Crushing behavior of circumferentially corrugated square tube with different cross inner ribs

Thin-Walled Structures 144 (2019) 106370

Contents lists available at ScienceDirect

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

Full length article

Crushing behavior of circumferentially corrugated square tube with different cross inner ribs

T

Zhixiang Lia,b,c, Wen Maa,b,c, Ping Xua,b,c,d, Shuguang Yaoa,b,c,∗ a

Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha, 410075, China Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Central South University, Changsha, 410075, China c National & Local Joint Engineering Research Center of Safety Technology for Rail Vehicle, Central South University, Changsha, 410075, China d State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410075, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Circumferentially corrugated square tube Crushing performance Experimental Theoretical Parametric study

In this paper, new designed circumferentially corrugated square tubes (CCSTs) with cosine and triangular corrugation cross-sectional profiles, respectively, are proposed for energy absorption. Three types of cross inner ribs, namely, crisscross, X shape and star shape, are introduced to induce the deformation of CCSTs. Quasi-static crushing test and finite element simulation were used to investigate the crushing performance of the CCSTs. The results showed that all triangular profile CCSTs can not deform stably, and only the crisscross inner rib can induce the cosine profile CCSTs to deform progressively. The performance of cosine profile CCSTs with crisscross inner rib (i.e., CTC_C) under different geometric parameters was therefore further studied. It was found that the CTC_C develops four different deformation modes, and the ordered progressive mode Ⅱ (OPM Ⅱ) showed better energy absorption efficiency than the other modes. Finally, a theoretical model based on Simplified Super Folding Element method was presented to predict the mean crushing force of CTC_C deforming with OPM Ⅱ. By comparing with the experimental and FE results, the theoretical model was proved to be very accurate.

1. Introduction Thin-walled tubes are widely used for energy absorption in many industry fields [1–3]. The crashworthiness performance of thin-walled tubes can be affected by many factors, such as material properties, wall thickness, cross-sectional configuration, and geometric imperfections, among others [4–7]. Among these factors, cross-sectional configuration is one of the most important one. To realize better energy absorption efficiency, various cross-sectional configurations have been introduced into thin-walled tubes by researchers [8–11]. Previous studies have indicated that most of the energy is absorbed by bending deformation and membrane deformation along the bending hinge line, especially in the corner region [12]. Therefore, increasing the number of corners is a good way to increase the energy absorption efficiency of thin-walled tubes. The polygonal configuration can realize the purpose of increasing of corners number for single cell columns. Increasing the number of corners of a polygon means that the number of edges needs to be increased. It was found that both the crushing stability and specific energy absorption (SEA) were improved significantly when the cross-sectional

configuration changes from triangle to hexagon, but seldom from hexagon to octagon [13]. As Tang reported, the angle between neighboring flanges should be 90–120° for the highest energy absorption efficiency [14]. Therefore, the effect of increasing energy absorption efficiency by changing the edge number of polygons is limited. Sun et al. [15] obtained a so-called “crisscross tube” by pushing the four corners of square tube inward along the diagonal lines, so that adds more corner elements while ensure the angle of each corner is 90° and the perimeter of the cross section remains unchanged. The results showed that this crisscross tube had better energy absorption efficiency than square tube. Tang [14] repeatedly used the same method as Sun for square tube, and obtained a concave profile column. On the other hand, if this method is applied to other polygons, concave profile columns with different corner angles can be obtained. For instance, Reddy et al. [16] realized some concave profile columns with the corner angle between 90° and 145°. They found that corner angle of 105° is the transition from stable deformation to unstable deformation, and the concave profile columns deforming with stable mode had better energy efficiency than polygonal columns. In addition, some special shaped cross-sectional configurations were

∗ Corresponding author. Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha, 410075, China. E-mail address: [email protected] (S. Yao).

https://doi.org/10.1016/j.tws.2019.106370 Received 3 June 2019; Received in revised form 31 July 2019; Accepted 19 August 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Nomenclature ρ E σy σu ν σ0 CCST SST CTC CTT ST_C ST_X ST_S CTC_C CTC_X CTC_S CTT_C CTT_X CTT_S

CCCT CCCT_C A N L0 Lc h t η OPM NPM UM MM IPCF MCF, Pm EA SEA 2H k M0

Density Young's modulus Yield stress Ultimate stress Poisson's ratio Flow stress Circumferentially corrugated square tubes Straight square tube CCST with cosine corrugation profiles CCST with triangular corrugation profiles triangular SST with crisscross inner rib SST with X shape inner rib SST with star shape inner rib CTC with crisscross inner rib CTC with X shape inner rib CTC with star shape inner rib CTT with crisscross inner rib CTT with X shape inner rib CTT with star shape inner rib

Circumferentially corrugated circular tube CCCT with crisscross inner rib Amplitude of corrugation Number of wave crests of one side Nominal side length of CCST Total length of sectional profile Length of tube Wall thickness of tube Ratio of amplitude and 1/4 wavelength of corrugation Ordered progressive mode Non-ordered progressive mode Unstable mode Mixed mode Initial peak crushing force Mean crushing force Energy absorption Specific energy absorption Wavelength of each folded wave Effective crushing distance coefficient Fully plastic bending moment per unit length

of corrugation geometry on the crushing behavior of woven roving glass fiber/epoxy laminated composite tube. They found that radial corrugation could significantly applicable as a stable and effective energy absorber. The sinusoidal corrugation also be added on tubes along the longitudinal direction. Liu [24] studied this kind of corrugated tube and found that the corrugation can make the deformation more stable and significantly reduce the initial peak force. All the researches mentioned above show that the multi-corner cross-sectional configuration can considerably improve the energy absorption efficiency of thin-walled tubes. However, the multi-corner configuration also caused the instability of deformation. The progressive deformations of most of the above multi-corner tubes were obtained with introduced some constraints into tubes. For instance, the progressive deformations of the crisscross tube [15] and the CCCT [19] were achieved by setting one or more slots to one end of the tubes. The steady deformation of the concave profile columns [14,25] were obtained through adding some bulkheads. In fact, besides the above methods, there are many other ways to induce the tubes to deform in stable and progressive mode. Such as

also adopted to realize the multi-corner configuration and improve the crashworthiness of thin-walled tubes. For example, Liu [17,18] proposed a thin-walled tube with star-shaped cross section, in which the convex corner angles are constant and equal to 90° while the concave corner angles differs with different corner number. They found that when the number of concave corners is less than 12, the SEA of starshaped tube was worse than that of polygons with the same perimeter, and when the number of concave corners was greater than 12, the SEA of star-shaped tube was only a little higher than polygonal tube. Wu et al. [19] introduced a Fourier series expansion, which is expressed as r (θ) = r0 [1 + c1⋅cos(4θ) + c2⋅cos(8θ)], to control the cross-sectional configuration of tubes. Different numbers of corners and degrees of corner angles can be achieved by changing the values of c1 and c2. The results showed that the collapse modes of Fourier varying section tubes are fairly sensitive to cross-sectional configuration. Deng [20,21] designed a circumferentially corrugated circular tube (CCCT) with a sinusoidal cross-section and analyzed its crashworthiness characteristics. It was found that the tube with number of corrugations equaled 6 had the best energy absorption. Abdewi et al. [22,23] investigated the effect

