Theoretical prediction and crashworthiness optimization of multi-cellsquare tubes under oblique impact loading

TrongNhan Tran a,c, Shujuan Hou a,b,n, Xu Han a,b,n, NhatTan Nguyen b,d, MinhQuang Chau c

a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, PR Chinab College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan 410082, 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, Hanoi, Vietnam

a r t i c l e i n f o

Article history:Received 7 April 2014Received in revised form22 August 2014Accepted 27 August 2014Available online 20 September 2014

Keywords:CrashworthinessTheoretical predictionMulti-objective optimizationMulti-cell tubeEnergy absorptionOblique impact loading

a b s t r a c t

Multi-cell square tubes under dynamic oblique impact loading were studied in our work. The theoreticalpredictions of mean crushing force, mean horizontal force, and mean bending moment were proposedby dividing the profile into basic angle elements based on a Simplified Super Folding Element (SSFE)theory. The formulas of an oblique impacting coefficient (λ) with a load angle of 151 were proposed basedon the geometric parameters, the inertia effect and the oblique loading angle by taking the effect ofoblique loading and dynamic crushing into account for aluminum alloy tubes. A new method wasproposed to find out a “knee point” from Pareto set with maximizing the reflex angle. The optimalconfigurations of multi-cell tubes were analyzed under axial and more than one oblique impact loadings.The results showed that the FE numerical results agreed well with the theoretical predictions.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Thin-walled tubes were widely used as energy absorber duringthe past two decades. Extensive efforts by Wierzbicki and Abramo-wicz [1], Abramowicz and Jones [2,3], Guillow et al. [4], DiPaolo et al.[5], Krolak et al. [6], were conducted to investigate the crushing andenergy absorption characteristics of the thin-walled tubes subjectedto axial impact loading by using experimental, theoretical andnumerical methods. For multi-cell tube, Wierzbicki and Abramowicz[1] concluded that the number of “angle” elements on cross-sectionof tube, to a certain extent, decided the effectiveness of the energyabsorption. The quasi-static axial crushing of single-cell, double-celland triple-cell hollow tubes and corresponding foam-filled tubeswere examined by Chen and Wierzbicki [7]. The work of Chen andWierzbicki [7] showed that the multi-cell tube increased the specificenergy absorption SEA by approximately 15%, compared to hollowtube. Therefore, it is necessary to design multi-cell thin-walled tubesas weight-efficient energy absorption components. In order to get asimplification to replace the kinematical admissible model of SuperFolding Element (SFE) theory, the Simplified Super Folding Element

(SSFE) theory was proposed by comprising three extensional trian-gular elements and three stationary hinge lines [1]. Assuming thateach panel and angle element has the same role, the theoreticalprediction of the mean crush force was deduced by dividing thecross-sectional tube into distinct panel section and basic angleelement. Kim [8], Jensen et al. [9], Karagiozova and Jone [10], Zhanget al. [11–15], Najafi and Rais-Rohani [16] and other authors haveinvestigated the multi-cell thin-walled tubes under axial impactloadings and made many valuable conclusions. The progressivecollapse of tubes under axial loadings was summarized by Karagiozovaand Alves [17]. Otherwise, the global bending was an undesirableenergy-dissipating mechanism. Alternatively, the desirable energy-dissipating mechanism was the stable and progressive wrinkle defor-mation. Kim and Reid [18] also proposed an approximate method topredict the bending collapse, the crumpling moment and the energyabsorption for tubes subjected to pure bending. Recently, Tran et al.[19] utilized the Simplified Super Folding Element (SSFE) theory toestimate the energy dissipation of angle elements in the theoreticalpredictions and crashworthiness optimization of multi-cell trian-gular tubes.

Nevertheless, multi-cell thin-walled tubes as an energy absorbernormally bear the oblique impact loading. At that time, the tubesare subjected to both axial force and bending moment. In case thetube experiences a global bending, the energy absorption wouldbe smaller [20]. Therefore, it is necessary to study the mechanical

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International Journal of Mechanical Sciences

http://dx.doi.org/10.1016/j.ijmecsci.2014.08.0270020-7403/& 2014 Elsevier Ltd. All rights reserved.

n 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).

International Journal of Mechanical Sciences 89 (2014) 177–193

property of tubes under oblique impact loadings. According to theaspects of numerical simulation, Han and Park [21] studied theoblique behavior of thin-walled square mild steel tube. Their studyshowed that the axial progressive collapse should be transferred toglobal bending collapse at the critical load angle. Reyes et al. [22,23]carried out extensive numerical and experimental analyses on thesquare tubes under quasi-static oblique loading. Their researchesconcluded that the energy absorption dropped sharply when loadangle exceeded the critical value. Nagel and Thambiratnam [24] alsoinvestigated square tubes under dynamic oblique impact loading.Their results made a conclusion that the impact velocity had nosignificant effect on the critical load angle. Qi et al. [25] investigatedthe crash behavior of the single-cell straight, the single-cell tapered,the multi-cell straight, and the multi-cell tapered tubes by thenumerical method. Their work showed that the MCT tube had thebest crashworthiness performance under oblique loading. Yang andQi [26] optimized the crashworthiness of the empty and foam-filledthin-walled square tubes under oblique impact loading. Song [27]also studied the windowed square tubes subjected to obliqueimpact loading by the numerical method. Song utilized the vari-ables for the sake of the load angle, the geometrical parameters ofwindow, and the impact velocity. In addition, a multi-objectiveoptimization design (MOD) method was employed for the crash-worthiness design of multi-cell thin-walled tubes [28–30].

