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Crashworthiness analysis and optimization of sinusoidal corrugation tube
Thin-Walled Structures 105 (2016) 121–134
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Full length article
Crashworthiness analysis and optimization of sinusoidal corrugation tube Shengyin Wu a, Guangyao Li a, Guangyong Sun a,n, Xin Wu b, Qing Li c a
State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha 410082, China Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA c School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia b
art ic l e i nf o
a b s t r a c t
Article history: Received 23 November 2015 Received in revised form 22 March 2016 Accepted 28 March 2016 Available online 14 April 2016
Thin-walled structures are widely used as energy absorbing devices for their proven advantages on lightweight and crashworthiness. However, conventional thin-walled structures often exhibit unstable collapse modes and excessive initial peak crushing force (IPCF) followed by undesirable fluctuation in force-displacement curves under impact loading. This paper introduces a novel tubal configuration, namely sinusoidal corrugation tube (SCT), to control the collapse mode, and minimize the IPCF and fluctuations. Through validating the finite element (FE) models established, the effects of wavelength, amplitude, thickness and diameter of SCTs on collapse mode and energy absorption were investigated. The results showed that SCTs can make the deformation mode more controllable and predictable, which can be transformed from a mixed mode to a ring mode by simply changing the wavelength and amplitude. Compared with the traditional straight circular tube, the IPCF is reduced appreciably. Furthermore, SCTs have lower fluctuation in the force–displacement curves than traditional straight circular tubes. Finally, a multiobjective optimization is conducted to obtain the optimized SCT configuration for maximizing specific energy absorption (SEA), minimizing IPCF under the constraint of fluctuation criterion. The optimal SCTs are of even more superior crashworthiness and great potential as an energy absorber. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Sinusoidal corrugation tube Collapse mode Energy absorption Multiobjective optimization
1. Introduction Thin-walled tubes are frequently used as energy absorbing devices in the vehicular frontal structures, attributable to their excellent energy absorption efficiency, lightweight and low cost for manufacturing. To better understand energy absorption of thin-walled tubes and fully explore their potential as energy absorbers, a large body of studies has been reported by utilizing analytical, experimental and numerical methods for various crosssectional configurations, such as circular, square, hexagonal, octagon, top-hat, etc. For example, Alexander [1] first derived an approximate formula for predicting the mean crushing force of circular tubes under axial loading; and then Abramowicz et al. [2] developed a more realistic theoretical model by introducing an effective plastic hinge length. Huh et al. [3] analyzed the collapse modes of square tubes for sequential crushing loads as well as deformation modes, numerically and experimentally. Tarlochan et al. [4] explored the crashworthiness of thin-walled tubes with n
Corresponding author. E-mail address: [email protected] (G. Sun).
http://dx.doi.org/10.1016/j.tws.2016.03.029 0263-8231/& 2016 Elsevier Ltd. All rights reserved.
several different cross-sectional shapes (e.g. circular, rectangle, square, hexagonal, octagonal and elliptical cross-sections) under dynamic loading. The previous studies demonstrated that crosssectional configurations have critical influences in energy absorption (EA) capacity of thin-walled columns. Further studies showed that circular tubes can more easily generate a stable progressive folding mode [5–8], which is often considered as an effectie way of energy absorption. For this reason, the study on crashworthiness of circular thin-walled tubes had drawn considerable attentions. Generally speaking, the collapse mode of circular tubes depends on both material properties and geometric dimensions, typically leading to three different modes: symmetric (concertina or ring) mode, non-symmetric (diamond or mixed) mode and Euler mode (global buckling), respectively [8]. Although circular tubes have great advantages on energy absorption and weight reduction, their collapse modes are easily affected by geometric dimension and manufacturing imperfections. Moreover, they are often associated with an excessive peak crushing force. In many cases, the collapse process of an ideal energy absorber is expected to be stable to dissipate the maximum kinetic energy. Since the high peak crushing force can lead to a
