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A relationship between progressive collapse and initial buckling for tubular structures under axial loading
International Journal of Mechanical Sciences 75 (2013) 200–211
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A relationship between progressive collapse and initial buckling for tubular structures under axial loading Jie Song a,n, Yufeng Zhou a, Fenglin Guo b a b
School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China
art ic l e i nf o
a b s t r a c t
Article history: Received 28 November 2012 Received in revised form 13 June 2013 Accepted 28 June 2013 Available online 10 July 2013
The progressive collapse of tubular structures under axial loading is a challenging problem in mechanics. Due to the nonlinearities in large plastic deformation, such a problem can only be solved case by case under the assumption of an appropriate collapse mechanism. In this paper, a relationship between the progressive collapse of an axially loaded tube and the initial buckling of its windowed counterpart is presented. Numerical investigation was performed on the axial crushing of triangular, square and pentagonal tubes and the initial buckling modes of the corresponding windowed tubes. Results show that at the critical symmetric buckling mode, the theoretical mean crushing force of the angle-shaped column in the windowed tube matches very well with the actual mean crushing force of the conventional one. This relationship is crucial to the development of a generalized method for progressive collapse without assuming collapse mechanism. Based on it, an empirical equation on the mean crushing force of axially loaded square tubes is presented. The mean crushing forces predicted by this equation are in good agreement with the experimental results and theoretical values. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Progressive collapse Axial loading Windowed tube Buckling mode
1. Introduction Large plastic deformation of thin-walled structures is a challenging problem in mechanics. Due to its high geometric, material and contact nonlinearity, as well as strong deformation-history dependency, most studies in this field are conducted through experimental and numerical approaches. For example, experimental studies on the collapse of axially loaded square tubes were performed by Abramowicz and Jones [1,2], Langseth and Hopperstad [3], and Jensen et al. [4]. In numerical approaches, the explicit time integration scheme is widely employed because of its effectiveness in convergence for highly nonlinear problems. However, the time step has to be sufficiently small to ensure numerical stability. For practical problems involving hundreds of thousand degrees of freedom, it may take hours or days to simulate the deformation. Theoretical studies on the thin-walled structures with large plastic deformation are limited to those in simple geometric shapes, such as circular/square tubes and spherical shells. Currently, the only method available is the limit analysis based on upper bound theorem, in which a collapse mechanism or a field of plastic flow has to be assumed. The load–displacement relationship is obtained through equilibrium condition or energy balance between external work and plastic dissipation. However, the method is very
n
Corresponding author. Tel.: +65 83590056. E-mail addresses: [email protected], [email protected] (J. Song).
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subtle as a good assumption of the collapse mechanism is essential to its success. Therefore, it can be only applied to structures with simple geometry and deformation mode, e.g., the ring mode of axially loaded circular tubes [5] and the bending collapse of square tubes [6]. Another drawback of this method is that it may not give a closed-form solution. In Ref. [6], for example, the so-called “rolling radius” was indeterminate using this method and had to be obtained from tests. Among various deformation forms of thin-walled structures, the progressive collapse of tubular structures from one end to the other with the formation of successive folding along its central axis has attracted many attentions. Because of its excellent collapse characteristics, e.g., long stroke distance and large plastic deformation, this form is widely used for energy absorption in automotive and construction industries. Theoretical analysis on the progressive collapse focuses mainly on circular and square tubes. The circular tube has two distinct collapse modes, i.e., ring and diamond mode. Alexander [5] first proposed an approximate model to predict the mean crushing force of the ring mode, which was later modified by Abramowicz and Jones [2,7] with the consideration of curvature of the folds and by Wierzbicki et al. [8] considering both inward and outward folding. The diamond mode is more complex, and its theoretical study is not so successful as that for the ring mode. Some studies have been made by Pugsley and Macaulay [9], Johnson et al. [10] and Singace [11]. The relationship between collapse mode and geometry of the circular tube, e.g., diameter-to-thickness ratio, was investigated in Refs. [12–14].
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progressive collapse of a tube and the initial buckling of its windowed counterpart is described. With this relationship, the mean crushing force of progressive collapse can be calculated without assuming the collapse mechanism, which may provide a novel strategy in the theoretical studies on progressive collapse and further the understanding to the collapse nature. The outline of the paper is as follows. First the geometry of windowed as well as corresponding conventional tubes is introduced, and the extensional collapse mode is discussed. Next, with an experimentally validated FE model, the relationship between the buckling modes of the windowed tube and the window size is described. Particular attentions will be paid to the so-called “critical symmetric buckling mode”. It illustrates that the theoretical mean crushing force of this mode of the windowed tube matches very well with the actual mean crushing force of the corresponding conventional tube. Then, based on the critical buckling mode, an empirical equation of the mean crushing force for axially loaded square tubes is presented. Mean forces predicted by this equation are compared with experimental results from open literature and calculated values using Wierzbicki and Abramowicz's model [1]. This paper ends with a conclusion and suggestions for future work.
For square tubes, the symmetric and the extensional modes are the most important ones. The former was theoretically studied by Wierzbicki and Abramowicz [15] with their super-folding element and later improved by Abramowicz and Jones [1] with consideration of the effective crushing distance, which is defined as the maximum crushing distance of a tube before being compactly compressed to explain the discrepancy between theoretical and experimental results. The extensional mode was studied by Abramowicz and Jones [2]. They presented a basic collapse element and derived the formula of the mean crushing force. The relationship between collapse mode and width-to-thickness ratio of the square tube was studied by Abramowicz and Jones [2]. More recently, axial crushing behavior of square tubes with various modifications, such as those with reinforcements [16] or origami patterns [17], has been investigated. Existing theoretical studies on the progressive collapse of tubular structures are based on the following two assumptions: (1) The material's plastic behavior can be simplified to a perfect plastic model with flow stress s0 . (2) The tube length has no effect on the characteristics of progressive collapse. These assumptions in general hold true for most prismatic tubes, and they are adopted in the present paper. For circular tubes, studies [12–14] have shown that the tube length may affect the tube's collapse mode. Recently, the windowing method proposed by Song et al. [18] has shown great potentials in controlling the tube's collapse behavior. By introducing windows to the tube wall, the collapse mode of the tube can be altered, resulting in a higher mean crushing force. In the present paper, a relationship between the
2. Geometry and extensional mode 2.1. Geometry of tube Regular triangular, square and pentagonal tubes are considered. Table 1 shows the geometry of the conventional tubes, in which M is the number of edges, c is width of edge in the middle plane, t is wall thickness, and Lc is tube length. Notation TRI/SQU/PEN-c-t is
Table 1 Geometry of conventional and windowed tubes. Cross-sectional shape
a
M
c (mm) a
t (mm) a
a
a
a
Lw (mm)
120 180 240
240t ¼ 3,
3
40.36 60 80
4
40.36a 60 80
3a, 2a, 1.5a, 1a, 0.5a 3, 2.25, 1.5 4, 3, 2
120 180 240
120
5
40.36a 60 80
3, 2a, 1.5a, 1a, 0.5a 3, 2.25, 1.5 4, 3, 2
120 180 240
120
Value also used for the windowed tube.
3 , 2 , 1.5 , 1 , 0.5 3, 2.25, 1.5 4, 3, 2
Lc (mm) a
, 120t ¼ 1,
2, 1.5 mm
0.5 mm
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J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
adopted to distinguish each tube. Symbols TRI, SQU and PEN represent triangular, square and pentagonal tubes, respectively. For each conventional polygon tube with c¼40.36 mm, except PEN-40.36-3, a corresponding windowed tube is considered. The windows are in rectangular shape, with width a and height b, one on each side. Each window is positioned at the center of the side. An example of windowed square tube is shown in Fig. 1. The leftover, also referred to as angle-shaped column, has width e ¼ ðcaÞ=2. The length of windowed tube Lw is identical to that of the corresponding conventional tube expect for triangular ones with t from 3 mm to 1.5 mm, where Lw ¼ 240 mm. The values of Lw are shown in Table 1.
in which M 0 ¼ s0 t 2 =4 is the fully plastic bending moment, and κ is the scale factor defining the effective crushing distance. The details of derivation of Eq. (1) can be found in Appendix. Allowing the window b varies, P m reaches minimum when pffiffiffiffiffiffiffiffiffiffiffiffiheight, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ 2Met ¼ MðcaÞt . In this case, we have rffiffiffiffiffiffiffiffiffiffiffi Pm 1 e 4π 2M þ πðM2Þ : ð2Þ ¼ κ t M0 Eq. (2) is also applicable for conventional tubes. In this case, the window width a vanishes and e ¼ c=2.