Most of the above studies just emphasized independentlynumerical simulations, theoretical analysis or experiments. In thispaper, theoretical predictions, numerical analysis and optimizationdesign were combined together for multi-cell square tubes under

oblique collapses. Based on the SSFE theory [19], theoreticalexpressions for mean crushing force of tubes under obliqueloading were derived by dividing the profile of tubes into thebasic angle elements (the right corner, 3-, T-shape, criss-cross,4-, 5- and 6-panel angle elements). The theoretical solutions fortube I, II and III were proposed for the calculation of the bendingresulted by the mean horizontal force and the mean bendingmoment. Dynamic finite element analyses were performed byANSYS/LS-DYNA. A new method was proposed to get a “kneepoint” from a Pareto set with maximizing the reflex angle ψ.

2. Theoretics

2.1. Theoretical prediction of multi-cell square tubes

For the predictions of collapse of thin-walled multi-cell squaretubes, the SSFE theory was applied to calculate the mean crushingforce [31]. In this theory, the variation of wavelength 2H and wallthickness for different lobes was ignored, which was assumed tobe constant respectively. Each panel (flange) and angle elementhad the same role during the collapse. The profiles of tubes weredivided into seven basic elements (right-corner, 3-, T-shape, Criss-cross, 4-, 5- and 6-panel angle element) to evaluate dissipatedbending and membrane energy during the collapse of a fold, asshown in Fig. 1.

Regarding tubes under an oblique impact loading, the equili-brium of the element is expressed via the principle of virtual work,

Nomenclature

2H wavelengtha side lengtht wall thicknessL0 tube lengthsb panel widthB sum of side-length and internal web lengthsd crushing displacementSEA specific energy absorptionEb, Em bending and membrane energyEA total strain energyη effective collapse coefficientμ rotation angle at bending hinge line

PCF initial peak crushing forceP(x) instantaneous crushing forceP mean crushing force at load angle αPa axial crushing forcePm mean crushing forcePh horizontal forceM bending momentM0 fully plastic bending momentλ oblique impacting coefficientψ reflex angleβ, ϕ angle formed by internal panelsα load angleσ0 flow stress of materialσy, σu yield strength and ultimate strength of material

Fig. 1. Cross-sectional geometry of multi-cell square tubes and typical angle elements.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193178

which is _Eext ¼ _Eint . Therefore, the external work from compressionto create a complete collapse of a single fold must be equal to thesum of energy dissipation in bending and membrane of all panels.Therefore,

Pm2H¼ 1ηEbþEmð Þ ð1Þ

where Pm, 2H, Eb and Em are, respectively, the mean crushing force,the wavelength, the bending energy and the membrane energy.η is the effective collapse coefficient. In fact, the panel of foldingelement after deformation is not entirely flattened as displayed inFig. 2. Hence, the available collapse displacement is smaller than2H. In this study, the value of η is taken as 0.75, that lies in therange of 0.7–0.75 [1].

2.1.1. The bending energy of tube structureIn this study, the SSFE theory was applied to calculate the

dissipated energy in bending of each panel. The basic foldingelement (BFE) includes the triangular elements and the bending

hinge lines (Figs. 2 and 3). According to the assumed kinematicsof the SSFE theory, the dissipated bending energy of each panelcan be calculated by summing up the energy dissipation at thebending hinge line, that is

Efb ¼ ∑m

i ¼ 1μM0bi ¼ ∑

m

i ¼ 12πM0b ð2Þ

where μ¼2π is the rotation angle at the bending hinge line.

α = 2π

Bending hinge line

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

Fig. 4. (a) Collapse mode of T-shape element, (b) Extensional element of T-shape element and (c) 3-panel angle element.

Fig. 3. Basic folding element: (a) Asymmetric mode [7], and (b) Symmetric (extensional) mode.

Fig. 5. Collapse mode and extensional elements of criss-cross angle element.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193 179

Since each panel (web) contributes the similar function andmulti-cell tubes are created by n panels (Fig. 1), the bendingenergy of multi-cell tubes derived from Eq. (2) is

Etubeb ¼ 2πM0nb¼ 2πM0B ð3Þwhere b and B are, respectively, the panel width and the sum ofside-length and internal web lengths.M0¼σ0t

2/4 is the fully plasticbending moment.

2.1.2. The membrane energy of Angle element2.1.2.1. The membrane energy of right-corner, 3-panel, T-shape andcriss-cross angle element. In order to get the membrane energy ofright-corner in the SSFE theory, the basic folding element (BFE) wasformed by using the triangular elements and the bending hinge lines(Fig. 3). Two possible collapse modes (asymmetric and symmetric)were proposed in the BFE. The asymmetric and symmetric modescan be referred as the quasi-inextensional and extensional modes,respectively. In regards to the asymmetric mode and the symmetricmode, the three triangular elements and two triangular elementswere both developed for each panel (web) after the deformation.Simultaneously, it was assumed that the role of each panel of multi-cell tubes was the same. Accordingly, the dissipated energy inmembrane Em of right-corner element in the case of asymmetricmode, during one wavelength crushing, could be calculated byintegrating the area of triangular elements (the shaded areas asshown in Fig. 3(a)). Then, we get

Easymm_r�c ¼Zsσ0tds¼ 4M0

H2

tð4Þ

Then, the energy dissipation in membrane of right cornerelement in the case of symmetric mode (Fig. 3(b)) was estimatedas

Esymm_r�c ¼Zsσ0tds¼ 8M0

H2

tð5Þ

The dissipation in membrane energy of 3-panel angle element(Fig. 4c) was estimated by Zhang et al. [32]. According to this work,the membrane energy of 3-panel angle element, during one

wavelength crushing, was

E3�panelm ¼ 4M0

H2

t1þ2 tan ϕ=2

� �� � ð6Þ

The structure of T-shape element was formed by a combinationof one right-corner element and one additional panel, as shown inFig. 4(a) and (b). Therefore, the dissipated energy in membrane ofT-shape element can be calculated by summing up the right-corner element's membrane energy and one additional panel'smembrane energy [31]. On the other hand, each panel oftendeforms through symmetric mode, and has the similar role inthe structural deformation. Consequently, the dissipated energy inmembrane of T-shape element, during one wavelength crushing, isthe sum of membrane energy absorbed by all three panels. Thenwe get

