Theoretical prediction and crashworthiness optimization of top-hat thin-walled structures under transverse loading

Thin-Walled Structures 144 (2019) 106261

Contents lists available at ScienceDirect

Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws

Full length article

Theoretical prediction and crashworthiness optimization of top-hat thinwalled structures under transverse loading

T

Libin Duana, Zhanpeng Dub,*, Haobin Jianga, Wei Xua, Zhanjiang Lic a

School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang, 212013, China State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China c Nanjing YueBoo Power System Co.,Ltd, Nanjing, 210019, China b

ARTICLE INFO

ABSTRACT

Keywords: Top-hat thin-walled structures Bending collapse Energy absorption Theoretical prediction Crashworthiness optimization

In this study, a theoretical model is developed to reveal the bending collapse of top-hat thin-walled structures by dividing a top-hat thin-walled structure into a top-hat element and flat-plate element. A theoretical formula is also developed to describe the bending deformation and energy-absorption of the structures. The theoretical model is capable of predicting the bending collapse and energy-absorption of top-hat thin-walled structures with different thickness and material specification. The accuracy and generality of the theoretical prediction model is validated by performing three-point bending tests and finite element simulations. Then, both theoretical prediction formulas and finite element analysis (FEA) based surrogate models are employed to perform the crashworthiness optimization of top-hat thin-walled structures. The results show that (i) the theoretical prediction model is capable of producing results that can be directly used to optimize the thicknesses, cross-sectional geometry, and material specifications for top-hat thin-walled structures, which will increase the efficiency and shorten the cycle time of crashworthiness design optimization for this type of structure; and (ii) steelaluminum hybrid top-hat thin-walled structure has a larger energy-absorption capacity than high-strength steel without exceeding the initial weight, whereas a lightweight design is more feasible with an aluminum alloy than with high-strength steel without sacrificing the energy absorption of the baseline design.

1. Introduction Thin-walled structures are widely employed as structural components in vehicle bodies owing to their advantages of high strength, good energy-absorption capacity, and light weight [1–5]. The experimental and theoretical studies on the collapse and energy-absorption mechanisms of thin-walled structures have been reported since the 1960s. Alexander [6] firstly conducted axial crushing tests and established the theoretical prediction model to describe the relationship between the mean impact force and energy absorption of circular section tubes. In a series of studies on the collapse and energy-absorption mechanisms of square and circular section structures subjected to axial crushing, Abramowicz and Jones [7–9] proposed the concept of effective crushing distance and modified the prediction formula for the mean impact force. Wierzbicki and Abramowicz [10] proposed the super folding element theory, which generates predictions that agree well with the corresponding experimental results when the effects of material hardening and effective crushing distance are taken into account. Sun et al. [11] conducted dynamic and quasi-static tests and

*

simulations to investigate the effects of thickness distributions on energy-absorption behavior of top-hat thin-walled structures. Zhang and Zhang [12] investigated the collapse behavior of multi-cell thin-walled structures of different cross-sectional sizes subjected to axial crushing, and derived a theoretical prediction formula to determine the mean crushing force. In comparative studies of the energy-absorption behaviors of multi-cell and single-cell structures under axial and oblique loading, Fang et al. [13] and Song et al. [14] found that multi-cell structures have a better energy-absorption capacity than single-cell structures. Tran et al. [15,16] studied the deformation behavior of multi-cell thin-walled square structures under axial and oblique impact loading, and developed simplified super folding element (SSFE) theorybased theoretical formulas to determine their mean crushing force and bending moment. Sun et al. [17] proposed an integer coded genetic algorithm and used it to optimize the topological distribution of multicelled web members for multi-cell structures. The above-mentioned studies are focused on energy-absorption mechanism and crashworthiness optimization of thin-walled structures under axial loading. Lateral impact is a major cause of transportation accidents and

Corresponding author. E-mail address: [email protected] (Z. Du).

https://doi.org/10.1016/j.tws.2019.106261 Received 11 November 2018; Received in revised form 25 April 2019; Accepted 12 June 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

vehicle passenger injuries. The bending deformation and energy-absorption behavior of the structural components of car bodies (e.g., Bpillars and sill beams) are major factors that affect the safety and passenger protection performance of vehicles subjected to lateral impact loading; Thus, improving the crashworthiness design of vehicles by studying the bending deformation and energy-absorption behavior of thin-walled structures is important. In an earlier theoretical and experimental study of the bending deformation behavior of rectangular and square-sectioned thin-walled structures, Kecman [18] proposed the concepts of fixed hinging line and rolling hinging line and developed a set of theoretical prediction formulas for the bending moments. Kim and Reid [19] improved the theoretical prediction formula for the bending moment of rectangular-sectioned thin-walled structures by using the theoretical model of Kecman. Kotelko et al. [20] conducted a series of tests to explore the bending deformation behavior of boxsectioned thin-walled structures at different post-failure stages, and established a theoretical model to describe their bending collapse. In a study of circular hollow tubes, Maduliat et al. [21] developed a theoretical prediction model to describe their bending collapse and energyabsorption mechanism, and conducted experiments to validate the accuracy of the model. Zhang et al. [22], and Wang et al. [23,24] conducted comparative analyses of the energy-absorption efficiencies of various multi-cell structures, and developed theoretical prediction formulas to describe the bending moments of multi-cell structures. Meanwhile, Du and Duan [25] investigated the collapse behavior of thin-walled structures with single-box multi-cell section, and a theoretical prediction formula of energy absorption is developed under threepoint bending loading. Although multi-cell thin-walled structures exhibit good energy-absorption capacity under bending loading, vehicle bodies mainly include top-hat-sectioned thin-walled structures, which have wider applicability. Therefore, the bending collapse behavior of top-hat thin-walled structures has received attention from various researchers. In this regard, Chen [26] investigated the effects of the rotation angle on the bending collapse and energy-absorption behavior of top-hat foam-filled thin-walled aluminum alloy structures. Sato et al. [27] conducted dynamic and quasi-static tests to investigate the effects of material properties on the bending resistance and energy-absorption behavior of top-hat thin-walled structures. Santosa [28] investigated the effects of the forming process on the bending resistance capacity of top-hat thin-walled structures. In consideration of the rapid development of numerical simulation techniques and optimization algorithms, Sun et al. [29] and Fang et al. [30] conducted crashing analysis and multiobjective optimization for thin-walled structures with functionally graded thickness (FGT). Later, Sun et al. [31], An et al. [32] and Xu et al. [33] studied the bending deformation mechanism and crashworthiness design optimization of FGT tubes with circular and square section, respectively. Duan et al. [34] established a parameterization model for the thickness distribution of top-hat structures with varying thickness, and investigated their bending deformation behavior and crashworthiness design optimization methods. Sun et al. [35] and Zhang et al. [36] employed the elemental energy density method to optimize the thickness distribution of top-hat thin-walled structures in order to improve their structural energy-absorption capacity under bending loading. Elmarakbi et al. [37] and Beik et al. [38] investigated the influence of cross-sectional geometry parameters on the energy-absorption behavior of S-shaped longitudinal members; both research groups improved the crashworthiness and energy-absorption capacity of the members by optimizing their cross-sectional geometry. Qiu et al. [39] employed the particle swarm algorithm to optimize the cross-sectional geometry and crashworthiness design of multi-cell tubes subjected to bending impact loading. Zuo et al. [40] optimized the cross-sectional geometry of tophat thin-walled structures formed by stamping, with the aim of improving their crashworthiness and energy-absorption capacity while also minimizing the structural weight. Wang et al. [41] performed implicit parameterization modeling and crashworthiness design