3. Numerical simulation 2.2. Extensional mode
3.1. FE modeling
The extensional mode of square tubes under axial loading, as shown in Fig. 2(a), has been theoretically studied by Abramowicz and Jones [2]. Fig. 2(b) shows the extensional mode of windowed square tube, in which both panels of the angle-shaped columns deform outward. This mode has been observed in the experimental study by Song et al. [18] on axially loaded windowed square tubes. The extensional modes of other polygon conventional and windowed tubes are similar to those in Fig. 2(a) and (b), respectively. In the present paper, the theoretical model proposed by Abramowicz and Jones [2] is extended to regular polygon windowed tubes. Following their analysis with a rigid-perfect plastic material with flow stress s0 , the mean crushing force, P m of extensional mode of a regular M-gon windowed tube is given by Pm 1 b e 2π þ 4πM þ πðM2Þ ; ð1Þ ¼ κ t b M0
FE code Abaqus/Explicit 6.11 was adopted for numerical simulation. The tubes were meshed with 4-node shell element, S4R with 7 integration points through the thickness. The element size was 1.5 mm 1.5 mm. The set-up of axial loading of the tube is shown in Fig. 3. The tube was put on the bottom plate, and the top plate, initially contacted with the tube, moved downward to crush the tube. No extra clamping or holding apparatus was used to constrain the tube's ends. Both top and bottom plates were modeled as rigid body. The crushing process was quasi-static. In Abaqus, it can be implemented with SMOOTH STEP sub-option by setting the time step more than 10 times larger than the tube's longest natural period [19]. For conventional tubes, the mean crushing force, P m is calculated as follows: Rd Pδd Ea ¼ 0 ; ð3Þ Pm ¼ d d in which P is crushing force, d is axial displacement, and Ea is the energy absorption of the tube up to that displacement. Finite sliding penalty formulation was used to model frictions between plates and tube walls and between tube walls themselves. The frictional coefficient was 0.3 for all the contact interactions. This value has been adopted in the study of Song et al. [18] on windowed tubes. The contact behavior in normal direction was modeled as “hard contact”. The material of tube was annealed mild steel with properties as follows: density ρ ¼ 7332:3 kg/m3, Young's modulus E ¼ 190:5 GPa, Poisson's ratio ν ¼ 0:3, yield stress sy ¼ 287:9 MPa, and ultimate stress su ¼ 506:9 MPa. The true stress–plastic strain data of the material is given in Table 2. The Von-Misses yield criterion and isotropic strain hardening were assumed, and strain rate effects were neglected. The plastic flow stress, s0 of the material was calculated by using [20] rffiffiffiffiffiffiffiffiffiffiffiffi sy su s0 ¼ ; ð4Þ 1þn
Fig. 1. A windowed square tube.
in which n is the strain hardening exponent. For the present material, n ¼0.22. To simulate the conventional tube collapsing in its natural mode, the whole tube was modeled, and small initial deflection of middle plane was introduced to mimic the inevitable imperfections that would be present in any real specimen. This was realized by superimposing the first 3 buckling modes of the tube onto the tube wall, each with certain amplitude. The amplitudes were proportional to the wall thickness. Their values are shown in Table 3. The values of amplitudes were determined through trial simulations. It will be shown in Section 3.2 that with these values FE simulation can achieve very good agreement with experimental and theoretical results.
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Fig. 2. Extensional modes of conventional and windowed square tubes: (a) conventional tube and (b) windowed tube.
P
Symmetric boundary condition
d Symmetric boundary condition
Top plate Tube
Fig. 4. FE model of a conventional square tube for the simulation of extensional mode.
Bottom plate
Fig. 3. Sketch of tube under axial loading.
Table 2 True stress–plastic strain of the mild steel. True stress (MPa) 287.91 335.43 385.38 425.38 476.66 506.93 Plastic strain 0 0.0319 0.0533 0.0793 0.137 0.205
Table 3 Amplitude of each buckling mode. Buckling mode Amplitude (mm)
1 0.01t
2 0.007t
3 0.004t
The extensional mode of the conventional tube was also simulated. In this case, only one corner of the tube was modeled, like the example of square tube shown in Fig. 4. Symmetric boundary condition was applied on the side edge of each panel. For each pair of nodes on the side edges that are at the same level, a constraint was imposed making them have the same axial displacement and out-of-plane rotation. No initial deflection was introduced to the tube wall. The windowed tube was also modeled as one corner, with symmetric boundary condition applied on the side edge of each panel. Although a windowed tube may not collapse in such symmetrical way, in the present study, only initial collapse of the windowed tube is considered, and it has been observed that deformation always concentrated on the angle-shaped columns while the rest of the tube had little deformation. Therefore, modeling the windowed tube as one corner should be accurate enough to investigate the initial collapse response. The initial
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deflection was introduced to each panel with the following expression: w ¼ w0 sin 1
πe πx 3πz sin sin ; c c Lw
ð5Þ
in which w0 is the maximum initial deflection of the corresponding conventional tube. The coordinate system of the panel is shown in Fig. 5(a). The wave numbers in x and z directions are identical to those of the first buckling mode of the conventional 1 tube. The factor sin πe=c was introduced to ensure that the angle-shaped column and the conventional tube had the same maximum deflection. The FE model of windowed tube with initial deflection is shown in Fig. 5(b).
3.2. Validation Axial crushing test on a square tube was conducted using Instron 5500 Universal Testing Machine. The tube had dimensions of Lc ¼ 120 mm, c ¼ 40:36 mm, and t ¼ 2:1 mm. It was fabricated from a mild steel block by wire-cut. Heat-treatment was performed before test to remove residual stress. The tube had material properties described in Section 3.1. The experimental set-up is shown in Fig. 3. Quasi-static crushing process was adopted with a loading speed of 1 mm/min.
The tube collapsed in symmetric mode. The deformed profiles of the tube from experiment and simulation are shown in Fig. 6. Table 4 shows the numerical, experimental and theoretical values of mean crushing force at displacement of 67% of the tube length. The theoretical value is calculated using the formula of Abramowicz and Jones [1] with a more precise constant factor as c1=3 Pm ¼ 52:91 : ð6Þ t M0 Since the progressive collapse of conventional square tube is well understood and good agreement between theoretical and experimental values of mean crushing force was achieved, the test was conducted only once. From Fig. 6 and Table 4, it is seen that the numerical results match very well with the experimental ones in both collapse mode and mean crushing force. Therefore the present FE model has been validated.
4. Results and discussion 4.1. Extensional mode The scale factor κ of effective crushing distance of the extensional mode of conventional tube is determined by substituting the numerical result of mean crushing force, calculated with Eq. (3), into
Symmetric boundary condition
Symmetric boundary condition
z
x
0
Fig. 5. FE model of the windowed tube with initial deflection: (a) local coordinate system of the panel and (b) FE model.
Fig. 6. Deformed profiles of tube SQU-40.36-2.1: (a) experiment and (b) simulation.
J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
Eq. (2). Table 5 shows the value of κ for each conventional tube. The mean crushing force of extensional mode is calculated at displacement of 50% of the tube length. The reason of using 50% instead of 67% is that at this displacement most tubes just developed two pure extensional folds, as shown in Fig. 7(a), so the mean crushing force of extensional mode, and also the value of κ, can be determined relatively accurately. If the tube has large width-tothickness ratio c/t, the extensional mode may be corrupted. For example, Fig. 7(b) shows the collapse of tube TRI-40.36-1 with perfect geometry at d¼37 mm. It can be seen that the tube has inextensional deformation below the first extensional layer. For the tube not following extensional mode exactly, it is assumed that its value of κ is the same as that of the tube with largest c/t ratio and pure extensional collapsing mode. Those values are shown in bold in Table 5. We adopted this assumption because it seems yielding consistent results as will be shown in Table 5, but it is still not clear why the assumption works. The values of κ of SQU-40.36-1.5 and PEN-40.36-1.5 calculated from numerical mean crushing forces are larger than those of SQU-40.36-2 and PEN-40.36-2, respectively. Such abnormality should be caused by numerical errors, because it is intuitively clear that κ must be a decreasing function of c/t, otherwise it Table 4 Numerical, experimental and theoretical values of mean crushing force of tube SQU-40.36-2.1 at d ¼80 mm.
Pm (kN)
Numerical
Experimental
Theoretical
54.12
54.78
54.06
205
would mean that for two tubes with the same width, the one with thicker wall could be crushed more deeply than that with thinner wall. The values of κ of SQU-40.36-1.5 and PEN-40.36-1.5 have been adjusted to those of SQU-40.36-2 and PEN-40.36-2, respectively. It should be noted that according to Abramowicz and Jones [1], κ is 0.77 for the extensional mode of conventional square tube. In the present study, however, κ varies with c/t and, in some cases, is significantly smaller than the value proposed by Abramowicz and Jones. This discrepancy will be addressed in the future.
4.2. Critical symmetric buckling mode Two buckling modes of the angle-shaped column in the windowed tube are observed, i.e., symmetric and anti-symmetric modes as shown in Fig. 8. The occurrence of each mode depends on both width and height of the window. Fig. 9 shows distribution of the two buckling modes with respect to width and height of the window for windowed tubes TRI-40.36-1, SQU-40.36-1 and PEN40.36-1. It is seen that for both SQU-40.36-1 and PEN-40.36-1, there exists a minimum width ac , where only symmetric buckling mode occurs. For TRI-40.36-1, there is a minimum width ac , where there is transition between the symmetric and anti-symmetric buckling mode in turn with the increase of window height. The characteristics of ac of other square, pentagonal and triangular windowed tubes with c¼40.36 mm are determined in the same way,pwhich are listed in Table 5. The buckling mode at a ¼ ac and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ Mðcac Þt is referred to as critical symmetric buckling mode,
Table 5 Numerical results and theoretical mean crushing forces of critical symmetric buckling mode of tubes. Tube
P m (numerical result, kN)
κ
ac (mm)
P m (critical symmetric buckling mode, kN)
TRI-40.36-3 TRI-40.36-2 TRI-40.36-1.5 TRI-40.36-1 TRI-40.36-0.5 TRI-60-3 TRI-60-2.25 TRI-60-1.5 TRI-80-4 TRI-80-3 TRI-80-2 SQU-40.36-3 SQU-40.36-2 SQU-40.36-1.5 SQU-40.36-1 SQU-40.36-0.5 SQU-60-3 SQU-60-2.25 SQU-60-1.5 SQU-80-4 SQU-80-3 SQU-80-2 PEN-40.36-3 PEN-40.36-2 PEN-40.36-1.5 PEN-40.36-1 PEN-40.36-0.5 PEN-60-3 PEN-60-2.25 PEN-60-1.5 PEN-80-4 PEN-80-3 PEN-80-2
92.42 37.71 23.64 9.99 3.07 82.22 50.15 23.02 158.05 84.20 42.12 113.42 49.09 28.38 14.19 4.28 108.46 63.19 30.67 190.41 110.89 52.77 145.23 68.00 41.49 17.76 5.09 153.76 79.25 38.37 254.92 140.42 66.90
0.546 0.609 0.646 0.646 0.646 0.635 0.686 0.686 0.646 0.646 0.646 0.568 0.614 0.614c 0.614 0.614 0.638 0.686 0.686 0.647 0.709 0.709 0.587 0.634 0.634c 0.634 0.634 0.648 0.693 0.693 0.656 0.709 0.709
16a 21a 24a 27a 31a 32b 36b 41b 42b 48b 54b 8a 17a 21a 26a 30a 26b 32b 39b 34b 42b 52b 0 7a 12a 19a 28a 11b 18b 29b 14b 24b 38b
92.92 40.24 22.61 11.08 3.26 85.36 47.35 22.85 150.45 89.45 43.71 121.75 51.98 30.64 14.31 4.25 110.96 60.62 28.47 195.81 105.13 48.96 149.21 67.76 40.39 19.00 5.09 147.75 82.64 38.51 260.49 143.73 67.48
a b c
Value determined from the corresponding windowed tube. Value determined from scaling. Adjusted value, the original value is 0.613 for SQU-40.36-1.5 and 0.582 for PEN-40.36-1.5.
