- Email: [email protected]
Crushing and theoretical analysis of multi-cell thin-walled triangular tubes under lateral loading
Thin–Walled Structures 115 (2017) 205–214
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Crushing and theoretical analysis of multi-cell thin-walled triangular tubes under lateral loading TrongNhan Trana,b, a b
MARK
⁎
Division of Computational Mechatronics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
A R T I C L E I N F O
A B S T R A C T
Keywords: Crushing force Lateral crushing behavior Theoretical analysis Multi-cell tube
This paper investigates the lateral crushing behavior of multi-cell thin-walled triangular tubes using experiments. More crushing modes are found in the lateral compression experiments of the multi-cell triangular tubes. The average crushing force, Pa, is governed by the plastic hinge lines. Based on experiments and the improved simplified super folding element (ISSFE) theory, theoretical models are proposed to predict average crushing force (Pa) in each stage. The formula of Pa is a function of flow stress of material, wall thickness, and length of tube. The results show that the theoretical solutions agree well with the experiment data.
1. Introduction Crushing behaviors of multi-cell thin-walled tubes, especially in cases of axial and oblique loading, have received enormous investigations due to the promising applications in energy absorber. Numerous literatures published are concerned with their experimental study, theoretical analysis and numerical investigation. Chen and Wierzbicki [1] was one of the firsts to investigate the multi-cell thin-walled tube. They have simplified the super folding element (SFE) theory [2] to study the crushing performance of the multi-cell tubes with right-corner angle element. Zhang et al. [3] also adopted the model of [1] to derive a theoretical solution for calculating the mean crushing force of multi-cell square tubes under the dynamic loading. Kim [4] used Chen and Wierzbicki's mode to study multi-cell tubes with four square elements at the corner. The method of [1] was also applied by Hanssen et al. [5] to predict the mean crushing force of complex aluminum extrusion. Najafi and Rais-Rohani [6] extended the SFE theory to investigate the crushing characteristics of multi-cell tubes with two different types of three-flange elements. An equation of closed form for prediction of the mean crushing force was also proposed. Alavi Nia et al. [7] carried out an investigation on the energy absorption characteristics of multi-cell square tubes. Sun et al. [8] investigated crushing mechanism of the hierarchical lattice structure under axial loading. Then Tran et al. [9,10] extended simplified super folding element (SSFE) theory to investigate the crushing characteristic of multi-cell tubes with different types of angle element. Concerning lateral crushing, Gupta et al. [11–13] studied the deformation and energy absorbing behaviors of rectangular and square
⁎
tubes under lateral compression through experiments and simulations. They figured out that the square tube absorbs more energy as compared to the rectangular tube of equal cross-sectional area, and the tube sections collapsed due to the formation of two sets of plastic hinges. Maduliat et al. [14] revealed the collapse behavior and energy absorption capability of hollow steel tubes under large deformation due to lateral impact load. In their work, analytical solutions for the collapse curve and in-plane rotation capacity was developed, and then used to simulate the large deformation behavior and energy absorption. Tran [15] investigated the crushing behavior of multi-cell tubes and the energy absorption in case lateral crushing was estimated through the bending and membrane energy. Additionally, the energy absorption behaviors of the hollow and nested tubes were also addressed by Baroutaji et al. [16–18], Eyvazian et al. [19], Wang et al. [20], Tran et al. [21], and Morris et al. [22,23] in lateral loading. Most of the above studies, however, emphasized axial crushing of the multi-cell tubes or lateral crushing of the hollow and nested tubes. The study on lateral crushing of the multi-cell thin-walled tubes is therefore quite rare. In this work, lateral crushing behaviors of multicell triangular tubes are investigated through experiment and theoretical analyses. To apply the SSFE theory to lateral loading, this theory will be improved and also added hypotheses. Based on the improved simplified super folding element (ISSFE) theory and experiment, theoretical expressions for each stage are proposed to predict the average crushing forces.
Correspondence address: Division of Computational Mechatronics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.tws.2017.02.027 Received 26 November 2016; Received in revised form 12 February 2017; Accepted 24 February 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 1. a) Tensile test; b) Typical stress-strain diagram.
2. Material properties and experimental setup
horizontal side of tubes. The process of tube type II is basically similar to that of tube type I. The corresponding load–displacement curves are shown in Fig. 5, respectively. The load–displacement curves of all the profiles show that the crushing load first reaches an initial peak, then drops rapidly and then slightly declines before rising again as a result of the new plastic hinge lines’ formation. However, the load-displacement of tube type II is different to that of tube type I. Shape of the loaddisplacement curve in second stage in which crushing force is greater is similar to that in first stage (Fig. 5b). Different stages of the lateral crushing at which crushing behavior recorded are marked on the corresponding load-displacement curves. Fig. 5 also shows that although the tube continued to deform under lateral crushing but the crushing force does not change much or decreases slightly in plateau region. In this plateau region, the area under the curve represents the ideal energy absorption zone. The average crushing force, Pa, in each stage is therefore defined as the equivalent constant force with a corresponding amount of stroke. As shown in Figs. 3–5, their deformation processes corresponding to the load-displacement. To investigate the crushing mechanism occurring during lateral crushing of multi-cell triangular tubes, it is convenient to roughly divide the deformation process into two principal stages: designated as stage A and stage B (Fig. 5). Deformation patterns of these tubes in Figs. 3 and 4 are hence used to raise the theoretical analysis of the crushing forces.
To find out the material properties, the specimen was carried out on UH-F500kNI SHIMADZU universal testing machine, as indicated in Fig. 1(a). Specimens were cut the same tubes made of steel plate CT3 as used for performing the lateral crushing tests. The engineering stress strain curve, used to determine the material properties, is recorded and shown in Fig. 1(b). The yield and ultimate stresses of this materials are σy=218 MPa and σu=316 MPa, respectively. Cross beam of an automotive bumper is generally subjected to the lateral crushing. Potential of using the multi-cell triangular structure as a new cross beam of automobile bumper is one of its functions. However, the investigation of the multi-cell triangular tubes under the lateral crushing is too infrequent. A new system of thin-walled triangular specimens with special cross-section is therefore presented in this research work (Fig. 2). To manufacture specimens of the multi-cell thin-walled triangular tubes, the component parts of tube are firstly cut from the steel plate. The outer wall is bent in the shape of equilateral triangle. Finally, they are welded together. The tubes’ specifications are depicted in Table 1. Crushing behaviors of triangular tubes in Fig. 2 were then studied in this article and experiments were performed on the same device at a loading rate of 5 mm/min. The specimens were freely placed between the upper and lower platen of the test machine. The upper platen moved downward and together with the apex of the tube after the impinging instant. The lateral compression process continued until the sides of the deforming tubes got in contact with each other. The load-displacement curves were recorded over the whole lateral crushing process with the automatic chart recorder of the machine. The deformed profiles of the specimens were taken at different stages of compression by camera. The typical progressive lateral collapse modes of the multi-cell triangular tube type I and II at different stages are displayed in Figs. 3 and 4, respectively. In spite of being the multi-cell triangular tubes, their lateral collapse behaviors are different from each other. This difference is due to structure of tube. Regarding the tube type I, the decline in force is dramatically larger after the elastic deformation phase. Then, rotations at plastic hinge lines control the deformation plateau before this process repeats. The formation of the second process is demonstrated by new moving plastic hinge lines developed on the
3. Lateral crushing analysis The analysis of the dissipated energy in bending and membrane for lateral crushing can be found in previous literature [15]. Therefore, the equilibrium of the work and energy is also expressed via the principle virtual work in case of tube under lateral crushing. By expanding this principle, the external energy work for a crushing equals to the total energy absorbed during crushing, Et, including bending and membrane energy, Eb and Em. That is
Pa δ = Et = Eb + Em
(1)
where Pa is the average crushing force. δ is the distance of the crushing. To apply the SSFE theory to lateral loading case, an improved approach is used in the following derivation. In this approach, instead of creating a model consisting of triangular elements with plastic hinge 206
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 2. Multi-cell triangular tubes subjected to lateral crushing: a) tube type I, b) tube type II.
process. Variations in thickness at different plastic hinge lines are ignored. Areas of the extensional membrane elements and the rotation angles at the plastic hinge lines are constant and equivalent during whole crushing process.
