Theoretical prediction and crashworthiness optimization of multi-cell triangular tubes

Thin-Walled Structures 82 (2014) 183–195

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Theoretical prediction and crashworthiness optimization of 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 China College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, Hunan, PR China c Faculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, Go Vap District, HCM City, Vietnam d Center for Mechanical Engineering, Hanoi University of Industry, Tu Liem District, Ha Noi, Vietnam b

art ic l e i nf o

a b s t r a c t

Article history: Received 2 January 2014 Received in revised form 13 March 2014 Accepted 26 March 2014

The triangular tubes with multi-cell were first studied on the aspects of theoretical prediction and crashworthiness optimization design under the impact loading. The tubes' profiles were divided into 2-, 3-, T-shapes, 4-, and 6-panel angle elements. The Simplified Super Folding Element theory was utilized to estimate the energy dissipation of angle elements. Based on the estimation, theoretical expressions 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 to find out the knee points from the Pareto solutions space. Finally, the theoretical prediction showed an excellent coincidence with the numerical optimal results, and also validated the efficiency of the crashworthiness optimization design method based on surrogate models. Crown Copyright & 2014 Published by Elsevier Ltd. All rights reserved.

Keywords: Crashworthiness Multiobjective optimization Triangular tube Multi-cell Energy absorption Impact loading

1. Introduction Thin-walled extrusions have been extensively applied in vehicle crashworthiness components to absorb impact energy in the past three decades. The tests and the theoretical expressions of square and circular tubes under axial quasi-static and dynamic loading cases were first described by Wierzbicki and Abramowicz [1] and Abramowicz 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] also did many researches on these aspects. Beside square and circular tubes, several other profiles were also studied on their quasi-static or dynamic responses, such as triangular tubes [9–12], hexagonal tubes [13], etc. The structural collapse modes of triangular and square tubes are different from those of circular tubes. Nevertheless, the crushing curves of force–displacement of triangular and square tubes are similar to those of circular tubes. The crushing curves of force– displacement of all the profiles show that the crushing force first reaches an initial peak, then drops down and then fluctuates around a value of the mean crushing force. The extensional deformation has

n Corresponding author at: College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan 410082, PR China. E-mail addresses: [email protected] (S. Hou), [email protected] (X. Han).

http://dx.doi.org/10.1016/j.tws.2014.03.019 0263-8231/Crown Copyright & 2014 Published by Elsevier Ltd. All rights reserved.

more dominant effect on the crushing responses while the quasiinextensional mode occurs normally [14]. According to studies by Wierzbicki and Abramowicz [1], the number of “angle” elements on cross-section of tube decided, to a certain extent, the effectiveness of energy absorption. As a matter of fact, it is necessary to design thin-walled multi-cell tubes for weight-efficient energy absorption. Chen and Wierzbicki [15] examined the axial crushing resistance of single-cell, doublecell and triple-cell hollow tubes, and the respective foam-filled tubes under the quasi-static axial loading. The Simplified Super Folding Element (SSFE) theory was applied to simplify SFE theory, and three extensional triangular elements and three stationary hinge lines were comprised instead of the kinematically admissible model of SFE [1]. The average folding wavelength and the theoretical expression of the mean crushing force were deduced by dividing the cross-sectional tube into distinct panel section and angle element, assuming that the roles of each panel and of angle element were at the same level. The work of Chen and Wierzbicki [15] showed that the multi-cell tube could increase the specific energy absorption SEA by approximately 15%, compared to the respective hollow tube. Kim [16] used Chen and Wierzbicki's model [15] to study multi-cell tubes with four square elements at the corner. The SEA of new multi-cell tube was reported to increase by 190%, compared to conventional square tube. Zhang et al. [17] also applied SSFE theory to derive a theoretical expression of the mean

