% of the initial length. Fig. 10shows the deformation process of three tubes at different times.Sometimes the exact value of the effective crushing distance onthe crushing force–displacement curve was not unique. Thecorresponding crushing force–displacement curves of three tubesare also shown in Fig. 11. After reaching the initial peak and beforerising steeply whenever the deformation capacity is exhausted atthe effective crushing distance, the crushing force fell sharply andthen fluctuated periodically and around the values of the meancrushing force in correspondence with the formation, and finallycompleted the collapse of folds one by one.

3.2. Crashworthiness optimization

For obtaining the response functions of SEA and PCF, a series of25 design sampling points (based on a and t) were selected in thedesign space to provide sampling design values for FEA andregression analysis of three types of tubes (Table 1) so as to obtainthe response surface of the SEA and PCF. Fig. 12 shows that theSEA's and PCF's RS of tube types I, II and III cases behavemonotonically over the design domain. In addition, the curves inFig. 13 illustrate the variation of SEA and Pm with changes inweight. Meanwhile, energy absorptions of the tube types I and IIIwere better than those of tube type II.

The Pareto sets for these three cross-sectional profiles wereobtained by changing the weight coefficient w in Eq. (32), and thePareto frontiers are plotted in Fig. 13. In fact, any point on thePareto frontier can be an optimum. As a result, some methodswere proposed to determine the best solution (knee point) whichhas a large trade-off value in comparison with other Pareto-optimal points. In this case, methods of [31,32] were utilized todetermine the knee region and the knee point, respectively. Theresults of expression (33) showed that Pareto solutions (Kneepoints) for tube types I, II and III were 0.7924, 0.7818 and 0.7773,respectively. The relative errors (REs) of FE simulation value andRS approximate value are summarized in Table 3. Therefore the FEsimulation value and RS approximate value at the Knee pointswere exactly close to each other. According to the relationshipamong the weighted average method and those of [31,32], theseoptimal results are plotted in Fig. 14.

4. Theoretical validation and discussion

The theoretical expressions of the mean crushing force of tubetypes I, II and III are proposed in Section 2.1.3. Nevertheless, theseexpressions were applied in axial quasi-static loading case inwhich the effect of dynamic crushing was not considered. In thedynamic loading case, dynamic amplification effects, consisting ofinertia and strain rate effects, were considered in the theoreticalequations above. In reality, the aluminum alloy is insensitive to thestrain rate effect that can be neglected. A dynamic enhancingcoefficient λ was suggested to take the inertia effect into account.It is not easy to find the accurate value for dynamic enhancingcoefficient, and this coefficient λ is even a variable for differentgeometric parameters as described by Langseth and Hopperstad[33] and Hanssen et al. [34]. According to these studies, thecoefficient λ was proposed in the range of 1.3–1.6 for AA6060 T4extruded tubes. These coefficients (λ) were simply set to 1.41,1.3 and 1.45 for tube types I, II and III, respectively. Thus, thetheoretical solution to tube type I is applied as follows:

Pdym:m� I ¼ λIPm� I ¼ λIπ0:5s0t1:5B

0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiFðγ;βÞ

p2η

ð34Þ

where

Fðγ;βÞ ¼ 6 cos γþ12 1þ 1cos β

� �

for tube type II

Pdym:m� II ¼ λIIPm� II ¼ λIIπ0:5s0t1:5B

0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGðϕ;βÞ

p2η

ð35Þ

where

Gðϕ;βÞ ¼ 32þ12 tan ðϕ=2Þþ 8cos β

:Fig. 15. Comparison between numerical prediction and theoretical prediction:(a) tube type I, (b) tube type II and (c) tube type III.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195192

For tube type III

Pdym:m� III ¼ λIIIPm� III ¼ λIIIπ0:5s0t1:5B

0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ ðγ;βÞ

p2η

ð36Þ

where

Q ðγ;βÞ ¼ 32þ6 cos γþ 32cos β

:

In Eqs. (34)–(36), s0 is a flow stress of material with power lawhardening, which is calculated [35] as

s0 ¼ffiffiffiffiffiffiffiffiffiffiffisysu

1þn

r¼ 0:106 ðGPaÞ ð37Þ

where sy and su denotes the yield strength and the ultimatestrength of the material, respectively, and n is the strain hardeningexponent.

Hence, the mean crushing forces were calculated by expres-sions (34)–(36) and these values were used to compare with thevalue of mean crushing force of tubes at a displacement of 60% ineach different case. Remarkably, this mean crushing force isdefined as an equivalent constant force with a correspondingamount of displacement. The deviations among theoretical equa-tions above and numerical predictions for all cases of three typesof tubes are listed in Table 2. For tube types I and II, the differencesof Eqs. (34) and (35) and numeric results were, respectively,ranging from �3.83% to 4.21% and from �4.26% to 4.84%. Withregard to tube type III, the dissimilarities between Eq. (36) andnumeric results varied in the range of �4.65% to 4.33%. Obviously,these differences belong to available range (Table 2). Continuously,Fig. 15 reveals the very close agreement between the theoreticalsolutions and the numerical predictions for all cases.