Table 1 Specifications of multi-cell triangular tubes type I and II. Specimen number
Thickness – t (mm)
Side length – B (mm)
Length – L (mm)
Type I
1 2 3 4 5 6
2
70
48
1 2 3 4 5 6
1.5
Type II
70
3.1. Energy absorption 3.1.1. Energy absorption in bending The ISSFE theory is applied to calculate the dissipated energy in crushing of tube. The plastic zone in Fig. 6 shows the rotation angle at plastic hinge line. According to the assumed kinematic of the ISSFE theory, the bending energy at one plastic hinge line is given by
50
Ebp − hi = αM0 L
(2) 2
where L is the length of tube and M0=σ0t /4 is the fully plastic bending moment of the panel. σ0 = σy σu is flow stress, 262.4 MPa adopted with ultimate stress σu=316 MPa. Since the deformation process of a tube is governed by the plastic hinge lines and the bending energy of the panel depends crucially upon the rotational angle α and the hinge line length. Therefore, the tube's bending energy derived from Eq. (2) is
lines as in [1], the basic folding element (BFE) proposed hereby includes the plastic zones developed during crushing process. Plastic zone consists of one plastic hinge line and two extensional membrane element (Fig. 6). The ISSFE theory is therefore applied to predict the crushing strength of multi-cell triangular tubes under lateral crushing. In this theory, contributions of the stationary and moving plastic hinge lines at plastic zone on the wall are the same during the deformation
n
Ebt =
i =1
207
n
∑ Ebpi = M0 L ∑ αi i =1
(3)
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 3. Typical progressive lateral crushing modes of multi-cell triangular tube type I.
evaluated by integrating the extensional membrane areas (shaded in Fig. 6). That is
where n is the number of the plastic hinge lines.
3.1.2. Energy absorption in membrane To determine the membrane energy at plastic hinge line following the ISSFE theory, the membrane element is used. There is only a possible collapse mode which is supposed in the establishment of plastic zone. As for this collapse mode, the two rectangular elements are developed for each plastic hinge line during deformation. The energy absorption due to membrane effect at one plastic hinge line can thus be
Emp − h =
∫s σ0 tds = 2σ0 ts
(4)
where the area of the extensional membrane element is assumed to be equal to εδ2. ε is the coefficient of the area. The tube's crushing process is controlled by the plastic hinge lines. Thus, the membrane energy of tube is given by 208
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 4. Typical progressive lateral crushing modes of multi-cell triangular tube type II. n
Emt =
n
∑ Emp −hi = 2σ0 t ∑ si i =1
i =1
stage A (Fig. 5a). Then, the bending at A, B, and C are due to the effect of the frictions between tube-two platens and the buckling of AB, AD, and AC. The influence of the bending at A, B, and C on crushing loading are quite small compared with that at E, F and G, and the dissipated energy at A, B and C is therefore determined based on the dissipated energy at E, F, and G. Then, the buckling at A, B, C, E, F, and G controls of the crushing force. As a result, moving plastic hinge lines E, F and G form and divide each side into two segments (Fig. 7(a)). Length ratios of the upper segment length to the whole length are nearly the same as one third. The upper and lower segments AE, EB, AF, FD, AG and GC rotate around the plastic hinge lines E, F, and G; the segments EA, GA, EB, and GC rotate around the stationary plastic hinges at A, B, and C;
(5)
3.2. Crushing analysis 3.2.1. Crushing analysis of multi-cell triangular tube type I The deformation process at any instant of tube type I is shown in Fig. 3. In stage A, the three sides fall into buckling simultaneously. The vertical and right inclined sides bend inwardly and another one outwardly. As mentioned above, although the tube continued to deform under lateral crushing but the crushing force does not change much in 209
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 5. Typical corresponding load–displacement curves: a) Tube type I; b) Tube type II.
rotation angles at plastic hinge lines E, F, G and at plastic hinge line A, B, C are equal to π/2 and π/12, respectively. At the end of stage A, the buckling of the vertical and inclined sides progress so much that their component segments become almost flat. Regarding to the stage B, its deformation shape is introduced in Fig. 7(b). The moving plastic hinge lines at E, F, and G continue to move to the right. Simultaneously, points A, B keep moving in the vertical direction. During this process, side AEB contact with side AFD. This contact contributes form two new plastic hinge lines that are moving and stationary lines at H and D, respectively. Formation of new plastic hinge lines makes crushing force rise to a higher plateau (Fig. 5(a)). Since sides FHD and BDC rotate inwardly around H and D, it is clearly that the energy absorptions of the plastic hinge lines at H and D are smaller than that of the plastic hinge lines at E, F, and G. This can be explained by the fact that the areas of extensional membrane elements on either side of the plastic hinge lines at H and D are smaller than that at E, F, and G. For the sake of calculation, the total area of the extensional membrane element at H or D is equivalent to a half of the extensional membrane elements’ total areas at E. The rotation angle at plastic hinge lines H and D are approximately equal to π/2 and π/6, respectively. Thus there are eight plastic hinge lines developed on all sides, three of them are stationary.
Fig. 6. Plastic zone of basic folding element.
3.2.2. Crushing analysis of multi-cell triangular tube type II With regards to the tube type II, Fig. 4 shows the crushing process at any instant. In first stage, all sides fall into buckling simultaneously. Two inclined sides, IMJ and ONK, bend inwardly while other ones, ION and JLK, outwardly. Similar to the deformation process of the multi-cell triangular tube type I, the impact of the bending at I, J, and K on crushing loading is relative small compared with that at M, L, N, and O. This result shows that the absorbed energies of the plastic hinge lines at I, J and K are smaller than, and calculated based on the energy absorption of the plastic hinge lines at M, L, N and O. Therefore, seven plastic hinge lines are developed at I, J, K, L, M, N, and O in which two lines at L and M divide each side into two segments (Fig. 8(a)). The buckling at I, J, K, M, L, N and O governs the crushing force. For the horizontal side, length ratios of each segment to the whole length approximate a half whereas this ratio is one four for inclined side IMJ. Accordingly, all segments rotate around the plastic hinge lines. The buckling of the horizontal and inclined sides progresses so much that their component segments become almost flat at the end of stage A. It is, however, impossible to give a precise determination of the rotation angle and the extensional membrane element's areas on either side of the plastic hinge lines. For the sake of convenience, the extensional membrane elements’ gross area on either side of the plastic hinge lines at M, N, O, and L are equal to εδ2=0.16δ2, respectively. Regarding each extensional membrane element at plastic hinges I, J, and K, its area is equal to one-third of εδ2. The plastic hinge lines at M, N, O, and L rotate
Fig. 7. Deformation shapes of the tube type I in: a) stage A; b) stage B.
whereas the segments on horizontal side, BD and DC have no rotation. It is obvious that the area of the extensional membrane element and rotation angle at plastic hinge line cannot determine exactly. For the sake of convenience, the areas of the extensional membrane elements at plastic hinge lines at E, F, and G are assumed to be equal to εδ2=0.14δ2, respectively and equal to one-third of εδ2 for each extensional membrane element at plastic hinge lines at A, B and C. In addition, 210
Thin–Walled Structures 115 (2017) 205–214
T. Tran
in crushing, Et, as
PatI − A δ = EttI − A = EbtI − A + EmtI − A Substituting
EbtI − A,
EmtI − A
(8)
back into Eq. (8) gives
7π PatI − A δ = M0 L + 8σ0 tεδ 2 4
(9)
The unknown parameter δ can be determined by applying the stationary condition of the crushing force as
∂ ⎛ PatI − A ⎞ ⎜ ⎟ = 0, ∂δ ⎝ M0 ⎠
(10)
which leads to
−7Lπ 32ε + =0⇒δ= 4δ 2 t
7Ltπ 128ε
(11)
To substitute the term δ of Eq. (11) into Eq. (9), the proposed expression of the average crushing force in stage A of tube type I is
PatI − A =
σ0 t 2 7π 128ε 7Lπt L + 8σ0 tε = 3.74σ0 L0.5π 0.5t1.5ε 0.5 4 128ε 4 7Lπt
3.3.2. Stage B Due to the formation of two new plastic hinge lines, crushing force rises to a higher plateau. Their rotation angles are, respectively, π/2 and π/6. With eight plastic hinge lines, the energy absorption due to bending of tube in stage B is evaluated by the following equation,
Fig. 8. Deformation shapes of the tube type II in: a) stage A; b) stage B.
through the angles of π/2 while the rotation angles at I, J, and K are π/ 12, respectively. For the stage B, plastic deformation mechanism continues to evolve, as shown in Fig. 8(b). Points I and O continue to move vertically until upper platen contacts with the plastic hinge line at N. Two sides IMJ and NPK bend inwardly whereas ION and JLK outwardly. Then, one new moving plastic hinge line develops at P. With the formation of new moving plastic hinge line, the crushing force quickly climbed up to a higher plateau (Fig. 5(b)). Referring to the moving plastic hinge line at M in stage A, the line at P form and develops in a similar way. The areas of the extensional membrane elements and rotation angle at the lines at P in stage B and M in stage A are therefore equivalent. This means that their energy dissipations by moving plastic hinge lines at P, M are the same. Finally, eight plastic hinge lines are formed during the crushing process of stage B.