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crush force of multi-cell square tubes under the dynamic impact loading. In Zhang's work, the cross-section of tube was divided into three basic angle elements, and the study also came to the contribution that plastic energy of each element type was dissipated through membrane action. It was assumed from the theoretical expression that the average wavelength for the dissimilar folds developed at corners. Thereafter, the SSFE theory was also adopted by Zhang et al. [18] to predict the mean crushing force of 3panel angle element. At the same time, the SFE theory was extended by Najafi and Rais-Rohani [19] to explore the crushing characteristics of multi-cell tubes with two different types of threepanel elements. A closed form expression of mean crushing force was also put forward by Najafi and Rais-Rohani [19]. Dynamic progressive buckling of thin-walled multi-cell tubes under axial impact loadings was studied by Jensen et al. [20] and Karagiozova and Jone [21]. Then, the structural dynamic progressive buckling under the axial loading was summarized by Karagiozova and Alves [22] from a phenomenological point of view. Consequently, the desirable energy-dissipating mechanism was a stable and progressive folding deformation pattern for the structural deformation. On the other hand, the global bending on a structure was an undesirable energy-dissipating mechanism mode. At the beginning, multi-cell tubes were mostly employed from the aspects of theoretical researches, such as by Kim [16] and Najafi and Rais-Rohani [19], Nowadays, either FE solutions [23] or surrogate models [13,24–26] developed appeared in the search field of multi-cell tubes under the impact loading. However, there is seldom a combination study of theory, numeric and optimal method for thin-walled multi-cell tubes. Above all, the axial crushing of tube types I, II, III was studied on both theoretical prediction and numerical optimization design in this paper. Based on the SSFE theory, theoretical expressions of the mean crushing force for the three types were derived. All the profiles studies in this paper were divided into 2-, 3-, T-shape, 4 and 6-panel angle elements. In order to obtain the optimal profiles under the crashworthiness criterion, dynamic finite element analysis code ANSYS/LS-DYNA was executed to simulate tubes and to obtain the numerical results at the design sampling points. The multiobjective optimization design was utilized to obtain the optimal configurations. Finally, the theoretical expressions are employed to validate the numerical optimal solutions.

wavelength 2H for different lobes was ignored. To analyze energy dissipation over the collapse of a fold, the triangular multi-cell thin-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, the internal and external energy dissipations are of equal rate ðE_ ext ¼ E_ int Þ. The external energy work for a complete single fold is equal to the sum of dissipated bending and membrane energy. That is 1 P m 2H ¼ ðEb þEm Þ

η

where Pm, 2H, Eb and Em respectively denote the mean crushing force, the length of the fold, the bending energy and the membrane energy, and η is the effective crushing distance coefficient. The panel of folding element after deformation is not completely flattened as shown in Fig. 2. Hence, the available crushing displacement is smaller than 2H. In this study, the value of η was taken as 0.7 since it was found between 0.7 and 0.8 [1]. 2.1.1. The bending energy of tube The SSFE theory was applied to calculate the dissipated energy in bending of each panel. Only three extensional and compressional triangular elements and three stationary hinge lines were used in SSFE theory, which was different from SFE theory. In SFE theory, a model was built with trapezoidal, toroidal, conical and cylindrical surfaces of moving hinge lines. In the work of Chen and Wierzbicki [15], the energy dissipation for bending of each panel Eb was estimated by summing up the energy dissipation at three stationary hinge lines. Then 4

Efb ¼ ∑ M 0 αi b i¼1

ð2Þ

where M 0 ¼ s0 t 2 =4 is the fully plastic bending moment of the panel, b is the sectional breadth and α denotes the rotation angle at the stationary hinge lines. In this case, the panel was supposed to completely flatten after the deformation of the wavelength 2H. Consequently, the four rotation angles α at three stationary hinge lines were π/2 one by one (as shown in Fig. 2). By applying Eq. (2), the bending energy at

2. Theoretics 2.1. Theoretical prediction of multi-cell triangular tube The SSFE theory was applied to solve the axial collapse of triangular multi-cell thin-walled tubes. In the SSFE theory, the wall thickness was assumed to be constant and the variation of

ð1Þ

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

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.

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stationary hinge lines of panel is obtained as Efb

¼ 2π M 0 b

ð3Þ

Since the role of structural each panel is similar and the multicell tube is constituted by m panels (as shown in Fig. 1), the energy dissipation for bending of multi-cell tube is inferred as ¼ 2π M 0 mb ¼ 2π M 0 B Etube b

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

2.1.2. The membrane energy of angle element 2.1.2.1. The membrane energy of 2-panel, 3-panel and T-shape angle element. The basic folding element (BFE) was created by using the triangular elements and the stationary hinge lines (Fig. 3) so as to calculate the membrane energy of right-corner through asymmetric or symmetric deformation mode in the SSFE theory. Two possible collapse modes of the asymmetric and symmetric deformation were supposed in the establishment of BFE. The symmetric and asymmetric modes came from the extensional and quasi-inextensional modes respectively. As for the asymmetric mode, the three triangular elements were developed for each web after the deformation. However, the symmetric mode had two triangular elements for each panel after the deformation. Thus for the asymmetric mode during one wavelength crushing, the energy dissipation in membrane Em of each panel was evaluated by integrating the area of triangular elements (shaded areas in Fig. 3a). Then Z 1 H2 ð5Þ Easym ¼ s0 tds ¼ s0 tH 2 ¼ 2M 0 mf 2 t s Each panel was assumed to have the similar contribution. For the asymmetric mode, the dissipated membrane energy of the right corner element is double of that in one single panel. Then H2 t