optimization of S-shaped longitudinal members. The studies mentioned above mainly attempted to improve the bending behavior and energy-absorption capacity of thin-walled structures by optimizing their thickness and cross-sectional geometry. However, there are few reports on the bending deformation behavior, energy-absorption mechanism, and crashworthiness design optimization of top-hat thin-walled structures that consider all of the following design variables: thickness, cross-sectional geometry, and material specification. The purpose of this paper is to explore the bending collapse behavior and energy absorption characteristics of top-hat thinwalled structures under transverse loading by using theoretical predictions and design optimization. For this reason, the work is organized as follows: Following the introduction, Section 2 develops a theoretical prediction model of top-hat thin-walled structure under three-point bending loading; Section 3 performs three-point bending tests and finite element simulations to validate the accuracy and generality of the theoretical prediction model; in Section 4 both theoretical prediction formulas and finite element analysis (FEA) are employed to perform the crashworthiness optimization of top-hat thin-walled structures, followed by the results and discussions; Conclusions are given in Section 5 to close the paper. 2. Theoretical prediction model for three-point bending collapse Although finite element analysis (FEA) has been widely applied to study the bending collapse and energy-absorption behavior of top-hat thin-walled structures, it is timely expensive to implement this technique at the conceptional design stage in the study of the crashworthiness of the thin-walled structural components of vehicle bodies when detailed geometrical data are not available. Thus, it is difficult to quickly obtain the optimum thicknesses, cross-sectional geometries, and material specifications of top-hat thin-walled structures. Therefore, establishing a theoretical prediction model to describe the bending collapse of top-hat thin-walled structures at the conceptual design stage can allow the energy-absorption characteristics of top-hat structures to be quickly obtained, and thus increase design efficiency and shorten the developmental cycle time. To predict the bending collapse and energy absorption of top-hat thin-walled structures with different thickness and material specification (i.e., the same materials with the same thickness, the same materials with different thickness, the different materials with the same thickness, and the different materials with the different thickness), each of the top-hat thin-walled structures investigated in this study is separated into a top-hat element and flat-plate element according to its unique deformation mode. Fig. 1 shows the theoretical prediction model for the bending collapse of top-hat thin-walled structure; as previously mentioned, this model is developed based on the bending collapse theory of Kecman [18]. Fig. 1(a) shows the theoretical model to describe the structures before bending deformation; l is the axial length of the top-hat thinwalled structure, a is the width of the flange, b is the height of the web, f is the width of the lip, that is the thickness of the top-hat element, tpl is the thickness of the flat-plate element, and D is the center point of the hinging line KL. A rectangular coordinate system is established; K is the reference point, and the hinging lines KL, KG, and KK1 correspond to axes x, y, and z, respectively. Fig. 1(b) shows the model that is developed to describe the bending collapse of the top-hat element. As the structure is axially symmetric, the theoretical model shown in Fig. 1(b) only describes one of the two symmetric halves of the top-hat element. As a top-hat thin-walled structure starts to collapse locally under a bending load, the hinging line BC moves downward. When Points A, B, and C are positioned along the same horizontal line, collapse is apparent in Areas CBEF and CBGH. As the bending collapse further develops, Point A moves away from Point B and approaches Point D from the y direction. During this time, the collapsed areas CBEF and CBGH come into contact and are pressed 2

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

Fig. 2. Schematic of the longitudinal section of the top-hat element after bending collapse.

2.1. Theoretical prediction formula for bending collapse and energy absorption Fig. 2 shows a schematic illustration of the longitudinal section of the top-hat element after bending collapse; θ is the plastic rotation angle of the top-hat element that satisfies the following equation: θ/ 2 = ρ. Assume KL = 2h, where h is equal to the smaller value between a and b according to the minimum energy absorption principle [18]. The coordinates of Point B can be obtained via the following equations: (1)

xB = h yB = b cos

b sin (2h

(2)

b sin p)

(3)

zB = 0

Fig. 3 shows the position of Point A in plane yz, which is perpendicular to the axis of the top-hat thin-walled beam. The following equation can be obtained in consideration of cross-sectional continuity: BA + AD = b , where

Fig. 1. Theoretical prediction model for three-point bending collapse of top-hat thin-walled structure: (a) theoretical model before bending deformation; (b) theoretical model of the top-hat element; (c) theoretical model of the flat-plate element.