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Fig. 7. Pure and corrupted extensional modes: (a) TRI-40.36-2 in pure extensional mode at d ¼ 50 mm and (b) TRI-40.36-1 in corrupted extensional mode at d ¼37 mm.
Fig. 8. Two buckling modes of the angle-shaped column in windowed tube: (a) symmetric mode and (b) anti-symmetric mode.
J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
Symmetric Anti-symmetric
Symmetric
Symmetric Anti-symmetric
Anti-symmetric
Symmetric
column also has significant deformation when the column is deeply crushed. Both effects will result in an overestimated mean crushing force and, hence, an underestimated κ value. Therefore, parameter κ of conventional tube is used as an approximate value. The special case is PEN-40.36-3, which collapses in extensional mode even with imperfection and ac ¼ 0. It is seen from Table 5 that the theoretical mean crushing force of critical buckling mode matches well with the actual value of the conventional tube. This shows the relationship between the progressive collapse behavior of a tube and the initial buckling of its windowed counterpart. The deformed profiles of TRI-40.36-2, SQU-40.36-2 and PEN-40.36-2 are shown in Fig. 10. It is found that only the square tube shows regular collapse pattern, but irregular folding for both the triangular and pentagonal tubes, which makes it very difficult to determine the mean crushing force from traditional approach. However, with the present relationship, it is possible to calculate the mean crushing force without assuming the collapse mechanism. There are still mathematical difficulties in theoretically studying the initial buckling of windowed tube. In the present study, the windowed tube with a ¼ ac buckles locally at its angle-shaped column in the plastic range. In order to theoretically determine the value of ac , one must solve the problem of the plastic buckling of an axially loaded angle column with elastic–plastic boundary condition at both ends, which is too complicated to be treated analytically. Nevertheless, the relationship discovered in the present paper is crucial to the development of a generalized method for progressive collapse. The value of ac for tubes with c¼60 mm and 80 mm could also be determined by finding the critical symmetric buckling mode of their windowed counterparts. Here, a more convenient approach is adopted. According to the second assumption, the tube length has no effect on the mean crushing force. Therefore, for two tubes, TUBE1 with L1 , c1 and t 1 and TUBE2 with L2 , c2 and t 2 , that have the same width-to-thickness ratio, the following relationship holds: c1 t1 1 ac1 ¼ ¼ ¼ ; s c2 t2 ac2
Fig. 9. Distribution of symmetric and anti-symmetric buckling modes with respect to width and height of window: (a) TRI-40.36-1; (b) SQU-40.36-1 and (c) PEN-40.36-1.
and the corresponding angle-shaped column is referred to as critical column. It is worth pointing out that the second transition from symmetric to anti-symmetric mode for windowed tubes TRI40.36-1.5 to TRI-40.36-3 occurs at b larger than 100 mm. Therefore, the length of those windowed tubes is 240 mm, which is long enough to determine the value of ac . 4.3. Mean crushing force Table 5 shows the numerical results of mean crushing force for conventional tubes with c ¼ 40:36 mm at displacement 67% of the tube length. The theoretical mean crushing forces of critical symmetric buckling mode of the corresponding windowed tubes are calculated by Eq. (2), in which e ¼ ðcac Þ=2 and κ takes the value of that of the corresponding conventional tube. The value of κ of the critical column is not determined using the same approach as that for the conventional tube because the crushing distance of the critical column is not long enough and the material around the
207
ð7Þ
in which s is the scale factor and ac1 and ac2 are the width of critical symmetric buckling mode of TUBE1 and TUBE2, respectively. The reasoning is as follows. First, a new unit of length, which is s times the original one, is applied to TUBE1. In this unit, TUBE1 has the same width and thickness in value as those of TUBE2, and with a new length sL1 , a new width of critical mode sac1 and a new flow stress s0 =s. Since tube length does not influence mean crushing force, the length of TUBE1 can be changed to L2 without affecting other parameters. Further applying a new unit of mass to TUBE1, which is s times the original one, the value of flow stress of TUBE1 is changed back to s0 . Now TUBE1 and TUBE2 are identical from the mathematical viewpoint, so their mean crushing forces must have the same value. According to the relationship between mean crushing force and width of critical symmetric buckling mode, the value of ac2 should be equal to sac1 . Table 5 shows the values of ac for tubes with c¼60 mm and 80 mm. The value of each tube is calculated by scaling the ac of the tube with c¼40.36 mm and having the same width-to-thickness ratio (for convenience, the decimal part of width, 0.36, is dropped in calculation). The numerical mean crushing force at displacement 67% of the tube length and theoretical mean crushing force of the critical symmetric buckling mode of these tubes are also presented. Again, good agreement between the two values is observed.
5. An empirical equation As an application and a verification of the relationship in Section 4, an empirical equation of the mean crushing force of
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Fig. 10. Deformed profiles of triangular, square and pentagonal tubes: (a) TRI-40.36-2; (b) SQU-40.36-2 and (c) PEN-40.36-2.
axial loaded square tube is presented and its predictions of mean crushing force are compared with experimental and theoretical results. Define effective half width of the tube, ec as ec ¼
cac ¼ f c; 2
ð8Þ
in which f is a dimensionless ratio function of ec and c. According to the two assumptions and due to its dimensionlessness, f must be a function of t=c. The expression of f is obtained through curve fitting with data points of tubes SQU-40.36-x in Table 5, which is given as follows: t f ¼ 4:419 þ 0:07209: c
ð9Þ
The result of the curve fitting is shown in Fig. 11. The factor of effective crushing distance κ is dimensionless, so it should depends only on t=c. However, Table 5 shows that the value of κ may be different for tubes with the same width-to-thickness ratio, e.g., those of SQU-60-3 and SQU-80-4. This discrepancy might be due to the numerical errors. To obtain the expression of κ, the data of tubes SQU-40.36-x are used. Numerical results of these tubes are most reliable as their width is the same as that of the tube in the validation test. The expression of κ is given by ( t 1:838 ct þ 0:7046 c ≥0:05 κ¼ :ð10Þ t 0:6135 c o0:05 Substituting Eqs. (9) and (10) into Eq. (2) and letting M ¼4, the empirical equation of the mean crushing force of square tube
Fig. 11. Curve fitting results of function f.
under axial loading is obtained as follows: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1442c=tþ8:838þ2π > < 8π 1:838t=cþ0:7046 Pm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ M0 > : 8π 0:1442c=tþ8:838þ2π 0:6135
t c ≥0:05 t c
o 0:05
:ð11Þ
Table 6 compares some experimental results from open literature on the mean crushing force of axially loaded square tubes with values calculated by Eqs. (6) and (11). The tubes are made of mild steel or aluminum. The loading condition is quasi-static and all the tubes collapse in symmetric mode. For tubes except those from Jensen et al. [4], the values of s0 are either taken from the
J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
209
Table 6 Comparison between mean crushing forces from experiment and calculated by Eqs. (6) and (11) for square tubes under quasi-static axial loading. Reference Specimen c (mm)
t (mm)
s0 (MPa)
Pm (kN) Experiment Eq. (6) Eq. (11) a
[2]
C9 D3 D4 E9 E10
17.09 18.18 18.13 36.45 36.45
1.00 0.91 0.91 1.65 1.63
414.20 413.30a 413.30a 315.60a 315.60a
12.97 11.72 12.08 34.40 35.55
14.11 12.28 12.27 31.89 31.25
15.74 12.89 12.90 32.71 31.97
[1]
24 34 131 139 141
49.30 49.34 37.11 37.10 37.06
1.63 1.64 1.15 1.16 1.18
328.50a 328.50a 330.50a 330.50a 330.50a
35.28 36.71 17.90 19.75 18.50
35.78 36.16 18.27 18.54 19.07
34.51 34.91 17.42 17.70 18.27
[3]
T4*-1.8-S T4*-2.0-S T4*-2.5-S
77.65 77.52 76.93
1.81 1.90 2.45
149.97b 159.08b 156.37b
21.80 25.80 37.90
22.75 26.15 39.17
20.76 24.04 37.54
[21]
SS RS
48.00 78.40
2.02 1.85
572.49b 572.49b
94.10 77.82
88.83 90.35
89.76 82.60
[4]
S3 S4 S6 S19
80.00 80.00 80.00 80.00
2.50 2.00 3.50 4.50
186.70c 186.70c 186.70c 186.70c
49.00 32.00 87.00 142.00
a b c
49.00 46.82 33.78 31.14 85.85 87.44 130.51 143.29
Value from the original reference. Value calculated by Eq. (4). Value retrieved from experimental mean crushing force of S3.
references or calculated by Eq. (4) with material properties obtained from the references. The value of s0 for tubes in Jensen et al. [4] is retrieved from the experimental mean crushing force of tube S3 using Eq. (6). It can be seen that Eq. (11) predicts mean crushing force close to the experimental value for most tubes. For S19 and RS, it achieves much higher accuracy than Eq. (6). The largest error of Eq. (11) occurs at tube C9, where it predicts a mean crushing force 21% higher than the experimental one. However, the absolute error is still fairly small in this case. The values of Eqs. (6) and (11) have good agreement for all the tubes with the largest difference being 12% for tube C9. Given the fact that Eq. (11) is only an empirical function based on numerical results, its accuracy has shown the potential of the relationship described in Section 4 being used as an alternative way to calculate the mean crushing force. Particular attentions are paid to tubes C9, D3, D4 and S19, where t=c 4 0:05. Fig. 12 shows the relationship between normalized mean crushing force and thickness-to-width ratio for these tubes. In this range, Eqs. (6) and (11) have different properties. Namely, Eq. (6) is a decreasing function of t=c while Eq. (11) is an increasing one. The relationship between normalized mean crushing force and thickness-to-width ratio for tubes D3, D4, and S19 agrees with that of Eq. (11), although this might be due to the fluctuation in experimental results. Also by comparing tubes C9 and S19, it can be found that they have close thickness-to-width ratios but their errors of mean crushing force predicted by Eq. (11) are quite different. It is worth pointing out that the ultimate stress was used as plastic flow stress in the study of Abramowicz and Jones [2]. For tube C9, this value might be a little higher so that Eq. (11) (and also Eq. (6)) overestimates the mean crushing force. Another possibility is that κ depends on parameters besides c and t, e.g., the local curvature of the tube wall. In this case, the first assumption may not hold as the local curvature depends on the tangential modulus of the material. This may also explain why tubes with the same thickness-to-width ratio have different values of κ in Table 5.