For the symmetric mode during one wavelength crushing, the dissipated energy in membrane of each panel was estimated by integrating the area of triangular elements (shaded areas in Fig. 3b). Then Z H2 Esym ð7Þ ¼ s0 tds ¼ s0 tH2 ¼ 4M0 mf t s

ð4Þ

where B is the sum of side and internal web lengths.

f Easym m_r  c ¼ 2E m ¼ 4M 0

185

ð6Þ

sym Esym m_r  c ¼ 2E m_f ¼ 8M 0

H2 t

ð8Þ

Concerning hollow tubes such as triangular, square, pentagonal and hexagonal, the cross-section profiles were formed by 2-panel angle elements. The crushing forces of hollow tubes were strengthened in turn from triangular tube to hexagonal tube. When the central angle φ varies from 301 to 1201, the crushing force of 2-panel angle elements has an appropriate increase [27]. Eq. (1) shows that the mean crushing force is a function of bending and membrane energy. Because the role of each structural panel is assumed to be similar, the bending energy of angle elements with the same numbers of panel will not change. Then, the mean crushing force will vary with respect to the membrane energy. Thus, the membrane energy of right corner element is bigger than that of 2-panel angle element with central angle of 601. From Fig. 4a, the membrane energy of a 2-panel angle, during one wavelength crushing, is calculated as  panel Eind:2 ¼ Easym m m_r  c cos γ ¼ 4M 0

H2 cos γ t

ð9Þ

The energy dissipation in membrane of 3-panel angle element was 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.

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energy of 3-panel angle element, during one wavelength crushing, is E3m panel ¼ 4M 0

H2 ð1 þ 2 tan ðϕ=2ÞÞ t

ð10Þ

T-shape element was created by one right-corner element and one additional panel as shown in Fig. 5. For this reason, the membrane energy of T-shape element was calculated by a sum between membrane energy of right-corner element for the symmetric 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, during one wavelength crushing, is more than triple each panel's membrane energy. Then ETm shape ¼ 3Esym ¼ 12M 0 mf

H2 t

membrane energy of 4-panel angle element was estimated by summing up the membrane energy of 2-panel angle element for symmetric mode and the membrane energy of two additional panels. It is not easy to give a precise calculation for membrane energy of two additional panels. In this case, it is too complicated for the SFE theory to be applied here. So, a simplified deformation model of two additional panels was proposed and the SSFE theory was utilized to solve this problem. Fig. 7b shows the membrane triangular elements developed at the corner line. Accordingly, the membrane energy Em of 4-panel angle element, during one wavelength crushing, is evaluated by integrating the triangular areas as   H2 1 1þ ð12Þ E4m panel ¼ 8M 0 t cos β

ð11Þ

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

2.1.2.3. The membrane energy of 6-panel angle element. As a symmetric structure and being composed of six panels, the dissipated energy in membrane of a 6-panel angle element was calculated by the sum of membrane energy absorbed by all 6 panels. Because of the symmetric structure and the similar contribution of each angle element, the 6-panel angle element was formed by two 4-panel angle elements (Fig. 8a). As a result, the membrane energy of 6-panel angle element, during one wavelength crushing, was estimated by the sum of membrane energy absorbed by two 4-panel angle elements. This is   H2 1 E6m panel ¼ 2E4panel 1 þ ¼ 16M ð13Þ 0 m t cos β

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

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

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187

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.

2.1.3. The mean crushing force in quasi-static case To build a quantitative prediction of crushing resistance of triangular multi-cell tubes and to find out how angle element affects the structure of tube, the theoretical solution of the mean crushing force of multi-cell tube was introduced. Fig. 1a shows the three independent 2-panel angle elements and three 4-panel angle elements form profile of tube type I. Substituting the terms of Eqs. (4), (9) and (12) into Eq. (1), the general theoretical equation of the mean crushing force of tube type I was obtained. This is

Rigid wall Lumped Mass

a

v = 10 m/s

L0 = 250 mm

Fig. 9. Schematic of the computational model.