Fig. 1. Sketch of CCSTs and STs with different inner ribs. 2

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Uniaxial tension test was adopted to obtain the material properties. Three specimens were prepared and the true stress-strain curves of the thee three tests are plotted in Fig. 4. By taking the average value of the three tests for each mechanical parameter, we obtained that the material had the properties of density ρ = 2700 kg/m3, Young's modulus E = 69.79 GPa, yield stress σy = 54.00 MPa, ultimate stress σu = 163.46 MPa and Poisson's ratio ν = 0.33.

adding holes [26], grooves [27], tendons [28], corrugations [29–32] and origami features [33–36] on the wall, or diaphragms [37] and ribs [13,38] inside the tube. However, while these methods cause progressive deformation, they reduce the energy absorption efficiency. Therefore, we should be more carefully in choosing these methods. Of course, there are also some methods, like adding inner ribs, can improve energy absorption efficiency while causing progressive deformation [8,39]. Motivated by the designing of the CCCT [20,21], we introduced cosine wave into the cross-section of square tube and proposed a circumferentially corrugated square tube (CCST) in our previous study [40]. The number of corners can be controlled by a parameter of the cosine expression. The crushing performance of the CCST was only investigated with finite element (FE) method and two slots were set on the wall to induce the tube to deform steadily. However, in practical application, it is very difficult to add slots on the wall of CCST because the existence of the corrugations. To this end, we considered adding inner ribs into CCST to make it deform steadily. In this paper, the crushing performance of CCSTs with cosine and triangular waved cross-sectional profiles are studied. Three types of cross inner ribs are considered to induce the deformation of the CCSTs. The crushing performance of the CCSTs are studied with the quasistatistic experiment and FE simulation. Then, the effect of geometric parameters on the crushing performance are conducted. Finally, a theoretical model is proposed to predict the mean crushing force of CCSTs.

2.3. Quasi-static crushing tests All the specimens were tested with quasi-static compression test using the test machine MTS Landmark. The loading velocity was chosen as 2 mm/min throughout the tests. The test sketch is displayed in Fig. 5. The force and displacement responses were recorded by computer. The deformation process was recorded by a camera placed directly in front of the test specimen. 2.4. Experimental results The final deformation shapes of the specimens are displayed in Fig. 6. It was found that the STs all deformed with ordered progressive mode (OPM). The deformation of ST with inner ribs were clearly steadier than that of ST. Moreover, ST_C and ST_S produced more folds than ST_X, which means that the crisscross and star shaped inner ribs had advantages in improving the energy absorption efficiency of square tube than X shape inner rib. The final deformation shapes of the four cosine profile CCSTs are displayed in Fig. 6(b). It can be seen that the CTC deformed with global buckling mode (GBM), which is one kind of unstable mode (UM). Apparently, multi-corner configuration increases the instability of tube deformation. The CTC_C and CTC_X deformed with OPM and UM, respectively. When X-type ribs was added into the CTC_C, the tube changed from CTC_C to CTC_S, and the deformation mode was changed from OPM to mixed mode (MM). It indicates that C-type inner rib can increase the deformation stability of CCST, while X_type inner ribs may increase the instability. Apparently, the C_type inner rib was better than X_type whereas X_tape rib creates bending mode. We noticed that for a cosine profile CCST, if its deformation was progressive, the deformation modes of the constitute corner elements located at the corners of the

2. Experiments 2.1. Geometry description In the present study, two kinds of corrugations, i.e., cosine and triangular, are introduced into the cross-section of square tube to generate the circumferentially corrugated square tubes (CCST). Three types of cross inner ribs, namely, crisscross shape (C-type), X shape (X-type), and star shape (S-type), are added to induce the deformation of CCST. The performance of straight square tube (ST) is also studied for comparison. The cross-sections of these tubes are listed in Fig. 1. Every tube is named with a code, as shown in Fig. 1. The CCST with cosine and triangular corrugation profiles are named with CTC and CTT, respectively. The letters “C”, “X” and “S” following underline represent crisscross, X and star shaped inner ribs, respectively. In addition, for the convenience of description, the corrugation tubes with and without inner ribs are collectively called as CCSTs, and the straight tube with and without inner ribs are collectively called as STs. The design of the cross-section profile of CTC is displayed in Fig. 2. The side in red is controlled with a cosine function as follows:

5 L y = A cos ⎛ πx ⎞ + 0 L 2 0 ⎝ ⎠ ⎜

⎟

(1)

where A is the amplitude. The nominal side length is L0 and x ∈ [−L0 /2, L0 /2] in the cosine function. The other three sides are obtained by performing circular array with the red side around the original point in the x-y plane. The cross-sectional profile of CTT can be obtained using the same way with the controlling parameters A and L0. For comparison, the nominal side lengths L0 and the height h of CTC, CTT and ST are equal, that is L0 = 40 mm and h = 100 mm. Twelve test specimens were prepared, as shown in Fig. 3. The test specimens are fabricated with the wire cutting electrical discharge, which is able to achieve precision of 0.05 mm. All specimens had the same wall thickness t of 1 mm. Besides, A = 3 mm for all CCSTs. 2.2. Material properties Fig. 2. Design of the cross-section of CTC.