ET�shapem ¼ 12M0

H2

tð7Þ

Being a symmetric structure and formed by four panels, theenergy dissipation in membrane of a criss-cross element can becalculated by summing up the membrane energy of all four panels(Fig. 5). Nevertheless, the criss-cross element is also created bytwo right-corner elements that deform in a symmetric mode.Moreover, each angle element contributes the similar role in thestructural deformation. Accordingly, the membrane energy ofcriss-cross element, during one wavelength crushing, can bedetermined by summing up the membrane energy of two right-corner elements in the case of symmetric mode, that is

Ec�cm ¼ 16M0

H2

tð8Þ

2.1.2.2. The membrane energy of 4-, 6- and 5- panel angleelement. The membrane energy of 4-panel angle element (Fig. 6)was analyzed by Tran et al. [31]. The structure of this angle elementwas constructed by two additional panels and one right-corner angleelement. Thus, energy dissipation in membrane of 4-panel angleelement was calculated by summing up the membrane energy of

Fig. 6. (a) Collapse mode of 4-panel angle element, and (b) Extensional elements.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193180

two additional panels and one right-corner angle element for asymmetric mode. Consequently, the membrane energy Em of 4-panelangle element, during one wavelength crushing, was evaluated byintegrating the triangular areas as shown in Fig. 6(b). Then we get

E4panelm ¼ Er�cm þ2Ea�panel

m ¼ 8M0H2

t1þ 1

cos β

� �ð9Þ

Because of the symmetric structure and similar role of eachpanel in the structure, the energy dissipation in membrane of6-panel angle element can be calculated by summing up themembrane energy of all six panels. However, the 6-panel angleelement was formed by a combination of two right cornerelements and two additional panels, as shown in Fig. 7. Inaddition, it was assumed that each angle element had the similarrole in structure. Thus, the dissipated membrane energy can becalculated by summing up the membrane energy of two rightcorner elements acting on two additional panels and two rightcorner elements. The deformation mode of right corner element in6-panel angle element was symmetric. Regarding the additionalpanel, it is too difficult to give a precise calculation for membraneenergy. Therefore, a simplified deformation model of additionalpanels was assumed and the SSFE theory was used to solve this

problem. Represented in Fig. 7(b), the areas of ILP and IFP weredefined to be the extensional elements of two additional panels.Hence, the dissipated energy in membrane of one additional panelof 6-panel angle element, during one wavelength crushing, wasestimated by integrating the extensional areas. This was

Ea�panelnm ¼

Zsσ0tds¼ σ0t

H2

cos β¼ 8M0

H2

t cos βð10Þ

Accordingly, the membrane energy of 6-panel angle element[19] was

E6�panelm ¼ 2Esymm_r�cþ2Ea�paneln

m ¼ 8M0H2

t2þ 2

cos β

� �ð11Þ

As mentioned above, each angle element contributes the samerole in the structural deformation. Thus, the 5-panel angle elementwas formed by a combination of one T-shape and one right cornerelement (Fig. 8). In this case, the deformation mode of right cornerelement in 5-panel angle element was a symmetric mode.Furthermore, the membrane energy of T-shape and right-cornerones in the case of symmetric mode can be determined. Hence, thedissipated membrane energy of 5-panel one was estimated by

Fig. 7. (a) 6-panel angle element, and (b) Extensional elements of additional panel [19].

Fig. 8. 5-panel angle element.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193 181

summing up the membrane energy of one T-shape element andone right-corner element. Then we get

E5�panelm ¼ ET�shape

m þEsymm_r�c ¼ 20M0H2

tð12Þ

2.1.3. The mean crushing forceThe theoretical expressions were developed for the mean

crushing forces of multi-cell tubes under oblique impact loading.The profile of tube type I was formed by a combination of fourright-corner elements in a asymmetric mode, two 4-panelangle elements, and two 5-panel ones, as shown in Fig. 1(a).To substitute Eqs. (3), (4), (9) and (12) into Eq. (1), the theoreticalexpression for mean crushing force of tube type I can be deter-mined as

Pm�I2Hη¼ Etubeb þ 4Easymm_r�cþ2E4�panelm þ2E5�panel

m

� � ¼ 2πM0Bþ16M0H2

tþ16M0

H2

t1þ 1

cos β

� �þ40M0

H2

tð13Þ

Fig. 9. Flowchart of the MOPSO method.

Fig. 10. Max reflex angle based on definition of a knee point.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193182

From Eq. (13), we can get another transformation, that is

Pm�Iη

M0¼ πB

HþH

t36þ 8

cos β

� ¼ πB

HþH

tF βð Þ ð14Þ

Applying the stationary condition of the mean crushing force∂Pm/∂H, the half-wavelength can be determined as

0¼ �πB

H2þF βð Þt

) H ¼ffiffiffiffiffiffiffiffiffiπBtF βð Þ

sð15Þ

To substitute the term H in Eq. (15) into Eq. (14), the proposedexpression of mean crushing force of tube type I is

Pm�I ¼πM0BηH

þM0Hηt

F βð Þ ¼ π0:5σ0t1:5B0:5

ffiffiffiffiffiffiffiffiffiF βð Þ

p2η

ð16Þ

where F βð Þ ¼ 36þð8= cos βÞ.As indicated in Fig. 1(b), the profile of tube type II was created

by a combination of two right-corner angle elements in anasymmetric mode, two 3-panel angle elements, two criss-crossangle elements and four 4-panel ones. To substitute terms in Eqs.(3), (4), (6), (8) and (9) into Eq. (1), the theoretical expression ofmean crushing force of tube type II is