Fig. 12. Relationship between normalized mean crushing force and thickness-towidth ratio for tubes C9, D3, D4 and S19.
6. Conclusion and recommendations A relationship between the progressive collapse of an axially loaded tube and the initial buckling of its windowed counterpart has been discovered. With an experimentally validated FE model, triangular, square and pentagonal tubes under quasi-static axial loading have been simulated, and the initial collapse behavior of the corresponding windowed tubes has also been investigated. Two buckling modes, symmetric and anti-symmetric, are observed in the angle-shaped column of the windowed tube, and there exists a critical symmetric buckling mode. It is found that at the critical symmetric buckling mode, the theoretical mean crushing force of the angle-shaped column in the windowed tube matches well with the actual one of the corresponding conventional tube. This relationship makes it possible to calculate the mean crushing force without assuming a collapse mechanism. As an application and a verification of the relationship, an empirical equation on the mean crushing force of axially loaded square tube has been presented. The values given by this equation match fairly well with the experimental results and theoretical values. In the future, experiments will be conducted on the tubes considered in the present paper to further verify this relationship, and the applicability range of width-to-thickness ratios of the tube to this relationship will be determined. The relationship will be extended to tubes with other cross-sectional shapes, in particular the circular tube. The two discrepancies on the factor of effective crushing distance of the extensional mode will be further investigated for explanation. So far the critical symmetric buckling mode can only be determined numerically. Due to inevitable errors in the numerical simulation, it is not sure whether the critical mode has systematic errors in estimating tube's mean crushing force, and why its characteristics in triangular tube are different from those in square and pentagonal ones.
Appendix A The mean crushing force of extensional mode of a regular M-gon windowed tube is calculated up to the point that the angleshaped columns are completely crushed. The derivation is based on the studies by Abramowicz and Jones [2] and Hayduk and Wierzbicki [22]. Fig. A1 shows the collapse mechanism of the angle-shaped column, in which α is the rotation angle of the panels and 2ψ 0 is the exterior angle of the column which remains constant during the crushing process. The energy dissipation of this mechanism consists of three parts, which are discussed as follows.
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From Figs. A1 and A2, the velocity, vc can be obtained as vc ¼
b _ ψ cos α α: 2 0
ðA:5Þ
Substituting Eq. (A.5) into Eq. (A.4) and integrating from α ¼ 01 to α ¼ 901, the energy dissipation of stretching is 2
E1 ¼ M 0 ψ 0
b : 2t
ðA:6Þ
(2). Energy dissipated in rebending of the conical surface in Fig. A2 Initially, the angle between two panels in the angle-shaped column is π2ψ 0 . The angle becomes zero after the conical surfaces are flattened in the final configuration. For one quarter of the mechanism shown in Fig. A2, the energy dissipation due to the rebending of conical surface is π b : ðA:7Þ E2 ¼ M 0 ψ 0 2 2
Fig. A1. Collapse mechanism of the angle-shaped column.
(3). Energy dissipated along horizontal plastic hinges in one quarter of the mechanism
R
Q
The bending angle is 901 at both horizontal hinges QR and OS. The corresponding energy dissipation is E3 ¼ M 0 πe:
P
ðA:8Þ
The energy dissipation of a single angle-shaped column is " # 2 π b b þ M 0 πe : þ M 0 ψ 0 ðA:9Þ E4 ¼ 4 M 0 ψ 0 2 2 2t
O
S
For a regular M-gon windowed tube, the number of angleshaped columns is M and ψ 0 ¼ π=M. Therefore, the total energy dissipation amounts to
Fig. A2. One quarter of the collapse mechanism in the crushing process.
2
(1). Energy dissipated in stretching in one quarter of the mechanism Consider one quarter of the collapse mechanism in its local coordinate system during the crushing process, as shown in Fig. A2. The boundary, γ between deforming and rigid regions is an arbitrary function gðxÞ, and s is the curvilinear coordinate in the circumferential direction with the origin on γ. Assuming that there is only velocity in s direction and it varies linearly with s, the following velocity field can be adopted: v¼
vc xs ðb=2ÞðgðxÞ þ βxÞ
0≤s≤gðxÞ þ βx;
ðA:1Þ
in which vc is the velocity of point P. Neglecting the shear strain rate and its contribution to the energy dissipation, the only non-vanishing strain rate is the normal strain rate in the circumferential direction, which is ε_ ¼
vc x : ðb=2ÞðgðxÞ þ βxÞ
ðA:2Þ
The rate of energy dissipation due to stretching is Z E_ 1 ¼ s0 t_εdA;
ðA:3Þ
A
in which A is the area of the deforming dA ¼ ðgðxÞ þ βxÞdx. Substituting Eq. (A.2) into Eq. (A.3) yields 1 E_ 1 ¼ s0 tbvc : 4
region
and
ðA:4Þ
E ¼ ME4 ¼ 2M 0 π
b þ M 0 πðM2Þb þ 4M 0 Mπe: t
ðA:10Þ
The work done by the external force to crush all the angleshaped columns in the windowed tube, with the consideration of effective crushing distance, is W ¼ P m bκ:
ðA:11Þ
Using the law of energy balance, Eq. (1) can be obtained. References [1] Abramowicz W, Jones N. Dynamic axial crushing of square tubes. International Journal of Impact Engineering 1984;2:179–208. [2] Abramowicz W, Jones N. Dynamic progressive buckling of circular and square tubes. International Journal of Impact Engineering 1986;4:243–70. [3] Langseth M, Hopperstad OS. Static and dynamic axial crushing of square thinwalled aluminium extrusions. International Journal of Impact Engineering 1996;18:949–68. [4] Jensen Ø, Langseth M, Hopperstad OS. Experimental investigations on the behaviour of short to long square aluminium tubes subjected to axial loading. International Journal of Impact Engineering 2004;30:973–1003. [5] Alexander JM. An approximate analysis of the collpase of thin cylindrical shells under axial loading. The Quarterly Journal of Mechanics and Applied Mathematics 1960;13:10–5. [6] Kecman D. Bending collapse of rectangular and square section tubes. International Journal of Mechanical Sciences 1983;25:623–36. [7] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. International Journal of Impact Engineering 1984;2:263–81. [8] Wierzbicki T, Bhat SU, Abramowicz W, Brodkin D. Alexander revisited—a two folding elements model of progressive crushing of tubes. International Journal of Solids and Structures 1992;29:3269–88. [9] Pugsley A, Macaulay M. The large-scale crumpling of thin cylindrical columns. The Quarterly Journal of Mechanics and Applied Mathematics 1960;13:1–9. [10] Johnson W, Soden PD, Al-Hassani STS. Inextensional collapse of thin-walled tubes under axial compression. The Journal of Strain Analysis for Engineering Design 1977;12:317–30.
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[11] Singace AA. Axial crushing analysis of tubes deforming in the multi-lobe mode. International Journal of Mechanical Sciences 1999;41:865–90. [12] Andrews KRF, England GL, Ghani E. Classification of the axial collapse of cylindrical tubes under quasi-static loading. International Journal of Mechanical Sciences 1983;25:687–96. [13] Guillow SR, Lu G, Grzebieta RH. Quasi-static axial compression of thin-walled circular aluminium tubes. International Journal of Mechanical Sciences 2001;43:2103–23. [14] Abdul-Latif A, Baleh R, Aboura Z. Some improvements on the energy absorbed in axial plastic collapse of hollow cylinders. International Journal of Solids and Structures 2006;43:1543–60. [15] Wierzbicki T, Abramowicz W. On the crushing mechanics of thin-walled structures. Journal of Applied Mechanics 1983;50:727–34. [16] Bambach MR, Elchalakani M. Plastic mechanism analysis of steel SHS strengthened with CFRP under large axial deformation. Thin-Walled Structures 2007;45:159–70.
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[17] Song J, Chen Y, Lu G. Axial crushing of thin-walled structures with origami patterns. Thin-Walled Structures 2012;54:65–71. [18] Song J, Chen Y, Lu G. Light-weight thin-walled structures with patterned windows under axial crushing. International Journal of Mechanical Sciences 2013;66:239–48. [19] Abaqus 6.11 documentation. Dassault Systèmes Simulia Corp., Providence, RI, USA; 2011. [20] Wierzbicki T, Abramowicz W. The mechanics of deep plastic collapse of thinwalled structures. In: Wierzbicki T, Jones N, editors. Structural failure. New York: Wiley; 1989. [21] Hsu SS, Jones N. Quasi-static and dynamic axial crushing of circular and square stainless steel tubes. In: Jones N, Brebbia CA, Rajendran AM, editors. Structures under shock and impact VII; 2002 (pp. 169–78). [22] Hayduk RJ, Wierzbicki T. Extensional collapse modes of structural members. Computers & Structures 1984;18:447–58.
Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A relationship between progressive collapse and initial buckling for tubular structures under axial loading Jie Song a,n, Yufeng Zhou a, Fenglin Guo b a b
School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China
art ic l e i nf o
a b s t r a c t
Article history: Received 28 November 2012 Received in revised form 13 June 2013 Accepted 28 June 2013 Available online 10 July 2013
The progressive collapse of tubular structures under axial loading is a challenging problem in mechanics. Due to the nonlinearities in large plastic deformation, such a problem can only be solved case by case under the assumption of an appropriate collapse mechanism. In this paper, a relationship between the progressive collapse of an axially loaded tube and the initial buckling of its windowed counterpart is presented. Numerical investigation was performed on the axial crushing of triangular, square and pentagonal tubes and the initial buckling modes of the corresponding windowed tubes. Results show that at the critical symmetric buckling mode, the theoretical mean crushing force of the angle-shaped column in the windowed tube matches very well with the actual mean crushing force of the conventional one. This relationship is crucial to the development of a generalized method for progressive collapse without assuming collapse mechanism. Based on it, an empirical equation on the mean crushing force of axially loaded square tubes is presented. The mean crushing forces predicted by this equation are in good agreement with the experimental results and theoretical values. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Progressive collapse Axial loading Windowed tube Buckling mode
1. Introduction Large plastic deformation of thin-walled structures is a challenging problem in mechanics. Due to its high geometric, material and contact nonlinearity, as well as strong deformation-history dependency, most studies in this field are conducted through experimental and numerical approaches. For example, experimental studies on the collapse of axially loaded square tubes were performed by Abramowicz and Jones [1,2], Langseth and Hopperstad [3], and Jensen et al. [4]. In numerical approaches, the explicit time integration scheme is widely employed because of its effectiveness in convergence for highly nonlinear problems. However, the time step has to be sufficiently small to ensure numerical stability. For practical problems involving hundreds of thousand degrees of freedom, it may take hours or days to simulate the deformation. Theoretical studies on the thin-walled structures with large plastic deformation are limited to those in simple geometric shapes, such as circular/square tubes and spherical shells. Currently, the only method available is the limit analysis based on upper bound theorem, in which a collapse mechanism or a field of plastic flow has to be assumed. The load–displacement relationship is obtained through equilibrium condition or energy balance between external work and plastic dissipation. However, the method is very
n
Corresponding author. Tel.: +65 83590056. E-mail addresses: [email protected], [email protected] (J. Song).
0020-7403/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmecsci.2013.06.016
subtle as a good assumption of the collapse mechanism is essential to its success. Therefore, it can be only applied to structures with simple geometry and deformation mode, e.g., the ring mode of axially loaded circular tubes [5] and the bending collapse of square tubes [6]. Another drawback of this method is that it may not give a closed-form solution. In Ref. [6], for example, the so-called “rolling radius” was indeterminate using this method and had to be obtained from tests. Among various deformation forms of thin-walled structures, the progressive collapse of tubular structures from one end to the other with the formation of successive folding along its central axis has attracted many attentions. Because of its excellent collapse characteristics, e.g., long stroke distance and large plastic deformation, this form is widely used for energy absorption in automotive and construction industries. Theoretical analysis on the progressive collapse focuses mainly on circular and square tubes. The circular tube has two distinct collapse modes, i.e., ring and diamond mode. Alexander [5] first proposed an approximate model to predict the mean crushing force of the ring mode, which was later modified by Abramowicz and Jones [2,7] with the consideration of curvature of the folds and by Wierzbicki et al. [8] considering both inward and outward folding. The diamond mode is more complex, and its theoretical study is not so successful as that for the ring mode. Some studies have been made by Pugsley and Macaulay [9], Johnson et al. [10] and Singace [11]. The relationship between collapse mode and geometry of the circular tube, e.g., diameter-to-thickness ratio, was investigated in Refs. [12–14].
J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
201
progressive collapse of a tube and the initial buckling of its windowed counterpart is described. With this relationship, the mean crushing force of progressive collapse can be calculated without assuming the collapse mechanism, which may provide a novel strategy in the theoretical studies on progressive collapse and further the understanding to the collapse nature. The outline of the paper is as follows. First the geometry of windowed as well as corresponding conventional tubes is introduced, and the extensional collapse mode is discussed. Next, with an experimentally validated FE model, the relationship between the buckling modes of the windowed tube and the window size is described. Particular attentions will be paid to the so-called “critical symmetric buckling mode”. It illustrates that the theoretical mean crushing force of this mode of the windowed tube matches very well with the actual mean crushing force of the corresponding conventional tube. Then, based on the critical buckling mode, an empirical equation of the mean crushing force for axially loaded square tubes is presented. Mean forces predicted by this equation are compared with experimental results from open literature and calculated values using Wierzbicki and Abramowicz's model [1]. This paper ends with a conclusion and suggestions for future work.
For square tubes, the symmetric and the extensional modes are the most important ones. The former was theoretically studied by Wierzbicki and Abramowicz [15] with their super-folding element and later improved by Abramowicz and Jones [1] with consideration of the effective crushing distance, which is defined as the maximum crushing distance of a tube before being compactly compressed to explain the discrepancy between theoretical and experimental results. The extensional mode was studied by Abramowicz and Jones [2]. They presented a basic collapse element and derived the formula of the mean crushing force. The relationship between collapse mode and width-to-thickness ratio of the square tube was studied by Abramowicz and Jones [2]. More recently, axial crushing behavior of square tubes with various modifications, such as those with reinforcements [16] or origami patterns [17], has been investigated. Existing theoretical studies on the progressive collapse of tubular structures are based on the following two assumptions: (1) The material's plastic behavior can be simplified to a perfect plastic model with flow stress s0 . (2) The tube length has no effect on the characteristics of progressive collapse. These assumptions in general hold true for most prismatic tubes, and they are adopted in the present paper. For circular tubes, studies [12–14] have shown that the tube length may affect the tube's collapse mode. Recently, the windowing method proposed by Song et al. [18] has shown great potentials in controlling the tube's collapse behavior. By introducing windows to the tube wall, the collapse mode of the tube can be altered, resulting in a higher mean crushing force. In the present paper, a relationship between the
2. Geometry and extensional mode 2.1. Geometry of tube Regular triangular, square and pentagonal tubes are considered. Table 1 shows the geometry of the conventional tubes, in which M is the number of edges, c is width of edge in the middle plane, t is wall thickness, and Lc is tube length. Notation TRI/SQU/PEN-c-t is
Table 1 Geometry of conventional and windowed tubes. Cross-sectional shape
a
M
c (mm) a
t (mm) a
a
a
a
Lw (mm)
120 180 240
240t ¼ 3,
3
40.36 60 80
4
40.36a 60 80
3a, 2a, 1.5a, 1a, 0.5a 3, 2.25, 1.5 4, 3, 2
120 180 240
120
5
40.36a 60 80
3, 2a, 1.5a, 1a, 0.5a 3, 2.25, 1.5 4, 3, 2
120 180 240
120
Value also used for the windowed tube.
3 , 2 , 1.5 , 1 , 0.5 3, 2.25, 1.5 4, 3, 2
Lc (mm) a
, 120t ¼ 1,
2, 1.5 mm
0.5 mm
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J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
adopted to distinguish each tube. Symbols TRI, SQU and PEN represent triangular, square and pentagonal tubes, respectively. For each conventional polygon tube with c¼40.36 mm, except PEN-40.36-3, a corresponding windowed tube is considered. The windows are in rectangular shape, with width a and height b, one on each side. Each window is positioned at the center of the side. An example of windowed square tube is shown in Fig. 1. The leftover, also referred to as angle-shaped column, has width e ¼ ðcaÞ=2. The length of windowed tube Lw is identical to that of the corresponding conventional tube expect for triangular ones with t from 3 mm to 1.5 mm, where Lw ¼ 240 mm. The values of Lw are shown in Table 1.
in which M 0 ¼ s0 t 2 =4 is the fully plastic bending moment, and κ is the scale factor defining the effective crushing distance. The details of derivation of Eq. (1) can be found in Appendix. Allowing the window b varies, P m reaches minimum when pffiffiffiffiffiffiffiffiffiffiffiffiheight, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ 2Met ¼ MðcaÞt . In this case, we have rffiffiffiffiffiffiffiffiffiffiffi Pm 1 e 4π 2M þ πðM2Þ : ð2Þ ¼ κ t M0 Eq. (2) is also applicable for conventional tubes. In this case, the window width a vanishes and e ¼ c=2.