BA = zA

(4)

yA = yB

(5)

b = zA +

yA2 + zA2

(6)

The following equation can be derived from Equations (2), (5) and (6):

z A = b sin2

against each other. As the plastic rotation angle θ increases, the mutual interference in the theoretical model intensifies, and the plastic hinging lines are broken. In the proposed theoretical model, failure is defined as the initiation of irregular plastic deformation. According to the bending collapse behavior of the top-hat element, the plastic hinging lines in the top-hat element are classified as either fixed or rolling hinging lines. Specifically, the lines of AK, AL, AB, BC, GH, EF, KL, KK1, and LL1 are fixed hinging lines, whereas the lines of AG and AE are rolling hinging lines, as shown in Fig. 1(b). Fig. 1(c) shows the model for the bending collapse of the flat-plate element. Like line segments KL and K1L1 in the top-hat element, line segments K2L2 and N2M2 in the flat-plate element are considered to remain straight and fixed in length. As the plastic rotation angle θ increases in this model, the hinging lines K2N2 and L2M2 become merges. The two hinging lines in the model for the bending collapse of the flatplate element, i.e., K2N2 and L2M2, are considered to be fixed hinging lines, as shown in Fig. 1(c).

h sin

+

b sin (2h

b sin ) cos

Fig. 3. Sectional view of Plane BAD.

3

(7)

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

The following equation can be obtained in consideration of the fact that GB = EB = h, and GK = EL = b, as shown in Fig. 1(b):

b sin h

= arcsin 1

where ρ is the rotation angle at hinging lines KK1 and LL1. From the geometric relationship shown in Fig. 3, it can be determined that the rotation angle ξ at hinging line KL satisfies the following equation:

(8)

When the two collapsed areas CBEF and CBGH come into contact with each other and mutual interference occurs in the top-hat element, the interference angle satisfies the following equation: J

= 2 arcsin

h

0.5that b

(9)

Ehat

= /2

arcsin 1

b sin h

1

= 2M0

a 2

hat

= M0

hat a

arcsin 1

2

(10)

b sin h

M0

hat

=

(11)

=

2

2 arcsin 1

Ehat

2

= M0

hat

2 arcsin × 1

Ehat

3

= M0

hat z A

2arc sin × 1

BAQ = arc sin

AB AQ

= arc sin

zA h

4

= 2M0

hat barc

sin

zA h

5

= 2M0

hat f

(24) (25)

n1 n2 n1

(26)

n2

= 2M0

hat

8

=2

2M0 r

(27)

KA

hat

r = r ( ) = 0.07

hz A 2M0 hat hz A = 2 r

(28)

70

h

(29)

The amount of energy absorbed by the top-hat element during bending collapse can then be expressed by the following equation:

(15)

8

Ehat ( ) = 2

Ehat i ( ) 1

(30)

2.3. Computation of the amount of energy absorbed by the flat-plate element

(16)

In the model for the bending collapse of the flat-plate element, there are only two fixed hinging lines, K2N2 and L2M2, as shown in Fig. 1(c). The amount of energy absorbed by the two hinging lines K2N2 and L2M2 can be obtained via the following equation:

(17)

Epl

The amount of energy absorbed by hinging lines KK1 and LL1 can be obtained via the following equation:

Ehat

y 2A + zA2

where r is the rolling radius of the plastic hinging line; its approximate value can be obtained via the following equation:

(14)

The amount of energy absorbed by hinging lines GK and EF can then be obtained via the following equation:

Ehat

7

Ehat

From the geometric relationship shown in Fig. 1(b), it can be determined that the rotation angle η at hinging lines GK and EL satisfies the following equation:

=

(23)

The amount of energy absorbed by rolling hinging lines AG and AE can then be obtained via the following equation:

The rotation angle of hinging line AB is equal to (π-2β), and its length can be obtained via Equation (7). The amount of energy absorbed by hinging line AB can then be obtained via the following equation:

b sin h

(21)

The amount of energy absorbed by fixed hinging lines GK and EL can then be obtained via the following equation:

(13)

b sin h

b cos , z A)

KL = (2h, 0,0)

= arc cos

The amount of energy absorbed by hinging line BC can be determined via the following equation:

a 2

(20)

(22)

Ehat

b sin h

zA yA

n1 = GA × KA , n2 = KA × KL

where 0 hat and that are the maximum flow stress and thickness of the top-hat element, respectively. The rotation angle at fixed hinging line BC satisfies the following equation:

2

2harc tan

KA = (0, yA , z A )

KA =

(12)

4

hat

By designating the normal vectors of Planes GKA and KAL as n1 and n2 , respectively, and the plastic rotation angle of hinging line AK as λ, the following equations can be derived:

where M0-hat is the plastic bending moment per unit length of the tophat element, and it can be obtained via the following equation: 2 0 hat that

= 2M0

GA = ( b sin p , yA

The amount of energy absorbed by hinging lines GH and EF can then be obtained via the following equation:

Ehat

6

From the geometric relationship shown in Fig. 1(b), it can be determined that the coordinates of Points A, G, K, and L are (0, yA, zA), (bsin ρ, bcos ρ, 0), (0, 0, 0), and (2h, 0, 0), respectively. The following equations can then be derived:

In the model for the bending collapse of the top-hat element, there are fixed hinging lines AK, AL, AB, BC, GH, EF, KL, KK1, and LL1, and rolling hinging lines AG and AE, as shown in Fig. 1(b). From Fig. 2, it can be determined that the rotation angle α of hinging lines GH and EF satisfies the following equation:

/2

(19)

The amount of energy absorbed by hinging line KL can then be obtained via the following equation:

2.2. Computation of the amount of energy absorbed by the top-hat element

=

zA yA

= arc tan

1

= 2M0

PL

2

a +f 2

(31)

where M0-pl is the plastic bending moment per unit length of the flatplate element; it can be obtained via the following equation:

(18) 4

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

Fig. 4. Typically top-hat thin-walled structures subjected to transverse loading: (a) front longitudinal beam; (b) B-pillar.

Fig. 5. Geometry description of the top-hat thin-walled structure: (a) top view; (b) side view.