Pm�II2Hη¼ Etubeb þ 2Easymm_r�cþ2E3�panelm þ2Ec�c

m þ4E4�panelm

� �¼ 2πM0Bþ2M0

H2

t40þ8 tan ϕ=2

� �þ 16cos β

� ð17Þ

From Eq. (17), we can get another transformation, that is

Pm�IIη

M0¼ πB

HþH

t40þ8 tan ϕ=2

� �þ 16cos β

� �¼ πB

HþH

tG ϕ; βð Þ ð18Þ

The half-wavelength can be received by using the stationarycondition of the mean crushing force ∂Pm/∂H, that is

0¼ �πB

H2þG ϕ; βð Þ

t) H¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiπBt

G ϕ; βð Þ

sð19Þ

To substitute the place of H in Eq. (19) back into Eq. (18), thetheoretical expression of mean crushing force of tube type II is

Pm�II ¼πM0LηH

þM0Hηt

G ϕ; βð Þ ¼ π0:5σ0t1:5B0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiG ϕ; βð Þ

p2η

ð20Þ

where G ϕ; βð Þ ¼ 40þ8 tan ðϕ=2Þþð16= cos βÞ.

The profile of tube type III was constituted by a combination offour 3-panel angle elements, two T-shape angle elements and one6-panel one. To substitute terms in Eqs. (3), (6), (7), (11) intoEq. (1), the theoretical expression for the mean crushing force oftube type III is

Pm�III2Hη¼ Etubeb þ 4E3�panelm þ2ET�shape

m þE6�panelm

� �

¼ 2πM0Bþ2M0H2

t28þ16 tan ϕ=2

� �þ 8cos β

� ð21Þ

Then, an alternative form of Eq. (21) is

Pm�IIIη

M0¼ πB

HþH

t28þ16 tan ϕ=2

� �þ 8cos β

� �¼ πB

HþH

tQ ϕ; βð Þ ð22Þ

Postulating the stationary condition of the mean crushing force∂Pm/∂H, the half-wavelength can be expressed as

0¼ �πB

H2þQ ðϕ; βÞ

t) H¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπBt

Q ðϕ; βÞ

sð23Þ

From Eqs. (23) and (22), the theoretical solution of the meancrushing force for tube type III is

Pm�III ¼πM0LηH

þM0Hηt

Q ϕ; βð Þ ¼ π0:5σ0t1:5B0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ ϕ; βð Þ

p2η

ð24Þ

where Q ϕ; βð Þ ¼ 28þ16 tan ðϕ;2Þþð8= cos βÞ.

2.2. Optimization design methodology

Among all the crashworthiness indicators, the energy-absorption is a vital analytical objective. It is expected that thetube can absorb as much energy as possible with a structural totalweight m [28]. Therefore, the specific energy absorption SEA isdefined as

SEA¼ EAm

ð25Þ

In Eq. (25), EA denotes the total strain energy during crushing,that can be determined from the crushing force–displacement

Fig. 11. Schematic of the computational model.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193 183

curve

EA ¼Z d

0P xð Þdx ð26Þ

where P(x) is the instantaneous crushing force.In addition, the initial peak crushing force (PCF) of multi-cell

square tubes can, to a certain extent, evaluate the impact char-acteristics. Meanwhile, the mean crushing force P at load angle α isalso defined to be another crashworthiness indicator [29]. Meancrushing force P can be formulated as

P ¼ EAd¼ 1d

Z d

0P xð Þdx ð27Þ

where d is the crushing displacement of tubes at a specific time.For the nonlinear contact-impact mechanics problems, it is

very difficult to obtain the analytical expressions for the crash-worthy indicators SEA and PCF. Therefore, the surrogate models

based on a response surface method (RSM) [28–30] are aneffective mathematical regression in the multi-objective optimiza-tion design of this kind of issues.

2.2.1. Multi-objective optimization design (MOD) based on surrogatemodels

For the safety of automobile industry, the smaller the PCF is, thelower the deceleration is. Nonetheless, an increase in SEA oftenleads the increase in PCF. On the other hand, SEA can interact withPCF. As a result, It is too difficult for us to have two concurrentlyoptimized objectives. We have to impose the optimum on a Paretoset. The multi-objective optimal problem of both maximizing SEAand minimizing PCF was defined by the multi-objective particleswarm optimization MOPSO methods [30]. A flowchart showingthe procedure of MOPSO is provided in Fig. 9.

Fig. 12. Deformation process of three tubes.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193184

The multi-objective optimal problem can be formulated as

Minimize 1=SEA;PCF� �

s:t: 1rtr2 mm80rar100 mm

8><>: ð28Þ

where t and a are the cross-sectional wall thickness and sidelength.

2.2.2. Knee pointIn many cases, the designer has to get a better solution (termed

as “knee point”) from the optimal solutions set based on theirrequirements. There are several methods which can get a “knee

point”, such as a minimum distance selection method to get a“knee point” from Pareto set [33]; the solution to find a “kneepoint” with maximum bend-angle [34]; or a modified multi-objective evolutionary algorithm [35]. However, if there is a great

Fig. 13. The crushing force–displacement curve of (a) tube I, (b) tube II and(c) tube III.