3. Numerical simulation 2.2. Extensional mode
3.1. FE modeling
The extensional mode of square tubes under axial loading, as shown in Fig. 2(a), has been theoretically studied by Abramowicz and Jones [2]. Fig. 2(b) shows the extensional mode of windowed square tube, in which both panels of the angle-shaped columns deform outward. This mode has been observed in the experimental study by Song et al. [18] on axially loaded windowed square tubes. The extensional modes of other polygon conventional and windowed tubes are similar to those in Fig. 2(a) and (b), respectively. In the present paper, the theoretical model proposed by Abramowicz and Jones [2] is extended to regular polygon windowed tubes. Following their analysis with a rigid-perfect plastic material with flow stress s0 , the mean crushing force, P m of extensional mode of a regular M-gon windowed tube is given by Pm 1 b e 2π þ 4πM þ πðM2Þ ; ð1Þ ¼ κ t b M0
FE code Abaqus/Explicit 6.11 was adopted for numerical simulation. The tubes were meshed with 4-node shell element, S4R with 7 integration points through the thickness. The element size was 1.5 mm 1.5 mm. The set-up of axial loading of the tube is shown in Fig. 3. The tube was put on the bottom plate, and the top plate, initially contacted with the tube, moved downward to crush the tube. No extra clamping or holding apparatus was used to constrain the tube's ends. Both top and bottom plates were modeled as rigid body. The crushing process was quasi-static. In Abaqus, it can be implemented with SMOOTH STEP sub-option by setting the time step more than 10 times larger than the tube's longest natural period [19]. For conventional tubes, the mean crushing force, P m is calculated as follows: Rd Pδd Ea ¼ 0 ; ð3Þ Pm ¼ d d in which P is crushing force, d is axial displacement, and Ea is the energy absorption of the tube up to that displacement. Finite sliding penalty formulation was used to model frictions between plates and tube walls and between tube walls themselves. The frictional coefficient was 0.3 for all the contact interactions. This value has been adopted in the study of Song et al. [18] on windowed tubes. The contact behavior in normal direction was modeled as “hard contact”. The material of tube was annealed mild steel with properties as follows: density ρ ¼ 7332:3 kg/m3, Young's modulus E ¼ 190:5 GPa, Poisson's ratio ν ¼ 0:3, yield stress sy ¼ 287:9 MPa, and ultimate stress su ¼ 506:9 MPa. The true stress–plastic strain data of the material is given in Table 2. The Von-Misses yield criterion and isotropic strain hardening were assumed, and strain rate effects were neglected. The plastic flow stress, s0 of the material was calculated by using [20] rffiffiffiffiffiffiffiffiffiffiffiffi sy su s0 ¼ ; ð4Þ 1þn
Fig. 1. A windowed square tube.
in which n is the strain hardening exponent. For the present material, n ¼0.22. To simulate the conventional tube collapsing in its natural mode, the whole tube was modeled, and small initial deflection of middle plane was introduced to mimic the inevitable imperfections that would be present in any real specimen. This was realized by superimposing the first 3 buckling modes of the tube onto the tube wall, each with certain amplitude. The amplitudes were proportional to the wall thickness. Their values are shown in Table 3. The values of amplitudes were determined through trial simulations. It will be shown in Section 3.2 that with these values FE simulation can achieve very good agreement with experimental and theoretical results.
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203
Fig. 2. Extensional modes of conventional and windowed square tubes: (a) conventional tube and (b) windowed tube.
P
Symmetric boundary condition
d Symmetric boundary condition
Top plate Tube
Fig. 4. FE model of a conventional square tube for the simulation of extensional mode.
Bottom plate
Fig. 3. Sketch of tube under axial loading.
Table 2 True stress–plastic strain of the mild steel. True stress (MPa) 287.91 335.43 385.38 425.38 476.66 506.93 Plastic strain 0 0.0319 0.0533 0.0793 0.137 0.205
Table 3 Amplitude of each buckling mode. Buckling mode Amplitude (mm)
1 0.01t
2 0.007t
3 0.004t
The extensional mode of the conventional tube was also simulated. In this case, only one corner of the tube was modeled, like the example of square tube shown in Fig. 4. Symmetric boundary condition was applied on the side edge of each panel. For each pair of nodes on the side edges that are at the same level, a constraint was imposed making them have the same axial displacement and out-of-plane rotation. No initial deflection was introduced to the tube wall. The windowed tube was also modeled as one corner, with symmetric boundary condition applied on the side edge of each panel. Although a windowed tube may not collapse in such symmetrical way, in the present study, only initial collapse of the windowed tube is considered, and it has been observed that deformation always concentrated on the angle-shaped columns while the rest of the tube had little deformation. Therefore, modeling the windowed tube as one corner should be accurate enough to investigate the initial collapse response. The initial
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deflection was introduced to each panel with the following expression: w ¼ w0 sin 1
πe πx 3πz sin sin ; c c Lw
ð5Þ
in which w0 is the maximum initial deflection of the corresponding conventional tube. The coordinate system of the panel is shown in Fig. 5(a). The wave numbers in x and z directions are identical to those of the first buckling mode of the conventional 1 tube. The factor sin πe=c was introduced to ensure that the angle-shaped column and the conventional tube had the same maximum deflection. The FE model of windowed tube with initial deflection is shown in Fig. 5(b).
3.2. Validation Axial crushing test on a square tube was conducted using Instron 5500 Universal Testing Machine. The tube had dimensions of Lc ¼ 120 mm, c ¼ 40:36 mm, and t ¼ 2:1 mm. It was fabricated from a mild steel block by wire-cut. Heat-treatment was performed before test to remove residual stress. The tube had material properties described in Section 3.1. The experimental set-up is shown in Fig. 3. Quasi-static crushing process was adopted with a loading speed of 1 mm/min.
The tube collapsed in symmetric mode. The deformed profiles of the tube from experiment and simulation are shown in Fig. 6. Table 4 shows the numerical, experimental and theoretical values of mean crushing force at displacement of 67% of the tube length. The theoretical value is calculated using the formula of Abramowicz and Jones [1] with a more precise constant factor as c1=3 Pm ¼ 52:91 : ð6Þ t M0 Since the progressive collapse of conventional square tube is well understood and good agreement between theoretical and experimental values of mean crushing force was achieved, the test was conducted only once. From Fig. 6 and Table 4, it is seen that the numerical results match very well with the experimental ones in both collapse mode and mean crushing force. Therefore the present FE model has been validated.
4. Results and discussion 4.1. Extensional mode The scale factor κ of effective crushing distance of the extensional mode of conventional tube is determined by substituting the numerical result of mean crushing force, calculated with Eq. (3), into
Symmetric boundary condition
Symmetric boundary condition
z
x
0
Fig. 5. FE model of the windowed tube with initial deflection: (a) local coordinate system of the panel and (b) FE model.
Fig. 6. Deformed profiles of tube SQU-40.36-2.1: (a) experiment and (b) simulation.
J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
Eq. (2). Table 5 shows the value of κ for each conventional tube. The mean crushing force of extensional mode is calculated at displacement of 50% of the tube length. The reason of using 50% instead of 67% is that at this displacement most tubes just developed two pure extensional folds, as shown in Fig. 7(a), so the mean crushing force of extensional mode, and also the value of κ, can be determined relatively accurately. If the tube has large width-tothickness ratio c/t, the extensional mode may be corrupted. For example, Fig. 7(b) shows the collapse of tube TRI-40.36-1 with perfect geometry at d¼37 mm. It can be seen that the tube has inextensional deformation below the first extensional layer. For the tube not following extensional mode exactly, it is assumed that its value of κ is the same as that of the tube with largest c/t ratio and pure extensional collapsing mode. Those values are shown in bold in Table 5. We adopted this assumption because it seems yielding consistent results as will be shown in Table 5, but it is still not clear why the assumption works. The values of κ of SQU-40.36-1.5 and PEN-40.36-1.5 calculated from numerical mean crushing forces are larger than those of SQU-40.36-2 and PEN-40.36-2, respectively. Such abnormality should be caused by numerical errors, because it is intuitively clear that κ must be a decreasing function of c/t, otherwise it Table 4 Numerical, experimental and theoretical values of mean crushing force of tube SQU-40.36-2.1 at d ¼80 mm.
Pm (kN)
Numerical
Experimental
Theoretical
54.12
54.78
54.06
205
would mean that for two tubes with the same width, the one with thicker wall could be crushed more deeply than that with thinner wall. The values of κ of SQU-40.36-1.5 and PEN-40.36-1.5 have been adjusted to those of SQU-40.36-2 and PEN-40.36-2, respectively. It should be noted that according to Abramowicz and Jones [1], κ is 0.77 for the extensional mode of conventional square tube. In the present study, however, κ varies with c/t and, in some cases, is significantly smaller than the value proposed by Abramowicz and Jones. This discrepancy will be addressed in the future.
4.2. Critical symmetric buckling mode Two buckling modes of the angle-shaped column in the windowed tube are observed, i.e., symmetric and anti-symmetric modes as shown in Fig. 8. The occurrence of each mode depends on both width and height of the window. Fig. 9 shows distribution of the two buckling modes with respect to width and height of the window for windowed tubes TRI-40.36-1, SQU-40.36-1 and PEN40.36-1. It is seen that for both SQU-40.36-1 and PEN-40.36-1, there exists a minimum width ac , where only symmetric buckling mode occurs. For TRI-40.36-1, there is a minimum width ac , where there is transition between the symmetric and anti-symmetric buckling mode in turn with the increase of window height. The characteristics of ac of other square, pentagonal and triangular windowed tubes with c¼40.36 mm are determined in the same way,pwhich are listed in Table 5. The buckling mode at a ¼ ac and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ Mðcac Þt is referred to as critical symmetric buckling mode,
Table 5 Numerical results and theoretical mean crushing forces of critical symmetric buckling mode of tubes. Tube
P m (numerical result, kN)
κ
ac (mm)
P m (critical symmetric buckling mode, kN)
TRI-40.36-3 TRI-40.36-2 TRI-40.36-1.5 TRI-40.36-1 TRI-40.36-0.5 TRI-60-3 TRI-60-2.25 TRI-60-1.5 TRI-80-4 TRI-80-3 TRI-80-2 SQU-40.36-3 SQU-40.36-2 SQU-40.36-1.5 SQU-40.36-1 SQU-40.36-0.5 SQU-60-3 SQU-60-2.25 SQU-60-1.5 SQU-80-4 SQU-80-3 SQU-80-2 PEN-40.36-3 PEN-40.36-2 PEN-40.36-1.5 PEN-40.36-1 PEN-40.36-0.5 PEN-60-3 PEN-60-2.25 PEN-60-1.5 PEN-80-4 PEN-80-3 PEN-80-2
92.42 37.71 23.64 9.99 3.07 82.22 50.15 23.02 158.05 84.20 42.12 113.42 49.09 28.38 14.19 4.28 108.46 63.19 30.67 190.41 110.89 52.77 145.23 68.00 41.49 17.76 5.09 153.76 79.25 38.37 254.92 140.42 66.90
0.546 0.609 0.646 0.646 0.646 0.635 0.686 0.686 0.646 0.646 0.646 0.568 0.614 0.614c 0.614 0.614 0.638 0.686 0.686 0.647 0.709 0.709 0.587 0.634 0.634c 0.634 0.634 0.648 0.693 0.693 0.656 0.709 0.709
16a 21a 24a 27a 31a 32b 36b 41b 42b 48b 54b 8a 17a 21a 26a 30a 26b 32b 39b 34b 42b 52b 0 7a 12a 19a 28a 11b 18b 29b 14b 24b 38b
92.92 40.24 22.61 11.08 3.26 85.36 47.35 22.85 150.45 89.45 43.71 121.75 51.98 30.64 14.31 4.25 110.96 60.62 28.47 195.81 105.13 48.96 149.21 67.76 40.39 19.00 5.09 147.75 82.64 38.51 260.49 143.73 67.48
a b c
Value determined from the corresponding windowed tube. Value determined from scaling. Adjusted value, the original value is 0.613 for SQU-40.36-1.5 and 0.582 for PEN-40.36-1.5.