8 ⎛ π π π π⎞ EbtI − B = M0 L ∑ αi = M0 L ⎜3 +3 + + ⎟ ⎝ 12 2 2 6⎠ i =1
8 ⎛ 1 1⎞ EmtI − B = 2σ0 t ∑ si = 2σ0 tεδ 2 ⎜3 + 3 + 2 ⎟ ⎝ 3 2⎠ i =1
(14)
According to ISSFE theory, the total internal energy absorption after crushing consists of the energy dissipated by plastic hinge lines formation Eb and the energy dissipated by membrane area Em. Since the internal energy should be equal to the corresponding average crushing force, this relationship can be expressed as
PatI − B δ = M0 L 3.3.1. Stage A There are six plastic hinge lines developed on all sides of tube. As mentioned above, the rotation angle at the plastic hinge lines E, F, and G is equal to π/2. In regards to plastic hinge lines at A, B, and C, rotation angle equals to π/12. The corresponding energy dissipation due to bending of tube in stage A is calculated as
29π + 10σ0 tεδ 2 12
(15)
Let ∂(Pa/M0)/∂δ=0. This leads to
−29Lπ 40ε + =0⇒δ= 12δ 2 t
29Lπt 480ε
(16)
Hence the average crushing force is of the following form
6
PatI − B = (6)
Eq. (5) shows that the membrane energy depends substantially on the area of extensional membrane elements at plastic hinge line. The extensional membrane elements on either side at moving and stationary plastic hinge lines are equally area. This energy is hence given as follow 6 ⎛ ⎞ 1 EmtI − A = 2σ0 t ∑ si = 2σ0 t ⎜3 × εδ 2 + 3εδ 2⎟ ⎝ ⎠ 3 i =1
(13)
As stated previously, the membrane element's area has a considerable effect on membrane energy. Two new plastic hinge lines have the total area of extensional membrane elements equal to the membrane elements’ area at plastic hinge line E. Consequently, the membrane energy, dissipated by the plastic hinge lines, is determined by,
3.3. Average crushing force for multi-cell triangular tube type I
⎛ π π⎞ EbtI − A = M0 L ∑ αi = M0 L ⎜3 +3 ⎟ ⎝ 12 2⎠ i =1
(12)
σ0 t 2 29π 480ε 29Lπt L + 10σ0 tε = 4.91σ0 L0.5π 0.5t1.5ε 0.5 4 12 29Lπt 480ε
(17)
3.4. Average crushing force for multi-cell triangular tube type II 3.4.1. Stage A As shown in Figs. 4 and 7(a), all plastic hinge lines in stage A developed on outer wall and at corner of tube. The rotation angles at plastic hinge lines M, N, O, and L are of π/2, respectively and a value of π/12 for rotation angles at plastic hinge line I, J, and K, respectively. In accordance with the ISSFE theory, the bending energy is determined based on the rotation angle. The energy absorption in bending of tube in stage A is
(7)
The deformation process in stage A of multi-cell tube type I is controlled by the moving plastic hinge lines. The total work done in causing the crushing, Paδ, should be equal to the total energy absorbed 211
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 9. Representative load-displacement curves of multi-cell triangular tubes: (a) TI-2, (b) TI-3, and (c) TI-4; (d) TII-2, (e) TII-3 and (f) TII-4. 7 ⎛ π π⎞ EbtII − A = M0 L ∑ αi = M0 L ⎜3 +4 ⎟ ⎝ 12 2⎠ i =1
which leads to (18)
−9Lπ 40ε + =0⇒δ= 4δ 2 t
The ISSFE theory also points out that the extensional membrane element at plastic hinge line affects significantly the membrane energy. Besides, the absorbed energies in the membrane at L, M, N, and O are the same as discussed above. The membrane energies at I, J and K are also equal. The energy dissipated by the membrane is thus given as
⎞ ⎛ 1 EmtII − A = 2σ0 t ∑ si = 2σ0 t ⎜3 εδ 2 + 4εδ 2⎟ ⎠ ⎝ 3 i =1
PatII − A =
(23)
3.4.2. Stage B In the ISSFE theory, the areas of the membrane elements and rotation angle are assumed to be constant and equivalent during the whole crushing process. The formation of moving plastic hinge line at P in stage B makes the force quickly climb up to a higher plateau. Since there are eight moving plastic hinge lines developed on tube during crushing process of stage B, the absorbed energy in bending and membrane of tube are respectively
(20)
Using the stationary condition gives
∂ ⎛ PatII − A ⎞ ⎜ ⎟ = 0, ∂δ ⎝ M0 ⎠
σ0 t 2 9π 160ε 9Lπt L + 10σ0 tε = 4.743σ0 L0.5π 0.5t1.5ε 0.5 4 160ε 4 9Lπt
(19)
Combining Eqs. (1), (18) and (19) yields
9π + 10σ0 tεδ 2 4
(22)
Substituting Eq. (22) back into Eq. (20), the average crushing force in stage A of tube II is obtained as
7
PatII − A δ = M0 L
9Ltπ 160ε
(21) 212
Thin–Walled Structures 115 (2017) 205–214
T. Tran 8 ⎛ π π⎞ EbtII − B = M0 L ∑ αi = M0 L ⎜3 +5 ⎟ ⎝ 12 2⎠ i =1
(24)
8 ⎞ ⎛ 1 EmtII − B = 2σ0 t ∑ si = 2σ0 t ⎜3 εδ 2 + 5εδ 2⎟ ⎠ ⎝ 3 i =1
(25)
Based on the kinematic analysis of a SBFE during crushing, the work of external force is equal to the energy dissipation by shell extension and plastic hinge lines formation of on the curved surfaces. Then,
PatII − B δ = M0 L
11π + 12σ0 tεδ 2 4
(26)
Tran et al. [10] pointed out that the independent parameter of the crushing mode would minimize the crushing force, or otherwise the differential equation ∂(Pa/M0)/∂δ is equal to zero, i.e,
−11Lπ 48ε + =0⇒δ= 4δ 2 t
11Ltπ 192ε
(27)
In can be seen that, on substituting the expression for the distance of the crushing (Eq. (27)) into Eq. (26), the average crushing force is
PatII − B =
σ0 t 2 11π 192ε 11Lπt L + 12σ0 tε = 5.74σ0 L0.5π 0.5t1.5ε 0.5 4 192ε 4 11Lπt
(28) Fig. 10. Tube type I: a) Traditional collapse mode; b) Exceptional collapse mode.
4. Comparison with experiment test and discussion The works of Fan et al. [24], Wang et al. [25], and Sun et al. [26] introduced theoretical analyses for single and multi-cell triangular tubes. However, these formulas are just for equilateral or isosceles triangular cell. Then, the expressions of the average crushing force for two types of tubes in case of lateral crushing are proposed in Section 3. The theoretical predictions in stage A, B are therefore calculated by expression (12), (17), (23), and (28) and these values were used to compare with the value of average crushing force of tubes, n Pa = ∑i =1 Pi / n , in each stage. The load-displacement curves from representative specimens of multi-cell triangular tubes are illustrated in Fig. 9. Given the values for the average crushing force, it is possible to conclude the difference between crushing force at stage A and B depends on the number of plastic hinge lines. The average crushing force obtained from theoretical and experimental results are listed in Table 2. It is revealed that the errors vary from −16.49% to −4.49% in stage A and from −2.92% to 9.52% in stage B for tube type I. Regarding to tube type II, these errors are, respectively from −12.91% to −7.16% in stage A and from −3.79% to 7.36% in stage B. Figs. 10 and 11 show that crushing mechanism of tube type I is Table 2 Tested and predicted average crushing forces. Specimen number
Average crushing force (kN)
Error (%) Fig. 11. Tube type II: a) Traditional collapse mode; b) Exceptional collapse mode.