Fig. 6. Experimental specimens: (a) specimen#01; (b) specimen#02; (c) specimen#03.

bending collapse can then be expressed by the following equation:

Table 1 The summary of test specimens. top-hat section

top-hat element

Epl ( ) = Epl

flat-plate element

1

(33)

2.4. Total energy absorption Specimens specimen#01 specimen#02 specimen#03

M0

pl

=

2 0 pl tpl

4

that (mm)

Mathat

tpl (mm)

Matpl

1.4 1.6 1.8

HSLA340 HSLA340 HSLA340

1.4 1.6 1.8

HSLA340 HSLA340 HSLA340

The total energy absorbed by a top-hat thin-walled structure during the three-point bending collapse can then be expressed as

Etotal ( ) = Ehat ( ) + Epl ( )

(34)

where Etotal ( ) is the total energy absorbed by a top-hat thin-walled structure. 3. Validation of finite element model and theoretical prediction model

(32)

where 0 pl and tpl are the maximum flow stress and thickness of the flat-plate element, respectively. The amount of energy absorbed by the flat-plate element during

3.1. Geometry description Top-hat thin-walled structures are commonly used in sill, front 5

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

Table 2 Chemical composition (Wt%) of HSLA340.

Table 3 Relative errors for theoretical predictions, FE simulations and tests.

Alloying elements

weight %

C Si Mn P S Alt

≤0.15 ≤0.6 ≤2.5 ≤0.04 ≤0.015 ≤2.0

Specimens

specimen#01 specimen#02 specimen#03

Test

FE simulation

Theory

EA (J)

EA (J)

Error (%)

EA (J)

Error (%)

599.03 788.37 997.01

605.01 782.81 982.23

0.998% 0.705% −1.48%

622.93 813.85 1029.7

3.98% 3.23% 3.27%

investigated material is summarized in Table 2. The Young's modulus E=210 GPa, Poisson's ratio = 0.3, Density ρ=7.83 × 103 kg/m3, initial yield stress s = 359MPa , and the ultimate stress b = 439MPa . In Table 1, that is the thickness of hat-shaped part, tpl is the thickness of flat plate, Mathat is the material of hat-shaped part and Matpl is the material of flat plate. The three-point bending loading for the test specimens are shown in Fig. 7. The bending tests are carried out on an INSTRON (MTS647) electronic universal testing machine. The specimens lay on two cylindrical supports. The diameter of the supports is 25 mm and the span L0 is 300 mm. A cylindrical punch with a diameter of 25 mm impacts on the mid-span of the specimens at a constant impact velocity of 5 mm/ min. The maximum punch displacement is set to be 60 mm. 3.3. Finite element modeling In this study, the finite element (FE) model of the top-hat thinwalled structure is established using commercial software LS-DYNA [42]. The specimens, supports and punch are modelled using Belytschko-Tsay reduced integration shell elements with five integration points throughout the thickness. The piecewise linear plasticity material law (Mat 24 in LS-DYNA) is chosen as the material model in the FE models. Because the deformation of supports and punch is very limited, they are modelled as rigid body. Stiffness-type hourglass control is used to avoid spurious zero energy modes. The contact algorithm among the punch, supports and the specimen is “surface-to-surface”. The “automatic single surface” contact is used to prescribe the specimen to avoid interpenetration. The FE model of top-hat thin-walled structure is shown in Fig. 8.

Fig. 7. Experimental setup of top-hat thin-walled structures.

longitudinal beam and B-pillar etc., which often collapse in bending mode (see Fig. 4). In this study, a top-hat section of 105 mm × 85 mm and 35 mm flange is selected as a ‘standard’ section, which is composed of two parts: one is a hat-shaped cross-section and the other is a flat plate. The two parts are joined by spot-welding along the center line of the flange width at intervals of 30 mm and a spot-weld diameter of approximately 6 mm. The corner radii at the top edges and near the weld flanges are approximately 4 mm, respectively. The total length L of the specimen is 400 mm. The geometry description of the top-hat thin-walled structure is shown in Fig. 5.

3.4. Validation of FE model and theoretical prediction model

3.2. Experimental set-up and test specimens

The theoretical predictions of the energy absorptions for the three specimens are calculated by Eq. (34) and these values are used to compare with the results of tests and FE simulations. The relative errors for theoretical predictions, FE simulations and tests are listed in Table 3. The energy absorption vs. rotation angle curves among the theoretical predictions, FE simulations and tests of the three specimens

The quasi-static bending tests are conducted on the top-hat thinwalled structures. In this work, three different specimens are considered to validate the simulation accuracy of the FE model, as shown in Fig. 6. The details of the specimens are listed in Table 1. The steel grade used in the specimens is HSLA340. The chemical composition of the

Fig. 8. FE model of top-hat thin-walled structure: (a) front view; (b) 3D view. 6

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

Fig. 10. Schematic of dimension parameters of the top-hat section.

are compared in Fig. 9. It is revealed from Table 3 and Fig. 9 that the theoretical predictions and the numerical values are in good agreement with the experimental values, and that the accuracy of the theoretical prediction model and the FE model are controllable within 5%. In addition, we can see from Fig. 9 that the final deformation patterns predicted by FE simulations agree well with the tests. Therefore, the established FE model is very accurate and can be used to replace the physical test in this paper. 3.5. Generality verification of the theoretical prediction model In this section, six top-hat thin-walled structures with different combination of cross section dimensions, thicknesses and materials are adopted to demonstrate the generality of the theoretical prediction model. Four materials, BLC, DP780, 6060-T6 and 6061-T6 commonly used in manufacturing of these top-hat thin-walled structures in vehicle industry, are selected for the analysis with properties listed in Table 4. To illustrate more clearly, the dimension parameters of the top-hat section is shown in Fig. 10. The details of the numerical top-hat thinwalled structures are given in Table 5. It includes the cases of a > b, a = b and a < b; the combinations between top-hat element and flatplate element under four different cases (i.e., the same materials with the same thickness, the same materials with different thickness, the different materials with the same thickness, and the different materials with the different thickness) are also given. The values of theoretical predictions are used to compare with the results of FE simulations. The compared relative errors between FE simulations and theoretical predictions are listed in Table 5. It is revealed from Table 5 that the theoretical predictions are in good agreement with the numerical values, and that the accuracy of the theoretical prediction model is controllable within 5%. In addition, the energy absorption vs. rotation angle curves between FE simulations and theoretical predictions are compared in Fig. 11. We can see from Fig. 11 that the energy absorption vs. rotation angle curves obtained by theoretical predictions agree well with the FE simulations. The aforementioned analysis demonstrates that the proposed theoretical prediction model shows high accuracy and generality to capture the energy absorption characteristics of top-hat thin-walled structures with combinations of cross section dimensions, thicknesses and materials. Fig. 9. Comparison of energy absorption vs. rotation angle curves: (a) specimen #01, (b) specimen #02, (c) specimen #03.

