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Energy absorption of axially crushed expanded metal tubes
Thin-Walled Structures 71 (2013) 134–146
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Energy absorption of axially crushed expanded metal tubes G. Martínez, C. Graciano n,1, P. Teixeira Universidad Simón Bolívar, Departamento de Mecánica Apdo. 89000, Caracas 1080-A, Venezuela
art ic l e i nf o
a b s t r a c t
Article history: Received 28 October 2012 Received in revised form 6 May 2013 Accepted 7 May 2013 Available online 17 July 2013
Expanded metal tubes have a great potential for energy absorbing applications. A study on the energy absorption capacity of axially compressed expanded metal tubes is conducted herein; the investigation is performed trough nonlinear finite element analyses. At first, the numerical models are validated with experimental results, thereafter a parametric study is carried out in order to investigate the effects of the length-to-diameter ratio, on both the peak forces and the energy absorption capacity of the tubes. The numerical results are also compared with those obtained using a mechanical model found in the literature. Finally, it is found that peak loads and energy absorption capacity depends on the number of expanded metal cells in the cross-section. The results also show that, concentrical expanded tubes could be an effective mean to enhance energy absorption capacity. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Expanded metal Finite element analysis Energy absorption Axial Crushing Ultimate strength
1. Introduction In the last three decades there has been a large amount of research projects looking for the best configuration for energy absorption purposes with metallic and non-metallic structural components. In some cases the failure mechanisms are described and hence mechanical models are developed to predict the strength. In 2001, Alghamdi [1] presented an overview covering four decades of research on collapsible impact energy absorbers under compressive loading. Olabi et al. [2] presented a second overview on metallic tubes used as energy absorbers under axial crushing, bending and oblique impact. Jones [3] performed a comparison between various energy absorbing systems by means of an energy-absorbing effectiveness factor. These reviews were conducted mainly on metallic thin-walled structural elements, but composites and non-metallic materials are also used in energy absorbing applications. In this sense, Lau et al. [4] conducted a survey on geometrical variables and failure mechanisms in composite energy absorbers. In order to understand the structural behavior of thin-walled elements three main approaches have been undertaken, i.e. experimental, numerical and analytical. Experimental testing is perhaps the most expensive approach in terms of materials and time consumption, nevertheless necessary for numerical model validation [5–10]. Regarding experimentation, for short and long aluminum tubes under axial crushing the transition between n
Corresponding author. Tel.: +57 4 425 51 66; fax: +57 4 425 51 75. E-mail addresses: [email protected], [email protected] (C. Graciano). 1 Present address: Facultad de Minas, Departamento de Ing. Civil, Universidad Nacional de Colombia, Sede Medellín, A.A. 75267 Medellín, Colombia. 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.05.003
global and progressive buckling has been investigated [5,6]. Gupta and Vankatesh [7] also conducted experimental and numerical studies on the axial compression of aluminum cylindrical shells, varying the diameter and wall thickness. Continuing this, Gupta [8] investigated the collapse mode of aluminum tubes with frusta geometry. Accordingly, the influence of the cross-section on the collapse response is an important issue when dealing with energy absorption, Nia and Hamendani [9] after studying the energy absorption capacity of tubes with various section shapes (circular, square, rectangular, hexagonal, triangular, pyramidal and conical) found that a circular tube has the higher energy absorption capacity. Using cross-sectional shapes with only convex polygons, Fan et al. [10] established an improvement in energy absorption when increasing the number of inward corners. Composite materials are an alternative for energy absorption applications, an increase on energy absorption capacity is observed when comparing the results for foam-filled tubes with those achieved with empty tubes subjected to axial compressive loads [11–14]. Materials with non-metallic matrixes such as glass and carbon fiber have also been investigated [15,16]. Summarizing, metallic energy absorbers under axial compressive loads fail in various modes, namely axial crushing, global and local buckling depending essentially on geometrical parameters (length and cross-sectional variables). On the contrary, energy absorbers built on composite matrix fail due to a combined mechanism characterized by yielding and fracture. Enhancements in computing power, over the last two decades, have reduced significantly time consumption and empowered engineers and researchers to explore by means of numerical analyses the nonlinear structural response of energy absorbing components [17–19]. Aimed at optimizing the energy-absorbing
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effectiveness factor, the shape of the structural component has also been an important factor in the literature. An important task to accomplish in the development of energy absorbing structures is to reduce the initial peak load exerted in the load–displacement response. It has been observed in both experimental and numerical studies. In this regard, buckling initiators in the form of initial geometric imperfections such as patterns [20], grooves [21,22], pulling strips [23,24], profile corners and indentations [25] have been implemented. Due to the complex nature of the failure observed experimentally and/or numerical few analytical models predicting the strength can be found in the literature. The main results from the previous review can be summarized as follows:
For energy absorption applications the structural response to
quasi-static loading should be stable with a reduced peak load, i.e. energy absorbed in a square like manner. The initial peak load, exerted in axial compression testing, can be significantly reduced by using buckling initiators. Circular geometry, instead of stiffened corners, allows a better energy-absorbing effectiveness. It is important to have in mind that, stiffen corners strengthened the tubes but not necessarily improves the postbuckling response.
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It seems that expanded metal tubes may satisfy all requirements to build an efficient energy absorber. Graciano et al. [26] investigated the axial collapse of round tubes made of expanded metal sheeting under compressive loading. From the test results, it was observed that the collapse mechanism depends on the orientation of the expanded metal cells, for α¼01 a mode characterized by a plastic collapse mechanism (Fig. 1b); and for α¼901 the tube failed by global
Top plate
D
L
Bottom plate
F Fig. 2. An expanded metal tube under axial compression.
Fig. 3. Schematic view of pattern cell.
Fig. 1. Initial and final configuration for experimental specimens with α¼ 01 and α¼901 (Graciano et al. [26]). (a) Initial a=0° (b) at failure a=0° (c) Initial a=90° (d) at failure a=90°.
Fig. 4. Stress–strain curve [29].
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buckling (Fig. 1d). A mechanical model resembling the failure mechanism for these two geometries was later presented by Graciano et al. [27]. It was found that the peak load depends on the number of expanded metal cells in the cross-section due to the formation of plastic hinges in the connecting nodes. The model presented by Graciano et al. [27] was derived from a limited number of experimental tests results, obtained from a particular geometry. However, the influence of some geometrical parameters such as the length, and hence the length-to-diameter ratio was disregarded. This paper is aimed at investigating the energy absorption response of axially compressed expanded metal tubes (Fig. 2). At first, a numerical model is built upon a nonlinear
Clamped
finite element approach; thereafter, the model is validated with experimental results and a parametric analysis is conducted in order to investigate in depth the influence of the length-to-diameter ratio on peak load and the absorbed energy. The results from these analyses are compared to those obtained using a failure mechanism model [27]. Finally, the numerical model is used to obtain the energy absorbing capacity of concentrical expanded metal.
2. Failure mechanism model Graciano et al. [27] propose the formation of four plastic hinges to describe the failure of expanded metal cells in tubes subjected to axial crushing. Accordingly, the cells in an expanded metal tube (Fig. 3) can be represented as beams within a structural frame. Considering an expanded metal cell, the strength is obtained when plastic hinges at nodes 1–4 (Fig. 3) are formed at each cell in the mid-section of the tube. Hence, the failure load P of such structure can be calculated trough the following equation: P¼
N P 2 i
ð1Þ
Displacement only in the vertical direction
Fig. 5. Boundary conditions in the numerical model for α ¼01.
Fig. 7. Comparison between load–displacement responses.