Theoretical prediction
Experiment data
A
B
A
B
A
B
Type I
1 2 3 4 5 6
12.75
16.74
13.54 15.27 13.98 13.35 13.85 13.59
15.67 18.52 17.23 17.24 15.86 15.28
−5.81 −16.49 −8.79 −4.49 −7.93 −6.21
6.82 −9.61 −2.86 −2.92 5.56 9.52
Type II
1 2 3 4 5 6
11.46
13.87
12.58 12.34 12.48 12.63 12.55 13.16
14.42 14.39 12.92 13.73 13.67 15.17
−8.89 −7.16 −8.15 −9.26 −8.68 −12.91
−3.79 −3.62 7.36 0.98 1.45 −8.59
different to that of tube type II. It is interesting that two new collapse modes for tube type I and II, shown in Figs. 10(b) and 11(b), are detected in the experiments. As regards tube type I, specimen TI-2 has seven plastic hinge lines while the traditional collapse mode has six (Fig. 9). Pa in stage A and B of TI-2 are 15.27 kN and 18.52 kN, respectively, greater than Pa of the other specimens, which show the errors between theoretical prediction and experimental data are −16.49% and −9.61% in stage A and B. Respecting tube type II, there are eight plastic hinge lines developed on classical collapse mode whereas the collapse mode of TII-6 has nine lines (Fig. 10). Pa in stage A and B of the rest of specimens are smaller than those of the specimen TII-6 that are 13.16 kN and 15.17 kN, respectively, which demonstrate the differences between theoretical solution and experimental result are 213
Thin–Walled Structures 115 (2017) 205–214
T. Tran
3. Triangular tubes with different cross-sections have different crushing mechanism. More crushing modes are detected in the lateral crushing behavior of the multi-cell triangular tubes. Two exceptional collapse mechanisms enlarging Pa of tube type I and II appear during lateral crushing. More detailed works are then needed to reveal these collapse mechanisms. References [1] W. Chen, T. Wierzbicki, Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption, Thin-Walled Struct. 39 (2001) 287–306. [2] T. Wierzbicki, W. Abramowicz, On the crushing mechanics of thin-walled structures, J. Appl. Mech. 50 (1983) 727–734. [3] X. Zhang, G. Cheng, H. Zhang, Theoretical prediction and numerical simulation of multi-cell square thin-walled structures, Thin-Walled Struct. 44 (2006) 1185–1191. [4] H.-S. Kim, New extruded multi-cell aluminum profile for maximum crash energy absorption and weight efficiency, Thin-Walled Struct. 40 (2002) 311–327. [5] A.G. Hanssen, A. Artelius, M. Langseth, Validation of the simplified super folding element theory applied for axial crushing of complex aluminium extrusions, Int. J. Crashworthiness 12 (2007) 591–596. [6] A. Najafi, M. Rais-Rohani, Mechanics of axial plastic collapse in multi-cell, multicorner crush tubes, Thin-Walled Struct. 49 (2011) 1–12. [7] A. Alavi Nia, M. Parsapour, An investigation on the energy absorption characteristics of multi-cell square tubes, Thin-Walled Struct. 68 (2013) 26–34. [8] F. Sun, C. Lai, H. Fan, D. Fang, Crushing mechanism of hierarchical lattice structure, Mech. Mater. 97 (2016) 164–183. [9] T. Tran, S. Hou, X. Han, N. Nguyen, M. Chau, Theoretical prediction and crashworthiness optimization of multi-cell square tubes under oblique impact loading, Int. J. Mech. Sci. 89 (2014) 177–193. [10] T. Tran, S. Hou, X. Han, M. Chau, Crushing analysis and numerical optimization of angle element structures under axial impact loading, Compos. Struct. 119 (2015) 422–435. [11] N.K. Gupta, G.S. Sekhon, P.K. Gupta, A study of lateral collapse of square and rectangular metallic tubes, Thin-Walled Struct. 39 (2001) 745–772. [12] N.K. Gupta, A. Khullar, Collapse load analysis of square and rectangular tubes subjected to transverse in-plane loading, Thin-Walled Struct. 21 (1995) 345–358. [13] N.K. Gupta, A. Khullar, Collapse of square and rectangular tubes in transverse loading, Arch. Appl. Mech. 63 (1993) 479–490. [14] S. Maduliat, T.D. Ngo, P. Tran, R. Lumantarna, Performance of hollow steel tube bollards under quasi-static and lateral impact load, Thin-Walled Struct. 88 (2015) 41–47. [15] T. Tran, Crushing analysis of multi-cell thin-walled rectangular and square tubes under lateral loading, Compos. Struct. 160 (2017) 734–747. [16] A. Baroutaji, M.D. Gilchrist, D. Smyth, A.G. Olabi, Crush analysis and multiobjective optimization design for circular tube under quasi-static lateral loading, Thin-Walled Struct. 86 (2015) 121–131. [17] A. Baroutaji, E. Morris, A.G. Olabi, Quasi-static response and multi-objective crashworthiness optimization of oblong tube under lateral loading, Thin-Walled Struct. 82 (2014) 262–277. [18] A. Baroutaji, M.D. Gilchrist, A.G. Quasi-static Olabi, impact and energy absorption of internally nested tubes subjected to lateral loading, Thin-Walled Struct. 98 (2016) 337–350. [19] A. Eyvazian, I. Akbarzadeh, M. Shakeri, Experimental study of corrugated tubes under lateral loadingmechanical engineers, Proc. Inst. Mech. Eng. Part L: J. Mater. Des. Appl. 226 (2012) 109–118. [20] H. Wang, J. Yang, H. Liu, Y. Sun, T.X. Yu, Internally nested circular tube system subjected to lateral impact loading, Thin-Walled Struct. 91 (2015) 72–81. [21] T.N. Tran, T.N.T. Ton, Lateral crushing behaviour and theoretical prediction of thinwalled rectangular and square tubes, Compos. Struct. 154 (2016) 374–384. [22] E. Morris, A.G. Olabi, M.S.J. Hashmi, Analysis of nested tube type energy absorbers with different indenters and exterior constraints, Thin-Walled Struct. 44 (2006) 872–885. [23] E. Morris, A.G. Olabi, M.S.J. Hashmi, Lateral crushing of circular and non-circular tube systems under quasi-static conditions, J. Mater. Process. Technol. 191 (2007) 132–135. [24] H. Fan, W. Hong, F. Sun, Y. Xu, F. Jin, Lateral compression behaviors of thin-walled equilateral triangular tubes, Int. J. Steel Struct. 15 (2015) 785–795. [25] P. Wang, Q. Zheng, H. Fan, F. Sun, F. Jin, Z. Qu, Quasi-static crushing behaviors and plastic analysis of thin-walled triangular tubes, J. Constr. Steel Res. 106 (2015) 35–43. [26] F. Sun, C. Lai, H. Fan, In-plane compression behavior and energy absorption of hierarchical triangular lattice structures, Mater. Des. 100 (2016) 280–290.
Fig. 12. Scatter diagram of crushing force: a) Tube type I; b) Tube type II.
−12.91% and −8.59%. Perhaps it is the exceptional collapse mechanisms that augment the Pa of tube type I and II. Since these phenomena are not prevailing in the experiment, more detailed tests are needed to reveal these collapse mechanism. Multi-cell triangular tube type I has greater average crushing force compared to tube type II in both stages A and B, as shown in Fig. 12 and Table 2. In addition, Fig. 12 also points out that the reasonable agreement can be observed. Accordingly, a very strong support is verified between the theoretical solutions and the experiment data in all cases. 5. Conclusion This work presents a combined experimental and theoretical study on the crushing behavior of multi-cell thin-walled triangular tubes under lateral crushing. Two types of multi-cell triangular tubes are performed using universal testing machine. It is concluded that: 1. The load-displacement curves are hence obtained. Their deformation patterns are examined and used to govern the theoretical analysis. The average crushing force is controlled by the number of the plastic hinge lines. 2. The theoretical solutions are developed based on the ISSFE theory for each tube type. These solutions show that the average crushing force is a function of flow stress of material, wall thickness, and length of tube. The average crushing force of the tube type I is bigger than that of the tube type II in both stages A and B. The predictions are consistent with the experimental results within acceptable errors.