P m  I 2H η

 panel ¼ Etube þ 3ðEind:2 þ E4m panel Þ b m

¼ 2π M 0 B þ 2M 0

   H2 1 6 cos γ þ 12 1 þ : t cos β

ð14Þ

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

Transforming Eq. (14), we obtain    Pm  I η πB H 1 πB H þ 6 cos γ þ 12 1 þ þ Fðγ ; βÞ ¼ ¼ H t H t M0 cos β

Pm  I ¼ ð15Þ

π B Fðγ ; βÞ H

þ 2

t

)H¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi π Bt Fðγ ; βÞ

ð17Þ

where

The half-wavelength was obtained under the stationary condition of the mean crushing force ∂P m =∂H ¼ 0, then 0¼ 

Fðγ ; βÞ π M0 B M0 H þ Fðγ ; βÞ ¼ π 0:5 s0 t 1:5 B0:5 ηH ηt 2η

ð16Þ

 Fðγ ; βÞ ¼ 6 cos γ þ 12 1 þ

 1 : cos β

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

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elements and one 6-panel angle element. Substituting the terms of Eqs. (4), (10), (11) and (13) into Eq. (1), the general theoretical expression of the mean crushing force of tube type II is obtained as P m  II 2H η ¼ Etube þ 3ðE3m panel þ ETm shape Þ þ E6m panel b ¼ 2π M 0 B þ 2M 0

  H2 8 32 þ 12 tan ðϕ=2Þ þ t cos β

ð18Þ

An alternative form of Eq. (18) is   P m  II η π B H 8 πB H þ 32 þ 12 tan ðϕ=2Þ þ þ Gðϕ; βÞ ¼ ¼ H t H t M0 cos β

ð19Þ

By using the stationary condition of the mean crushing force, the half-wavelength can be obtained as ∂P m =∂H ¼ 0. Then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π B Gðϕ; βÞ π Bt )H¼ 0¼  2þ ð20Þ t Gðϕ; β Þ H

Fig. 10. Deformation process of three tubes.

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Substituting Eq. (20) back into Eq. (19), the equation of the mean crushing force for tube type II under the quasi-static loading is obtained as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π M0 B M0 H Gðϕ; β Þ P m  II ¼ þ Gðϕ; β Þ ¼ π 0:5 s0 t 1:5 B0:5 ð21Þ 2η ηH ηt where Gðϕ; β Þ ¼ 32 þ 12 tan ðϕ=2Þ þ

8 : cos β

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

189

and one 6-panel angle element. Substituting the terms of Eqs. (4), (9), (12) and (13) into Eq. (1), the general theoretical solution to the mean crushing force of tube type III is obtained as P m  III 2H η ¼ Etube þ 3E2m panel þ 6E4m panel þ E6m panel b   H2 32 32 þ 6 cos γ þ ¼ 2π M 0 B þ 2M 0 t cos β

ð22Þ

Transforming Eq. (22), we obtain   P m  III η π B H 32 πB H þ 32 þ6 cos γ þ þ Q ðγ ; β Þ ¼ ¼ H t H t M0 cos β

ð23Þ

By using the stationary condition of the mean crushing force, the half-wavelength is expressed as ∂P m =∂H ¼ 0. Then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π B Q ðγ ; β Þ π Bt )H¼ ð24Þ 0¼  2þ t Q ðγ ; β Þ H Substituting Eq. (24) into Eq. (23), the mean crushing force of tube type III under quasi-static loading is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ðγ ; βÞ π M0 B M0 H P m  III ¼ þ Q ðγ ; βÞ ¼ π 0:5 s0 t 1:5 B0:5 ð25Þ ηH ηt 2η where Q ðγ ; β Þ ¼ 32 þ6 cos γ þ

32 : cos β

2.2. Optimization design methodology Among all the indicators of crashworthiness optimization design, the vital analytical objective was the energy-absorption. Hence, in order to estimate the energy absorption of structural unit mass m, specific energy absorption (SEA) was formulated as SEA ¼

EA m

ð26Þ

In fact, a higher SEA indicates a better capability of energy absorption. In Eq. (26), the total strain energy during crushing is Table 1 Design matrix of three types of tube for crashworthiness. n

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

t (mm) a (mm) Tube type I

1 1.3 1.6 1.9 2.2 1 1.3 1.6 1.9 2.2 1 1.3 1.6 1.9 2.2 1 1.3 1.6 1.9 2.2 1 1.3 1.6 1.9 2.2

80 80 80 80 80 85 85 85 85 85 90 90 90 90 90 95 95 95 95 95 100 100 100 100 100

Tube type II

Tube type III

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

PCF (kN) SEA (kJ/kg)

PCF (kN)