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Fig. 7. Pure and corrupted extensional modes: (a) TRI-40.36-2 in pure extensional mode at d ¼ 50 mm and (b) TRI-40.36-1 in corrupted extensional mode at d ¼37 mm.
Fig. 8. Two buckling modes of the angle-shaped column in windowed tube: (a) symmetric mode and (b) anti-symmetric mode.
J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
Symmetric Anti-symmetric
Symmetric
Symmetric Anti-symmetric
Anti-symmetric
Symmetric
column also has significant deformation when the column is deeply crushed. Both effects will result in an overestimated mean crushing force and, hence, an underestimated κ value. Therefore, parameter κ of conventional tube is used as an approximate value. The special case is PEN-40.36-3, which collapses in extensional mode even with imperfection and ac ¼ 0. It is seen from Table 5 that the theoretical mean crushing force of critical buckling mode matches well with the actual value of the conventional tube. This shows the relationship between the progressive collapse behavior of a tube and the initial buckling of its windowed counterpart. The deformed profiles of TRI-40.36-2, SQU-40.36-2 and PEN-40.36-2 are shown in Fig. 10. It is found that only the square tube shows regular collapse pattern, but irregular folding for both the triangular and pentagonal tubes, which makes it very difficult to determine the mean crushing force from traditional approach. However, with the present relationship, it is possible to calculate the mean crushing force without assuming the collapse mechanism. There are still mathematical difficulties in theoretically studying the initial buckling of windowed tube. In the present study, the windowed tube with a ¼ ac buckles locally at its angle-shaped column in the plastic range. In order to theoretically determine the value of ac , one must solve the problem of the plastic buckling of an axially loaded angle column with elastic–plastic boundary condition at both ends, which is too complicated to be treated analytically. Nevertheless, the relationship discovered in the present paper is crucial to the development of a generalized method for progressive collapse. The value of ac for tubes with c¼60 mm and 80 mm could also be determined by finding the critical symmetric buckling mode of their windowed counterparts. Here, a more convenient approach is adopted. According to the second assumption, the tube length has no effect on the mean crushing force. Therefore, for two tubes, TUBE1 with L1 , c1 and t 1 and TUBE2 with L2 , c2 and t 2 , that have the same width-to-thickness ratio, the following relationship holds: c1 t1 1 ac1 ¼ ¼ ¼ ; s c2 t2 ac2
Fig. 9. Distribution of symmetric and anti-symmetric buckling modes with respect to width and height of window: (a) TRI-40.36-1; (b) SQU-40.36-1 and (c) PEN-40.36-1.
and the corresponding angle-shaped column is referred to as critical column. It is worth pointing out that the second transition from symmetric to anti-symmetric mode for windowed tubes TRI40.36-1.5 to TRI-40.36-3 occurs at b larger than 100 mm. Therefore, the length of those windowed tubes is 240 mm, which is long enough to determine the value of ac . 4.3. Mean crushing force Table 5 shows the numerical results of mean crushing force for conventional tubes with c ¼ 40:36 mm at displacement 67% of the tube length. The theoretical mean crushing forces of critical symmetric buckling mode of the corresponding windowed tubes are calculated by Eq. (2), in which e ¼ ðcac Þ=2 and κ takes the value of that of the corresponding conventional tube. The value of κ of the critical column is not determined using the same approach as that for the conventional tube because the crushing distance of the critical column is not long enough and the material around the
207
ð7Þ
in which s is the scale factor and ac1 and ac2 are the width of critical symmetric buckling mode of TUBE1 and TUBE2, respectively. The reasoning is as follows. First, a new unit of length, which is s times the original one, is applied to TUBE1. In this unit, TUBE1 has the same width and thickness in value as those of TUBE2, and with a new length sL1 , a new width of critical mode sac1 and a new flow stress s0 =s. Since tube length does not influence mean crushing force, the length of TUBE1 can be changed to L2 without affecting other parameters. Further applying a new unit of mass to TUBE1, which is s times the original one, the value of flow stress of TUBE1 is changed back to s0 . Now TUBE1 and TUBE2 are identical from the mathematical viewpoint, so their mean crushing forces must have the same value. According to the relationship between mean crushing force and width of critical symmetric buckling mode, the value of ac2 should be equal to sac1 . Table 5 shows the values of ac for tubes with c¼60 mm and 80 mm. The value of each tube is calculated by scaling the ac of the tube with c¼40.36 mm and having the same width-to-thickness ratio (for convenience, the decimal part of width, 0.36, is dropped in calculation). The numerical mean crushing force at displacement 67% of the tube length and theoretical mean crushing force of the critical symmetric buckling mode of these tubes are also presented. Again, good agreement between the two values is observed.
5. An empirical equation As an application and a verification of the relationship in Section 4, an empirical equation of the mean crushing force of
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Fig. 10. Deformed profiles of triangular, square and pentagonal tubes: (a) TRI-40.36-2; (b) SQU-40.36-2 and (c) PEN-40.36-2.
axial loaded square tube is presented and its predictions of mean crushing force are compared with experimental and theoretical results. Define effective half width of the tube, ec as ec ¼
cac ¼ f c; 2
ð8Þ
in which f is a dimensionless ratio function of ec and c. According to the two assumptions and due to its dimensionlessness, f must be a function of t=c. The expression of f is obtained through curve fitting with data points of tubes SQU-40.36-x in Table 5, which is given as follows: t f ¼ 4:419 þ 0:07209: c
ð9Þ
The result of the curve fitting is shown in Fig. 11. The factor of effective crushing distance κ is dimensionless, so it should depends only on t=c. However, Table 5 shows that the value of κ may be different for tubes with the same width-to-thickness ratio, e.g., those of SQU-60-3 and SQU-80-4. This discrepancy might be due to the numerical errors. To obtain the expression of κ, the data of tubes SQU-40.36-x are used. Numerical results of these tubes are most reliable as their width is the same as that of the tube in the validation test. The expression of κ is given by ( t 1:838 ct þ 0:7046 c ≥0:05 κ¼ :ð10Þ t 0:6135 c o0:05 Substituting Eqs. (9) and (10) into Eq. (2) and letting M ¼4, the empirical equation of the mean crushing force of square tube
Fig. 11. Curve fitting results of function f.
under axial loading is obtained as follows: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:1442c=tþ8:838þ2π > < 8π 1:838t=cþ0:7046 Pm pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ M0 > : 8π 0:1442c=tþ8:838þ2π 0:6135
t c ≥0:05 t c
o 0:05
:ð11Þ
Table 6 compares some experimental results from open literature on the mean crushing force of axially loaded square tubes with values calculated by Eqs. (6) and (11). The tubes are made of mild steel or aluminum. The loading condition is quasi-static and all the tubes collapse in symmetric mode. For tubes except those from Jensen et al. [4], the values of s0 are either taken from the
J. Song et al. / International Journal of Mechanical Sciences 75 (2013) 200–211
209
Table 6 Comparison between mean crushing forces from experiment and calculated by Eqs. (6) and (11) for square tubes under quasi-static axial loading. Reference Specimen c (mm)
t (mm)
s0 (MPa)
Pm (kN) Experiment Eq. (6) Eq. (11) a
[2]
C9 D3 D4 E9 E10
17.09 18.18 18.13 36.45 36.45
1.00 0.91 0.91 1.65 1.63
414.20 413.30a 413.30a 315.60a 315.60a
12.97 11.72 12.08 34.40 35.55
14.11 12.28 12.27 31.89 31.25
15.74 12.89 12.90 32.71 31.97
[1]
24 34 131 139 141
49.30 49.34 37.11 37.10 37.06
1.63 1.64 1.15 1.16 1.18
328.50a 328.50a 330.50a 330.50a 330.50a
35.28 36.71 17.90 19.75 18.50
35.78 36.16 18.27 18.54 19.07
34.51 34.91 17.42 17.70 18.27
[3]
T4*-1.8-S T4*-2.0-S T4*-2.5-S
77.65 77.52 76.93
1.81 1.90 2.45
149.97b 159.08b 156.37b
21.80 25.80 37.90
22.75 26.15 39.17
20.76 24.04 37.54
[21]
SS RS
48.00 78.40
2.02 1.85
572.49b 572.49b
94.10 77.82
88.83 90.35
89.76 82.60
[4]
S3 S4 S6 S19
80.00 80.00 80.00 80.00
2.50 2.00 3.50 4.50
186.70c 186.70c 186.70c 186.70c
49.00 32.00 87.00 142.00
a b c
49.00 46.82 33.78 31.14 85.85 87.44 130.51 143.29
Value from the original reference. Value calculated by Eq. (4). Value retrieved from experimental mean crushing force of S3.
references or calculated by Eq. (4) with material properties obtained from the references. The value of s0 for tubes in Jensen et al. [4] is retrieved from the experimental mean crushing force of tube S3 using Eq. (6). It can be seen that Eq. (11) predicts mean crushing force close to the experimental value for most tubes. For S19 and RS, it achieves much higher accuracy than Eq. (6). The largest error of Eq. (11) occurs at tube C9, where it predicts a mean crushing force 21% higher than the experimental one. However, the absolute error is still fairly small in this case. The values of Eqs. (6) and (11) have good agreement for all the tubes with the largest difference being 12% for tube C9. Given the fact that Eq. (11) is only an empirical function based on numerical results, its accuracy has shown the potential of the relationship described in Section 4 being used as an alternative way to calculate the mean crushing force. Particular attentions are paid to tubes C9, D3, D4 and S19, where t=c 4 0:05. Fig. 12 shows the relationship between normalized mean crushing force and thickness-to-width ratio for these tubes. In this range, Eqs. (6) and (11) have different properties. Namely, Eq. (6) is a decreasing function of t=c while Eq. (11) is an increasing one. The relationship between normalized mean crushing force and thickness-to-width ratio for tubes D3, D4, and S19 agrees with that of Eq. (11), although this might be due to the fluctuation in experimental results. Also by comparing tubes C9 and S19, it can be found that they have close thickness-to-width ratios but their errors of mean crushing force predicted by Eq. (11) are quite different. It is worth pointing out that the ultimate stress was used as plastic flow stress in the study of Abramowicz and Jones [2]. For tube C9, this value might be a little higher so that Eq. (11) (and also Eq. (6)) overestimates the mean crushing force. Another possibility is that κ depends on parameters besides c and t, e.g., the local curvature of the tube wall. In this case, the first assumption may not hold as the local curvature depends on the tangential modulus of the material. This may also explain why tubes with the same thickness-to-width ratio have different values of κ in Table 5.