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large deceleration, which increases probability of damage and passenger’s injury, severe initial peak force should be avoided. To reduce the initial peak force and enhance the stability of the crushing process, there have been many exploratory studies. For example, Arameh et al. [9] investigated the effect of corrugations on the crushing behaviors such as energy absorption and failure mode of circular aluminum tubes experimentally. The results showed that the corrugated tubes can make the collapse mode more predictable and controllable. From this point, corrugated tube was considered a good candidate for a controllable energy absorption device. Singace et al. [10] also studied the energy absorption characteristics of corrugated tubes experimentally, in which the corrugations are introduced to force the plastic deformation to occur at predetermined intervals along the tube generator for improving the uniformity of the load-displacement behavior under axial loading. Chen and Ozaki [11] used the finite element method (FEM) to investigate the nonlinear elastoplastic behavior of circular tubes with corrugated profiles. Their results showed that the corrugations are one of the dominant factors to influence the deformation mode and fluctuations in the load-displacement curves of circular tubes. Song et al. [12] introduced the origami patterns to thin-walled tubes to minimize the initial peak force and subsequent fluctuations. Their numerical results showed that the patterned tubes exhibit a lower initial peak force and more uniform crushing load. To facilitate the axisymmetric collapse mode in frusta tube, Rezvani and Damghani [13] presented a new design by cutting the circumferential grooves from the outer and inner surfaces of the frusta at different positions in the tube. A series of experimental tests showed that the grooves can stabilize the collapse mode and control energy absorption of the frusta tubes. Daneshi and Hosseinipour [14,15] investigated the buckling mode and energy absorption capacity of grooved tubes and further presented a theoretical formulation for predicting the energy absorption and mean crushing load of these tubes under axial loading. The results showed that the load-displacement curve and energy absorption could be controlled by introducing the grooves with different distances. It was found that these grooves can stabilize the deformation behaviors. In summary, these corrugated tubes have a great potential to reduce the initial peak force and improve the deformation stability, which has been attracting more and more attention. To the authors’ best knowledge, however, there is rather limited research to systematically quantify the influence of the wavelength, amplitude, thickness and diameter of SCTs on the collapse mode, initial peak force and energy absorption. To explore the energy absorption mechanism of SCTs and provide some insights into the design of the SCTs, this study aimed to investigate the parametric effects of wavelength, amplitude, thickness and basic diameter of SCTs on the deformation mechanisms, distribution area of different collapse modes, initial peak crushing force (IPCF), mean crushing force, undulation of load-carrying capacity (ULC), absorbed energy (EA) and specific energy absorption (SEA). It will show that SCT parameters can significantly affect the uniformity of the load-displacement curve, as well as the controllability and predictability of collapse modes. To obtain the optimal configuration of SCTs, the Non-domain Sorting Genetic Algorithm II (NSGA-II) [16] that is integrated with surrogate models was utilized to seek the optimal wavelength, amplitude, thickness and basic diameter of SCTs. Of some common surrogates, such as radial basis function (RBF) [17], response surface method (RSM) [18], Kriging (KRG) [19] and support vector machine (SVM) [20], the radial basis function (RBF) surrogate models have proven effective for such highly nonlinear problems as crashworthiness [21,22]. Thus, the RBF surrogate modeling technique is adopted in this study. In the optimization process, the IPCF is minimized and SEA is maximized simultaneously, the load-
carrying capacity is constrained within a level of 0.2. The results demonstrated that the SCTs can provide a better crashworthiness than corresponding straight circular tubes.
2. Crashworthiness criteria and material properties 2.1. Crashworthiness criteria Crashing criteria are essential to evaluate the crashworthiness performance. In literature, different criteria have been adopted in different structures [23,24]. In this paper, the following criteria are utilized to evaluate the crashworthiness characteristics of SCT structures.