The material used on the test tubes is aluminum alloy AA6061O. 3

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Fig. 3. Test specimens.

mode of the original corner element can only be in-extensional mode [41]. For a cosine profile CCST, adding a X_type inner rib will make the constitute corner elements located at the corners of the square profile deform with in-extensional mode rather than extensional mode, therefore, the CTT_C can not deform progressively. The deformations of all the four triangular profile CCSTs were not stable and controllable, as displayed in Fig. 6(c). In addition, as the triangular profile CCSTs contain sharp corners, the stress concentration was created and the fracture toughness of aluminium tubes was lowered. Therefore, the fractures were found in the four triangular profile CCSTs, as shown in Fig. 6(c). Apparently, the fracture of the sharp corners also affected the deformation of tubes, which makes the deformation of CTT_C was not as stable and orderly as that of CTC_C. The force-displacement responses of these testing specimens are plotted in Fig. 8. It was found that the trends of force-displacement curves were highly related to the deformation modes. An initial peak crushing force (IPCF) occurred at the beginning of deformation for all tubes. After IPCF, the forces of different deformation modes exhibited different characteristics. The deformation modes of ST, ST_C, ST_X, ST_S and CT_C were OPM, their forces all developed a plateau value after IPCF. The plateau stage consisted of multiple peaks and valleys, which corresponds to the generation of folds in deformation. The more folds, the more peaks and valleys of force curve, and the smaller the fluctuation of the plateau force. For UM, like the deformations of CTC, CTT,

Fig. 4. True stress-strain curve of AA6061O.

square profile must be extensional mode, as shown in Fig. 7(a), and the deformation mode of the constitute corner elements located at the sides of the square profile must be in-extensional mode, as shown in Fig. 7(b). When adding an additional plate at the middle plane of a constitute corner element, the constitute element becomes a 3-panel element. If a 3-panel element develops progressive deformation, the deformation 4

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Fig. 5. Sketch of crushing test.

Fig. 6. Deformation shapes of test specimens.

Fig. 7. Progressive deformation of CTC_C (a) extensional mode of the constitute corner elements located at the corners of the square profile and (b) in-extensional mode of the constitute corner elements located at the sides of the square profile.

5

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Fig. 8. Experimental force-displacement curves of test specimens (a) without rib, (b) with crisscross rib, (c) with X shape rib and (d) with star shape rib.

Fig. 9. Values of crashworthiness indicators of test tubes (a) IPCF, (b) MCF, (c) EA and (d) SEA. 6

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

IPCF is the initial peak crushing force in the compression process. A larger IPCF always brings a larger acceleration, which is harmful to occupants. Therefore, a smaller IPCF is always preferable in designing an energy absorber. EA is the amount of energy absorption during the whole deformation process. It is calculated by integrating the force-displacement curve: s

EA =

∫ F (s) ds 0

(2)

where F(s) represents the loading force function of the displacement s. SEA specifies the energy absorption per unit mass and is defined as:

SEA =

EA m

(3)

where m is the mass of the energy-absorbing structure. SEA indicates the energy absorbing efficiency of structures, therefore, a higher the SEA is always be pursued by designers. MCF is the mean crushing force and defined as the ratio of EA to crushing distance s:

Fig. 10. FE modeling of CCST under axial crushing.

CTC_X, and CTT_X, the force dropped sharply after IPCF to a small value, and then there were no plateau stage, as shown in Fig. 8(a) and (c). This kind of force characteristics was not good for energy absorption. The force of MM can be found in CTT_C, CTC_S and CTT_S, as shown in Fig. 7(b) and (d). It is clearly that the stability of force in MM lies between OPM and UM. From the point of view of deformation and force characteristics, the OPM exhibited the best competitiveness in energy absorption. Some important crashworthiness indicators should be defined to further analyze and compare the crashworthiness performance of these test tubes. Four crashworthiness indicators, namely, initial peak crushing force (IPCF), mean crushing force (MCF), energy absorption (EA) and specific energy absorption (SEA) are adopted in the present study. The four parameters are defined as follows.

MCF =

EA s

(4)

The values of these crashworthiness indicators of the test tubes are shown in Fig. 9. It is observed that almost every indicator of CCSTs was larger than that of SST with the same inner rib. Adding the three inner ribs all increase values of the four crashworthiness indicators both for CCST and SST, and X-type inner rib had the weakest influence on the crashworthiness performance among the three considered inner ribs. For STs, the maximum value of each indicator occurred on ST_S. For cosine profile CCSTs, the maximum values of IPCF, MCF and EA all appeared on CTS_S, while the maximum SEA appeared on CTC_C. It was

Fig. 11. Comparison of deformation shapes between test and FE simulation. 7

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

emphatically discussed in later section.

because that only CTC_C developed progressive deformation mode among the four CCSTs. The triangular profile CCSTs had the almost same conditions as cosine profile CCSTs. The maximum SEA of triangular profile CCSTs appeared on CTT_C because its deformation was closest to OPM among the four tubes. In addition, in the 12 test tubes, the SEA of CTC_C was the largest, that was 15.72 kJ/kg, which was 23.58% larger than the maximum SEA of SST (12.72 kJ/kg). Apparently, CTC_C had the best energy absorption efficiency among the test tubes. By analyzing the deformation, the force-displacement curves and crashworthiness indicators of the 12 tubes, we can obtain that the CTC_C performed better than others in crashworthiness, because it had the best energy absorption efficiency, and controllable and stable deformation. Therefore, the crushing behaviors of CTC_C will be

3. Numerical model 3.1. Finite element (FE) modeling FE analysis was also performed to investigate the axial crushing problem of CCST. The FE simulation was conducted in the nonlinear explicit FE analysis program LS_DYNA. The FE model consisted of a moving plate, a fixed plate and a test tube, and the test tube was sandwiched between the two plates, as displayed in Fig. 10. The test tube was modeled with the Belytschko-Tsay shell element with one center integration point and five through thickness integration points. By performing the mesh convergence analysis, the element size of

Fig. 12. Comparison of force-displacement responses between test and FE simulation: (a) ST, (b) ST_C, (c) ST_X, (d)ST_S, (e) CTC, (f) CTC_C, (g) CTC_X, and (h)CTC_S. 8

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

traditional tubes will certainly increase the axial stiffness, thereby increase the instability of deformation. For square section, the corrugations on the adjacent sides can deform in the same direction (Fig. 13(b)) with in-extensional mode. Therefore, the corrugated square can deform progressively. However, for circular section, the corrugations can not deform in the same direction and the tube can only deform with global buckling. Therefore, the corrugated circular tube can not deform progressively and stably. In general, the corrugated square tube had obvious advantages in energy absorption versus corrugated circular tube.

1 mm × 1 mm was found to be able to ensure the accuracy of numerical results. Two contact algorithms were adopted in the FE models. The “AUTOMATIC_SURFACE_TO_SURFACE” contact algorithm was used to account for the contacts between the two plates and the test tube, and the “AUTOMATIC_SINGLE_SURFACE” was defined to simulate the selfcontact of the test tube. The coefficients of friction for both types of contacts were set to 0.3 [40]. In addition, the material model, MAT_PIECEWISE_LINEAR_PLASTICITY (Mat_24) was selected from material library in Ls_dyna to model the test tube. The moving and fixed plates were modeled with Mat_rigid (Mat_20) to make sure they would not show any deformation in simulation, as in the tests. On the premise that kinetic energy does not exceed 5% of internal energy throughout the simulation, the loading velocity was chosen as 3 m/s to improve the computational efficiency.