Y Z X
File
G _ 0 _I x J Number of columns Number of files Orientation
Z Y X
Fig. 6. Finite element model. (a) a=0°, (b) a=90° and (c) Individual cell.
Fig. 8. Nomenclature used in the numerical analysis.
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Fig. 9. Schematic view of the geometries used in the parametric analysis for α ¼ 01.
Fig. 10. Schematic view of the geometries used in the parametric analysis for α¼ 901.
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In Eq. (1), N is the total number of nodes in a circumferential row at the mid-section of the tube; and Pi is the load under which four plastic hinges form in a single cell and can be calculated as Pi ¼
8M P l1
ð2Þ
Substituting the expression for the plastic moment in a strand MP ¼Sy wt2/4 in Eq. (2), we get
Pi ¼
2Sy wt 2 l1
Fig. 11. Load–displacement response for various lengths (α¼ 01).
ð3Þ
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where t is the strand thickness; w is the strand width, l1 is the major axis and Sy is the yield stress. This value of Pi represents a lower-bound solution for the ultimate load of a single cell. These Table 1 Numerical results for α¼ 01. Model I
G_0_
J
139
formulae can be either used for expanded metal tubes with cell oriented at 01 or 901. A detailed derivation of the previous formulation can be found in [27].
3. Numerical model
Ppeak-col (N)
Ppeak (N)
Ptheo Eq. (1) (N)
r Eq. (4)
Eabs Eq.(5) SEA Eq. (6) (kJ) (kJ/kg)
8
3 4 5 6 7 8 10
285.16 280.45 274.54 270.11 267.88 265.93 260.55
855.48 1121.80 1372.70 1620.66 1875.16 2127.44 2605.50
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.86 0.88 0.89 0.90 0.91 0.93
100.60 131.12 161.10 190.38 220.56 250.21 307.52
0.760 0.743 0.730 0.719 0.714 0.709 0.697
10
3 4 5 6 7 8 10
284.28 278.78 273.47 270.22 268.41 265.32 262.71
852.84 1115.12 1367.35 1621.32 1878.87 2122.56 2627.10
723.21 964.29 1205.36 1446.43 687.50 1928.57 2410.71
0.85 0.86 0.88 0.89 0.90 0.91 0.92
126.40 164.62 201.78 239.33 277.65 313.55 384.09
0.764 0.746 0.732 0.723 0.719 0.711 0.696
12
3 4 5 6 7 8 10
283.63 279.46 272.90 269.47 268.35 264.57 260.49
850.89 1117.84 1364.50 1616.82 1878.45 2116.56 2604.90
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.86 0.88 0.89 0.90 0.91 0.93
150.34 196.57 240.01 284.40 330.45 373.81 459.46
0.757 0.742 0.725 0.716 0.713 0.706 0.694
14
3 4 5 6 7 8 10
283.77 278.43 273.37 270.03 268.27 264.73 259.77
851.31 1113.72 1366.85 1620.18 1877.89 2117.84 2597.70
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.87 0.88 0.89 0.90 0.91 0.93
175.34 229.53 280.24 332.27 385.40 435.45 537.52
0.757 0.743 0.726 0.717 0.713 0.705 0.696
16
3 4 5 6 7 8 10
283.56 278.43 273.03 269.29 267.97 264.77 260.60
850.68 1113.72 1365.15 1615.74 1875.79 2118.16 2606.00
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.87 0.88 0.90 0.90 0.91 0.93
200.38 262.66 319.89 378.82 439.82 497.22 614.52
0.757 0.744 0.725 0.715 0.712 0.704 0.696
As in the experiments, the geometry used to model the expanded metal tubes used was length L¼ 400 mm, diameter D¼120 mm, strand thickness t¼3 mm, strand width w¼3.2 mm, a major axis l1 ¼89.6 mm and minor axis l2 ¼44.2 mm. The nonlinear finite element code ANSYS 12 [28] was used to predict the response of the expanded metal tubes subjected to axial crushing. The expanded metal cells in the tube were modeled with 3D solid elements (SOLID 187), with 10 nodes and a quadratic displacement behavior, making them suitable for modeling plasticity and large strain analyses. Fig. 4 shows the actual stress–strain curve used to model the material; the data was obtained from experimental testing of the base material [29]. Accordingly, the material used was ASTM A-569 with yield strength Sy ¼250 MPa, ultimate strength Su ¼ 385 MPa, Young's modulus E ¼205 GPa and Poisson ratio v ¼ 0.3. The material plasticity is accounted for with a multilinear isotropic hardening model, generated from the Ramberg–Osgood fit to the actual stress–strain curve of ASTM A-569, since no cyclic response is addressed in this study. Each configuration was subjected to a quasi-static analysis simulation, where a load-step control strategy to solve the nonlinear incremental constitutive problem due to plasticity and contact is adopted. Initially, the load is divided into 100 steps, and the convergence rate of the Newton–Raphson scheme adjusts the load increment step to optimize solution time in order to achieve convergence. Clamped boundary conditions were applied at the top and bottom of the tubes (Fig. 5). Prescribed displacement conditions were applied on a rigid plane at the bottom end until the geometry is displaced one-third of its original length (L/3). In order to model contact within the cells during deformation, an edge to edge rough contact with an infinite frictional coefficient is defined at the strands and nodes due to the uncertainties in the real contact behavior.
Fig. 12. Structural response for models with α ¼01.
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As seen in Fig. 5, the expanded metal tubes exhibit a cyclic symmetry around the longitudinal axis. Therefore, each column can be seen as a periodical pattern as shown in Fig. 6a for α¼ 01, and Fig. 6b for α¼901. In this way a radial segment of each model can be solved, saving computational time. The symmetry condition is defined by planes that cut the geometry in the radial direction, where deformation normal to the symmetry plane is not allowed. The numerical model was validated using the load–displacement response for a round tube with α¼01 and the same dimensions as those tested in [26]. A convergence analysis was performed leading to a model with 3 mm elements, Fig. 6c shows the mesh for a single cell obtained from a path conforming meshing algorithm. Comparing the load–displacement responses obtained with the FE model and the experimental data (Fig. 7), a good agreement between these is observed. Furthermore, the difference between these curves is the relative unstability of the experimental data when compared to the numerical results. Concerning the parametric analysis in the next section, the ratio between the theoretical prediction and the numerical value for the peak load is calculated: r¼
P theo P peak
D¼300 mm. i.e. from J ¼4 to 20 columns. For each configuration, the length was also modified, according to the number of files from L ¼392 mm (I ¼4 files) to L¼ 784 mm (I ¼8 files). Figs. 9 and 10 show schematic views of the geometries used in the computational study, with its corresponding nomenclature. These figures show the cell orientation (01 or 901), the number of files I (length) and the number of columns J (diameter). In summary, a total of 75 models were defined for various L/D ratios for the two chosen cell orientations (35 for α¼01, and 40 for α¼901).