4. Crashworthiness design optimization Apart from the aforementioned theoretical analysis, the FEA could act as a generic tool for evaluating the crashworthiness of the top-hat thin-walled structures. In this section, theoretical formulas and FEA based surrogate models are used to solve the same optimization problems for comparison. The flowchart of these two optimization procedures is shown in Fig. 12.

Table 4 Material properties. Material

Young's modulus (GPa)

Poisson ratio

Density (kg/m3)

Yield strength (MPa)

Ultimate stress (MPa)

BLC DP780 6060-T6 6061-T6

210 210 70 70

0.3 0.3 0.3 0.3

7.83 × 103 7.83 × 103 2.7 × 103 2.7 × 103

183 394 208 276

395 848 248 341

4.1. Design responses and variables As a weight-efficient energy absorber, the top-hat thin-walled structure is expected to have higher energy absorption capacity to reduce the kinetic energy transmitting to the occupants. Besides, the 7

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

Table 5 Details of the numerical top-hat thin-walled structures. No.

Section dimensions (a × b × f) (mm)

that (mm)

tpl (mm)

Mathat

Matpl

Sim. (J)

Theory (J)

Error (%)

1 2 3 4 5 6 7 8

110 × 80 × 40 60 × 30 × 15 100 × 100 × 35 70 × 80 × 30 60 × 90 × 20 50 × 100 × 25 107 × 73 × 35 65 × 65 × 35

1.3 1.0 1.0 1.8 0.8 1.8 1.7 1.2

1.3 1.0 1.3 1.8 0.8 0.8 0.8 0.8

BLC DP780 BLC 6060-T6 6061-T6 6060-T6 BLC 6061-T6

BLC DP780 DP780 6060-T6 6061-T6 6061-T6 6060-T6 DP780

449.31 228.10 331.51 489.95 138.52 534.45 693.11 280.21

463.71 235.62 344.56 513.98 144.06 549.61 712.73 267.38

3.20% 3.29% 3.93% 4.9% 3.99% 2.83% 2.83% −4.57%

smaller the weight of the top-hat thin-walled structure is, the lower cost for the structure will be. Therefore, the energy absorption and the weight of the top-hat thin-walled structure are chosen as the output responses, and they are represented by EA and Mass, respectively. To maximize the weight reduction and improve energy absorber efficiency, combinations of shape, size and material optimization seems to be a promising way. In this paper, the selection of design variables is based on design requirements and the number of design parameters (optimization time). Specifically, six design variables including 2 thickness variables, 2 material type variables and 2 cross-sectional shape variables are defined in the top-hat thin-walled structure. The details of the design variables are listed in Table 6.

According to Section 2, the theoretical solutions of energy absorption for the top-hat thin-walled structure can be derived in Eq. (34). 4.3.2. Response functions of numerical surrogate-based optimization The surrogate model-based methods have been widely adopted to improve computational efficiency for large-scale engineering problems [47–49]. In the numerical surrogate-based optimization procedure, εsupport vector regression (ε-SVR) technique [46,50–54] is employed to formulate the sophisticated output responses. Firstly, 120 sampling points are generated using the optimal Latin hypercube sampling (OLHS) technique [55,56]. Then, the output responses are calculated using commercial code LS-DYNA. Thirdly, the ε-SVR models are constructed to approximate the output functions. After surrogate models are constructed, their accuracy should be assessed. Thus, 20 samples are newly generated to validate the prediction accuracy of the constructed ε-SVR models. Herein, three numerical estimators, namely R-square (R2), relative average absolute error (RAAE) and relative maximum absolute error (RMAE), are used. R2 and RAAE are used to measure the overall accuracy of the model in the design space, while RMAE is used to measure the local accuracy of the model [46,57,58]. The results of the accuracy assessment for the ε-SVR models are listed in Table 8. It can be concluded from Table 8 that the constructed ε-SVR models are reasonably accurate and they can be used to substitute the high-fidelity FE model of the top-hat thin-walled structure.

4.2. Definition of optimization problem In the optimization models, EA is a key indicator to evaluate the energy absorption capacity of thin-walled structure [43]. For this reason, EA should be selected as an objective function and expected to be maximized. Besides, the weight of the top-hat thin-walled structure is also an important indicator of the cost of energy absorber. Thus, Mass would be as another objective function and expected to be minimized. Considering real requirements, the multi-objective optimization (MOO) model for the light-weight and crashworthiness design of the top-hat thin-walled structure can be formulated as

min s. t.

{EA, Mass} 50mm a 120mm 50mm b 120mm 0.7mm that 2mm 0.7mm tpl 2mm Mathat Matpl

5. Results and discussions

[DP780,HSLA340, BLC, 6060 [DP780,HSLA340, BLC, 6060

T6,6061 T6,6061

T6] T6]

To compare the results of the theoretical-formulas based optimization and the numerical surrogate-based optimization, their Pareto optimal frontiers are plotted in Fig. 13. In order to quantitatively compare the performances of light-weight and crashworthiness for the optimized top-hat thin-walled structures and the initial counterpart, the best compromise solution, the optimal solutions of Mass and EA are selected from the Pareto optimal frontier of the numerical surrogate-based optimization, as shown in Fig. 13. The minimum distance selection method (MDSM) [59] is adopted to find the best compromise solution from the Pareto optimal set (i.e. the optimum point A). The optimal solution of Mass (i.e. the optimum point B) can be selected from the intersection between the Pareto optimal frontier and the horizontal line of EA =782.81 J. The optimal solution of EA (i.e. the optimum point C) can be selected from the intersection between the Pareto optimal frontier and the vertical line of Mass = 2.58 kg. The detailed design parameters between baseline and optimal solutions are compared in Table 9. It can be found from Table 9 that the thicknesses, cross-sectional geometry, and material specifications of the top-hat thin-walled structures are significantly improved after design optimization. The results and relative errors of the selected optimal solutions are listed in Table 10. We can see that the optimal solutions generated from the ε-SVRs and theoretical predictions have sufficient accuracy when compared to the results of FEA. It is revealed from Table 10 that the results of ε-SVRs and theoretical predictions are in good agreement with