4. Further analysis for CTC_C As mentioned above that the CTC_C has better crashing performance than other CCSTs, so the parametric analysis for the tube is conducted in this section. The effects of two geometric parameters, namely, the wall thickness (t) and a dimensionless parameter η (η is defined as shown in Fig. 2 and eq. (5)) were studied. The same levels and values of the two parameters are adopted for CTC_C, as shown in Table 2. Finally, a total of 25 tubes with different geometric parameters were produced.

3.2. Validation of FE model The simulations for STs and cosine profile CCSTs were used to validate the FE models. The triangular profile CCSTs were not used because the fractures were found in these tubes, while in FE model the failure of material was not considered. The deformation shapes and force-displacement responses between tests and FE simulations were compared respectively for the 12 test specimens to validate the FE model, as shown in Figs. 11 and 12. It can be seen that both deformation shapes and force-displacement responses had good agreement except for CTC_S. The CTC_S deformed with MM both in test and FE simulation. However, in FE simulation, there were not so many uncontrollable factors as in experiment, and the deformation mode of CTC_S was inherently uncontrollable. Therefore, the results in FE simulation was more ideal than that in experiment, leading to the deformation in FE simulation was steadier than that in test. This made the force curve of FE simulation was larger than that of test. In addition, the crashworthiness indicators were also calculated and compared between test and simulation, as displayed in Table 1. The maximum difference for each indicator appeared on CTC_S, and the difference of each index was larger than 15% for CTC_S. While the difference of every indicator for other tubes are all relatively small and within an acceptable range. Generally speaking, the FE models had high accuracies.

η=

A × (N − 0.5) A A = = = 0.25A 40/((N − 0.5) × 4) 10 C

(5)

4.1. Deformation modes The deformation shapes of CTC_C with different geometric parameters are displayed in Fig. 15. The deformation can be classified into four types [40], namely, ordered progressive mode Ⅰ (OPM Ⅰ), ordered progressive mode Ⅱ (OPM Ⅱ), non-ordered progressive mode (NPM), and unstable mode (UM), as shown in Fig. 16. And then the deformation chart was obtained and exhibited in Fig. 17. It was found from Fig. 16(a) and (b) that in OPM Ⅰ and OPM Ⅱ, both the outer tubes and the inner ribs developed stable deformation, but there were still some differences. Fig. 18 lists the deformation shapes of each constitute element in OPM Ⅰ and OPM Ⅱ. It can be seen that whatever the geometric parameters were, the CTC_C can be divided into four types of constitute elements (elements ①-④ in Fig. 18). In OPM Ⅱ, the four types of constitute elements deformed separately with different modes. However, in OPM Ⅰ, a new element, namely element ⑤, was

3.3. Comparison with circumferentially corrugated circular tube (CCCT)

Table 1 Comparisons of crashworthiness indicators between tests and FE simulations.

To show the advantages in crashworthiness of CCST, besides comparing with the traditional square tube in the previous section, we also made a comparison with the circumferentially corrugated circular tube (CCCT). The CCST in the present study had 12 outward corrugations, so, the CCCT was also endowed with 12 outward corrugations, as shown in Fig. 13. The nominal diameter of CCCT equaled to the nominal side length of CCST, that is 40 mm; the amplitude of CCCT also equals to that of CCST, that is 3 mm. In this section, the CTC and CTC_C are selected to compare with CCCT and CCCT_C. Fig. 14(a) displays the comparisons of deformation and force-displacement curves between CTC and CCCT. It can be seen that the deformation and force curves of both the two tubes were unstable. Apparently, neither of the CTC and CCCT had good crushing performance. Fig. 14(b) shows the comparison between CTC_C and CCCT_C. In crushing process, the CTC_C developed a few folds, which were orderly, controllable, and repeatable. The force curve of CTC_C formed a plateau stage and had only a small fluctuation. However, the deformation of CCCT_C was not that stable as CTC_C. The CCCT_C developed only 2 folds in global buckling mode. It made the force curve fluctuate greatly. Therefore, the CTC_C performed better than CCCT_C in crashworthiness. The differences between corrugated circular and square tubes due to the different deformation modes. Adding corrugations on the profiles of

Tube ST

ST_C

ST_X

ST_S

CTC

CTC _C

CTC _X

CTC _S

9

Exp FE Difference Exp FE Difference Exp FE Difference Exp FE Difference Exp FE Difference Exp FE Difference Exp FE Difference Exp FE Difference

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

IPCF (kN)

MCF (kN)

EA (kJ)

SEA (kJ/kg)

8.28 8.00 3.38 24.73 24.52 0.85 19.20 20.91 8.91 34.51 35.92 4.09 18.34 19.85 8.23 34.78 33.96 3.60 34.75 30.78 11.42 37.31 46.26 23.99

3.73 4.03 8.04 13.19 14.00 6.14 9.65 9.61 0.41 20.20 21.89 8.37 5.70 6.37 11.75 21.04 21.32 1.33 16.79 17.38 3.51 25.73 29.91 16.25

0.22 0.24 9.09 0.79 0.84 6.33 0.53 0.58 9.43 1.21 1.31 8.26 0.34 0.38 11.76 1.26 1.28 1.59 1.01 1.04 2.97 1.54 1.79 16.23

5.18 5.60 8.11 12.23 12.98 6.13 7.20 7.83 8.75 12.72 13.79 8.41 6.14 6.91 12.54 15.72 15.92 1.27 11.77 12.12 2.97 13.93 16.18 16.15

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Table 2 Levels and values of each parameter for CTC_C. Parameters

Levels

Values

t η

5 5

0.5, 0.75, 1.0, 1.25, 1.5 0.25, 0.5, 0.75, 1.0, 1.25

when η = 0.25. This was because the smaller η was, the closer the crosssectional profile of CTC_C was to that of ST_C. In addition, we found that smaller η and t will make it more difficult for OPM Ⅱ to appear. The NPM is shown in Fig. 16 (c). It was observed that both the deformations of outer tube and inner rib were not ordered. Apparently, the NPM was the non-ordered form of OPM Ⅰ or OPM Ⅱ. This mode mainly occurred when the wall thickness t was small or when the deformation mode transited from OPM Ⅰ to OPM Ⅱ with the increase of η (Fig. 17). The UM is displayed in Fig. 16(d). It is clearly that global buckling occurred on the outer tube, and the inner rib deformed unstably. The UM occurs mainly when both η and t were large, as shown in Fig. 17. This illustrated that the increase in η and t will increase the instability of the crushing resistance of CTC_C. By comparing the four deformation modes in Fig. 15, it is clearly that the deformation of tube in OPM Ⅱ was the most stable and controllable, which made it more competitive in energy absorption than other modes.