Table 1Design matrix of thin-walled structures 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 12.914 69.331 14.135 73.385 12.758 70.252 1.25 80 14.174 88.068 15.806 93.419 13.719 89.2383 1.5 80 15.024 107.114 17.04 113.662 14.954 108.4024 1.75 80 16.169 125.74 18.355 133.311 16.054 126.945 2 80 17.035 143.022 19.372 151.463 16.632 144.0846 1 85 12.374 74.315 14.481 78.791 11.871 75.2527 1.25 85 14.04 94.116 15.805 99.808 13.383 95.5248 1.5 85 14.814 114.467 16.928 121.127 13.476 116.2179 1.75 85 15.938 134.261 18.094 141.625 14.731 136.207

10 2 85 16.416 152.533 18.371 161.44 15.232 155.09111 1 90 11.272 78.93 13.214 83.498 11.232 79.82612 1.25 90 13.27 99.949 14.392 105.921 12.35 101.49813 1.5 90 14.25 121.595 15.219 128.896 13.47 123.62514 1.75 90 14.538 142.777 16.223 151.251 14.004 145.18415 2 90 15.492 162.284 17.18 171.731 14.539 165.00916 1 95 11.003 83.773 12.638 88.596 10.719 84.55317 1.25 95 12.662 106.08 13.619 112.382 11.487 107.51718 1.5 95 13.424 129.035 14.915 136.743 12.735 130.82219 1.75 95 14.238 151.289 16.008 160.211 13.844 153.27820 2 95 14.918 171.756 16.752 181.684 14.602 173.87221 1 100 10.412 88.422 12.337 93.609 10.108 89.31222 1.25 100 11.755 111.748 13.554 118.335 11.307 112.923 1.5 100 13.003 136.175 14.821 143.939 12.25 137.45624 1.75 100 13.519 159.917 15.649 168.569 13.278 161.12625 2 100 13.681 181.906 16.114 191.301 14.146 182.973

Fig. 14. Response surface of (a) PCF; and (b) SEA.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193 185

deviation among the orders of magnitude of different objectives,these methods seem ineffective. In our study, a new method wasdeveloped to determine a “knee point” by maximizing the reflexangle ψ, as shown in Fig. 10. Mathematically, the formula is givenas

Maximize ψ ¼ 3601�δ

s:t: cos δ¼ a1a2þb1b2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21þb21

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22þb22

q ð29Þ

where δ is the internal angle formed by two lines f1 and f2.

3. Numerical simulation and crashworthiness optimization

3.1. Numerical simulation

For the multi-objective optimization design, the effects ofgeometric parameters (the cross-sectional wall thickness t and sidelength a) were studied by numerical simulations. Thus, FE modelwas performed by using ANSYS/LS-DYNA to simulate the multi-cell

square tubes under the oblique impact loading. The structures werethin-walled square tubes with 6, 8 and 6 cells, as shown in Fig. 1.The cross-sectional side length a (80 mmrar100 mm) and wallthickness t (1 mmrtr2 mm) were defined to be design variables.The tube lengths L0 were defined to be a constant of 250 mm.

The tube was modelled by using Belytschko-Tsay four-nodeshell elements with five integration points in the element plane.The material AA6060 T4 of tube was modelled by using materialmodel #24 (Mat_Piecewise_Linear_Plasticity) with mechanicalproperties as: Young's modulus E¼68,200 MPa, initial yield stressσy¼80 MPa, the ultimate stress σu¼173 MPa, poisson's rationυ¼0.3 and the power law exponent n¼0.23 [36]. Since aluminumis insensitive to the strain rate effect, this effect is neglected in thefinite element analysis. An automatic single surface contact wasused for the self-contact among the shell elements to avoidinterpenetration of folding generated during the axial collapse.Simultaneously, an automatic node to surface contact between thethin-walled column and rigid-wall was defined to simulate thereal contact. A friction coefficient of 0.3 is utilized for all contactinteraction. To generate enough kinetic energy as applied invehicle crashing, a lumped mass of 500 kg was attached to the

Fig. 15. Relative error curves for all the response surfaces at all the design sampling points.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193186

end of the tube, whereas another end impacts on an oblique rigidwall with an initial velocity of 10 m/s. The load angle α is of 151 inthis study [25]. The three multi-cell sections are not axial-symmetric and therefore the oblique loading from different direc-tions should be different. The schematic of the computationalmodel is shown in Fig. 11. In this paper, we just predicted thetheoretical solution for the loading condition as shown in Fig. 11.

As shown in Fig. 1, all tubes are multi-cell structures. Havingthe same length, same side-lengh, and same wall thickness, tube Iand II are the same weight, while tube III is the heaviest one.Fig. 12 shows the deformation process of three tubes at differentperiods of time. There is no transition from progressive to globalbending collapse mode and no global bending collapse modeappearing. Fig. 13 shows three corresponding crushing force–displacement curves of tube types I, II and III under oblique impactloading. The crushing force, after touching the initial peak, falls sharply and then fluctuates periodically around the mean values

of crushing force corresponding to the formation. Finally, itcompletes the collapse of folds one by one. In addition, a decreaseof mean crushing force in crushing force–displacement curve doesnot appear. This can explain that there is no evidence of bendingphenomenon in these multi-cell thin-walled tubes under obliqueimpact loading, as shown in Fig. 12. Fig. 13 also shows that theexact value of the effective crushing distance on the crushingforce–displacement curve is not unique.

3.2. Crashworthiness optimization

To construct the quartic fitting models for SEA and PCF, a seriesof 25 design sampling points were selected in the design space, asshown in Table 1. The crashworthiness analyses were performedon those models to get response surfaces of the SEA and PCF.Accordingly, the response surfaces of SEA and PCF for tubes I, IIand III are plotted in Fig. 14, where the response surfaces of SEAand PCF behaved monotonically over the design domain. SEA andPCF were inclined to a smaller side-length and a thinner wallthickness. Of all the six RS functions, the maximum fitting REinterval is (�2.66%; 2.68%), which is considered acceptable. Fig. 15shows relative error curves for all the response surfaces at all thedesign sampling points.