5. Results and discussions 5.1. Models with α ¼01 Fig. 11 shows the load–displacement responses for the numerical models with α¼ 01 (G_0_I J). As seen from Fig. 11a–g, all models within this group exhibit a very stable load–displacement
ð4Þ
where Ptheo is calculated using Eq. (1), and Ppeak is computed numerically. As mentioned above, one-third of the total length is crushed, therefore the energy absorbed Eabs is calculated by integrating the load–displacement curves: Z L=3 Eabs ¼ P dx ð5Þ 0
Finally, the specific energy absorbed (SEA) is calculated by dividing the energy absorbed with its weight in the following equation, accordingly W is the weight of the one-third of the model: SEA ¼
Eabs W
ð6Þ
4. Parametric analysis Fig. 13. Average load–displacement response for various diameters α ¼01.
In this section a parametric analysis is conducted in order to investigate the influence of the length-to-diameter ratio (L/D) on the energy absorption capacity of expanded metal tubes under axial compression. Only two cell orientations were considered in the study, i.e. α ¼01 and α ¼901. After examining the load– displacement response for each configuration the following parameters were calculated: peak load per column (Ppeak-col); peak load (Ppeak); theoretical load prediction (Ptheo); the ratio between the theoretical and the numerical peak load (r); energy absorbed (Eabs); and the specific energy absorbed (SEA). Theses parameters are calculated according to the formulae presented in the previous section. Fig. 8 shows the nomenclature used in the parametric analysis, where I and J are the number of files and columns, respectively. Varying the length-to-diameter ratio (L/D), the study was divided into two stages: initially modifying L maintaining D constant and second varying D maintaining L constant. For α ¼01, the geometries were created according to its diameters; the range in the study goes from D ¼91 mm to D¼ 300 mm, i.e. from J¼ 3 to 10 columns. For each configuration, the length was also modified, according to the number of files, from L ¼ 383 mm (I ¼8) to L¼766 mm (I ¼16). Correspondingly, for α ¼901, the geometries were also created according to its diameters; the range in the study goes from D ¼ 61 mm to
Fig. 14. Average absorbed energy for various diameters α ¼01.
G. Martínez et al. / Thin-Walled Structures 71 (2013) 134–146
response, similar to the experimental results obtained in [26], which is desirable for energy absorbing systems. It is worth mentioning that the slope in the load–displacement responses is always ascending and positive, therefore it was necessary to define a criterion to report the peak load (Ppeak) in Table 1. It was established that a change of 0.1% in the slope was
141
necessary to achieve Ppeak. Regarding the failure mechanism of the tubes with this configuration, all the expanded metal cells folded stably and progressively during the crushing process as observed in Fig. 12. Table 1 shows the numerical results for the tubes with α ¼01, Ppeak is computed from the load–displacement responses in Fig. 11, Ptheo is calculated using Eq.(1); the ratio between peak load
Fig. 15. Load–displacement responses (α¼ 901).
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Fig. 16. Structural response for models with α ¼ 901.
Table 2 Numerical results for α¼ 901. Model
I
J
Ppeak-col (N)
Ppeak (N)
Ptheo Eq.(1) (N)
r Eq.(4)
G_90_
4
4 6 8 10 12 14 16 20
556.90 530.66 513.53 497.65 487.30 476.93 469.81 457.74
2227.60 3183.96 4108.24 4976.50 5847.60 6677.02 7516.96 9154.80
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.88 0.92 0.95 0.98 1.00 1.02 1.04 1.07
247.24 372.45 431.19 457.86 505.05 551.42 588.78 674.20
1.401 1.407 1.221 1.038 0.954 0.893 0.834 0.764
5
4 6 8 10 12 14 16 20
560.63 537.16 512.42 498.27 486.20 473.64 462.82 446.71
2242.52 3222.96 4099.36 4982.70 5834.40 6630.96 7405.12 8934.20
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.95 0.98 1.01 1.03 1.06 1.09
320.7 474.82 506.88 572.44 614.76 665.70 695.28 784.77
1.453 1.435 1.149 1.038 0.929 0.862 0.788 0.711
6
4 6 8 10 12 14 16 20
560.65 536.78 537.13 506.56 491.95 479.47 468.38 448.76
2242.60 3220.68 4297.04 5065.60 5903.40 6712.58 7494.08 8975.20
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.91 0.96 0.99 1.02 1.04 1.09
376.24 543.42 680.20 743.46 745.02 742.15 781.73 871.56
1.421 1.368 1.284 1.123 0.938 0.801 0.738 0.658
7
4 6 8 10 12 14 16 20
559.77 536.98 515.99 507.97 502.10 490.14 477.37 457.03
2239.08 3221.88 4127.92 5079.70 6025.20 6861.96 7637.92 9140.60
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.95 0.96 0.97 1.00 1.02 1.07
439.72 647.21 760.28 833.45 888.89 937.13 880.11 995.28
1.423 1.397 1.231 1.079 0.959 0.867 0.712 0.644
8
4 6 8 10 12 14 16 20
559.54 535.71 516.98 504.20 493.84 484.03 476.53 463.77
2238.16 3214.26 4135.84 5042.00 5926.08 6776.42 7624.48 9275.40
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.95 0.97 0.99 1.01 1.03 1.05
500.23 736.22 882.75 953.67 1012.26 1026.20 1084.95 1145.67
1.417 1.390 1.250 1.081 0.956 0.831 0.768 0.649
Eabs Eq.(5) (kJ)
SEA Eq. (6) (kJ/kg)
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and the theoretical prediction r is calculated using Eq.(4); Eabs is computed from the load–displacements response in Fig. 11 applying Eq. (5); and the SEA is calculated using Eq.(6). From the load–displacement responses (Fig. 11) and the results reported in Table 1, it is observed that after increasing the lengths, Ppeak remain almost constant for a given number of columns. In Fig. 11a, the peak load for model G_0_8 3 (L/D ¼4.2; Ppeak ¼855.48 N) is only 0.56% greater than the one for model G_0_16 3 (L/D ¼8.4; Ppeak ¼850.68 N). These results are similar for all models investigated in this section, therefore Fig. 13 summarizes the average load–displacement responses for the tubes with α ¼01. It is observed that the peak loads increase proportionally with the number of columns, i.e. the number of cells in the cross-section. This result was also expected after using Eq. (1), which shows a linear dependency between the strength and the number of cells across the tube. These findings also demonstrate that the mechanism model presented in [27] can be used to predict the peak load for expanded metal tubes with similar boundary conditions. It is interesting to see in Table 1 that the peak load per column (Ppeak-col) should remain constant for the models with the same number of files; however this is reduced when increasing the number of columns. Comparing the peak load for model G_0_8 3 (L/D ¼4.2; Ppeak ¼285.16 N) with the one for model G_0_8 10 (L/D ¼1.3; Ppeak ¼260.55 N), the former is 9% higher than the latter. This variation in the results is observed for all models in Table 1, and is ought to curvature effects. Fig. 14 plots the energy–displacement curves obtained in the analyses. It indicates that for a particular length, increments in the number of columns proportionally increase the energy absorbing performance of the expanded metal tube. Moreover, it can be seen in Table 1, where the energy absorbed for a tube with 10 columns (G_0_I 10) is 300% greater than for a tube with 3 columns (G_0_I 3). Looking at the SEA, it remains almost constant due to the proportionality between the energy absorbed and the number of columns. There is just a small difference of 1% that was explained by curvature effects. This finding is consistent with the conclusion of numerous studies that for a given cross-section the energy absorbing capacity of the structural component is proportional to the diameter of the cylindrical tube, and independent to its length.
The peak loads increase proportionally to the number of columns, however in the postpeak region the curves converge for large displacements. This behavior is quite different than the one for the tubes with α ¼01, and can be explained after observing the failure mode characterized by local instability on the strands. In a similar way as for α¼ 01, in Table 2, the peak loads and the energy absorbed remain almost constant after varying the length. Thus, the peak forces also increase proportionally to the diameter of the tube. The energy absorption capacity (Fig. 18) is more sensitive to a variation in the number of columns; as the number of cell increases in the circumferential direction so this capacity increases. For the given crushing length (one-third of the total length), the absorbed energy (Eabs) increases with the displacement and is proportional to the number of columns. However, the
Fig. 17. Average load–displacement response for various diameters α¼ 901.