Fig. 13. Cross sections of CCCT and CTC.

formed, and the element ⑤ consists of 1 element ① and 4 elements ②. The adjacent two elements ② on the same side always deform in the same direction, which makes the deformation of element ⑤ look like as a whole. Actually, the OPM Ⅰ was very similar to the deformation mode of ST_C (Fig. 6). From Fig. 17 we can see that the OPM Ⅰ only occurred

4.2. Force-displacement characteristics The force-displacement curves of CTC_C with different parameters η and t are plotted in Fig. 19. It is evident that the force level increases

Fig. 14. Comparison between (a) CCCT and CTC, (b) CCCT_C and CTC_C. 10

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Fig. 15. Deformation shapes of CTC_C.

Fig. 16. Classification of Deformation modes: (a) OPM Ⅰ [CTC_C with η = 0.5 and t = 1.25 mm], (b) OPM Ⅱ [CTC_C with η = 1.0 and t = 1.25 mm], (c) NPM [CTC_C with η = 1.0 and t = 0.5 mm], and (d) UM [CTC_C with η = 1.0 and t = 1.5 mm]. 11

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

increase rate decreased a lot because the deformation mode changes from OPM Ⅱ to NPM or UM (Fig. 17). The increase of η also caused the increase of MCF to some extent. When t = 0.5 mm, with the change of η, the change of MCF was very small, while with increase of t, the effect of η on MCF became more and more obvious. It can also be observed that when t ≤ 0.75 mm, η had a greater influence MCF; while when t ≥ 0.75 mm, the MCF had a very small change for different η. Fig. 22 exhibits the SEA change of CTC_C for different η and t. The effects of η and t on SEA were a little more complex, and closely related to whether the deformation mode changes or not with the change of η or t. The SEA increased exponentially with increasing t when η = 0.25 for CTC_C. For other situations, before t = 1.25 mm, the SEA increased roughly linearly with increasing t; however, after t = 1.25 mm, the SEA decreased for most η values; while for a small number of η values, the SEA still increased, but the increase rate was lower than that before t = 1.25 mm. The decrease in SEA after t = 1.25 mm was because that the deformation mode changed from OPM Ⅱ to UM, such as when η = 0.75, 1.0 and 1.25. The decrease in increase rate was mainly because that the deformation mode changed from OPM Ⅱ to NPM, such as when η = 0.5. From Fig. 22, we can also find that with the increase in η, the SEA increased first and then decreased. The turning point often occurred when the first OPM Ⅱ appeared. For example, when t = 1 mm, the fist OPM Ⅱ appeared when η = 0.5 for CTC_C. Therefore, for CTC_C, the SEA increases as η increased from 0.25 to 0.5, while decreased as η increases from 0.5 to 1.25. In addition, the maximum SEA appeared when η = 0.75 and t = 1.25 mm, and the deformation mode was OPM Ⅱ. The maximum SEA of CTC_C was 19.45 kJ/kg. It was 20.81% larger than that of ST_C with the same wall thickness (16.10 kJ/kg), and 8.42% larger than the maximum SEA of ST_C (17.49 kJ/kg). This illustrated that the CTC_C had good energy absorption efficiency.

Fig. 17. Distribution of deformation modes for CTC_C.

with the increasing t. However, when the t was greater than 1.25 mm, the deformation process was basically no longer stable and orderly (Fig. 15), which resulted the increase rate of force level decreased. For example, the force of CTC_C with η = 1.0 increased obviously when t increases from 0.5 to 1.25 mm, but because the deformation mode was UM when t = 1.5 mm, the plateau force at t = 1.5 mm was smaller than that at t = 1.25 mm. Moreover, whatever the deformation mode was, the increase of t will lead to the increase of IPCF. In addition, it is found that when the deformation mode was OPM Ⅱ, the tubes with large t and η produced large fluctuations in force-displacement responses.

4.3. Effects on crashworthiness indicators Fig. 20 presents the effects of η and t on IPCF of CTC_C. The increases in η and t both resulted the increase in IPCF, and the increase trends were almost linear relative to η and t. The effects of η and t on MCF were also analyzed and are shown in Fig. 21. When η = 0.25 and 0.5, the MCF increased linearly with increasing t. When η ≥ 0.75, with the increase in t, the MCF increased exponentially before t = 1.25 mm, while after t = 1.25 mm, the

5. Theoretical analysis for CTC_C deforming in OPM Ⅱ As described in previous sections, the CTC_C deforming in OPM Ⅱ had superior energy absorption performance. So, in this section, a theoretical model based on Simplified Super Folding Element (SSFE) method [12,39,42,43] was presented to predict the mean crushing force

Fig. 18. Deformation shapes of each constitute element in (a) OPM Ⅰ and (b) OPM Ⅱ. 12

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

Fig. 19. Force-displacement curves of CTC_C with different geometric parameters: (a) η = 0.25, (b) η = 0.5, (c) η = 0.75, (d) η = 1, and (e) η = 1.25.

Fig. 22. Effects on SEA of CTC_C.

Fig. 20. Effect on IPCF of CTC_C.

of CTC_C. In the SSFE theory, the external work is considered to be dissipated with the form of bending energy and membrane energy during crushing. Supposing that the wavelength of each folded wave is 2H, the energy balance conditions can be expressed as:

2HPm k = Wbending + Wmembrane

(6)

where Pm, Wbending and Wmembrane are mean crushing force, bending energy and membrane energy, respectively. It should be noted that the folds are often not completely compacted, and the wave length of a fold is therefore smaller than 2H. The effective crushing distance coefficient k is introduced to take this condition into consideration. According to Abramowicz and Wierzbicki [12,44], the value of k is between 0.7 and 0.75. While recently studies for multi-cell columns by Zhang and Zhang [8,43] show that the parameter k can vary between 0.65 and 0.75. In the present study, a mean value of k, i.e., k = 0.7, is adopted for all tubes.

Fig. 21. Effects on MCF of CTC_C.

13

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

5.1. Bending energy

Table 3 Validation of theoretical model for CTC_C deforming with OPM Ⅱ.