The Pareto frontiers of 1/SEA vs. PCF for three types of tubes areplotted in Fig. 16. At the same time, the Pareto frontiers showevenly distributed solution points in the Pareto space. In fact, anypoint on Pareto frontiers can be an optimum, and a range ofoptimal solutions are supplied for the decision-maker. That is whyseveral solutions are suggested to detect the best result or a Kneepoint, which have a large trade-off value in comparison with otherPareto-optimal points. Accordingly, a method for identifying Kneepoint with maximizing the reflex angle ψ was proposed as shownin Fig. 10. The results of Eq. (29) show that Knee points withmaximizing reflex angle ψ for tube types I, II and III are 183.2581,185.7081 and 184.5371, respectively as shown in Fig. 16. Derivingfrom the results of Eq. (32), the optimal design variables of multi-cell square sections for tube types I, II and III under oblique loadingare presented in Table 2. The relative errors REs between FEnumerical results and RS fitting values are also summarized inTable 2. The REs show that RS fitting has a high precision.

4. Validation and discussion

Fig. 17 shows the schematic of force acting on tube at load angleof α. In this figure, the crushing force P under oblique impactloading composes of the axial crushing force Pa and the horizontalforce Ph. The Pa and Ph cause the progressive collapse and thebending of tubes, respectively. Consequently, the relationship

Fig. 16. Pareto fronts of 1/SEA vs. Peak crushing force for (a) tube I, (b) tube II and(c) tube III.

Table 2Optimal results.

Type ofcross-section

Terms Optimal designvariables (mm)

SEA(kJ/kN)

PCF(kN)

Type I Approximate value t¼1.4, a¼80 14.378 98.523FE analysis value 14.279 98.779RE 0.693 �0.259

Type II Approximate value t¼1.43, a¼80 16.746 107.631FE analysis value 16.607 107.093RE 0.837 0.502

Type III Approximate value t¼1.48, a¼80 14.095 106.681FE analysis value 14.151 106.532RE �0.396 0.140

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193 187

between P and Ph is

Ph ¼ P sin α ð30Þ

and the bending moment is generated by the horizontal force

M¼ Phx¼ Ph L0�a�t

2 tan α

� �� ð31Þ

In Section 2.1.3, the theoretical expressions of mean crushingforce in Eqs. (16), (20), and (24) have been created for tubes I,II and III. Nevertheless, these expressions are applied under quasi-static loading cases, where the effects of the dynamic crushing andoblique loading angle are not taken into account. With respect to adynamic loading, dynamic amplification effects including inertiaand strain rate effects should also be taken into account. In reality,the aluminum alloy is not sensitive to strain rate; thus, strain rateeffect can be ignored. A dynamic enhancing coefficient wasbrought in and inertia effect were taken into account [37,38]. Thiscoefficient is a variable for different geometric parameters of tubestructure as described by Langseth and Hopperstad [39] andHanssen et al. [40]. According to their works, the value of thiscoefficient is in the range of 1.3–1.6 for AA6060 T4 extruded tubesunder the axial dynamic loading. Regarding the oblique impactloading, the influence of load angle must be taken into considera-tion. Thus, an oblique impacting coefficient λ was proposed toexpress the relationship among the inertia effect, geometricparameters and oblique loading angle under the oblique impactloading. Obviously, it is too difficult to determine the precise valuefor oblique impacting coefficient λ. In case of load angles exceeding101, value of λ for tubes under oblique loading is relatively lower.Basically, the larger a load angle is, the smaller λ is. Based on thegeometric parameters, the inertia effect and the oblique loadingangle, the following formulas for λ at load angle of 151 are given as

λI ¼ ð10nt=aþ1:24Þn cos α ð32Þ

for tube type I;

λII ¼ ð10nt=aþ1:17Þn cos α ð33Þ

for tube type II; and

λIII ¼ ð10nt=aþ1:1Þn cos α ð34Þ

for tube type III.Thereby, the theoretical solution for tube type I under oblique

impact loading is,

PI ¼ λIPm�I ¼ λIπ0:5σ0t1:5B

0:5

ffiffiffiffiffiffiffiffiffiFðβÞ

p2η

ð35Þ

where F βð Þ ¼ 36þð8= cos βÞ.From Eqs. (30), (31), (32) and (35), the mean horizontal force

and mean bending moment can be predicted respectively as

Pmh�I ¼ PI sin α ð36Þ

MmI ¼ Pm

h�I L0�a�t

2 tan α

� �� ¼ PI sin α L0�

a�t2 tan α

� �� ð37Þ

The mean crushing force at load angle 151 for tube type II is

PII ¼ λIIPm�II ¼ λIIπ0:5σ0t1:5B

0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiG ϕ; βð Þ

p2η

ð38Þ

where G ϕ; βð Þ ¼ 40þ8 tan ðϕ=2Þþð16= cos βÞ.Combining Eqs. (30), (31), (33), and (38), the mean horizontal

force and mean bending moment for tube type II are given as

Pmh�II ¼ PII sin α ð39Þ

MmII ¼ Pm

h�II L0�a�t

2 tan α

� �� ¼ PII sin α L0�

a�t2 tan α

� �� ð40Þ

Fig. 17. Schematic of forces acting on tube.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193188

For tube type III, the form of mean crushing force at load angle151 is

PIII ¼ λIIIPm�IIIπ0:5σ0t1:5B

0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ ϕ; βð Þ

p2η

ð41Þ

where Q ϕ; βð Þ ¼ 28þ16 tan ðϕ=2Þþð8= cos βÞ.To substitute Eqs. (34) and (41) back into Eqs. (30) and (31), the

theoretical expressions of mean horizontal force and mean bend-ing moment are given as

Pmh�III ¼ PIII sin α ð42Þ

MmIII ¼ Pm

h�III L0�a�t

2 tan α

� �� ¼ PIII sin α L0�

a�t2 tan α

� �� ð43Þ

The theoretical values of P for multi-cell tubes are determinedat the displacement of 50% in each specific case. These values aredefined as the equivalent constant force with a correspondingamount of displacement. In Eqs. (35), (38) and (41), σ0 is the flowstress of material with power law hardening estimated by anenergy equivalent stress [41]:

σ0 ¼ffiffiffiffiffiffiffiffiffiffiffiσyσu1þn

r¼ 0:106 GPað Þ ð44Þ

where σy and σu denote the yield strength and the ultimatestrength of the material, respectively; and n is the strain hardeningexponent.