5.2. Models with α¼ 901 Fig. 15 shows the load–displacement responses for the numerical models with α ¼901 (G_90_I J), the behavior observed in these curves is rather unstable, nevertheless the peak loads increase considerably with respect to the models with α¼ 01. For model G_90_4 8, Fig. 16 shows both the deformed and undeformed pattern, the failure mechanism is characterized by global outward buckling of the walls until the strands achieves the plastic moment in the connecting nodes. Thereafter, self-contact within the cells due to the collapse in other rows causes peak increments in the load–displacement responses (Fig. 15a–e). After increasing the number of columns (Fig. 15f–h), the behavior become more stable in the postpeak region and the load fluctuations almost disappear. Table 2 shows the numerical results for the tubes with α¼901. From the load–displacement responses (Fig. 15) and the results reported in Table 2, it is observed that after increasing the lengths, Ppeak remain approximately constant while keeping the number of columns. In Fig. 15a, the peak load for model G_90_4 4 (L/D ¼6.4; Ppeak ¼556.90 N) is only 0.47% smaller than the one for model G_0_8 4 (L/D ¼12.8; Ppeak ¼559.54 N). These results are similar for all models investigated in this section with α¼901, hence Fig. 17 summarizes the average load–displacement responses.
143
Fig. 18. Average absorbed energy for various diameters α ¼901.
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Table 3 Numerical results for concentrical tubes. Numerical Model
Concentric tubes
Ppeak (kN)
Ptheo Eq.(1) (kN)
r Eq.(4)
Eabs Eq.(5) (kJ)
SEA Eq. (6) (kJ/kg)
Ppeak ¼Pϕ8
2120
1929
0.91
0.314
0.889
Ppeak ¼ Pϕ7+Pϕ8
3996
3616
0.90
0.591
0.893
Ppeak ¼ Pϕ6+Pϕ7+Pϕ8
5616
5063
0.90
0.830
0.896
Ppeak ¼ Pϕ5+ϕ6+Pϕ7+Pϕ8
6981
6268
0.90
1.032
0.899
Ppeak ¼Pϕ4+Pϕ5+Pϕ6+Pϕ7+Pϕ8
8093
7232
0.89
1.197
0.904
Ppeak ¼Pϕ3+Pϕ4+Pϕ5+Pϕ6+Pϕ7+Pϕ8
8945
7955
0.89
1.323
0.908
behavior is not as uniform as for α¼01, increasing the number of columns significantly increases the energy absorbed, and this is obviously due to the increased amount of material available for plastic deformation and subsequent energy absorption. It is important to have in mind that the numerical model was built upon a radial symmetry, in Table 2 the effect of the curvature on the strength of a single column is clearly observed. The peak load per column (Ppeak-col) for a tube with D ¼61 mm (G_90_I 4) is 22% greater than the corresponding D ¼300 mm (G_90_I 20). As a rule, the peak load increases with the number of columns for the models analyzed in Table 2. In spite of this, the peak load per column is reduced attributable to curvature effects. For small diameters the curvature is larger, hence strengthening the expanded metal tube. The performance of the expanded metal tube by means of the specific energy absorption parameter SEA is very similar among the different configurations of α ¼01 tubes, since cell curvature has an small effect in the load response, and the increase in the absorbed energy is due to the increased amount of material available for plastic deformation as mentioned above. This leads to conclude that the SEA parameter depends of plastic hinges density more than geometric configuration, for this case. Comparing the performance of equivalent specimens by SEA parameter, this is for tubes of α¼901 and α¼ 01 of same volume, the firsts in general behaves better than α ¼01 tubes, due to the different failure mechanism. 5.3. Combined models In the previous sections, it was observed that for models with α¼01 the structural response is more stable and controlled than
for models with α¼901. In this section, a parametric study is conducted using various tubes placed concentrically with α¼01. The strength of the tubes can be significantly increased by using a larger number of cells, which is a natural conclusion after the analyses conducted herein. Table 3 shows the results for a combined geometry (concentric tubes), it is shown that both the load and the energy absorbed increase linearly with the number of columns. This is explained as follows, as the number of cells increases so the amount of material for plastic deformation increases. A small difference is observed due to curvature effects caused by diameter increments. Fig. 19 shows the deformation pattern for the concentric tubes. Comparing the predicted values Ptheo for the ultimate strength using Eq. (1), with the values obtained numerically Ptotal, the maximum difference is about 11%. Also, the SEA parameter remains almost constant, independently of the configuration, as obtained before for the single tubes. The results show that, it is possible to make arrays of expanded metal tubes in order to absorb a given amount of energy, which in many ways could fit other geometries or fill spaces.
6. Conclusions Structural components made with expanded metal sheets have a great potential in energy absorbing applications. From the numerical analyses conducted herein, it was found that the peak load and consequently the energy absorption capacity of expanded metal tubes increase linearly with its diameter and the length has little influence on the structural response. The peak load can be theoretically predicted with accuracy, although the results show
G. Martínez et al. / Thin-Walled Structures 71 (2013) 134–146
145
Fig. 19. Structural response for concentrical models with α ¼01.
that the curvature of the tube may affect the results. The peak load per column increases with the curvature of the model. It was found that the use of concentrical tubes could be an effective mean to enhance the energy absorption capacity and response of such tubes, without increasing the occupied space by the structural component. In general, the results draw attention to the advantages of using expanded metal tubes for energy absorption under axial crushing. References [1] Alghamdi AAA. Collapsible impact energy absorbers: an overview. ThinWalled Structures 2001;39:189–213. [2] Olabi AG, Morris E, Hashmi MSJ. Metallic tube type energy absorbers: a synopsis. Thin-Walled Structures 2007;45(7-8):706–26. [3] Jones N. Energy-absorbing effectiveness factor. International Journal of Impact Engineering 2010;37:754–65. [4] Lau STW, Said MR, Yaakob MY. On the effect of geometrical designs and failure modes in composite axial crushing: a literature review. Composite Structures 2012;94:803–12. [5] Teramoto SS, Alves M. Buckling transition of axially impacted open shells. International Journal of Impact Engineering 2004;30:1241–60. [6] Jensen Ø, Langseth M, Hopperstad OS. Experimental investigations on the behaviour of short to long square aluminum tubes subjected to axial loading. International Journal of Impact Engineering 2004;30:973–1003. [7] Gupta NK, Venkatesh. A study of the influence of diameter and wall thickness of cylindrical tubes on their axial collapse. Thin-Walled Structures 2006;44: 290–300. [8] Gupta PK. A study on mode of collapse of varying wall thickness metallic frusta subjected to axial compression. Thin-Walled Structures 2008;46: 561–71. [9] Nia AA, Hamedani JH. Comparative analysis of energy absorption and deformations of thin walled tubes with various section geometries. ThinWalled Structures 2010;48:946–54.