The bending energy Wbending can be calculated as the sum of dissipated energies at stationary hinge lines. Therefore, the bending energy is expressed as [8]:

Wbending = 2πM0 Lc

Parameters t = 0.75 mm Pm_TH (kN) Pm_FE (kN) Difference (%) t = 1.0 mm Pm_TH (kN) Pm_FE (kN) Difference (%) Pm_EXP (kN) Difference (%) t = 1.25 mm Pm_TH (kN) Pm_FE (kN) Difference (%)

(7)

where Lc is the total length of sectional profile, M0 is the fully plastic bending moment per unit length and can be calculated as

1 M0 = σ0⋅t 2 4

(8)

t is wall thickness. σ0 is the flow stress of the material and can be calculated as:

σ0 =

1 (σy + σu ) 2

(9)

According to section 2.2, σ0 equals 108.73 MPa. inext Mcorner =

5.2. Membrane energy

η = 1.25

η = 1.5

12.46 13.52 7.84

13.01 12.76 1.96

13.59 12.12 11.75

14.26 12.51 13.98

19.3 21.34 9.56 21.04 8.26

20.15 21.31 5.44

21.52 21.06 2.18

22.85 20.62 10.81

28.32 32.55 12.99

30.23 31.8 4.94

32.09 31.79 0.94

tan(θ /2) 4M0 H 2 ⋅ t (tan(θ /2) + 0.05/tan(θ /2))/1.1

(11)

5.2.2. Three-panel element The 3-panel element can be considered as consisting of a corner element with a central angle 2ϕ and an additional plate inserted at the symmetry plane of the corner element, as shown in Fig. 23. According to Zhang [41], the membrane energy dissipated by the additional plate is

Madditional (ϕ) =

8M0 H 2 ⋅tan(ϕ/2) t

(12)

And the membrane energy absorbed by a 3-panel element therefore can be calculated by sum of the corner element and the additional plate [41]:

M3 − panel (ϕ) =

1.1⋅tan(ϕ) 4M0 H 2 ⎛ + 2 tan(ϕ/2) ⎞⎟ ⋅⎜ t ⎝ tan(ϕ) + 0.05/tan(ϕ) ⎠

(13)

5.2.3. Criss-cross element The crisscross element can develop four collapse modes, however, the most common modes is the mode Ⅱ [46] (more details about the four collapse modes please refer to Ref. [46]). The crisscross element in the present study also deforms in mode Ⅱ, as displayed in Fig. 18. If the panel width of criss-cross element is B, the membrane energy dissipated by criss-cross element with mode Ⅱ is expressed as [46]:

5.2.1. Corner element The corner element can develop two deformation modes, i.e., extensional and in-extensional mode [12]. In the present study, the corner elements at the vertex regions deform with extensional mode, while the ones at sides deform with in-extensional mode. According to Zhang's model [41,45], the membrane energy dissipated by corner with extensional and in-extensional modes respectively are:

4M0 H 2 ⋅(π − θ) t

η = 1.0

where, θ is the central angle of corner.

The cross-sectional profile of CTC_C is complex. In order to accurately consider the membrane energy, the profile should be divided into several constituent elements, as displayed in Fig. 23, and then the total membrane energy is calculated by summing up the membrane energy of every constitute element. The CTC_C has 4 kinds of constitute elements, namely, shell corner Ⅰ, shell corner Ⅱ, 3-panel shell and crisscross elements, as shown in Fig. 23. The membrane energy of every constitute element is highly related the angle between two adjacent panels. For shell corner Ⅰ, Ⅱ and 3-panel shell elements, it is hard to determine the angles between two adjacent panels because some panels are shells. However, it is known that the membrane energy is primarily absorbed in the intersection region of a constituent element and in a very small region, the difference between shells and plates should not be very large. Therefore, Zhang [43] proposed that the shell types elements can be equivalent plane elements by drawing the tangent plane of the shell from the intersection line, as shown in Fig. 23. With this method, the shell corner Ⅰ and Ⅱ elements can be equivalent to corner elements with central angles of θ, and 90°, respectively, and the 3-panel shell element is equivalent to 3-panel element with ϕ = 90°.

ext Mcorner (θ) =

η = 0.75

Mcriss − cross = (10)

8M0 H 2 ⎛ 4 2 ⎞ ⎟ ⋅⎜1 + t (B / t )0.5 ⎠ ⎝

Fig. 23. Constituent elements of CTC_C. 14

(14)

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

5.3. Mean crushing force prediction for CTC_C

Scos ine = The shell corner Ⅰ element deforms with extensional mode in OPM Ⅱ. Its central angle can be calculated as

θshell

=

corner Ι

(15)

θshell

=

corner Ι

( ⋅η ) , − 2 arctan ( ⋅η ) ,

⎨ 3π ⎩2

π 2

π 2

η≤

2 π

η>

2 π

CTC _ C Mbending

ext Mshell

( (

( )) ( ))

η≤

2 π

4M0 t

(−

π 2

i=

+ 2 arctan

4M0 t

(

π 2

− 2 arctan

η>

2 π

H=

( ⋅η ) ) π 2

and

ΙΙ

( ⋅η ) ) . π 2

inext = Mcorner (180°) =

Pm =

4.4M0 H 2 = j⋅H 2 t

(18)

Here, j = The 3-panel shell element equivalent to a shell element with ϕ = 90°. The membrane energy 3-panel shell element is therefore calculated as shell

= M3 − panel (90°) =

12.4M0 H 2 = m ⋅H 2 t

12.4M0 . t

where m = For the crisscross element, the B = (20 + 4⋅η)/2 = 10 + 2⋅η, therefore we have

Mcriss − cross (η) = And l =

panel

(1 +

4 2 ((10 + 2 ⋅ η) / t )0.5

inext = Ns − c _ ΙΙ ⋅Mshell corner

⋅M3 − panel

shell

ΙΙ

width

corner Ι

+ Nc⋅Mcriss − cross

+ N3 − p _ s (21)

where Ns-c_Ⅱ, Ns-c_Ⅰ, N3-p_s and Nc are the number of shell corner Ⅱ, shell corner Ⅰ, 3-panel shell and crisscross elements in CTC_C. In the present study, Ns-c_Ⅱ, Ns-c_Ⅰ, N3-p_s and Nc are 16, 4, 4 and 1, respectively. Thus, eq. (21) is further written as

⎧ (16j + 4i + 4m + h)⋅H 2, CTC _ C Mmembrane = ⎨ (16j + 4i′ + 4m + h)⋅H 2, ⎩

η≤ η>

2 π 2 π

∫x

x1

1 + f (x )′2 dx

0

0.5 CTC _ C ⎧ (Mbending /(16j + 4i + 4m + l)) , _ 0.5 CTC C ⎨ (Mbending /(16j + 4i′ + 4m + l)) , ⎩

2 π 2 > π

η≤ η

(28)

1 0.5 CTC _ C ⎧ k ⋅(Mbending ⋅(16j + 4i + 4m + l)) , 0.5 CTC _ C ⎨ 1 ⋅(Mbending ⋅(16j + 4i′ + 4m + l)) , ⎩k

η≤ η>

2 π 2 π

(29)

Conflicts of interest (22)

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Moreover, to obtain the bending energy of CTC_C, the parameter Lc is must be calculated first. We know that for a curve, if it can be expressed as a function f = f (x ) , the length of the curve in the region [x0, x1] can be calculated as

S=

(27)

(1) The experimental results showed that all triangular profile CCSTs deform unstably, and only the crisscross inner rib can induce the cosine profile CCSTs to deform progressively. (2) Under different geometric parameters, the cosine profile CCST with crisscross inner rib (i.e., CTC_C) develops four different deformation modes, and the ordered progressive mode Ⅱ (OPM Ⅱ) showed best performance in crashworthiness because the deformation mode of CTC_C with the maximum SEA was OPM Ⅱ. (3) A theoretical model based on Simplified Super Folding Element was derived to predict the mean crushing force of CTC_C and CTT_C deforming with OPM Ⅱ. The results showed that the model had good accuracy.