Consequently, Eqs. (35), (38), and (41) are used to calculateP for all cases at load angle 151. The relative errors between FEnumerical results and theoretical predictions for all cases of threetubes are listed in Table 3. For tube I, the relative errors betweenEq. (35) and FE numerical results are ranging from �3.51% to4.51%. For tube II, the relative errors between Eq. (38) and FEnumerical results are from �2.36% to 1.43%. For tube III, those arefrom �2.29% to 3.98%. The relative errors are acceptable. Fig. 18shows that there are a very close agreement between the theore-tical predictions and the FE numerical results in these cases.

Fig. 18. Differences of FE numerical results and theoretical predictions: (a) tube I;(b) tube II and (c) tube III.

Table 3Relative errors among FE numerical results and theoretical predictions for three types of tube.

n Tube type I Tube type II Tube type III

FE Num. result P(kN)

Theo. prediction P(kN)

Diff.(%)

FE Num. result P(kN)

Theo. prediction P(kN)

Diff.(%)

FE Num. result P(kN)

Theo. prediction P(kN)

Diff.(%)

1 27.723 28.108 1.39 32.978 33.450 1.43 26.316 27.363 3.982 40.472 39.998 �1.17 47.958 47.670 �0.60 40.005 39.090 �2.293 53.831 53.482 �0.65 64.049 64.831 1.22 51.56 52.474 1.774 71.135 68.486 �3.72 82.66 82.852 0.23 68.516 67.463 �1.545 85.431 84.948 �0.57 103.756 101.666 �2.01 84.141 84.017 �0.156 27.989 28.836 3.03 34.142 34.306 0.48 27.078 28.049 3.597 39.876 40.996 2.81 48.253 48.840 1.22 40.448 40.025 �1.058 56.764 54.771 �3.51 65.957 65.538 �0.63 52.349 53.673 2.539 70.214 70.583 0.53 85.15 83.716 �1.68 69.16 68.939 �0.32

10 88.213 86.870 �1.52 106.289 103.905 �2.24 85.262 85.779 0.6111 28.574 29.547 3.40 34.297 35.141 2.46 27.825 28.720 3.2212 41.07 41.972 2.20 49.362 49.984 1.26 40.974 40.940 �0.0813 56.482 56.031 �0.80 66.316 66.814 0.75 52.924 54.848 3.6414 71.696 71.646 �0.07 87.045 85.842 �1.38 67.845 70.386 3.7515 89.487 88.752 �0.82 108.125 106.099 �1.87 86.014 87.509 1.7416 28.936 30.241 4.51 35.819 35.958 0.39 28.497 29.375 3.0817 41.687 42.926 2.97 50.528 51.103 1.14 41.518 41.836 0.7718 58.206 57.265 �1.62 69.054 68.259 �1.15 54.137 56.001 3.4419 72.511 70.682 �2.52 89.107 87.832 �1.43 69.988 71.807 2.6020 90.557 90.596 0.04 110.863 108.251 �2.36 86.657 89.210 2.9521 29.863 30.920 3.54 36.307 36.756 1.24 28.968 30.017 3.6222 42.832 43.859 2.40 51.663 52.199 1.04 42.515 42.714 0.4723 57.66 58.473 1.41 70.244 69.675 �0.81 55.068 57.131 3.7524 73.877 74.677 1.08 89.982 90.087 0.12 72.652 73.203 0.7625 91.278 92.405 1.23 111.905 110.363 �1.38 88.164 90.882 3.08

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193 189

Concerning oblique impact loading, the automobile industrystipulates a load angle of 301 for an impact condition on bumpersystem [22]. Thus, the tubes are subjected to both axial andhorizontal forces in an oblique impact condition. If the tubeexperiences a global bending, the energy absorption will be lower.Then, both axial and horizontal forces will be transferred to therest of the structure. It is necessary to understand the mechanismof tubes under oblique impact loadings. To validate the theoreticalpredictions and to consider influences of load angle on SEA and P,the optimal tubes were used for analysis under different loadingcases, that included axial and oblique dynamic loading. The loadangles are set to be 01, 151 and 301, respectively.

Based on Table 3, Eqs. (32)–(43) were adopted to calculate themean crushing force, the mean horizontal force, and the meanbending moment at load angle 151 for three optimal tubes. Thedeformation results and crushing force–displacement curves ofthree tubes at different load angles are shown in Figs. 19–21. Atload angles 01 and 151, all three optimal tubes show progressivewrinkle and no evidence of bending, as shown in Fig. 19(a) and (b),Fig. 20(a) and (b) and Fig. 21(a) and (b). All three optimal tubeswere bent in the case of load angle 301, as shown in Figs. 19(c), 20(c) and 21(c). Regarding the optimal tube type I with 6 cells, thecross-sectional side-length and wall thickness are 80 mm and1.4 mm, respectively. The mean crushing force obtained from FEanalysis is 46.815 kN. Obviously, the sum of side length andinternal web length of cross-section B is 603.831 mm. Substitutingitems into Eqs. (32) and (35), the theoretical prediction of mean

crushing force at load angle 151 is

PI�151 ¼ 1:366� π0:5 � 0:106� 1:11:5 � 603:8310:5 6:8782� 0:75

¼ 47:902 kNð Þ ð45ÞFor optimal tube type I, the difference between FE numerical

result and theoretical prediction is 2.32% obtained from Eq. (45).The theoretical expression shows a good agreement with FEnumerical result. Then, the mean horizontal force and meanbending moment for optimal tube type I under oblique impactloading can be estimated from Eqs. (36), (37). Then we get