[10] Fan Z, Lu G, Liu K. Quasi-static axial compression of thin-walled tubes with different cross-sectional shapes. Engineering Structures, in press, http://dx. doi.org/10.1016/j.engstruct.2011.09.020. [11] Meguid SA, Attia MS, Monfort A. On the crush behaviour of ultralight foamfilled structures. Materials and Design 2004;25:183–9. [12] Zhang Z, Liu S, Tang Z. Comparisons of honeycomb sandwich and foam-filled cylindrical columns under axial crushing loads. Thin-Walled Structures 2011;49:1071–9. [13] Ghamarian A, Zarei HR, Abadi MT. Experimental and numerical crashworthiness investigation of empty and foam-filled end-capped conical tubes. ThinWalled Structures 2011;49:1312–9. [14] Li Z, Yu J, Guo L. Deformation and energy absorption of aluminum foam-filled tubes subjected to oblique loading. International Journal of Mechanical Sciences 2012;54:48–56. [15] Han H, Taheri F, Pegg N, Lu Y. A numerical study on the axial crushing response of hybrid pultruded and 7 451 braided tubes. Composite Structures 2007;80:253–64. [16] Huang J, Wang X. Numerical and experimental investigations on the axial crushing response of composite tubes. Composite Structures 2009;91: 222–8. [17] Markiewicz E, Ducrocq P, Drazetic P. An inverse approach to determine the constitutive model parameters from axial crushing of thin-walled square tubes. International Journal of Impact Engineering 1998;21(6):433–49. [18] Nagel GM, Thambiratnam DP. Computer simulation and energy absorption of tapered thin-walled rectangular tubes. Thin-Walled Structures 2005;43: 1225–42. [19] Tai YS, Huang MY, Hu HT. Axial compression and energy absorption characteristics of high-strength thin-walled cylinders under impact load. Theoretical and Applied Fracture Mechanics 2010;53:1–8. [20] Zhang X, Cheng G, You Z, Zhang H. Energy absorption of axially compressed thin-walled square tubes with patterns. Thin-Walled Structures 2007;45 (9):737–46. [21] Di Paolo BP, Monteiro PJM, Gronsky R. Quasi-static axial crush response of a thin-wall stainless steel box component. International Journal of Solids and Structures 2004;41:3707–33. [22] Zhang X, Huh H. Energy absorption of longitudinally grooved square tubes under axial compression. Thin-Walled Structures 2009;47:1469–77.
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[23] Zhang XW, Tian QD, Yu TX. Axial crushing of circular tubes with buckling initiators. Thin-Walled Structures 2009;47(6–7):788–97. [24] Zhang XW, Su H, Yu TX. Energy absorption of an axially crushed square tube with a buckling initiator. International Journal of Impact Engineering 2009;36:402–17. [25] Tang Z, Liu S, Zhang Z. Energy absorption properties of non-convex multicorner thin-walled columns. Thin-Walled Structures 2012;51:112–20. [26] Graciano C, Martínez G, Smith D. Experimental investigation on the axial collapse of expanded metal tubes. Thin-Walled Structures 2009;47:953–61.
[27] Graciano C, Martínez G, Gutierrez A. Failure mechanism of expanded metal tubes under axial crushing. Thin-Walled Structures 2012;51:20–4. [28] ANSYS, Inc, ANSYS Release 12.0 Elements Reference, USA; 2009. [29] Sanchez, R. Determinación de las Propiedades Mecánicas de Láminas de Metal Expandido, Master Thesis, Coordinación de Postgrado en Ingeniería Mecánica, Universidad Simón Bolívar, Caracas, Venezuela; 2005 [In Spanish].
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Energy absorption of axially crushed expanded metal tubes G. Martínez, C. Graciano n,1, P. Teixeira Universidad Simón Bolívar, Departamento de Mecánica Apdo. 89000, Caracas 1080-A, Venezuela
art ic l e i nf o
a b s t r a c t
Article history: Received 28 October 2012 Received in revised form 6 May 2013 Accepted 7 May 2013 Available online 17 July 2013
Expanded metal tubes have a great potential for energy absorbing applications. A study on the energy absorption capacity of axially compressed expanded metal tubes is conducted herein; the investigation is performed trough nonlinear finite element analyses. At first, the numerical models are validated with experimental results, thereafter a parametric study is carried out in order to investigate the effects of the length-to-diameter ratio, on both the peak forces and the energy absorption capacity of the tubes. The numerical results are also compared with those obtained using a mechanical model found in the literature. Finally, it is found that peak loads and energy absorption capacity depends on the number of expanded metal cells in the cross-section. The results also show that, concentrical expanded tubes could be an effective mean to enhance energy absorption capacity. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Expanded metal Finite element analysis Energy absorption Axial Crushing Ultimate strength
1. Introduction In the last three decades there has been a large amount of research projects looking for the best configuration for energy absorption purposes with metallic and non-metallic structural components. In some cases the failure mechanisms are described and hence mechanical models are developed to predict the strength. In 2001, Alghamdi [1] presented an overview covering four decades of research on collapsible impact energy absorbers under compressive loading. Olabi et al. [2] presented a second overview on metallic tubes used as energy absorbers under axial crushing, bending and oblique impact. Jones [3] performed a comparison between various energy absorbing systems by means of an energy-absorbing effectiveness factor. These reviews were conducted mainly on metallic thin-walled structural elements, but composites and non-metallic materials are also used in energy absorbing applications. In this sense, Lau et al. [4] conducted a survey on geometrical variables and failure mechanisms in composite energy absorbers. In order to understand the structural behavior of thin-walled elements three main approaches have been undertaken, i.e. experimental, numerical and analytical. Experimental testing is perhaps the most expensive approach in terms of materials and time consumption, nevertheless necessary for numerical model validation [5–10]. Regarding experimentation, for short and long aluminum tubes under axial crushing the transition between n
Corresponding author. Tel.: +57 4 425 51 66; fax: +57 4 425 51 75. E-mail addresses: [email protected], [email protected] (C. Graciano). 1 Present address: Facultad de Minas, Departamento de Ing. Civil, Universidad Nacional de Colombia, Sede Medellín, A.A. 75267 Medellín, Colombia. 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.05.003
global and progressive buckling has been investigated [5,6]. Gupta and Vankatesh [7] also conducted experimental and numerical studies on the axial compression of aluminum cylindrical shells, varying the diameter and wall thickness. Continuing this, Gupta [8] investigated the collapse mode of aluminum tubes with frusta geometry. Accordingly, the influence of the cross-section on the collapse response is an important issue when dealing with energy absorption, Nia and Hamendani [9] after studying the energy absorption capacity of tubes with various section shapes (circular, square, rectangular, hexagonal, triangular, pyramidal and conical) found that a circular tube has the higher energy absorption capacity. Using cross-sectional shapes with only convex polygons, Fan et al. [10] established an improvement in energy absorption when increasing the number of inward corners. Composite materials are an alternative for energy absorption applications, an increase on energy absorption capacity is observed when comparing the results for foam-filled tubes with those achieved with empty tubes subjected to axial compressive loads [11–14]. Materials with non-metallic matrixes such as glass and carbon fiber have also been investigated [15,16]. Summarizing, metallic energy absorbers under axial compressive loads fail in various modes, namely axial crushing, global and local buckling depending essentially on geometrical parameters (length and cross-sectional variables). On the contrary, energy absorbers built on composite matrix fail due to a combined mechanism characterized by yielding and fracture. Enhancements in computing power, over the last two decades, have reduced significantly time consumption and empowered engineers and researchers to explore by means of numerical analyses the nonlinear structural response of energy absorbing components [17–19]. Aimed at optimizing the energy-absorbing
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effectiveness factor, the shape of the structural component has also been an important factor in the literature. An important task to accomplish in the development of energy absorbing structures is to reduce the initial peak load exerted in the load–displacement response. It has been observed in both experimental and numerical studies. In this regard, buckling initiators in the form of initial geometric imperfections such as patterns [20], grooves [21,22], pulling strips [23,24], profile corners and indentations [25] have been implemented. Due to the complex nature of the failure observed experimentally and/or numerical few analytical models predicting the strength can be found in the literature. The main results from the previous review can be summarized as follows:
For energy absorption applications the structural response to
quasi-static loading should be stable with a reduced peak load, i.e. energy absorbed in a square like manner. The initial peak load, exerted in axial compression testing, can be significantly reduced by using buckling initiators. Circular geometry, instead of stiffened corners, allows a better energy-absorbing effectiveness. It is important to have in mind that, stiffen corners strengthened the tubes but not necessarily improves the postbuckling response.