Initial Peak Crushing Force (IPCF)
Typically, thin-walled tubes hit a peak force in an early stage under crushing. The force at this point is known as initial peak force. As a device for energy adsorption in real life, it may cause severe injury or damage when the initial peak force is overly high [25]. For this reason, Initial Peak Crushing Force (IPCF) should be minimized or constrained to a safe level. Energy Absorption (EA) Energy absorption has been widely used to evaluate the crashing characteristic of structure, which indicates the amount of absorbed energy during crashing process; it can be calculated mathematically as follows:
∫0
EA (d) =
F (x) dx
(1)
where F (x ) is the axial impact force as a function of displacement x during crashing and d denotes the effective deformation distance, which was set to be 110 mm over the initial tube length of 150 mm (so that the final length of crushed tube is 40 mm) in this study. Specific Energy Absorption (SEA) To compare the energy absorbed capability of different materials and structures, the specific energy absorption (SEA) is often used to evaluate the capability of energy absorption by taking into account the mass factor, defined by:
SEA ( d)=
d
EA (d) m
(2)
where m is the mass of energy absorber. As a potential device for energy absorption, the larger the SEA, the higher the mass efficiency of energy absorber [26]. Mean Crushing Force ( Pm ) Mean crush force is the average value of the compressive force of the energy absorber through total effective deformation, given as: d
Pm =
∫ F (x) dx EA (d) = 0 d d
(3)
Undulation of Load-carrying Capacity (ULC) In order to estimate the stability of energy absorption mathematically, Xiang et al. [27] introduced the undulation of load-carrying capacity (ULC) to evaluate the stability of load, as: d
ULC =
∫0 | F (x) − Pm | dx d
∫0 F (x) dx
(4)
which indicates that when the deviation between peak and trough is too great, ULC is high as shown in Fig. 1; in which case the forcedisplacement curves is considered less effective for energy
S. Wu et al. / Thin-Walled Structures 105 (2016) 121–134
as the amplitude A approaches to zero, as shown in Fig. 3(b). Therefore, the straight tube can be regarded as a special case of SCTs.
50
2.3. Material properties
Force (kN)
40
The tube material is aluminum alloy 6063-T6, whose chemical composition (in weight percent) is Si-0.20–0.60, Fe-0.35, Cu-0.10, Mn-0.10, Mg-0.45–0.90, Cr-0.10, Zn-0.10, Ti-0.10 and the remaining major composition is Al. The stress-strain curve of the material was obtained by using the tensile tests with specimen dimensions as specified in ASTM standard B557M. The tensile experiment was performed in the MTS 5457 material testing machine with computer control and data acquisition system under a constant speed of 2 mm/ min. The specimen geometry with the unit of mm and the corresponding stress-strain curve of Al6063-T6 are shown in Fig. 4, and the obtained material properties are summarized in Table 1. The real strain rate in crashing scenarios would be two orders of magnitude higher than this testing condition, and based on the reported strain rate sensitivity of aluminum alloy [28], the flow stress is slightly higher; thus, the FE models neglected the rate-dependency. To model the tubes made of such an aluminum alloy, the piece-wise linear plasticity material (material model 24 in LS-DYNA) was utilized.
30
Peak
20
Mean crushing force
10
Trough 0 0
20
40
60
80
Displacement (mm) Fig. 1. Typical force-displacement curve.
absorption. On contrary, if ULC is small, the force-displacement curves become a rectangular shape and the energy-displacement curve is approaching to a straight line approximatively, indicating better energy absorption efficiency. 2.2. Geometric configuration In this study, the configuration of a sinusoidal corrugation tube (SCT) is shown in Fig. 2. The following function was utilized to describe the corrugation profile of the tube walls:
y=
123
⎛ 2π ⎞ D + A sin ⎜ x⎟ ⎝W ⎠ 2
(5)
where A is the amplitude, W is the wavelength (the vertical distance), D is the mean diameter. This structure is used for energy absorption in the front side rail of a passenger car, and the length of energy absorbing box is often 100–200 mm, so this paper chose L ¼150 mm as the total length of this tube. When the amplitude keeps a constant value A¼ 1.0 mm, the profile will tend to be a straight line with increasing wavelength W, as shown in Fig. 3(a). Similarly, the profile also becomes straight line
3. Finite element modeling and validation 3.1. Finite element model A representative 3D experimental schematic diagram for SCT is displayed in Fig. 5a. The tube specimen was placed in between two flat steel plates in the testing machine and it was compressed at a constant speed (1000 mm/s). During the test, the top plate of the machine moved downward to compress the specimen vertically by the displacement control. The real-time loading and compressive displacement were acquired automatically. When the compressive displacement reached 110 mm, the specimen was unloaded and the experiment stopped. The explicit finite element code LS-DYNA was used to simulate the experimental process. The tube wall was meshed using the 4-node quadrilateral Belytschko-Lin-Tsay shell elements with reduced integration. Five integration points were used across the thickness to capture the local bending behavior, and one integration point was used in the plane of the element. The stiffness-type hourglass control was employed to eliminate spurious zero energy
Fig. 2. The profile of sinusoidal corrugation.