(20)

).

ext + Ns − c _ Ι ⋅Mshell

η>

2 π 2 π

This paper proposes new designed circumferentially corrugated square tubes (CCSTs) with cosine and triangular corrugation crosssectional profiles. Three cross inner ribs, i.e., crisscross, X shape and star shape, were introduced to induce the deformation and improve the energy absorption performance of CCSTs. The crushing performances of the CCSTs were investigated experimentally, numerically and theoretically. Some important conclusions were drawn as follows:

Then the total membrane energy of CTC_C is obtained as CTC _ C Mmembrane

η≤

6. Conclusion

(19)

8M0 H 2 ⎛ 4 2 2 ⎞ ⋅⎜1 + ⎟ = l⋅H t ((10 + 2⋅η)/ t )0.5 ⎠ ⎝

8M0 ⋅ t

(26)

Using eq. (29), the theoretical values of Pm of CTC_C with different geometric parameters deforming with OPM Ⅱ were obtained and listed in Table 3. By comparing with FE simulation, we found that the maximum difference was 13.98%, which can be acceptable. Overall, the theoretical model had good accuracy.

4.4M0 . t

M3 − panel

= 8πM0 (Scos ine + 4η + 20)

Submitting eq. (28)–(27), we obtain

i′ =

The shell corner element Ⅱ deforms with in-extensional mode in OPM Ⅱ. Its central angle is 180°, and therefore its membrane energy is expressed as inext Mshell corner

(25)

The half wave length H can be obtained from the stationary condition of mean force as

(17) Introducing

= 4Scos ine + 16η + 80

⎧ (16j + 4i + 4m + l)⋅H 2, CTC _ C 2HPm k = Mbending + ⎨ (16j + 4i′ + 4m + l)⋅H 2, ⎩

(16)

⎧ 4M0 H π − 2 arctan π ⋅η = i⋅H 2, ⎪ t 2 2 = ⎨ 4M0 H2 − π + 2 arctan π ⋅η = i′⋅H 2, ⎪ t 2 2 ⎩

corner Ι

(24)

Submitting eqs. (21) and (26) to eq. (6), we have

Then the membrane energy absorbed by a shell corner element Ⅰ is 2

2

πη π 1 + ⎛ ⋅cos ⎛ x ⎞ ⎞ dx 2 8 ⎠⎠ ⎝ ⎝

The bending energy of CTC_C is then be got as

The central angle of shell corner Ⅰ should also be inferior angle, therefore, eq. (15) is rewritten as

⎧ π2 + 2 arctan

40

So, the total length of the sectional profile Lc of CTC_C is

_ LcCTC C

3π π − 2 arctan ⎛ ⋅η ⎞ 2 ⎝2 ⎠

∫0

Acknowledgements The present work was supported by the National Key Research Project of China (grant number 2016YFB1200404-04), the Fundamental Research Funds for the Central Universities of Central

(23)

So, one side length of CTC_C is 15

Thin-Walled Structures 144 (2019) 106370

Z. Li, et al.

South University (grant number 2019zzts269), the National Natural Science Foundation of China (grant number 51675537) and the Project of the State Key Laboratory of High Performance Complex Manufacturing (grant number ZZYJKT2018-09).

137 (2019) 303–317. [22] E.F. Abdewi, S. Sulaiman, A.M.S. Hamouda, E. Mahdi, Quasi-static axial and lateral crushing of radial corrugated composite tubes, Thin-Walled Struct. 46 (2008) 320–332. [23] E.F. Abdewi, S. Sulaiman, A.M.S. Hamouda, E. Mahdi, Effect of geometry on the crushing behaviour of laminated corrugated composite tubes, J. Mater. Process. Technol. 172 (2006) 394–399. [24] Z. Liu, W. Hao, J. Xie, J. Lu, R. Huang, Z. Wang, Axial-impact buckling modes and energy absorption properties of thin-walled corrugated tubes with sinusoidal patterns, Thin-Walled Struct. 94 (2015) 410–423. [25] S. Liu, Z. Tong, Z. Tang, Y. Liu, Z. Zhang, Bionic design modification of non-convex multi-corner thin-walled columns for improving energy absorption through adding bulkheads, Thin-Walled Struct. 88 (2015) 70–81. [26] H. Nikkhah, F. Guo, Y. Chew, J. Bai, J. Song, P. Wang, The effect of different shapes of holes on the crushing characteristics of aluminum square windowed tubes under dynamic axial loading, Thin-Walled Struct. 119 (2017) 412–420. [27] A. Asanjarani, S.H. Dibajian, A. Mahdian, Multi-objective crashworthiness optimization of tapered thin-walled square tubes with indentations, Thin-Walled Struct. 116 (2017) 26–36. [28] B.C. Chen, M. Zou, G.M. Liu, J.F. Song, H.X. Wang, Experimental study on energy absorption of bionic tubes inspired by bamboo structures under axial crushing, Int. J. Impact Eng. 115 (2018) 48–57. [29] S.E. Alkhatib, F. Tarlochan, A. Eyvazian, Collapse behavior of thin-walled corrugated tapered tubes, Eng. Struct. 150 (2017) 674–692. [30] A. Eyvazian, M.K. Habibi, A.M. Hamouda, R. Hedayati, Axial crushing behavior and energy absorption efficiency of corrugated tubes, Mater. Des. 54 (1980–2015) (2014) 1028–1038. [31] A. Eyvazian, T.N. Tran, A.M. Hamouda, Experimental and theoretical studies on axially crushed corrugated metal tubes, Int. J. Non-Linear Mech. 101 (2018) 86–94. [32] N.S. Ha, G. Lu, X. Xiang, High energy absorption efficiency of thin-walled conical corrugation tubes mimicking coconut tree configuration, Int. J. Mech. Sci. 148 (2018) 409–421. [33] K. Yang, S. Xu, S. Zhou, Y.M. Xie, Multi-objective optimization of multi-cell tubes with origami patterns for energy absorption, Thin-Walled Struct. 123 (2018) 100–113. [34] K. Yang, S. Xu, J. Shen, S. Zhou, Y.M. Xie, Energy absorption of thin-walled tubes with pre-folded origami patterns: numerical simulation and experimental verification, Thin-Walled Struct. 103 (2016) 33–44. [35] K. Yang, S. Xu, S. Zhou, J. Shen, Y.M. Xie, Design of dimpled tubular structures for energy absorption, Thin-Walled Struct. 112 (2017) 31–40. [36] J. Ma, Z. You, Energy absorption of thin-walled square tubes with a prefolded origami pattern—Part I: geometry and numerical simulation, J. Appl. Mech. (2013) 81. [37] P. Xu, K. Xu, S. Yao, C. Yang, Q. Huang, H. Zhao, J. Xing, Parameter study and multi-objective optimisation of an axisymmetric rectangular tube with diaphragms for subways, Thin-Walled Struct. 136 (2019) 186–199. [38] S. Tabacu, Analysis of circular tubes with rectangular multi-cell insert under oblique impact loads, Thin-Walled Struct. 106 (2016) 129–147. [39] X. Zhang, H. Zhang, Some problems on the axial crushing of multi-cells, Int. J. Mech. Sci. 103 (2015) 30–39. [40] Z. Li, S. Yao, W. Ma, P. Xu, Energy-absorption characteristics of a circumferentially corrugated square tube with a cosine profile, Thin-Walled Struct. (2019) 135. [41] X. Zhang, H. Zhang, Numerical and theoretical studies on energy absorption of three-panel angle elements, Int. J. Impact Eng. 46 (2012) 23–40. [42] G. Lu, T. Yu, Energy Absorption of Structures and Materials, CRC Press, 2003. [43] X. Zhang, H. Zhang, Axial crushing of circular multi-cell columns, Int. J. Impact Eng. 65 (2014) 110–125. [44] W. Abramowicz, T. Wierzbicki, Axial crushing of multicorner sheet metal columns, J. Appl. Mech. 56 (1989) 113–120. [45] X. Zhang, H. Zhang, The crush resistance of four-panel angle elements, Int. J. Impact Eng. 78 (2015) 81–97. [46] X. Zhang, H. Zhang, Theoretical and numerical investigation on the crush resistance of rhombic and kagome honeycombs, Compos. Struct. 96 (2013) 143–152.