135
It seems that expanded metal tubes may satisfy all requirements to build an efficient energy absorber. Graciano et al. [26] investigated the axial collapse of round tubes made of expanded metal sheeting under compressive loading. From the test results, it was observed that the collapse mechanism depends on the orientation of the expanded metal cells, for α¼01 a mode characterized by a plastic collapse mechanism (Fig. 1b); and for α¼901 the tube failed by global
Top plate
D
L
Bottom plate
F Fig. 2. An expanded metal tube under axial compression.
Fig. 3. Schematic view of pattern cell.
Fig. 1. Initial and final configuration for experimental specimens with α¼ 01 and α¼901 (Graciano et al. [26]). (a) Initial a=0° (b) at failure a=0° (c) Initial a=90° (d) at failure a=90°.
Fig. 4. Stress–strain curve [29].
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buckling (Fig. 1d). A mechanical model resembling the failure mechanism for these two geometries was later presented by Graciano et al. [27]. It was found that the peak load depends on the number of expanded metal cells in the cross-section due to the formation of plastic hinges in the connecting nodes. The model presented by Graciano et al. [27] was derived from a limited number of experimental tests results, obtained from a particular geometry. However, the influence of some geometrical parameters such as the length, and hence the length-to-diameter ratio was disregarded. This paper is aimed at investigating the energy absorption response of axially compressed expanded metal tubes (Fig. 2). At first, a numerical model is built upon a nonlinear
Clamped
finite element approach; thereafter, the model is validated with experimental results and a parametric analysis is conducted in order to investigate in depth the influence of the length-to-diameter ratio on peak load and the absorbed energy. The results from these analyses are compared to those obtained using a failure mechanism model [27]. Finally, the numerical model is used to obtain the energy absorbing capacity of concentrical expanded metal.
2. Failure mechanism model Graciano et al. [27] propose the formation of four plastic hinges to describe the failure of expanded metal cells in tubes subjected to axial crushing. Accordingly, the cells in an expanded metal tube (Fig. 3) can be represented as beams within a structural frame. Considering an expanded metal cell, the strength is obtained when plastic hinges at nodes 1–4 (Fig. 3) are formed at each cell in the mid-section of the tube. Hence, the failure load P of such structure can be calculated trough the following equation: P¼
N P 2 i
ð1Þ
Displacement only in the vertical direction
Fig. 5. Boundary conditions in the numerical model for α ¼01.
Fig. 7. Comparison between load–displacement responses.
Y Z X
File
G _ 0 _I x J Number of columns Number of files Orientation
Z Y X
Fig. 6. Finite element model. (a) a=0°, (b) a=90° and (c) Individual cell.
Fig. 8. Nomenclature used in the numerical analysis.
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Fig. 9. Schematic view of the geometries used in the parametric analysis for α ¼ 01.
Fig. 10. Schematic view of the geometries used in the parametric analysis for α¼ 901.
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In Eq. (1), N is the total number of nodes in a circumferential row at the mid-section of the tube; and Pi is the load under which four plastic hinges form in a single cell and can be calculated as Pi ¼
8M P l1
ð2Þ
Substituting the expression for the plastic moment in a strand MP ¼Sy wt2/4 in Eq. (2), we get
Pi ¼
2Sy wt 2 l1
Fig. 11. Load–displacement response for various lengths (α¼ 01).
ð3Þ
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where t is the strand thickness; w is the strand width, l1 is the major axis and Sy is the yield stress. This value of Pi represents a lower-bound solution for the ultimate load of a single cell. These Table 1 Numerical results for α¼ 01. Model I
G_0_
J
139
formulae can be either used for expanded metal tubes with cell oriented at 01 or 901. A detailed derivation of the previous formulation can be found in [27].
3. Numerical model
Ppeak-col (N)
Ppeak (N)
Ptheo Eq. (1) (N)
r Eq. (4)
Eabs Eq.(5) SEA Eq. (6) (kJ) (kJ/kg)
8
3 4 5 6 7 8 10
285.16 280.45 274.54 270.11 267.88 265.93 260.55
855.48 1121.80 1372.70 1620.66 1875.16 2127.44 2605.50
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.86 0.88 0.89 0.90 0.91 0.93
100.60 131.12 161.10 190.38 220.56 250.21 307.52
0.760 0.743 0.730 0.719 0.714 0.709 0.697
10
3 4 5 6 7 8 10
284.28 278.78 273.47 270.22 268.41 265.32 262.71
852.84 1115.12 1367.35 1621.32 1878.87 2122.56 2627.10
723.21 964.29 1205.36 1446.43 687.50 1928.57 2410.71
0.85 0.86 0.88 0.89 0.90 0.91 0.92
126.40 164.62 201.78 239.33 277.65 313.55 384.09
0.764 0.746 0.732 0.723 0.719 0.711 0.696
12
3 4 5 6 7 8 10
283.63 279.46 272.90 269.47 268.35 264.57 260.49
850.89 1117.84 1364.50 1616.82 1878.45 2116.56 2604.90
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.86 0.88 0.89 0.90 0.91 0.93
150.34 196.57 240.01 284.40 330.45 373.81 459.46
0.757 0.742 0.725 0.716 0.713 0.706 0.694
14
3 4 5 6 7 8 10
283.77 278.43 273.37 270.03 268.27 264.73 259.77
851.31 1113.72 1366.85 1620.18 1877.89 2117.84 2597.70
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.87 0.88 0.89 0.90 0.91 0.93
175.34 229.53 280.24 332.27 385.40 435.45 537.52
0.757 0.743 0.726 0.717 0.713 0.705 0.696
16
3 4 5 6 7 8 10
283.56 278.43 273.03 269.29 267.97 264.77 260.60
850.68 1113.72 1365.15 1615.74 1875.79 2118.16 2606.00
723.21 964.29 1205.36 1446.43 1687.50 1928.57 2410.71
0.85 0.87 0.88 0.90 0.90 0.91 0.93
200.38 262.66 319.89 378.82 439.82 497.22 614.52
0.757 0.744 0.725 0.715 0.712 0.704 0.696
As in the experiments, the geometry used to model the expanded metal tubes used was length L¼ 400 mm, diameter D¼120 mm, strand thickness t¼3 mm, strand width w¼3.2 mm, a major axis l1 ¼89.6 mm and minor axis l2 ¼44.2 mm. The nonlinear finite element code ANSYS 12 [28] was used to predict the response of the expanded metal tubes subjected to axial crushing. The expanded metal cells in the tube were modeled with 3D solid elements (SOLID 187), with 10 nodes and a quadratic displacement behavior, making them suitable for modeling plasticity and large strain analyses. Fig. 4 shows the actual stress–strain curve used to model the material; the data was obtained from experimental testing of the base material [29]. Accordingly, the material used was ASTM A-569 with yield strength Sy ¼250 MPa, ultimate strength Su ¼ 385 MPa, Young's modulus E ¼205 GPa and Poisson ratio v ¼ 0.3. The material plasticity is accounted for with a multilinear isotropic hardening model, generated from the Ramberg–Osgood fit to the actual stress–strain curve of ASTM A-569, since no cyclic response is addressed in this study. Each configuration was subjected to a quasi-static analysis simulation, where a load-step control strategy to solve the nonlinear incremental constitutive problem due to plasticity and contact is adopted. Initially, the load is divided into 100 steps, and the convergence rate of the Newton–Raphson scheme adjusts the load increment step to optimize solution time in order to achieve convergence. Clamped boundary conditions were applied at the top and bottom of the tubes (Fig. 5). Prescribed displacement conditions were applied on a rigid plane at the bottom end until the geometry is displaced one-third of its original length (L/3). In order to model contact within the cells during deformation, an edge to edge rough contact with an infinite frictional coefficient is defined at the strands and nodes due to the uncertainties in the real contact behavior.