(35)

In this study, the Pareto optimal frontier of the optimization problem (35) is obtained by non-dominated sorting genetic algorithm-II (NSGA-II) [44]. The NSGA-II algorithm is very useful for its in fast convergence and well-distributed Pareto frontier. It has also been successfully employed in crashworthiness applications [45], such as variable-rolled-blank thin-walled structures [34,46]. The detailed parameters of the NSGA-II algorithm are summarized in Table 7. 4.3. Response functions 4.3.1. Response functions of theoretical formulas-based optimization The main difference between the theoretical-formulas based optimization and the numerical surrogate-based optimization is the assessment of response functions (i.e., objective functions). In the theoretical-formulas based optimization, the mass function can be calculated as

Masstheo . = L [

hat that (a

+ 2b + 2f ) +

pl tpl (a

+ 2f )]

(36)

where ρhat is the density of the hat-shaped part, ρpl is the density of the flat plate, f is the width of the flange with 35 mm, and L is the total length of the top-hat thin-walled structure with 400 mm. 8

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

Fig. 11. Comparison of energy absorption vs. rotation angle curves between FE simulations and theoretical predictions: (a) No. 1; (b) No. 2; (c) No. 3; (d) No. 4; (e) No. 5; (f) No. 6; (g) No. 7; (h) No. 8.

the numerical values, and that the accuracy of the constructed ε-SVRs and theoretical prediction model are controllable within 5%. In this study, the results of FEA under the selected optimal solutions are chosen to indicate the performance improvements before and after optimization (see, Table 11). According to Table 11, the following results can be summarized: (1) The reduction percentage in weight and improvement percentage in energy absorption by optimal solution A

are 36.78% and 63.68%, respectively; (2) The optimal solution B improves the energy absorption by 249.51% without exceeding the initial weight, improving it from the initial 782.81 J–2736.02 J; (3) The optimal solution C achieves a maximum weight reduction of 69.22% without sacrificing the energy absorption of the baseline design. In summary, steel-aluminum hybrid top-hat thin-walled structure has a larger energy-absorption capacity than high-strength steel 9

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

Fig. 12. Flowchart of the two optimization procedures. Table 6 Description of design variables. Variable name

Symbol

Lower bound

Upper bound

Width of cross section Height of cross section Thickness of hat-shaped part Thickness of flat plate Material of hat-shaped part Material of flat plate

a b that tpl Mathat Matpl

50 mm 120 mm 50 mm 120 mm 0.7 mm 2.0 mm 0.7 mm 2.0 mm [DP780, HSLA340, BLC, 6060-T6, 6061-T6] [DP780, HSLA340, BLC, 6060-T6, 6061-T6]

Baseline design 105 mm 85 mm 1.6 mm 1.6 mm HSLA340 HSLA340

Table 7 Settings of the NSGA-II algorithm. NSGA-II parameter name

Values

Population size Crossover Probability Mutation Probability Crossover Distribution Index Mutation Distribution Index

40 0.9 0.9 10 20

Table 8 Accuracy assessment of the ε-SVR models. Responses

R2

RAAE

RMAE

Masssim. EA sim.

0.9819 0.9781

0.0550 0.0523

0.3300 0.4186

without exceeding the initial weight, whereas a lightweight design is more feasible with an aluminum alloy than with high-strength steel without sacrificing the energy absorption of the baseline design.

Fig. 13. Comparison of Pareto optimal frontiers from theoretical-formulas based optimization and surrogate-based optimization.

(1) A theoretical model is developed to reveal the bending collapse of top-hat thin-walled structures by dividing a top-hat thin-walled structure into a top-hat element and flat-plate element. A theoretical formula is also developed to describe the bending deformation and energy-absorption of the structures. The theoretical model is capable of predicting the bending collapse and energy-absorption of

6. Conclusions In this paper, theoretical prediction and crashworthiness optimization of top-hat thin-walled structures under transverse loading is investigated. Some interesting findings and contributions are summarized as follows: 10

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al.

2018M640460). The authors also wish to thank Scientific Research Grant of Jiangsu University (No. 17JDG037). This work was supported by the high-performance computing platform of Jiangsu University.

Table 9 Comparison of design parameters between baseline and optimal solutions. Description

a (mm)

b (mm)

that (mm)

tpl (mm)