214
Contents lists available at ScienceDirect
Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Full length article
Crushing and theoretical analysis of multi-cell thin-walled triangular tubes under lateral loading TrongNhan Trana,b, a b
MARK
⁎
Division of Computational Mechatronics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
A R T I C L E I N F O
A B S T R A C T
Keywords: Crushing force Lateral crushing behavior Theoretical analysis Multi-cell tube
This paper investigates the lateral crushing behavior of multi-cell thin-walled triangular tubes using experiments. More crushing modes are found in the lateral compression experiments of the multi-cell triangular tubes. The average crushing force, Pa, is governed by the plastic hinge lines. Based on experiments and the improved simplified super folding element (ISSFE) theory, theoretical models are proposed to predict average crushing force (Pa) in each stage. The formula of Pa is a function of flow stress of material, wall thickness, and length of tube. The results show that the theoretical solutions agree well with the experiment data.
1. Introduction Crushing behaviors of multi-cell thin-walled tubes, especially in cases of axial and oblique loading, have received enormous investigations due to the promising applications in energy absorber. Numerous literatures published are concerned with their experimental study, theoretical analysis and numerical investigation. Chen and Wierzbicki [1] was one of the firsts to investigate the multi-cell thin-walled tube. They have simplified the super folding element (SFE) theory [2] to study the crushing performance of the multi-cell tubes with right-corner angle element. Zhang et al. [3] also adopted the model of [1] to derive a theoretical solution for calculating the mean crushing force of multi-cell square tubes under the dynamic loading. Kim [4] used Chen and Wierzbicki's mode to study multi-cell tubes with four square elements at the corner. The method of [1] was also applied by Hanssen et al. [5] to predict the mean crushing force of complex aluminum extrusion. Najafi and Rais-Rohani [6] extended the SFE theory to investigate the crushing characteristics of multi-cell tubes with two different types of three-flange elements. An equation of closed form for prediction of the mean crushing force was also proposed. Alavi Nia et al. [7] carried out an investigation on the energy absorption characteristics of multi-cell square tubes. Sun et al. [8] investigated crushing mechanism of the hierarchical lattice structure under axial loading. Then Tran et al. [9,10] extended simplified super folding element (SSFE) theory to investigate the crushing characteristic of multi-cell tubes with different types of angle element. Concerning lateral crushing, Gupta et al. [11–13] studied the deformation and energy absorbing behaviors of rectangular and square
⁎
tubes under lateral compression through experiments and simulations. They figured out that the square tube absorbs more energy as compared to the rectangular tube of equal cross-sectional area, and the tube sections collapsed due to the formation of two sets of plastic hinges. Maduliat et al. [14] revealed the collapse behavior and energy absorption capability of hollow steel tubes under large deformation due to lateral impact load. In their work, analytical solutions for the collapse curve and in-plane rotation capacity was developed, and then used to simulate the large deformation behavior and energy absorption. Tran [15] investigated the crushing behavior of multi-cell tubes and the energy absorption in case lateral crushing was estimated through the bending and membrane energy. Additionally, the energy absorption behaviors of the hollow and nested tubes were also addressed by Baroutaji et al. [16–18], Eyvazian et al. [19], Wang et al. [20], Tran et al. [21], and Morris et al. [22,23] in lateral loading. Most of the above studies, however, emphasized axial crushing of the multi-cell tubes or lateral crushing of the hollow and nested tubes. The study on lateral crushing of the multi-cell thin-walled tubes is therefore quite rare. In this work, lateral crushing behaviors of multicell triangular tubes are investigated through experiment and theoretical analyses. To apply the SSFE theory to lateral loading, this theory will be improved and also added hypotheses. Based on the improved simplified super folding element (ISSFE) theory and experiment, theoretical expressions for each stage are proposed to predict the average crushing forces.
Correspondence address: Division of Computational Mechatronics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.tws.2017.02.027 Received 26 November 2016; Received in revised form 12 February 2017; Accepted 24 February 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 1. a) Tensile test; b) Typical stress-strain diagram.
2. Material properties and experimental setup
horizontal side of tubes. The process of tube type II is basically similar to that of tube type I. The corresponding load–displacement curves are shown in Fig. 5, respectively. The load–displacement curves of all the profiles show that the crushing load first reaches an initial peak, then drops rapidly and then slightly declines before rising again as a result of the new plastic hinge lines’ formation. However, the load-displacement of tube type II is different to that of tube type I. Shape of the loaddisplacement curve in second stage in which crushing force is greater is similar to that in first stage (Fig. 5b). Different stages of the lateral crushing at which crushing behavior recorded are marked on the corresponding load-displacement curves. Fig. 5 also shows that although the tube continued to deform under lateral crushing but the crushing force does not change much or decreases slightly in plateau region. In this plateau region, the area under the curve represents the ideal energy absorption zone. The average crushing force, Pa, in each stage is therefore defined as the equivalent constant force with a corresponding amount of stroke. As shown in Figs. 3–5, their deformation processes corresponding to the load-displacement. To investigate the crushing mechanism occurring during lateral crushing of multi-cell triangular tubes, it is convenient to roughly divide the deformation process into two principal stages: designated as stage A and stage B (Fig. 5). Deformation patterns of these tubes in Figs. 3 and 4 are hence used to raise the theoretical analysis of the crushing forces.
To find out the material properties, the specimen was carried out on UH-F500kNI SHIMADZU universal testing machine, as indicated in Fig. 1(a). Specimens were cut the same tubes made of steel plate CT3 as used for performing the lateral crushing tests. The engineering stress strain curve, used to determine the material properties, is recorded and shown in Fig. 1(b). The yield and ultimate stresses of this materials are σy=218 MPa and σu=316 MPa, respectively. Cross beam of an automotive bumper is generally subjected to the lateral crushing. Potential of using the multi-cell triangular structure as a new cross beam of automobile bumper is one of its functions. However, the investigation of the multi-cell triangular tubes under the lateral crushing is too infrequent. A new system of thin-walled triangular specimens with special cross-section is therefore presented in this research work (Fig. 2). To manufacture specimens of the multi-cell thin-walled triangular tubes, the component parts of tube are firstly cut from the steel plate. The outer wall is bent in the shape of equilateral triangle. Finally, they are welded together. The tubes’ specifications are depicted in Table 1. Crushing behaviors of triangular tubes in Fig. 2 were then studied in this article and experiments were performed on the same device at a loading rate of 5 mm/min. The specimens were freely placed between the upper and lower platen of the test machine. The upper platen moved downward and together with the apex of the tube after the impinging instant. The lateral compression process continued until the sides of the deforming tubes got in contact with each other. The load-displacement curves were recorded over the whole lateral crushing process with the automatic chart recorder of the machine. The deformed profiles of the specimens were taken at different stages of compression by camera. The typical progressive lateral collapse modes of the multi-cell triangular tube type I and II at different stages are displayed in Figs. 3 and 4, respectively. In spite of being the multi-cell triangular tubes, their lateral collapse behaviors are different from each other. This difference is due to structure of tube. Regarding the tube type I, the decline in force is dramatically larger after the elastic deformation phase. Then, rotations at plastic hinge lines control the deformation plateau before this process repeats. The formation of the second process is demonstrated by new moving plastic hinge lines developed on the
3. Lateral crushing analysis The analysis of the dissipated energy in bending and membrane for lateral crushing can be found in previous literature [15]. Therefore, the equilibrium of the work and energy is also expressed via the principle virtual work in case of tube under lateral crushing. By expanding this principle, the external energy work for a crushing equals to the total energy absorbed during crushing, Et, including bending and membrane energy, Eb and Em. That is
Pa δ = Et = Eb + Em
(1)
where Pa is the average crushing force. δ is the distance of the crushing. To apply the SSFE theory to lateral loading case, an improved approach is used in the following derivation. In this approach, instead of creating a model consisting of triangular elements with plastic hinge 206
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 2. Multi-cell triangular tubes subjected to lateral crushing: a) tube type I, b) tube type II.
process. Variations in thickness at different plastic hinge lines are ignored. Areas of the extensional membrane elements and the rotation angles at the plastic hinge lines are constant and equivalent during whole crushing process.