Fig. 12. Structural response for models with α ¼01.
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As seen in Fig. 5, the expanded metal tubes exhibit a cyclic symmetry around the longitudinal axis. Therefore, each column can be seen as a periodical pattern as shown in Fig. 6a for α¼ 01, and Fig. 6b for α¼901. In this way a radial segment of each model can be solved, saving computational time. The symmetry condition is defined by planes that cut the geometry in the radial direction, where deformation normal to the symmetry plane is not allowed. The numerical model was validated using the load–displacement response for a round tube with α¼01 and the same dimensions as those tested in [26]. A convergence analysis was performed leading to a model with 3 mm elements, Fig. 6c shows the mesh for a single cell obtained from a path conforming meshing algorithm. Comparing the load–displacement responses obtained with the FE model and the experimental data (Fig. 7), a good agreement between these is observed. Furthermore, the difference between these curves is the relative unstability of the experimental data when compared to the numerical results. Concerning the parametric analysis in the next section, the ratio between the theoretical prediction and the numerical value for the peak load is calculated: r¼
P theo P peak
D¼300 mm. i.e. from J ¼4 to 20 columns. For each configuration, the length was also modified, according to the number of files from L ¼392 mm (I ¼4 files) to L¼ 784 mm (I ¼8 files). Figs. 9 and 10 show schematic views of the geometries used in the computational study, with its corresponding nomenclature. These figures show the cell orientation (01 or 901), the number of files I (length) and the number of columns J (diameter). In summary, a total of 75 models were defined for various L/D ratios for the two chosen cell orientations (35 for α¼01, and 40 for α¼901).
5. Results and discussions 5.1. Models with α ¼01 Fig. 11 shows the load–displacement responses for the numerical models with α¼ 01 (G_0_I J). As seen from Fig. 11a–g, all models within this group exhibit a very stable load–displacement
ð4Þ
where Ptheo is calculated using Eq. (1), and Ppeak is computed numerically. As mentioned above, one-third of the total length is crushed, therefore the energy absorbed Eabs is calculated by integrating the load–displacement curves: Z L=3 Eabs ¼ P dx ð5Þ 0
Finally, the specific energy absorbed (SEA) is calculated by dividing the energy absorbed with its weight in the following equation, accordingly W is the weight of the one-third of the model: SEA ¼
Eabs W
ð6Þ
4. Parametric analysis Fig. 13. Average load–displacement response for various diameters α ¼01.
In this section a parametric analysis is conducted in order to investigate the influence of the length-to-diameter ratio (L/D) on the energy absorption capacity of expanded metal tubes under axial compression. Only two cell orientations were considered in the study, i.e. α ¼01 and α ¼901. After examining the load– displacement response for each configuration the following parameters were calculated: peak load per column (Ppeak-col); peak load (Ppeak); theoretical load prediction (Ptheo); the ratio between the theoretical and the numerical peak load (r); energy absorbed (Eabs); and the specific energy absorbed (SEA). Theses parameters are calculated according to the formulae presented in the previous section. Fig. 8 shows the nomenclature used in the parametric analysis, where I and J are the number of files and columns, respectively. Varying the length-to-diameter ratio (L/D), the study was divided into two stages: initially modifying L maintaining D constant and second varying D maintaining L constant. For α ¼01, the geometries were created according to its diameters; the range in the study goes from D ¼91 mm to D¼ 300 mm, i.e. from J¼ 3 to 10 columns. For each configuration, the length was also modified, according to the number of files, from L ¼ 383 mm (I ¼8) to L¼766 mm (I ¼16). Correspondingly, for α ¼901, the geometries were also created according to its diameters; the range in the study goes from D ¼ 61 mm to
Fig. 14. Average absorbed energy for various diameters α ¼01.
G. Martínez et al. / Thin-Walled Structures 71 (2013) 134–146
response, similar to the experimental results obtained in [26], which is desirable for energy absorbing systems. It is worth mentioning that the slope in the load–displacement responses is always ascending and positive, therefore it was necessary to define a criterion to report the peak load (Ppeak) in Table 1. It was established that a change of 0.1% in the slope was
141
necessary to achieve Ppeak. Regarding the failure mechanism of the tubes with this configuration, all the expanded metal cells folded stably and progressively during the crushing process as observed in Fig. 12. Table 1 shows the numerical results for the tubes with α ¼01, Ppeak is computed from the load–displacement responses in Fig. 11, Ptheo is calculated using Eq.(1); the ratio between peak load
Fig. 15. Load–displacement responses (α¼ 901).
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Fig. 16. Structural response for models with α ¼ 901.
Table 2 Numerical results for α¼ 901. Model
I
J
Ppeak-col (N)
Ppeak (N)
Ptheo Eq.(1) (N)
r Eq.(4)
G_90_
4
4 6 8 10 12 14 16 20
556.90 530.66 513.53 497.65 487.30 476.93 469.81 457.74
2227.60 3183.96 4108.24 4976.50 5847.60 6677.02 7516.96 9154.80
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.88 0.92 0.95 0.98 1.00 1.02 1.04 1.07
247.24 372.45 431.19 457.86 505.05 551.42 588.78 674.20
1.401 1.407 1.221 1.038 0.954 0.893 0.834 0.764
5
4 6 8 10 12 14 16 20
560.63 537.16 512.42 498.27 486.20 473.64 462.82 446.71
2242.52 3222.96 4099.36 4982.70 5834.40 6630.96 7405.12 8934.20
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.95 0.98 1.01 1.03 1.06 1.09
320.7 474.82 506.88 572.44 614.76 665.70 695.28 784.77
1.453 1.435 1.149 1.038 0.929 0.862 0.788 0.711
6
4 6 8 10 12 14 16 20
560.65 536.78 537.13 506.56 491.95 479.47 468.38 448.76
2242.60 3220.68 4297.04 5065.60 5903.40 6712.58 7494.08 8975.20
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.91 0.96 0.99 1.02 1.04 1.09
376.24 543.42 680.20 743.46 745.02 742.15 781.73 871.56
1.421 1.368 1.284 1.123 0.938 0.801 0.738 0.658
7
4 6 8 10 12 14 16 20
559.77 536.98 515.99 507.97 502.10 490.14 477.37 457.03
2239.08 3221.88 4127.92 5079.70 6025.20 6861.96 7637.92 9140.60
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.95 0.96 0.97 1.00 1.02 1.07
439.72 647.21 760.28 833.45 888.89 937.13 880.11 995.28
1.423 1.397 1.231 1.079 0.959 0.867 0.712 0.644
8
4 6 8 10 12 14 16 20
559.54 535.71 516.98 504.20 493.84 484.03 476.53 463.77
2238.16 3214.26 4135.84 5042.00 5926.08 6776.42 7624.48 9275.40
1954.75 2932.13 3909.50 4886.88 5864.25 6841.63 7819.00 9773.76
0.87 0.91 0.95 0.97 0.99 1.01 1.03 1.05
500.23 736.22 882.75 953.67 1012.26 1026.20 1084.95 1145.67
1.417 1.390 1.250 1.081 0.956 0.831 0.768 0.649
Eabs Eq.(5) (kJ)
SEA Eq. (6) (kJ/kg)
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and the theoretical prediction r is calculated using Eq.(4); Eabs is computed from the load–displacements response in Fig. 11 applying Eq. (5); and the SEA is calculated using Eq.(6). From the load–displacement responses (Fig. 11) and the results reported in Table 1, it is observed that after increasing the lengths, Ppeak remain almost constant for a given number of columns. In Fig. 11a, the peak load for model G_0_8 3 (L/D ¼4.2; Ppeak ¼855.48 N) is only 0.56% greater than the one for model G_0_16 3 (L/D ¼8.4; Ppeak ¼850.68 N). These results are similar for all models investigated in this section, therefore Fig. 13 summarizes the average load–displacement responses for the tubes with α ¼01. It is observed that the peak loads increase proportionally with the number of columns, i.e. the number of cells in the cross-section. This result was also expected after using Eq. (1), which shows a linear dependency between the strength and the number of cells across the tube. These findings also demonstrate that the mechanism model presented in [27] can be used to predict the peak load for expanded metal tubes with similar boundary conditions. It is interesting to see in Table 1 that the peak load per column (Ppeak-col) should remain constant for the models with the same number of files; however this is reduced when increasing the number of columns. Comparing the peak load for model G_0_8 3 (L/D ¼4.2; Ppeak ¼285.16 N) with the one for model G_0_8 10 (L/D ¼1.3; Ppeak ¼260.55 N), the former is 9% higher than the latter. This variation in the results is observed for all models in Table 1, and is ought to curvature effects. Fig. 14 plots the energy–displacement curves obtained in the analyses. It indicates that for a particular length, increments in the number of columns proportionally increase the energy absorbing performance of the expanded metal tube. Moreover, it can be seen in Table 1, where the energy absorbed for a tube with 10 columns (G_0_I 10) is 300% greater than for a tube with 3 columns (G_0_I 3). Looking at the SEA, it remains almost constant due to the proportionality between the energy absorbed and the number of columns. There is just a small difference of 1% that was explained by curvature effects. This finding is consistent with the conclusion of numerous studies that for a given cross-section the energy absorbing capacity of the structural component is proportional to the diameter of the cylindrical tube, and independent to its length.