Mathat

Matpl

Baseline design Optimum A Optimum B Optimum C

105 74.06 111.27 72.01

85 99.06 97.31 115.70

1.6 1.44 1.83 1.68

1.6 1.04 1.88 0.91

HSLA340 DP780 DP780 6061-T6

HSLA340 6061-T6 6061-T6 6061-T6

References [1] L.B. Duan, et al., An efficient lightweight design strategy for body-in-white based on implicit parameterization technique, Struct. Multidiscip. Optim. 55 (5) (2017) 1927–1943. [2] A. Baroutaji, M. Sajjia, A.G. Olabi, On the crashworthiness performance of thinwalled energy absorbers: recent advances and future developments, Thin-Walled Struct. 118 (2017) 137–163. [3] J. Fang, et al., On design optimization for structural crashworthiness and its state of the art, Struct. Multidiscip. Optim. 55 (3) (2017) 1091–1119. [4] F. Xu, X. Zhang, H. Zhang, A review on functionally graded structures and materials for energy absorption, Eng. Struct. 171 (2018) 309–325. [5] J. Fang, et al., On hierarchical honeycombs under out-of-plane crushing, Int. J. Solids Struct. 135 (2017) 1–13. [6] A. JM, An approximate the collapse of thin cylindrical shells under axial loading, Journal of Mechanics & Applied Mathematics 13 (1) (1960) 10–15. [7] The effective crushing distance in axially compressed thin-walled metal columns, Int. J. Impact Eng. 1 (3) (1983) 309–317. [8] W. Abramowicz, N. Jones, Dynamic axial crushing of square tubes, Int. J. Impact Eng. 2 (2) (1984) 179–208. [9] W. Abramowicz, N. Jones, Dynamic progressive buckling of circular and square tubes, Int. J. Impact Eng. 4 (4) (1986) 243–270. [10] T. Wierzbicki, W. Abramowicz, On the crushing mechanics of thin-walled structures, J. Appl. Mech. 50 (4a) (1983) 727–734. [11] G.Y. Sun, et al., Multi-objective and multi-case reliability-based design optimization for tailor rolled blank (TRB) structures, Struct. Multidiscip. Optim. 55 (5) (2016) 1899–1916. [12] X. Zhang, H. Zhang, Energy absorption of multi-cell stub columns under axial compression, Thin-Walled Struct. 68 (2013) 156–163. [13] J.G. Fang, et al., On design of multi-cell tubes under axial and oblique impact loads, Thin-Walled Struct. 95 (2015) 115–126. [14] J. Song, F.L. Guo, A comparative study on the windowed and multi-cell square tubes under axial and oblique loading, Thin-Walled Struct. 66 (2013) 9–14. [15] T. Tran, et al., Theoretical prediction and crashworthiness optimization of multi-cell square tubes under oblique impact loading, Int. J. Mech. Sci. 89 (2014) 177–193. [16] T. Tran, et al., Theoretical prediction and crashworthiness optimization of multi-cell triangular tubes, Thin-Walled Struct. 82 (2014) 183–195. [17] G. Sun, et al., Configurational optimization of multi-cell topologies for multiple oblique loads, Struct. Multidiscip. Optim. 57 (2) (2017) 469–488. [18] K. D, Bending collapse of rectangular and square section tubes, Int. J. Mech. Sci. 25 (9) (1983) 623–636. [19] T.H. Kim, S.R. Reid, Bending collapse of thin-walled rectangular section columns, Comput. Struct. 79 (20–21) (2001) 1897–1911. [20] M. Kotelko, T.H. Lim, J. Rhodes, Post-failure behaviour of box section beams under pure bending (an experimental study), Thin-Walled Struct. 38 (2) (2000) 179–194. [21] S. Maduliat, T.D. Ngo, P. Tran, Energy absorption of steel hollow tubes under bending, Proceedings of the Institution of Civil Engineers-Structures and Buildings 168 (12) (2015) 930–942. [22] X. Zhang, G. Cheng, H. Zhang, Theoretical prediction and numerical simulation of multi-cell square thin-walled structures, Thin-Walled Struct. 44 (11) (2006) 1185–1191. [23] Z. Wang, Z.H. Li, X. Zhang, Bending resistance of thin-walled multi-cell square tubes, Thin-Walled Struct. 107 (2016) 287–299. [24] Z. Wang, X. Zhang, Z.H. Li, Bending collapse of multi-cell tubes, Int. J. Mech. Sci. 134 (2017) 445–459. [25] Z.P. Du, et al., Theoretical prediction and crashworthiness optimization of thinwalled structures with single-box multi-cell section under three-point bending loading, Int. J. Mech. Sci. 157–158 (2019) 703–714. [26] W.G. Chen, Experimental and numerical study on bending collapse of aluminum foam-filled hat profiles, Int. J. Solids Struct. 38 (44–45) (2001) 7919–7944. [27] K. Sato, et al., Effect of material properties of advanced high strength steels on bending crash performance of hat-shaped structure, Int. J. Impact Eng. 54 (2013) 1–10. [28] S.S P, N.V.N.V., L. ea Gunawan, The influence of forming effects on the bending crush behavior of top-hat thin-walled beams, 25 (2016) 63–72. [29] G.Y. Sun, et al., Crashing analysis and multiobjective optimization for thin-walled structures with functionally graded thickness, Int. J. Impact Eng. 64 (2014) 62–74. [30] J.G. Fang, et al., Dynamic crashing behavior of new extrudable multi-cell tubes with a functionally graded thickness, Int. J. Mech. Sci. 103 (2015) 63–73. [31] G.Y. Sun, et al., Dynamical bending analysis and optimization design for functionally graded thickness (FGT) tube, Int. J. Impact Eng. 78 (2015) 128–137. [32] X.Z. An, et al., Crashworthiness design for foam-filled thin-walled structures with functionally lateral graded thickness sheets, Thin-Walled Struct. 91 (2015) 63–71. [33] F.X. Xu, X.J. Wan, Y.S. Chen, Design optimization of thin-walled circular tubular structures with graded thickness under later impact loading, Int. J. Automot. Technol. 18 (3) (2017) 439–449. [34] L. Duan, et al., Bending analysis and design optimisation of tailor-rolled blank thinwalled structures with top-hat sections, Int. J. Crashworthiness 22 (3) (2017) 227–242. [35] G.Y. Sun, et al., Crashworthiness optimization of automotive parts with tailor rolled

Table 10 Comparison of the results and relative errors of the selected optimal solutions. Description

Symbol

FEA

ε-SVR

Error (%)

Theory

Error (%)

Optimum A

EA (J) Mass (kg) EA (J) Mass (kg) EA (J) Mass (kg)

1281.36 1.63 2736.02 2.56 789.32 0.79

1294.43 1.55 2845.94 2.58 782.81 0.76

1.02% −4.90% 4.01% 0.74% −0.82% −3.79%

1302.01 1.69 2708.60 2.53 799.21 0.81

1.61% 3.61% −1.00% −1.17% 1.25% 2.64%

Optimum B Optimum C

Table 11 Performance improvements before and after optimization. Description

EA (J)

Mass (kg)

Baseline design Optimum A Improvement (%) Optimum B Improvement (%) Optimum C Improvement (%)