18.262 39.362 20.226 52.274 22.774 65.287 23.854 77.82 24.732 88.936 17.635 42.151 19.491 55.941 21.079 69.78 22.231 82.904 23.891 94.376 16.689 48.183 18.63 63.406 19.883 80.385 21.425 97.058 22.916 111.899 16.092 50.984 17.793 67.207 19.279 85.237 20.865 102.754 22.276 117.81 15.569 50.155 17.083 66.443 19.26 83.029 20.628 98.954 21.053 113.035

49.572 65.965 82.467 98.125 111.611 53.133 70.384 88.192 104.633 118.899 60.979 69.095 99.84 120.552 138.568 65.978 83.194 105.95 128.907 148.295 63.148 83.857 104.85 124.628 142.093

52.592 69.599 85.801 101.135 114.541 56.038 74.202 92.462 109.979 125.572 64.544 85.478 107.641 128.512 146.48 68.196 90.665 114.112 135.856 153.825 66.649 87.852 110.016 130.951 157.552

17.356 19.876 21.86 22.003 23.107 16.842 19.272 20.924 21.602 22.693 16.559 18.9 20.273 20.679 21.524 15.253 16.954 18.229 19.146 19.71 14.398 16.139 17.748 18.55 18.715

23.339 25.227 28.199 29.092 29.644 22.753 24.273 26.388 27.564 28.107 21.338 24.05 25.536 26.272 26.42 20.932 22.371 24.579 25.306 25.948 19.337 20.011 22.979 23.92 24.152

190

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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.

estimated as Z d PðxÞdx EA ¼ 0

ð27Þ

where P(x) is the instantaneous crushing force. In addition, the initial peak crushing force (PCF) of multi-cell thin-walled tube was used for estimating the impact characteristics. Another crashworthiness indicator is the mean crushing force Pm which is computed by Z E 1 d PðxÞdx ð28Þ Pm ¼ A ¼ d d 0 where d is the crushing displacement at a specific time. 2.2.1. Response Surface Method (RSM) A typical surrogate modelling technique was considered appropriate in the multivariate optimization process involving material, geometrical nonlinearities and contact-impact loading nonlinearities. The primary concept of RSM was applied to the construction of regression functions for crashworthiness indicators by using the function values at the design sampling points. The mathematical

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

expression of RSM is expressed as m

~ ¼ ∑ β ψ ðxÞ yðxÞ  yðxÞ i i i¼1

ð29Þ

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

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

coefficient. Taking n dimensional problem for example, the full linear polynomial basis function is 1; x1 ; x2 ; …; xn

ð30Þ

and the full quartic polynomial basis function is expressed as 1; x1 ; x2 ; :::; xn ; x21 ; x1 x2 ; :::; x1 xn ; :::; x2n ; x31 ; x21 x2 ; :::; x21 xn ; x1 x22 ; :::; x1 x2n ; :::; x3n ; x41 ; x31 x2 ; :::; x31 xn ; x21 x22 ; :::; x21 x2n ; :::; x1 x32 ; :::; x1 x3n ; :::; x4n

191

mathematically given as Maximize θðx; xL ; xR Þ ¼ θL  θR

ð33Þ

where θL ¼ arctanf 2 ðxL Þ  f 2 ðxÞ=f 1 ðxÞ  f 1 ðxL Þ and θR ¼ arctanf 2 ðxÞ  f 2 ðxR Þ=f 1 ðxR Þ  f 1 ðxÞ are the left and right bend-angles of x.

ð31Þ

The full quartic polynomial basis function was proved to be a better choice for the regression analysis [13,24–26]. The quartic response surface models were consequently adopted in this study. 2.2.2. Multi-objective optimization With two objectives of SEA and PCF, the multiobjective optimization problem for minimizing PCF and maximizing SEA was defined by the linear weighted average methods (LWAM) [24]. Then, the mathematical definition for the crashworthiness optimization in terms of the LWAM is given as 8 SEAn > Minimize Fðt; aÞ ¼ wPCFðt;aÞ > PCFn þ ð1  wÞSEAðt;aÞ > > < s:t w A ½0; 1 ð32Þ > 1 r t r 2:2 mm > > > : 80 r a r 100 mm where SEAn and PCFn are the given normalizing values for each cross-sectional profile. 2.2.3. Knee point In some certain cases, the designer must choose the most preferred solution (termed as “knee point”) from optimal solutions to meet their requirement. Several methods were proposed to determine a “knee point” from Pareto set such as Turevsky and Suresh [28] and Sun et al. [29]. However, if there is a great deviation among the orders of magnitude of different objectives, these methods [29] seem to be less effective. A modified multi-objective evolutionary algorithm suggested by Branke et al. [30] was utilized to seek the knee regions. Deb and Gupta [31] have recently suggested a solution to find a “knee point” with maximum bend-angle, which is