References [1] S. Xie, H. Zhou, Impact characteristics of a composite energy absorbing bearing structure for railway vehicles, Compos. B Eng. 67 (2014) 455–463. [2] J. Yan, S. Yao, P. Xu, Y. Peng, H. Shao, S. Zhao, Theoretical prediction and numerical studies of expanding circular tubes as energy absorbers, Int. J. Mech. Sci. 105 (2016) 206–214. [3] S. Yao, Z. Li, J. Yan, P. Xu, Y. Peng, Analysis and parameters optimization of an expanding energy-absorbing structure for a rail vehicle coupler, Thin-Walled Struct. 125 (2018) 129–139. [4] P. Xu, J. Xing, S. Yao, C. Yang, K. Chen, B. Li, Energy distribution analysis and multi-objective optimization of a gradual energy-absorbing structure for subway vehicles, Thin-Walled Struct. 115 (2017) 255–263. [5] H.H. Tsang, S. Raza, Impact energy absorption of bio-inspired tubular sections with structural hierarchy, Compos. Struct. 195 (2018) 199–210. [6] Q. Estrada, D. Szwedowicz, A. Rodriguez-Mendez, O.A. Gómez-Vargas, M. EliasEspinosa, J. Silva-Aceves, Energy absorption performance of concentric and multicell profiles involving damage evolution criteria, Thin-Walled Struct. 124 (2018) 218–234. [7] A. Jusuf, T. Dirgantara, L. Gunawan, I.S. Putra, Crashworthiness analysis of multicell prismatic structures, Int. J. Impact Eng. 78 (2015) 34–50. [8] X. Zhang, H. Zhang, Energy absorption of multi-cell stub columns under axial compression, Thin-Walled Struct. 68 (2013) 156–163. [9] X. Zhang, K. Leng, H. Zhang, Axial crushing of embedded multi-cell tubes, Int. J. Mech. Sci. 131–132 (2017) 459–470. [10] L. Zhang, Z. Bai, F. Bai, Crashworthiness design for bio-inspired multi-cell tubes with quadrilateral, hexagonal and octagonal sections, Thin-Walled Struct. 122 (2018) 42–51. [11] Y. Xiang, T. Yu, L. Yang, Comparative analysis of energy absorption capacity of polygonal tubes, multi-cell tubes and honeycombs by utilizing key performance indicators, Mater. Des. 89 (2016) 689–696. [12] T. Wierzbicki, W. Abramowicz, On the crushing mechanics of thin-walled structures, J. Appl. Mech. 50 (1983) 727–734. [13] A. Alavi Nia, M. Parsapour, Comparative analysis of energy absorption capacity of simple and multi-cell thin-walled tubes with triangular, square, hexagonal and octagonal sections, Thin-Walled Struct. 74 (2014) 155–165. [14] Z. Tang, S. Liu, Z. Zhang, Energy absorption properties of non-convex multi-corner thin-walled columns, Thin-Walled Struct. 51 (2012) 112–120. [15] G. Sun, T. Pang, J. Fang, G. Li, Q. Li, Parameterization of criss-cross configurations for multiobjective crashworthiness optimization, Int. J. Mech. Sci. 124–125 (2017) 145–157. [16] S. Reddy, M. Abbasi, M. Fard, Multi-cornered thin-walled sheet metal members for enhanced crashworthiness and occupant protection, Thin-Walled Struct. 94 (2015) 56–66. [17] W. Liu, Z. Lin, N. Wang, X. Deng, Dynamic performances of thin-walled tubes with star-shaped cross section under axial impact, Thin-Walled Struct. 100 (2016) 25–37. [18] W. Liu, Z. Lin, J. He, N. Wang, X. Deng, Crushing behavior and multi-objective optimization on the crashworthiness of sandwich structure with star-shaped tube in the center, Thin-Walled Struct. 108 (2016) 205–214. [19] S. Wu, G. Sun, X. Wu, G. Li, Q. Li, Crashworthiness analysis and optimization of fourier varying section tubes, Int. J. Non-Linear Mech. 92 (2017) 41–58. [20] X. Deng, W. Liu, L. Jin, On the crashworthiness analysis and design of a lateral corrugated tube with a sinusoidal cross-section, Int. J. Mech. Sci. 141 (2018) 330–340. [21] X. Deng, W. Liu, Multi-objective optimization of thin-walled sandwich tubes with lateral corrugated tubes in the middle for energy absorption, Thin-Walled Struct.

16