782.81 1281.36 63.68% 2736.02 249.51% 789.32 0.83%

2.58 1.63 −36.78% 2.56 −0.73% 0.79 −69.22%

top-hat thin-walled structures with the following thickness and material specification: the same materials with the same thickness, the same materials with different thickness, the different materials with the same thickness, and the different materials with the different thickness. The accuracy and generality of the theoretical prediction model is validated by performing three-point bending tests and finite element simulations. (2) Both theoretical prediction formulas and finite element analysis (FEA) are employed to perform the crashworthiness optimization of top-hat thin-walled structures. The results show that (i) the Pareto optimum frontier obtained via the theoretical prediction model agrees well with that obtained via a finite element method, thereby confirming the accuracy and general applicability of the theoretical prediction model; (ii) the theoretical prediction model is capable of producing results that can be directly used to optimize the thicknesses, cross-sectional geometry, and material specifications for top-hat thin-walled structures, which will increase the efficiency and shorten the cycle time of crashworthiness design optimization for this type of structure; (iii) steel-aluminum hybrid top-hat thinwalled structure has a larger energy-absorption capacity than highstrength steel without exceeding the initial weight, whereas a lightweight design is more feasible with an aluminum alloy than with high-strength steel without sacrificing the energy absorption of the baseline design. To sum up, the performances of light-weight and crashworthiness of the steel-aluminum hybrid top-hat thinwalled structures are more superior to the high-strength steel. Acknowledgments The authors would like to thank the support of National Natural Science Foundation of China (Grant No. 51805221) and Research Project funded by China Postdoctoral Science Foundation (No. 11

Thin-Walled Structures 144 (2019) 106261

L. Duan, et al. blank, Eng. Struct. 169 (2018) 201–215. [36] H.L. Zhang, et al., Bending characteristics of top-hat structures through tailor rolled blank (TRB) process, Thin-Walled Struct. 123 (2018) 420–440. [37] A. Elmarakbi, Y.X. Long, J. MacIntyre, Crash analysis and energy absorption characteristics of S-shaped longitudinal members, Thin-Walled Struct. 68 (2013) 65–74. [38] V. Beik, M. Fard, R. Jazar, Crashworthiness of tapered thin-walled S-shaped structures, Thin-Walled Struct. 102 (2016) 139–147. [39] N. Qiu, et al., Topological design of multi-cell hexagonal tubes under axial and lateral loading cases using a modified particle swarm algorithm, Appl. Math. Model. 53 (2018) 567–583. [40] J.T. Bai, Y.W. Li, W.J. Zuo, Cross-sectional shape optimisation for thin-walled beam crashworthiness with stamping constraints using genetic algorithm, Int. J. Veh. Des. 73 (1–3) (2017) 76–95. [41] K.F. Cai, D.F. Wang, Optimizing the design of automotive S-rail using grey relational analysis coupled with grey entropy measurement to improve crashworthiness, Struct. Multidiscip. Optim. 56 (6) (2017) 1539–1553. [42] Halquist, J., LS-DYNA Keyword User's Manual Version 971. 2007, Livermore, CA: Livermore Software Technology Corporation. [43] F.X. Xu, et al., Multi-response optimization design of tailor-welded blank (TWB) thin-walled structures using Taguchi-based gray relational analysis, Thin-Walled Struct. 131 (2018) 286–296. [44] K. Deb, et al., A fast and elitist multiobjective genetic algorithm: NSGA-II. Evolutionary Computation, IEEE Transactions 6 (2002) 182–197. [45] X.T. Liao, et al., Multiobjective optimization for crash safety design of vehicles using stepwise regression model, Struct. Multidiscip. Optim. 35 (6) (2008) 561–569. [46] L.B. Duan, et al., Design optimization of tailor-rolled blank thin-walled structures based on epsilon-support vector regression technique and genetic algorithm, Eng. Optim. 49 (7) (2017) 1148–1165. [47] N.-C. Xiao, M.J. Zuo, C. Zhou, A new adaptive sequential sampling mehtod to construct surrogate models for efficient reliability analysis, Reliab. Eng. Syst. Saf. 1

(69) (2018) 330–338. [48] N.-C. Xiao, M.J. Zuo, W. Guo, Efficient reliability analysis based on adaptive sequential sampling design and cross-validation, Appl. Math. Model. 58 (2018) 404–420. [49] N.-C. Xiao, et al., A novel reliability method for structural systems with truncated random variables, Struct. Saf. 50 (2014) 57–65. [50] S.M. Clarke, J.H. Griebsch, T.W. Simpson, Analysis of support vector regression for approximation of complex engineering analyses, J. Mech. Des. 127 (6) (2005) 1077–1087. [51] L.B. Duan, et al., Crashworthiness design of vehicle structure with tailor rolled blank, Struct. Multidiscip. Optim. 53 (2) (2016) 321–338. [52] X.G. Song, et al., Crashworthiness optimization of foam-filled tapered thin-walled structure using multiple surrogate models, Struct. Multidiscip. Optim. 47 (2) (2013) 221–231. [53] L. Duan, et al., Multi-objective reliability-based design optimization for the VRBVCS FLB under front-impact collision, Struct. Multidiscip. Optim. 59 (5) (2019) 1789–1812. [54] L. Duan, et al., Parametric modeling and multiobjective crashworthiness design optimization of a new front longitudinal beam, Struct. Multidiscip. Optim. 59 (5) (2019) 1835–1851. [55] P. JS, Optimal Latin-hypercube designs for computer experiments, J. Stat. Plan. Inference 39 (1) (1994) 95–111. [56] M.D. Shields, J.X. Zhang, The generalization of Latin hypercube sampling, Reliab. Eng. Syst. Saf. 148 (2016) 96–108. [57] L.B. Duan, et al., Multi-objective system reliability-based optimization method for design of a fully parametric concept car body, Eng. Optim. 49 (7) (2017) 1247–1263. [58] Z. Li, et al., Crashworthiness analysis and multi-objective design optimization of a novel lotus root filled tube (LFT), Struct. Multidiscip. Optim. 57 (2) (2017) 865–875. [59] G. Sun, et al., Crashworthiness design of vehicle by using multiobjective robust optimization, Struct. Multidiscip. Optim. 44 (1) (2010) 99–110.

12