Table 1 Specifications of multi-cell triangular tubes type I and II. Specimen number
Thickness – t (mm)
Side length – B (mm)
Length – L (mm)
Type I
1 2 3 4 5 6
2
70
48
1 2 3 4 5 6
1.5
Type II
70
3.1. Energy absorption 3.1.1. Energy absorption in bending The ISSFE theory is applied to calculate the dissipated energy in crushing of tube. The plastic zone in Fig. 6 shows the rotation angle at plastic hinge line. According to the assumed kinematic of the ISSFE theory, the bending energy at one plastic hinge line is given by
50
Ebp − hi = αM0 L
(2) 2
where L is the length of tube and M0=σ0t /4 is the fully plastic bending moment of the panel. σ0 = σy σu is flow stress, 262.4 MPa adopted with ultimate stress σu=316 MPa. Since the deformation process of a tube is governed by the plastic hinge lines and the bending energy of the panel depends crucially upon the rotational angle α and the hinge line length. Therefore, the tube's bending energy derived from Eq. (2) is
lines as in [1], the basic folding element (BFE) proposed hereby includes the plastic zones developed during crushing process. Plastic zone consists of one plastic hinge line and two extensional membrane element (Fig. 6). The ISSFE theory is therefore applied to predict the crushing strength of multi-cell triangular tubes under lateral crushing. In this theory, contributions of the stationary and moving plastic hinge lines at plastic zone on the wall are the same during the deformation
n
Ebt =
i =1
207
n
∑ Ebpi = M0 L ∑ αi i =1
(3)
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 3. Typical progressive lateral crushing modes of multi-cell triangular tube type I.
evaluated by integrating the extensional membrane areas (shaded in Fig. 6). That is
where n is the number of the plastic hinge lines.
3.1.2. Energy absorption in membrane To determine the membrane energy at plastic hinge line following the ISSFE theory, the membrane element is used. There is only a possible collapse mode which is supposed in the establishment of plastic zone. As for this collapse mode, the two rectangular elements are developed for each plastic hinge line during deformation. The energy absorption due to membrane effect at one plastic hinge line can thus be
Emp − h =
∫s σ0 tds = 2σ0 ts
(4)
where the area of the extensional membrane element is assumed to be equal to εδ2. ε is the coefficient of the area. The tube's crushing process is controlled by the plastic hinge lines. Thus, the membrane energy of tube is given by 208
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 4. Typical progressive lateral crushing modes of multi-cell triangular tube type II. n
Emt =
n
∑ Emp −hi = 2σ0 t ∑ si i =1
i =1
stage A (Fig. 5a). Then, the bending at A, B, and C are due to the effect of the frictions between tube-two platens and the buckling of AB, AD, and AC. The influence of the bending at A, B, and C on crushing loading are quite small compared with that at E, F and G, and the dissipated energy at A, B and C is therefore determined based on the dissipated energy at E, F, and G. Then, the buckling at A, B, C, E, F, and G controls of the crushing force. As a result, moving plastic hinge lines E, F and G form and divide each side into two segments (Fig. 7(a)). Length ratios of the upper segment length to the whole length are nearly the same as one third. The upper and lower segments AE, EB, AF, FD, AG and GC rotate around the plastic hinge lines E, F, and G; the segments EA, GA, EB, and GC rotate around the stationary plastic hinges at A, B, and C;
(5)
3.2. Crushing analysis 3.2.1. Crushing analysis of multi-cell triangular tube type I The deformation process at any instant of tube type I is shown in Fig. 3. In stage A, the three sides fall into buckling simultaneously. The vertical and right inclined sides bend inwardly and another one outwardly. As mentioned above, although the tube continued to deform under lateral crushing but the crushing force does not change much in 209
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 5. Typical corresponding load–displacement curves: a) Tube type I; b) Tube type II.
rotation angles at plastic hinge lines E, F, G and at plastic hinge line A, B, C are equal to π/2 and π/12, respectively. At the end of stage A, the buckling of the vertical and inclined sides progress so much that their component segments become almost flat. Regarding to the stage B, its deformation shape is introduced in Fig. 7(b). The moving plastic hinge lines at E, F, and G continue to move to the right. Simultaneously, points A, B keep moving in the vertical direction. During this process, side AEB contact with side AFD. This contact contributes form two new plastic hinge lines that are moving and stationary lines at H and D, respectively. Formation of new plastic hinge lines makes crushing force rise to a higher plateau (Fig. 5(a)). Since sides FHD and BDC rotate inwardly around H and D, it is clearly that the energy absorptions of the plastic hinge lines at H and D are smaller than that of the plastic hinge lines at E, F, and G. This can be explained by the fact that the areas of extensional membrane elements on either side of the plastic hinge lines at H and D are smaller than that at E, F, and G. For the sake of calculation, the total area of the extensional membrane element at H or D is equivalent to a half of the extensional membrane elements’ total areas at E. The rotation angle at plastic hinge lines H and D are approximately equal to π/2 and π/6, respectively. Thus there are eight plastic hinge lines developed on all sides, three of them are stationary.
Fig. 6. Plastic zone of basic folding element.
3.2.2. Crushing analysis of multi-cell triangular tube type II With regards to the tube type II, Fig. 4 shows the crushing process at any instant. In first stage, all sides fall into buckling simultaneously. Two inclined sides, IMJ and ONK, bend inwardly while other ones, ION and JLK, outwardly. Similar to the deformation process of the multi-cell triangular tube type I, the impact of the bending at I, J, and K on crushing loading is relative small compared with that at M, L, N, and O. This result shows that the absorbed energies of the plastic hinge lines at I, J and K are smaller than, and calculated based on the energy absorption of the plastic hinge lines at M, L, N and O. Therefore, seven plastic hinge lines are developed at I, J, K, L, M, N, and O in which two lines at L and M divide each side into two segments (Fig. 8(a)). The buckling at I, J, K, M, L, N and O governs the crushing force. For the horizontal side, length ratios of each segment to the whole length approximate a half whereas this ratio is one four for inclined side IMJ. Accordingly, all segments rotate around the plastic hinge lines. The buckling of the horizontal and inclined sides progresses so much that their component segments become almost flat at the end of stage A. It is, however, impossible to give a precise determination of the rotation angle and the extensional membrane element's areas on either side of the plastic hinge lines. For the sake of convenience, the extensional membrane elements’ gross area on either side of the plastic hinge lines at M, N, O, and L are equal to εδ2=0.16δ2, respectively. Regarding each extensional membrane element at plastic hinges I, J, and K, its area is equal to one-third of εδ2. The plastic hinge lines at M, N, O, and L rotate
Fig. 7. Deformation shapes of the tube type I in: a) stage A; b) stage B.
whereas the segments on horizontal side, BD and DC have no rotation. It is obvious that the area of the extensional membrane element and rotation angle at plastic hinge line cannot determine exactly. For the sake of convenience, the areas of the extensional membrane elements at plastic hinge lines at E, F, and G are assumed to be equal to εδ2=0.14δ2, respectively and equal to one-third of εδ2 for each extensional membrane element at plastic hinge lines at A, B and C. In addition, 210
Thin–Walled Structures 115 (2017) 205–214
T. Tran
in crushing, Et, as
PatI − A δ = EttI − A = EbtI − A + EmtI − A Substituting
EbtI − A,
EmtI − A
(8)
back into Eq. (8) gives
7π PatI − A δ = M0 L + 8σ0 tεδ 2 4
(9)
The unknown parameter δ can be determined by applying the stationary condition of the crushing force as
∂ ⎛ PatI − A ⎞ ⎜ ⎟ = 0, ∂δ ⎝ M0 ⎠
(10)
which leads to
−7Lπ 32ε + =0⇒δ= 4δ 2 t
7Ltπ 128ε
(11)
To substitute the term δ of Eq. (11) into Eq. (9), the proposed expression of the average crushing force in stage A of tube type I is
PatI − A =
σ0 t 2 7π 128ε 7Lπt L + 8σ0 tε = 3.74σ0 L0.5π 0.5t1.5ε 0.5 4 128ε 4 7Lπt
3.3.2. Stage B Due to the formation of two new plastic hinge lines, crushing force rises to a higher plateau. Their rotation angles are, respectively, π/2 and π/6. With eight plastic hinge lines, the energy absorption due to bending of tube in stage B is evaluated by the following equation,
Fig. 8. Deformation shapes of the tube type II in: a) stage A; b) stage B.
through the angles of π/2 while the rotation angles at I, J, and K are π/ 12, respectively. For the stage B, plastic deformation mechanism continues to evolve, as shown in Fig. 8(b). Points I and O continue to move vertically until upper platen contacts with the plastic hinge line at N. Two sides IMJ and NPK bend inwardly whereas ION and JLK outwardly. Then, one new moving plastic hinge line develops at P. With the formation of new moving plastic hinge line, the crushing force quickly climbed up to a higher plateau (Fig. 5(b)). Referring to the moving plastic hinge line at M in stage A, the line at P form and develops in a similar way. The areas of the extensional membrane elements and rotation angle at the lines at P in stage B and M in stage A are therefore equivalent. This means that their energy dissipations by moving plastic hinge lines at P, M are the same. Finally, eight plastic hinge lines are formed during the crushing process of stage B.