3. Numerical simulation and crashworthiness optimization 3.1. Numeric simulation In this section, the FE model was carried out by ANSYS/LS-DYNA to simulate the triangular multi-cell thin-walled tubes subjected to axial dynamic loading with 4, 6 and 9 cells (as shown in Fig. 1). The side-length a of the cross-sections and the thickness t were chosen to 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 the Belytschko–Tsay four-node shell element with the optimal mesh density of 2.5  2.5 mm. The material AA6060 T4 was modelled with material model #24 (Mat_Piecewise_Linear_Plasticity) with mechanical properties: Young's modulus E ¼68,200 MPa, initial yield stress sy ¼80 MPa, ultimate stress su ¼ 173 MPa, Poisson's ration υ ¼0.3, and power law exponent n ¼ 0.23 [32]. Since the aluminum was insensitive to the strain rate effect, this effect was neglected in the finite element analysis. An automatic node to surface contact between thin-walled tube and rigid wall was defined to simulate the real contact. Alternatively, an automatic single surface contact algorithm was utilized for the self-contact among the shell elements to avoid interpenetration of folding generated during the axial collapse. In the contact definition, a friction coefficient of 0.3 among all surfaces was employed. To generate enough kinetic energy, one end of tube was attached with a lumped mass of 500 kg whereas another end impacted onto a rigid wall with an initial velocity of 10 m/s. The schematic of the computational model is shown in Fig. 9. All of the tubes were axial symmetric structures. Despite the same length, same side-length and same thickness, the three tubes

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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. (%)

22.23 34.107 46.497 61.379 77.295 23.173 34.142 47.275 62.465 79.196 23.787 35.095 48.408 64.341 81.044 24.151 35.645 49.676 66.14 82.19 24.819 36.812 51.437 67.423 83.731

23.016 34.028 46.344 59.817 74.337 23.736 35.098 47.809 61.717 76.710 24.435 36.137 49.230 63.561 79.013 25.115 37.146 50.611 65.352 81.250 25.776 38.129 51.956 67.096 83.427

3.54  0.23  0.33  2.54  3.83 2.43 2.80 1.13  1.20  3.14 2.73 2.97 1.70  1.21  2.51 3.99 4.21 1.88  1.19  1.14 3.86 3.58 1.01  0.49  0.36

25.894 41.1 55.331 72.1 91.64 26.75 42.39 58.3 74.137 94.125 27.482 43.184 59.954 76.129 95.039 28.264 43.859 60.278 77.219 95.897 29.109 44.984 60.994 79.107 96.597

27.148 40.143 54.680 70.589 87.738 27.996 41.404 56.406 72.827 90.533 28.820 42.627 58.080 74.998 93.244 29.620 43.816 59.707 77.108 95.879 30.400 44.974 61.291 79.162 98.443

4.84  2.33  1.18  2.10  4.26 4.66  2.33  3.25  1.77  3.82 4.87  1.29  3.13  1.49  1.89 4.80  0.10  0.95  0.14  0.02 4.44  0.02 0.49 0.07 1.91

39.973 60.35 85.345 111.642 139.519 41.129 61.05 87.9 113.899 144.17 42.716 62.715 89.541 117.161 148.092 43.862 63.768 91.482 120.81 151.03 45.268 65.507 93.192 122.495 153.514

41.221 60.954 83.028 107.183 133.223 42.510 62.868 85.648 110.581 137.467 43.760 64.726 88.190 113.878 141.583 44.976 66.531 90.660 117.082 145.584 46.160 68.290 93.066 120.201 149.477

3.12 1.00  2.72  3.99  4.51 3.36 2.98  2.56  2.91  4.65 2.44 3.21  1.51  2.80  4.39 2.54 4.33  0.90  3.09  3.61 1.97 4.25  0.14  1.87  2.63

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were different in weight. Tube I is the lightest one while tube III is the heaviest. The axial crushing of multi-cell tubes was presented with a displacement equal to about 70% of the initial length. Fig. 10 shows the deformation process of three tubes at different times. Sometimes the exact value of the effective crushing distance on the crushing force–displacement curve was not unique. The corresponding crushing force–displacement curves of three tubes are also shown in Fig. 11. After reaching the initial peak and before rising steeply whenever the deformation capacity is exhausted at the effective crushing distance, the crushing force fell sharply and then fluctuated periodically and around the values of the mean crushing force in correspondence with the formation, and finally completed the collapse of folds one by one.