Fig. 12. Relationship between normalized mean crushing force and thickness-towidth ratio for tubes C9, D3, D4 and S19.
6. Conclusion and recommendations A relationship between the progressive collapse of an axially loaded tube and the initial buckling of its windowed counterpart has been discovered. With an experimentally validated FE model, triangular, square and pentagonal tubes under quasi-static axial loading have been simulated, and the initial collapse behavior of the corresponding windowed tubes has also been investigated. Two buckling modes, symmetric and anti-symmetric, are observed in the angle-shaped column of the windowed tube, and there exists a critical symmetric buckling mode. It is found that at the critical symmetric buckling mode, the theoretical mean crushing force of the angle-shaped column in the windowed tube matches well with the actual one of the corresponding conventional tube. This relationship makes it possible to calculate the mean crushing force without assuming a collapse mechanism. As an application and a verification of the relationship, an empirical equation on the mean crushing force of axially loaded square tube has been presented. The values given by this equation match fairly well with the experimental results and theoretical values. In the future, experiments will be conducted on the tubes considered in the present paper to further verify this relationship, and the applicability range of width-to-thickness ratios of the tube to this relationship will be determined. The relationship will be extended to tubes with other cross-sectional shapes, in particular the circular tube. The two discrepancies on the factor of effective crushing distance of the extensional mode will be further investigated for explanation. So far the critical symmetric buckling mode can only be determined numerically. Due to inevitable errors in the numerical simulation, it is not sure whether the critical mode has systematic errors in estimating tube's mean crushing force, and why its characteristics in triangular tube are different from those in square and pentagonal ones.
Appendix A The mean crushing force of extensional mode of a regular M-gon windowed tube is calculated up to the point that the angleshaped columns are completely crushed. The derivation is based on the studies by Abramowicz and Jones [2] and Hayduk and Wierzbicki [22]. Fig. A1 shows the collapse mechanism of the angle-shaped column, in which α is the rotation angle of the panels and 2ψ 0 is the exterior angle of the column which remains constant during the crushing process. The energy dissipation of this mechanism consists of three parts, which are discussed as follows.
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From Figs. A1 and A2, the velocity, vc can be obtained as vc ¼
b _ ψ cos α α: 2 0
ðA:5Þ
Substituting Eq. (A.5) into Eq. (A.4) and integrating from α ¼ 01 to α ¼ 901, the energy dissipation of stretching is 2
E1 ¼ M 0 ψ 0
b : 2t
ðA:6Þ
(2). Energy dissipated in rebending of the conical surface in Fig. A2 Initially, the angle between two panels in the angle-shaped column is π2ψ 0 . The angle becomes zero after the conical surfaces are flattened in the final configuration. For one quarter of the mechanism shown in Fig. A2, the energy dissipation due to the rebending of conical surface is π b : ðA:7Þ E2 ¼ M 0 ψ 0 2 2
Fig. A1. Collapse mechanism of the angle-shaped column.
(3). Energy dissipated along horizontal plastic hinges in one quarter of the mechanism
R
Q
The bending angle is 901 at both horizontal hinges QR and OS. The corresponding energy dissipation is E3 ¼ M 0 πe:
P
ðA:8Þ
The energy dissipation of a single angle-shaped column is " # 2 π b b þ M 0 πe : þ M 0 ψ 0 ðA:9Þ E4 ¼ 4 M 0 ψ 0 2 2 2t
O
S
For a regular M-gon windowed tube, the number of angleshaped columns is M and ψ 0 ¼ π=M. Therefore, the total energy dissipation amounts to
Fig. A2. One quarter of the collapse mechanism in the crushing process.
2
(1). Energy dissipated in stretching in one quarter of the mechanism Consider one quarter of the collapse mechanism in its local coordinate system during the crushing process, as shown in Fig. A2. The boundary, γ between deforming and rigid regions is an arbitrary function gðxÞ, and s is the curvilinear coordinate in the circumferential direction with the origin on γ. Assuming that there is only velocity in s direction and it varies linearly with s, the following velocity field can be adopted: v¼
vc xs ðb=2ÞðgðxÞ þ βxÞ
0≤s≤gðxÞ þ βx;
ðA:1Þ
in which vc is the velocity of point P. Neglecting the shear strain rate and its contribution to the energy dissipation, the only non-vanishing strain rate is the normal strain rate in the circumferential direction, which is ε_ ¼
vc x : ðb=2ÞðgðxÞ þ βxÞ
ðA:2Þ
The rate of energy dissipation due to stretching is Z E_ 1 ¼ s0 t_εdA;
ðA:3Þ
A
in which A is the area of the deforming dA ¼ ðgðxÞ þ βxÞdx. Substituting Eq. (A.2) into Eq. (A.3) yields 1 E_ 1 ¼ s0 tbvc : 4
region
and
ðA:4Þ
E ¼ ME4 ¼ 2M 0 π
b þ M 0 πðM2Þb þ 4M 0 Mπe: t
ðA:10Þ
The work done by the external force to crush all the angleshaped columns in the windowed tube, with the consideration of effective crushing distance, is W ¼ P m bκ:
ðA:11Þ
Using the law of energy balance, Eq. (1) can be obtained. References [1] Abramowicz W, Jones N. Dynamic axial crushing of square tubes. International Journal of Impact Engineering 1984;2:179–208. [2] Abramowicz W, Jones N. Dynamic progressive buckling of circular and square tubes. International Journal of Impact Engineering 1986;4:243–70. [3] Langseth M, Hopperstad OS. Static and dynamic axial crushing of square thinwalled aluminium extrusions. International Journal of Impact Engineering 1996;18:949–68. [4] Jensen Ø, Langseth M, Hopperstad OS. Experimental investigations on the behaviour of short to long square aluminium tubes subjected to axial loading. International Journal of Impact Engineering 2004;30:973–1003. [5] Alexander JM. An approximate analysis of the collpase of thin cylindrical shells under axial loading. The Quarterly Journal of Mechanics and Applied Mathematics 1960;13:10–5. [6] Kecman D. Bending collapse of rectangular and square section tubes. International Journal of Mechanical Sciences 1983;25:623–36. [7] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. International Journal of Impact Engineering 1984;2:263–81. [8] Wierzbicki T, Bhat SU, Abramowicz W, Brodkin D. Alexander revisited—a two folding elements model of progressive crushing of tubes. International Journal of Solids and Structures 1992;29:3269–88. [9] Pugsley A, Macaulay M. The large-scale crumpling of thin cylindrical columns. The Quarterly Journal of Mechanics and Applied Mathematics 1960;13:1–9. [10] Johnson W, Soden PD, Al-Hassani STS. Inextensional collapse of thin-walled tubes under axial compression. The Journal of Strain Analysis for Engineering Design 1977;12:317–30.
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[11] Singace AA. Axial crushing analysis of tubes deforming in the multi-lobe mode. International Journal of Mechanical Sciences 1999;41:865–90. [12] Andrews KRF, England GL, Ghani E. Classification of the axial collapse of cylindrical tubes under quasi-static loading. International Journal of Mechanical Sciences 1983;25:687–96. [13] Guillow SR, Lu G, Grzebieta RH. Quasi-static axial compression of thin-walled circular aluminium tubes. International Journal of Mechanical Sciences 2001;43:2103–23. [14] Abdul-Latif A, Baleh R, Aboura Z. Some improvements on the energy absorbed in axial plastic collapse of hollow cylinders. International Journal of Solids and Structures 2006;43:1543–60. [15] Wierzbicki T, Abramowicz W. On the crushing mechanics of thin-walled structures. Journal of Applied Mechanics 1983;50:727–34. [16] Bambach MR, Elchalakani M. Plastic mechanism analysis of steel SHS strengthened with CFRP under large axial deformation. Thin-Walled Structures 2007;45:159–70.
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[17] Song J, Chen Y, Lu G. Axial crushing of thin-walled structures with origami patterns. Thin-Walled Structures 2012;54:65–71. [18] Song J, Chen Y, Lu G. Light-weight thin-walled structures with patterned windows under axial crushing. International Journal of Mechanical Sciences 2013;66:239–48. [19] Abaqus 6.11 documentation. Dassault Systèmes Simulia Corp., Providence, RI, USA; 2011. [20] Wierzbicki T, Abramowicz W. The mechanics of deep plastic collapse of thinwalled structures. In: Wierzbicki T, Jones N, editors. Structural failure. New York: Wiley; 1989. [21] Hsu SS, Jones N. Quasi-static and dynamic axial crushing of circular and square stainless steel tubes. In: Jones N, Brebbia CA, Rajendran AM, editors. Structures under shock and impact VII; 2002 (pp. 169–78). [22] Hayduk RJ, Wierzbicki T. Extensional collapse modes of structural members. Computers & Structures 1984;18:447–58.