The peak loads increase proportionally to the number of columns, however in the postpeak region the curves converge for large displacements. This behavior is quite different than the one for the tubes with α ¼01, and can be explained after observing the failure mode characterized by local instability on the strands. In a similar way as for α¼ 01, in Table 2, the peak loads and the energy absorbed remain almost constant after varying the length. Thus, the peak forces also increase proportionally to the diameter of the tube. The energy absorption capacity (Fig. 18) is more sensitive to a variation in the number of columns; as the number of cell increases in the circumferential direction so this capacity increases. For the given crushing length (one-third of the total length), the absorbed energy (Eabs) increases with the displacement and is proportional to the number of columns. However, the
Fig. 17. Average load–displacement response for various diameters α¼ 901.
5.2. Models with α¼ 901 Fig. 15 shows the load–displacement responses for the numerical models with α ¼901 (G_90_I J), the behavior observed in these curves is rather unstable, nevertheless the peak loads increase considerably with respect to the models with α¼ 01. For model G_90_4 8, Fig. 16 shows both the deformed and undeformed pattern, the failure mechanism is characterized by global outward buckling of the walls until the strands achieves the plastic moment in the connecting nodes. Thereafter, self-contact within the cells due to the collapse in other rows causes peak increments in the load–displacement responses (Fig. 15a–e). After increasing the number of columns (Fig. 15f–h), the behavior become more stable in the postpeak region and the load fluctuations almost disappear. Table 2 shows the numerical results for the tubes with α¼901. From the load–displacement responses (Fig. 15) and the results reported in Table 2, it is observed that after increasing the lengths, Ppeak remain approximately constant while keeping the number of columns. In Fig. 15a, the peak load for model G_90_4 4 (L/D ¼6.4; Ppeak ¼556.90 N) is only 0.47% smaller than the one for model G_0_8 4 (L/D ¼12.8; Ppeak ¼559.54 N). These results are similar for all models investigated in this section with α¼901, hence Fig. 17 summarizes the average load–displacement responses.
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Fig. 18. Average absorbed energy for various diameters α ¼901.
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Table 3 Numerical results for concentrical tubes. Numerical Model
Concentric tubes
Ppeak (kN)
Ptheo Eq.(1) (kN)
r Eq.(4)
Eabs Eq.(5) (kJ)
SEA Eq. (6) (kJ/kg)
Ppeak ¼Pϕ8
2120
1929
0.91
0.314
0.889
Ppeak ¼ Pϕ7+Pϕ8
3996
3616
0.90
0.591
0.893
Ppeak ¼ Pϕ6+Pϕ7+Pϕ8
5616
5063
0.90
0.830
0.896
Ppeak ¼ Pϕ5+ϕ6+Pϕ7+Pϕ8
6981
6268
0.90
1.032
0.899
Ppeak ¼Pϕ4+Pϕ5+Pϕ6+Pϕ7+Pϕ8
8093
7232
0.89
1.197
0.904
Ppeak ¼Pϕ3+Pϕ4+Pϕ5+Pϕ6+Pϕ7+Pϕ8
8945
7955
0.89
1.323
0.908
behavior is not as uniform as for α¼01, increasing the number of columns significantly increases the energy absorbed, and this is obviously due to the increased amount of material available for plastic deformation and subsequent energy absorption. It is important to have in mind that the numerical model was built upon a radial symmetry, in Table 2 the effect of the curvature on the strength of a single column is clearly observed. The peak load per column (Ppeak-col) for a tube with D ¼61 mm (G_90_I 4) is 22% greater than the corresponding D ¼300 mm (G_90_I 20). As a rule, the peak load increases with the number of columns for the models analyzed in Table 2. In spite of this, the peak load per column is reduced attributable to curvature effects. For small diameters the curvature is larger, hence strengthening the expanded metal tube. The performance of the expanded metal tube by means of the specific energy absorption parameter SEA is very similar among the different configurations of α ¼01 tubes, since cell curvature has an small effect in the load response, and the increase in the absorbed energy is due to the increased amount of material available for plastic deformation as mentioned above. This leads to conclude that the SEA parameter depends of plastic hinges density more than geometric configuration, for this case. Comparing the performance of equivalent specimens by SEA parameter, this is for tubes of α¼901 and α¼ 01 of same volume, the firsts in general behaves better than α ¼01 tubes, due to the different failure mechanism. 5.3. Combined models In the previous sections, it was observed that for models with α¼01 the structural response is more stable and controlled than
for models with α¼901. In this section, a parametric study is conducted using various tubes placed concentrically with α¼01. The strength of the tubes can be significantly increased by using a larger number of cells, which is a natural conclusion after the analyses conducted herein. Table 3 shows the results for a combined geometry (concentric tubes), it is shown that both the load and the energy absorbed increase linearly with the number of columns. This is explained as follows, as the number of cells increases so the amount of material for plastic deformation increases. A small difference is observed due to curvature effects caused by diameter increments. Fig. 19 shows the deformation pattern for the concentric tubes. Comparing the predicted values Ptheo for the ultimate strength using Eq. (1), with the values obtained numerically Ptotal, the maximum difference is about 11%. Also, the SEA parameter remains almost constant, independently of the configuration, as obtained before for the single tubes. The results show that, it is possible to make arrays of expanded metal tubes in order to absorb a given amount of energy, which in many ways could fit other geometries or fill spaces.
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Fig. 19. Structural response for concentrical models with α ¼01.
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