3.2. Crashworthiness optimization For obtaining the response functions of SEA and PCF, a series of 25 design sampling points (based on a and t) were selected in the design space to provide sampling design values for FEA and regression analysis of three types of tubes (Table 1) so as to obtain the response surface of the SEA and PCF. Fig. 12 shows that the SEA's and PCF's RS of tube types I, II and III cases behave monotonically over the design domain. In addition, the curves in Fig. 13 illustrate the variation of SEA and Pm with changes in weight. Meanwhile, energy absorptions of the tube types I and III were better than those of tube type II. The Pareto sets for these three cross-sectional profiles were obtained by changing the weight coefficient w in Eq. (32), and the Pareto frontiers are plotted in Fig. 13. In fact, any point on the Pareto frontier can be an optimum. As a result, some methods were proposed to determine the best solution (knee point) which has a large trade-off value in comparison with other Paretooptimal points. In this case, methods of [31,32] were utilized to determine the knee region and the knee point, respectively. The results of expression (33) showed that Pareto solutions (Knee points) for tube types I, II and III were 0.7924, 0.7818 and 0.7773, respectively. The relative errors (REs) of FE simulation value and RS approximate value are summarized in Table 3. Therefore the FE simulation value and RS approximate value at the Knee points were exactly close to each other. According to the relationship among the weighted average method and those of [31,32], these optimal results are plotted in Fig. 14.

4. Theoretical validation and discussion The theoretical expressions of the mean crushing force of tube types I, II and III are proposed in Section 2.1.3. Nevertheless, these expressions were applied in axial quasi-static loading case in which the effect of dynamic crushing was not considered. In the dynamic loading case, dynamic amplification effects, consisting of inertia and strain rate effects, were considered in the theoretical equations above. In reality, the aluminum alloy is insensitive to the strain rate effect that can be neglected. A dynamic enhancing coefficient λ was suggested to take the inertia effect into account. It is not easy to find the accurate value for dynamic enhancing coefficient, and this coefficient λ is even a variable for different geometric parameters as described by Langseth and Hopperstad [33] and Hanssen et al. [34]. According to these studies, the coefficient λ was proposed in the range of 1.3–1.6 for AA6060 T4 extruded tubes. These coefficients (λ) were simply set to 1.41, 1.3 and 1.45 for tube types I, II and III, respectively. Thus, the theoretical solution to tube type I is applied as follows: 0:5 P dym: s0 t 1:5 B0:5 m  I ¼ λI P m  I ¼ λI π

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fðγ ; βÞ 2η

where

 Fðγ ; βÞ ¼ 6 cos γ þ 12 1 þ

1 cos β



for tube type II 0:5 s0 t 1:5 B0:5 P dym: m  II ¼ λII P m  II ¼ λII π

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gðϕ; β Þ 2η

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

Gðϕ; βÞ ¼ 32 þ 12 tan ðϕ=2Þ þ

ð34Þ

8 : cos β

ð35Þ

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

For tube type III

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ðγ ; β Þ 0:5 1:5 0:5 ¼ λ P ¼ λ π s t B P dym: III m  III III 0 m  III 2η

ð36Þ

s0 ¼

where Q ðγ ; βÞ ¼ 32 þ 6 cos γ þ

32 : cos β

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

Terms

Optimal design variables (mm)

SEA (kJ/kN)

PCF (kN)

Type I

Approximate value FE analysis value RE Approximate value FE analysis value RE Approximate value FE analysis value RE

t ¼1.23, a ¼80

19.897 19.728 0.856 25.282 25.717  1.691 25.100 24.800 1.210

49.315 49.088 0.462 63.436 63.056 0.603 63.967 64.267  0.467

Type II

Type III

t ¼1.25, a ¼80

t ¼1.21, a¼ 80

193

rffiffiffiffiffiffiffiffiffiffiffi

sy su

1þn

¼ 0:106 ðGPaÞ

ð37Þ

where sy and su denotes the yield strength and the ultimate strength of the material, respectively, and n is the strain hardening exponent. Hence, the mean crushing forces were calculated by expressions (34)–(36) and these values were used to compare with the value of mean crushing force of tubes at a displacement of 60% in each different case. Remarkably, this mean crushing force is defined as an equivalent constant force with a corresponding amount of displacement. The deviations among theoretical equations above and numerical predictions for all cases of three types of tubes are listed in Table 2. For tube types I and II, the differences of Eqs. (34) and (35) and numeric results were, respectively, ranging from  3.83% to 4.21% and from  4.26% to 4.84%. With regard to tube type III, the dissimilarities between Eq. (36) and numeric 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 theoretical solutions and the numerical predictions for all cases.