8 ⎛ π π π π⎞ EbtI − B = M0 L ∑ αi = M0 L ⎜3 +3 + + ⎟ ⎝ 12 2 2 6⎠ i =1
8 ⎛ 1 1⎞ EmtI − B = 2σ0 t ∑ si = 2σ0 tεδ 2 ⎜3 + 3 + 2 ⎟ ⎝ 3 2⎠ i =1
(14)
According to ISSFE theory, the total internal energy absorption after crushing consists of the energy dissipated by plastic hinge lines formation Eb and the energy dissipated by membrane area Em. Since the internal energy should be equal to the corresponding average crushing force, this relationship can be expressed as
PatI − B δ = M0 L 3.3.1. Stage A There are six plastic hinge lines developed on all sides of tube. As mentioned above, the rotation angle at the plastic hinge lines E, F, and G is equal to π/2. In regards to plastic hinge lines at A, B, and C, rotation angle equals to π/12. The corresponding energy dissipation due to bending of tube in stage A is calculated as
29π + 10σ0 tεδ 2 12
(15)
Let ∂(Pa/M0)/∂δ=0. This leads to
−29Lπ 40ε + =0⇒δ= 12δ 2 t
29Lπt 480ε
(16)
Hence the average crushing force is of the following form
6
PatI − B = (6)
Eq. (5) shows that the membrane energy depends substantially on the area of extensional membrane elements at plastic hinge line. The extensional membrane elements on either side at moving and stationary plastic hinge lines are equally area. This energy is hence given as follow 6 ⎛ ⎞ 1 EmtI − A = 2σ0 t ∑ si = 2σ0 t ⎜3 × εδ 2 + 3εδ 2⎟ ⎝ ⎠ 3 i =1
(13)
As stated previously, the membrane element's area has a considerable effect on membrane energy. Two new plastic hinge lines have the total area of extensional membrane elements equal to the membrane elements’ area at plastic hinge line E. Consequently, the membrane energy, dissipated by the plastic hinge lines, is determined by,
3.3. Average crushing force for multi-cell triangular tube type I
⎛ π π⎞ EbtI − A = M0 L ∑ αi = M0 L ⎜3 +3 ⎟ ⎝ 12 2⎠ i =1
(12)
σ0 t 2 29π 480ε 29Lπt L + 10σ0 tε = 4.91σ0 L0.5π 0.5t1.5ε 0.5 4 12 29Lπt 480ε
(17)
3.4. Average crushing force for multi-cell triangular tube type II 3.4.1. Stage A As shown in Figs. 4 and 7(a), all plastic hinge lines in stage A developed on outer wall and at corner of tube. The rotation angles at plastic hinge lines M, N, O, and L are of π/2, respectively and a value of π/12 for rotation angles at plastic hinge line I, J, and K, respectively. In accordance with the ISSFE theory, the bending energy is determined based on the rotation angle. The energy absorption in bending of tube in stage A is
(7)
The deformation process in stage A of multi-cell tube type I is controlled by the moving plastic hinge lines. The total work done in causing the crushing, Paδ, should be equal to the total energy absorbed 211
Thin–Walled Structures 115 (2017) 205–214
T. Tran
Fig. 9. Representative load-displacement curves of multi-cell triangular tubes: (a) TI-2, (b) TI-3, and (c) TI-4; (d) TII-2, (e) TII-3 and (f) TII-4. 7 ⎛ π π⎞ EbtII − A = M0 L ∑ αi = M0 L ⎜3 +4 ⎟ ⎝ 12 2⎠ i =1
which leads to (18)
−9Lπ 40ε + =0⇒δ= 4δ 2 t
The ISSFE theory also points out that the extensional membrane element at plastic hinge line affects significantly the membrane energy. Besides, the absorbed energies in the membrane at L, M, N, and O are the same as discussed above. The membrane energies at I, J and K are also equal. The energy dissipated by the membrane is thus given as
⎞ ⎛ 1 EmtII − A = 2σ0 t ∑ si = 2σ0 t ⎜3 εδ 2 + 4εδ 2⎟ ⎠ ⎝ 3 i =1
PatII − A =
(23)
3.4.2. Stage B In the ISSFE theory, the areas of the membrane elements and rotation angle are assumed to be constant and equivalent during the whole crushing process. The formation of moving plastic hinge line at P in stage B makes the force quickly climb up to a higher plateau. Since there are eight moving plastic hinge lines developed on tube during crushing process of stage B, the absorbed energy in bending and membrane of tube are respectively
(20)
Using the stationary condition gives
∂ ⎛ PatII − A ⎞ ⎜ ⎟ = 0, ∂δ ⎝ M0 ⎠
σ0 t 2 9π 160ε 9Lπt L + 10σ0 tε = 4.743σ0 L0.5π 0.5t1.5ε 0.5 4 160ε 4 9Lπt
(19)
Combining Eqs. (1), (18) and (19) yields
9π + 10σ0 tεδ 2 4
(22)
Substituting Eq. (22) back into Eq. (20), the average crushing force in stage A of tube II is obtained as
7
PatII − A δ = M0 L
9Ltπ 160ε
(21) 212
Thin–Walled Structures 115 (2017) 205–214
T. Tran 8 ⎛ π π⎞ EbtII − B = M0 L ∑ αi = M0 L ⎜3 +5 ⎟ ⎝ 12 2⎠ i =1
(24)
8 ⎞ ⎛ 1 EmtII − B = 2σ0 t ∑ si = 2σ0 t ⎜3 εδ 2 + 5εδ 2⎟ ⎠ ⎝ 3 i =1
(25)
Based on the kinematic analysis of a SBFE during crushing, the work of external force is equal to the energy dissipation by shell extension and plastic hinge lines formation of on the curved surfaces. Then,
PatII − B δ = M0 L
11π + 12σ0 tεδ 2 4
(26)
Tran et al. [10] pointed out that the independent parameter of the crushing mode would minimize the crushing force, or otherwise the differential equation ∂(Pa/M0)/∂δ is equal to zero, i.e,
−11Lπ 48ε + =0⇒δ= 4δ 2 t
11Ltπ 192ε
(27)
In can be seen that, on substituting the expression for the distance of the crushing (Eq. (27)) into Eq. (26), the average crushing force is
PatII − B =
σ0 t 2 11π 192ε 11Lπt L + 12σ0 tε = 5.74σ0 L0.5π 0.5t1.5ε 0.5 4 192ε 4 11Lπt
(28) Fig. 10. Tube type I: a) Traditional collapse mode; b) Exceptional collapse mode.
4. Comparison with experiment test and discussion The works of Fan et al. [24], Wang et al. [25], and Sun et al. [26] introduced theoretical analyses for single and multi-cell triangular tubes. However, these formulas are just for equilateral or isosceles triangular cell. Then, the expressions of the average crushing force for two types of tubes in case of lateral crushing are proposed in Section 3. The theoretical predictions in stage A, B are therefore calculated by expression (12), (17), (23), and (28) and these values were used to compare with the value of average crushing force of tubes, n Pa = ∑i =1 Pi / n , in each stage. The load-displacement curves from representative specimens of multi-cell triangular tubes are illustrated in Fig. 9. Given the values for the average crushing force, it is possible to conclude the difference between crushing force at stage A and B depends on the number of plastic hinge lines. The average crushing force obtained from theoretical and experimental results are listed in Table 2. It is revealed that the errors vary from −16.49% to −4.49% in stage A and from −2.92% to 9.52% in stage B for tube type I. Regarding to tube type II, these errors are, respectively from −12.91% to −7.16% in stage A and from −3.79% to 7.36% in stage B. Figs. 10 and 11 show that crushing mechanism of tube type I is Table 2 Tested and predicted average crushing forces. Specimen number
Average crushing force (kN)
Error (%) Fig. 11. Tube type II: a) Traditional collapse mode; b) Exceptional collapse mode.