Fig. 16. (a) Deformation result and (b) crushing force–displacement curve of tube I.

Table 3Optimal results by using the method of Deb and Gupta (Knee point).

Type ofcross-section

Terms Optimal designvariables (mm)

SEA(kJ/kN)

PCF(kN)

Type I Approximate value t¼1.23, a¼80 19.897 49.315FE analysis value 19.728 49.088RE 0.856 0.462

Type II Approximate value t¼1.25, a¼80 25.282 63.436FE analysis value 25.717 63.056RE �1.691 0.603

Type III Approximate value t¼1.21, a¼80 25.100 63.967FE analysis value 24.800 64.267RE 1.210 �0.467

Fig. 17. (a) Deformation result and (b) crushing force–displacement curve of tube II.

T. Tran et al. / Thin-Walled Structures 82 (2014) 183–195 193

Deriving from the optimal results of the method of Deb andGupta (Table 3), the optimal triangular sections of tube types I, IIand III were considered in this analysis. The deformation resultsand crushing force–displacement curves of three tubes are pre-sented in Figs. 16–18. For the optimal tube type I with 4 cells inTable 3, the side-length and the wall thickness were of 80 mm and1.23 mm, respectively (as shown in Fig. 16). The mean crushingforce obtained from FE analysis is 30.43 kN. Visibly, with thisprofile, the sum of side-length and of internal web length B is of352.62 mm. Substituting items into Eq. (34), the theoretical pre-diction of mean crushing force is

Pdym:m� I ¼ 1:41� π0:5 � 0:106� 1:231:5

�352:620:5 6:462� 0:7

¼ 31:335 ðkNÞ ð38Þ

With 6 cells, the optimal tube type II in Table 3 has the side-length of 80 mm and the wall thickness of 1.25 mm (Fig. 17). Themean crushing force received from FE analysis was 37.01 kN.Simultaneously, for this optimal tube, the sum of side-length andinternal web length B is 439.06 mm. To substitute items intoEq. (35), the theoretical prediction of mean crushing force is

Pdym:m� II ¼ 1:3� π0:5 � 0:106� 1:421:5

�439:060:5 7:412� 0:7

¼ 37:864 ðkNÞ ð39Þ

As listed in Table 3, optimal tube type III has 9 cells (Fig. 18).The width and the wall thickness of this cross-section arerespectively of 80 mm and 1.21 mm. Then, the mean crushingforce in FE analysis was 53.67 kN. As a matter of course, theparameter of profile of tube III is B¼439.376 mm. Replacing itemsinto Eq. (36), the theoretical prediction of mean crushing force is

Pdym:m� III ¼ 1:45� π0:5 � 0:106� 1:361:5

�439:3760:510:0892� 0:7

¼ 54:903 ðkNÞ ð40Þ

From the results above, Eq. (36) were adopted to calculate themean crushing force for three optimal tubes. Subsequently, thedifferences between numerical predictions and theoretical solu-tions for optimal tube types I, II and III were respectively of 2.97%,2.3% and 2.05%. These differences show that the proposed equationsare appropriate to the numerical predictions. In addition, the stableand progressive folding deformation patterns that are developed inall the three types of tube are the desirable energy-dissipatingmechanism.

5. Conclusions

The profiles of three types of tubes were divided into the basicelements: 2-, 3-, T-shape, 4- and 6-panel angle element. Based onthe Simplified Super Folding Element theory, theoretical expres-sions of the mean crushing force were proposed for the threetypes of triangular multi-cell thin-walled tubes under the axialcrushing loading. Numerical simulations of tubes under the axialdynamic impact loading were also carried out, and a dynamicenhancement coefficient was introduced to account for the inertiaeffects of aluminum alloy AA6060 T4. Numerical results showedthat tube types I and III were better than tube type II in the aspectof energy absorption. Simultaneously, the stable and progressivefolding deformation patterns appeared for all the three typesof tubes.

The two RS models of PCF and SEA for each tube were con-structed. Pareto sets were obtained by using the linear weightedaverage methods (LWAM). In this paper, the Pareto solutions of threetypes of tubes were identified to seek out the knee points. Therelative errors between RS approximate value and FE analysis valueat the Knee points were obtained and those were also acceptable.Finally, the theoretical expressions excellently agreed with thenumerical results, and simultaneously validated the efficiency ofthe crashworthiness optimization design method based on thesurrogate models and the numerical analysis techniques.

Acknowledgments

The financial supports from National Natural Science Founda-tion of China (Nos. 11232004 and 51175160), New Century Excel-lent Talents Program in University (NCET-12-0168) and HunanProvincial Natural Science Foundation (12JJ7001) are gratefullyacknowledged. Moreover, Joint Center for Intelligent New EnergyVehicle is also gratefully acknowledged.

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