Pmh�I�151 ¼ 47:902� 0:258¼ 12:398 kNð Þ ð46Þ

MmI�151 ¼ 12:398 0:25� 0:08�0:0011

2� 0:267

� �� ¼ 1:305 kN mð Þ ð47Þ

For the optimal tube type II, the cross-sectional side-length andthe wall thickness are 80 mm and 1.43 mm, respectively. Moreover,the mean crushing force obtained from FE analysis is 58.815 kN.As a matter of course, the sum of side length and internal weblength B is 635.47 mm. To substitute items into Eqs. (33), (38), thetheoretical prediction of mean crushing force for optimal tube typeII under oblique impact loading is

PII�151 ¼ 1:302� π0:5 � 0:106� 1:431:5 � 635:470:5 8:4042� 0:75

¼ 59:114 kNð Þ ð48Þ

Fig. 19. Final deformation of optimal tube type I at: (a) 01, (b) 151, (c) 301and (d) Crushing/mean crushing force-displacement curve.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193190

The variance between FE numerical result and theoreticalexpression is 0.509% for optimal tube type II. This variance revealsthe agreement between the proposed expression and the FEsimulation. Accordingly, the mean horizontal force and meanbending moment for optimal tube type II under oblique impactcan be measured from Eqs. (39), (40). Then we get

Pmh�II�151 ¼ 59:114� 0:258¼ 15:3 kNð Þ ð49Þ

MmII�151 ¼ 15:3 0:25� 0:08�0:0012

2� 0:267

� �� ¼ 1:537 kNmð Þ ð50Þ

The optimal tube type III has the side-length of 80 mm, and thewall thickness of 1.48 mm. FE numerical result for the meancrushing forces is 48.827 kN. Concurrently, for this optimal tube,the sum of side length and internal web length B is of608.269 mm. To substitute items into Eqs. (34) and (41), thetheoretical prediction of mean crushing force at load angle of 151 is

PIII�151 ¼ 1:231� π0:5 � 0:106� 1:481:5

�608:2690:5 7:432� 0:75

¼ 48:946 kNð Þ ð51Þ

The discrepancy between FE numerical results and theoreticalpredictions for optimal tube type III is 1.36%. Consequently, thisdiscrepancy reveals the agreement between the proposed expres-sion and the FE numerical results. The mean horizontal force andmean bending moment of optimal tube type III under oblique

impact can be predicted from Eqs. (42), (43). Then we get

Pmh�III�151 ¼ 48:946� 0:258¼ 12:668 kNð Þ ð52Þ

MmIII�151 ¼ 12:668 0:25� 0:08�0:0011

2x0:267

� �� ¼ 1:297 kN mð Þ ð53Þ

The values of mean horizontal force and mean bending momentfor three optimal tubes under oblique loading are also obtained fromEqs. (46), (47), (49), (50), (52), and (53). In addition, Fig. 22 (a) and(b) shows an influence of load angle on P and SEA. P and SEAdecrease slightly in the rang of 0–151 load angle, then fall sharplywhen the load angle exceeds 151. Consequently, the global bendingalways dominates tubes under oblique impact at load angle of 301.

5. Conclusions

The SSFE theory was used to get the theoretical expressions offorces and moment for thin-walled square multi-cell tubes underoblique impact loading. The profiles of tubes were divided intoseveral basic elements (the right-corner, 3-, T-shape, Criss-cross,4-, 5- and 6-panel angle element). The oblique impacting coeffi-cients (λ) were proposed based on the geometric parameters, theinertia effect and the oblique loading angle by considering theeffect of oblique loading and dynamic crushing for aluminum alloyAA6060 T4. The theoretical expression of the mean horizontalforce and the mean bending moment was also created. Numericalsimulations were also implemented for tubes under dynamic

Fig. 20. Final deformation of optimal tube type II at: (a) 01, (b) 151, (c) 301 and (d) Crushing/mean crushing force-displacement curve.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193 191

impact loadings. The theoretical predictions agreed well with theFE numerical results. FE numerical results showed that the energyabsorption of tube types I and II were better than tube III.

Response surfaces of PCF and SEA were constructed for threetubes. Pareto sets were obtained by the MOPSO method. A newmethod was introduced to find out a “knee point” from a Pareto

set with maximizing reflex angle ψ. The relative errors between RSfitting values and FE numerical results were acceptable. For theoptimal tube configurations, effects of load angle on P and SEAwere also considered under axial and oblique impact loading. Forthe optimal configurations at the knee points, the theoreticalpredictions agreed well with FE numerical results.

Fig. 21. Final deformation of optimal tube type III at: (a) 01, (b) 151, (c) 301 and (d) Crushing/mean crushing force-displacement curve.

Fig. 22. Effects of load angle on (a) P (b) SEA.

T. Tran et al. / International Journal of Mechanical Sciences 89 (2014) 177–193192

Acknowledgments

The financial supports from National Natural Science Foundationof China (Nos. 11232004, 11372106), New Century Excellent TalentsProgram in University (NCET-12-0168) and Hunan Provincial Nat-ural Science Foundation (12JJ7001) are gratefully acknowledged.Moreover, Joint Center for Intelligent New Energy Vehicle is alsogratefully acknowledged.

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