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

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

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Fig. 18. (a) Deformation result and (b) crushing force–displacement curve of tube III.

Deriving from the optimal results of the method of Deb and Gupta (Table 3), the optimal triangular sections of tube types I, II and III were considered in this analysis. The deformation results and crushing force–displacement curves of three tubes are presented in Figs. 16–18. For the optimal tube type I with 4 cells in Table 3, the side-length and the wall thickness were of 80 mm and 1.23 mm, respectively (as shown in Fig. 16). The mean crushing force obtained from FE analysis is 30.43 kN. Visibly, with this profile, the sum of side-length and of internal web length B is of 352.62 mm. Substituting items into Eq. (34), the theoretical prediction of mean crushing force is 0:5 P dym:  0:106  1:231:5 m  I ¼ 1:41  π

352:620:5

6:46 ¼ 31:335 ðkNÞ 2  0:7

ð38Þ

With 6 cells, the optimal tube type II in Table 3 has the sidelength of 80 mm and the wall thickness of 1.25 mm (Fig. 17). The mean crushing force received from FE analysis was 37.01 kN. Simultaneously, for this optimal tube, the sum of side-length and internal web length B is 439.06 mm. To substitute items into Eq. (35), the theoretical prediction of mean crushing force is 0:5 P dym:  0:106  1:421:5 m  II ¼ 1:3  π

439:060:5

7:41 ¼ 37:864 ðkNÞ 2  0:7

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 are respectively of 80 mm and 1.21 mm. Then, the mean crushing force in FE analysis was 53.67 kN. As a matter of course, the parameter of profile of tube III is B¼ 439.376 mm. Replacing items into Eq. (36), the theoretical prediction of mean crushing force is 0:5 P dym:  0:106  1:361:5 m  III ¼ 1:45  π

439:3760:5

10:089 ¼ 54:903 ðkNÞ 2  0:7

ð40Þ

From the results above, Eq. (36) were adopted to calculate the mean crushing force for three optimal tubes. Subsequently, the differences between numerical predictions and theoretical solutions for optimal tube types I, II and III were respectively of 2.97%, 2.3% and 2.05%. These differences show that the proposed equations are appropriate to the numerical predictions. In addition, the stable and progressive folding deformation patterns that are developed in all the three types of tube are the desirable energy-dissipating mechanism.

5. Conclusions The profiles of three types of tubes were divided into the basic elements: 2-, 3-, T-shape, 4- and 6-panel angle element. Based on the Simplified Super Folding Element theory, theoretical expressions of the mean crushing force were proposed for the three types of triangular multi-cell thin-walled tubes under the axial crushing loading. Numerical simulations of tubes under the axial dynamic impact loading were also carried out, and a dynamic enhancement coefficient was introduced to account for the inertia effects of aluminum alloy AA6060 T4. Numerical results showed that tube types I and III were better than tube type II in the aspect of energy absorption. Simultaneously, the stable and progressive folding deformation patterns appeared for all the three types of tubes. The two RS models of PCF and SEA for each tube were constructed. Pareto sets were obtained by using the linear weighted average methods (LWAM). In this paper, the Pareto solutions of three types of tubes were identified to seek out the knee points. The relative errors between RS approximate value and FE analysis value at the Knee points were obtained and those were also acceptable. Finally, the theoretical expressions excellently agreed with the numerical results, and simultaneously validated the efficiency of the crashworthiness optimization design method based on the surrogate models and the numerical analysis techniques.

Acknowledgments The financial supports from National Natural Science Foundation of China (Nos. 11232004 and 51175160), New Century Excellent Talents Program in University (NCET-12-0168) and Hunan Provincial Natural Science Foundation (12JJ7001) are gratefully acknowledged. Moreover, Joint Center for Intelligent New Energy Vehicle is also gratefully acknowledged. References [1] Wierzbicki T, Abramowicz W. On the crushing mechanics of thin-walled structures. J Appl Mech 1983;50:727–34. [2] Abramowicz W, Jones N. Dynamic axial crushing of square tubes. Int J Impact Eng 1984;2:179–208. [3] DiPaolo BP, Monteiro PJM, Gronsky R. Quasi-static axial crush response of a thin-wall, stainless steel box component. Int J Solids Struct 2004;41:3707–33. [4] DiPaolo BP, Tom JG. Effects of ambient temperature on a quasi-static axialcrush configuration response of thin-wall, steel box components. Thin-Walled Struct 2009;47:984–97.

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