Theoretical prediction
Experiment data
A
B
A
B
A
B
Type I
1 2 3 4 5 6
12.75
16.74
13.54 15.27 13.98 13.35 13.85 13.59
15.67 18.52 17.23 17.24 15.86 15.28
−5.81 −16.49 −8.79 −4.49 −7.93 −6.21
6.82 −9.61 −2.86 −2.92 5.56 9.52
Type II
1 2 3 4 5 6
11.46
13.87
12.58 12.34 12.48 12.63 12.55 13.16
14.42 14.39 12.92 13.73 13.67 15.17
−8.89 −7.16 −8.15 −9.26 −8.68 −12.91
−3.79 −3.62 7.36 0.98 1.45 −8.59
different to that of tube type II. It is interesting that two new collapse modes for tube type I and II, shown in Figs. 10(b) and 11(b), are detected in the experiments. As regards tube type I, specimen TI-2 has seven plastic hinge lines while the traditional collapse mode has six (Fig. 9). Pa in stage A and B of TI-2 are 15.27 kN and 18.52 kN, respectively, greater than Pa of the other specimens, which show the errors between theoretical prediction and experimental data are −16.49% and −9.61% in stage A and B. Respecting tube type II, there are eight plastic hinge lines developed on classical collapse mode whereas the collapse mode of TII-6 has nine lines (Fig. 10). Pa in stage A and B of the rest of specimens are smaller than those of the specimen TII-6 that are 13.16 kN and 15.17 kN, respectively, which demonstrate the differences between theoretical solution and experimental result are 213
Thin–Walled Structures 115 (2017) 205–214
T. Tran
3. Triangular tubes with different cross-sections have different crushing mechanism. More crushing modes are detected in the lateral crushing behavior of the multi-cell triangular tubes. Two exceptional collapse mechanisms enlarging Pa of tube type I and II appear during lateral crushing. More detailed works are then needed to reveal these collapse mechanisms. References [1] W. Chen, T. Wierzbicki, Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption, Thin-Walled Struct. 39 (2001) 287–306. [2] T. Wierzbicki, W. Abramowicz, On the crushing mechanics of thin-walled structures, J. Appl. Mech. 50 (1983) 727–734. [3] X. Zhang, G. Cheng, H. Zhang, Theoretical prediction and numerical simulation of multi-cell square thin-walled structures, Thin-Walled Struct. 44 (2006) 1185–1191. [4] H.-S. Kim, New extruded multi-cell aluminum profile for maximum crash energy absorption and weight efficiency, Thin-Walled Struct. 40 (2002) 311–327. [5] A.G. Hanssen, A. Artelius, M. Langseth, Validation of the simplified super folding element theory applied for axial crushing of complex aluminium extrusions, Int. J. Crashworthiness 12 (2007) 591–596. [6] A. Najafi, M. Rais-Rohani, Mechanics of axial plastic collapse in multi-cell, multicorner crush tubes, Thin-Walled Struct. 49 (2011) 1–12. [7] A. Alavi Nia, M. Parsapour, An investigation on the energy absorption characteristics of multi-cell square tubes, Thin-Walled Struct. 68 (2013) 26–34. [8] F. Sun, C. Lai, H. Fan, D. Fang, Crushing mechanism of hierarchical lattice structure, Mech. Mater. 97 (2016) 164–183. [9] T. Tran, S. Hou, X. Han, N. Nguyen, M. Chau, Theoretical prediction and crashworthiness optimization of multi-cell square tubes under oblique impact loading, Int. J. Mech. Sci. 89 (2014) 177–193. [10] T. Tran, S. Hou, X. Han, M. Chau, Crushing analysis and numerical optimization of angle element structures under axial impact loading, Compos. Struct. 119 (2015) 422–435. [11] N.K. Gupta, G.S. Sekhon, P.K. Gupta, A study of lateral collapse of square and rectangular metallic tubes, Thin-Walled Struct. 39 (2001) 745–772. [12] N.K. Gupta, A. Khullar, Collapse load analysis of square and rectangular tubes subjected to transverse in-plane loading, Thin-Walled Struct. 21 (1995) 345–358. [13] N.K. Gupta, A. Khullar, Collapse of square and rectangular tubes in transverse loading, Arch. Appl. Mech. 63 (1993) 479–490. [14] S. Maduliat, T.D. Ngo, P. Tran, R. Lumantarna, Performance of hollow steel tube bollards under quasi-static and lateral impact load, Thin-Walled Struct. 88 (2015) 41–47. [15] T. Tran, Crushing analysis of multi-cell thin-walled rectangular and square tubes under lateral loading, Compos. Struct. 160 (2017) 734–747. [16] A. Baroutaji, M.D. Gilchrist, D. Smyth, A.G. Olabi, Crush analysis and multiobjective optimization design for circular tube under quasi-static lateral loading, Thin-Walled Struct. 86 (2015) 121–131. [17] A. Baroutaji, E. Morris, A.G. Olabi, Quasi-static response and multi-objective crashworthiness optimization of oblong tube under lateral loading, Thin-Walled Struct. 82 (2014) 262–277. [18] A. Baroutaji, M.D. Gilchrist, A.G. Quasi-static Olabi, impact and energy absorption of internally nested tubes subjected to lateral loading, Thin-Walled Struct. 98 (2016) 337–350. [19] A. Eyvazian, I. Akbarzadeh, M. Shakeri, Experimental study of corrugated tubes under lateral loadingmechanical engineers, Proc. Inst. Mech. Eng. Part L: J. Mater. Des. Appl. 226 (2012) 109–118. [20] H. Wang, J. Yang, H. Liu, Y. Sun, T.X. Yu, Internally nested circular tube system subjected to lateral impact loading, Thin-Walled Struct. 91 (2015) 72–81. [21] T.N. Tran, T.N.T. Ton, Lateral crushing behaviour and theoretical prediction of thinwalled rectangular and square tubes, Compos. Struct. 154 (2016) 374–384. [22] E. Morris, A.G. Olabi, M.S.J. Hashmi, Analysis of nested tube type energy absorbers with different indenters and exterior constraints, Thin-Walled Struct. 44 (2006) 872–885. [23] E. Morris, A.G. Olabi, M.S.J. Hashmi, Lateral crushing of circular and non-circular tube systems under quasi-static conditions, J. Mater. Process. Technol. 191 (2007) 132–135. [24] H. Fan, W. Hong, F. Sun, Y. Xu, F. Jin, Lateral compression behaviors of thin-walled equilateral triangular tubes, Int. J. Steel Struct. 15 (2015) 785–795. [25] P. Wang, Q. Zheng, H. Fan, F. Sun, F. Jin, Z. Qu, Quasi-static crushing behaviors and plastic analysis of thin-walled triangular tubes, J. Constr. Steel Res. 106 (2015) 35–43. [26] F. Sun, C. Lai, H. Fan, In-plane compression behavior and energy absorption of hierarchical triangular lattice structures, Mater. Des. 100 (2016) 280–290.
Fig. 12. Scatter diagram of crushing force: a) Tube type I; b) Tube type II.
−12.91% and −8.59%. Perhaps it is the exceptional collapse mechanisms that augment the Pa of tube type I and II. Since these phenomena are not prevailing in the experiment, more detailed tests are needed to reveal these collapse mechanism. Multi-cell triangular tube type I has greater average crushing force compared to tube type II in both stages A and B, as shown in Fig. 12 and Table 2. In addition, Fig. 12 also points out that the reasonable agreement can be observed. Accordingly, a very strong support is verified between the theoretical solutions and the experiment data in all cases. 5. Conclusion This work presents a combined experimental and theoretical study on the crushing behavior of multi-cell thin-walled triangular tubes under lateral crushing. Two types of multi-cell triangular tubes are performed using universal testing machine. It is concluded that: 1. The load-displacement curves are hence obtained. Their deformation patterns are examined and used to govern the theoretical analysis. The average crushing force is controlled by the number of the plastic hinge lines. 2. The theoretical solutions are developed based on the ISSFE theory for each tube type. These solutions show that the average crushing force is a function of flow stress of material, wall thickness, and length of tube. The average crushing force of the tube type I is bigger than that of the tube type II in both stages A and B. The predictions are consistent with the experimental results within acceptable errors.
214