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W=6
W=8
W=12
W=15
W=20
W=30
W=∞
A=0
A=0.2
A=0.4
A=0.8
A=1.2
A=1.5
A=2.0
Fig. 3. The profiles of SCTs with various wavelengths W or amplitudes A at W ¼ 1 and/or A ¼ 0 the profile reduces to a straight tube: (a) A ¼ 1 mm; (b) W ¼10 mm.
Fig. 4. Axial tensile test set-up and specimen: (a) MTS 5457material testing machine; (b) geometry dimension of specimen;(c) a specimen after failure; (d) engineering stress versus engineering strain curve.
Table 1 Material properties of the aluminum alloy 6063-T6. Density
Young's modulus
2700 kg/m^3 65,000 MPa
Poisson’s ratio
Yield stress Ultimate tensile stress
0.33
145 MPa
161 MPa
models commonly arising when such reduced integration elements are used. As the steel plate did not deform much during compression process, thus it was treated as a rigid body using the 8 node brick elements. The interface between the specimen and the two steel plates was modeled using a ‘node-to-surface’ contact. The ‘automatic single surface’ contact was applied to the column wall to avoid interpenetration of folding generated during axial collapse. The dynamic and static coefficients of friction of
0.2 and 0.3 were respectively used as suggested in [29]. Several simulations were first run to conduct a convergence test and finally the element size of 0.8 mm 0.8 mm was determined for its acceptable accuracy and computational cost. Following the above description, the final FE model is shown in Fig. 5b. To investigate the effect of the wavelength and amplitude of SCTs on the collapse mode and energy absorption characteristics, a series of SCTs with different wavelengths and amplitudes were considered as summarized in Table 2, in which the amplitude ranges from 0.1 to 2.0 mm, and the wavelength ranges from 4 to 30 mm, while the tube wall thickness t¼1.4 mm and length L¼150 mm remain unchanged. 3.2. Validation of the FE model From Eq. (5), it can be easily found that the SCTs will become a simple straight circular tube when amplitude A¼0 mm, so the
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125
Fig. 5. Model of SCT: (a) A representative 3D experimental schematic diagram for SCT; (b) the final FE model for SCT. Table 2 Wavelengths and amplitudes of various models.
2st
1st
4st
3st
5st
7st
6 st
8st
9 st
11st
10 st
12st
13st
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
0.1
4 5 6 8 10 12 15 20 25 30
0.2
4 5 6 8 10 12 15 20 25 30
0.3
4 5 6 8 10 12 15 20 25 30
0.4
4 5 6 8 10 12 15 20 25 30
0.5
4 5 6 8 10 12 15 20 25 30
0.8
4 5 6 8 10 12 15 20 25 30
1.0
4 5 6 8 10 12 15 20 25 30
1.2
/ / 6 8 10 12 15 20 25 30
1.4
/ / 6 8 10 12 15 20 25 30
1.5
/ / 6 8 10 12 15 20 25 30
1.6
/ / 6 8 10 12 15 20 25 30
1.8
/ / 6 8 10 12 15 20 25 30
2.0
/ / 6 8 10 12 15 20 25 30
Fig. 6. Quasi-static axial compressive test for the straight column.
straight tube can be regarded as a special SCT. The FE model was thus validated using a simple straight circular tube with the diameter of 60 mm, height of 150 mm and thickness of 1.4 mm under
axial compression at a constant velocity of 2 mm/min. To improve the computing efficiency, the tube is compressed by a rigid plane with the velocity of 1000 mm/s in the FE model for quasi-static
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compare with the numerical results. From the literature [31], the mean crushing force of circular tubes can be expressed as follows:
⎛ D ⎞0.32 Fm = 72. 3⎜ ⎟ ⎝t⎠ MP
(6)
where Fm is the mean crushing force which can be calculated by Eq. (3). D is the diameter of circular tube, t is the wall thickness, MP 1 is the full plastic moment, calculated by MP = 4 σ0 t 2 (where σ0 is a characteristic stress of material). The comparison of the mean crushing force versus displacement between FE simulation and the analytical solution is plotted in Fig. 8. Obviously, the numerical result well agrees with the theoretical result. Therefore, the FE model is of an acceptable accuracy and can be used to analyze the crashing characteristics of sinusoidal corrugation tubes under quasi-static axial loading.
Fig. 7. Comparison of force-displacement curves in the experiment and FE simulation.
Fig. 8. Mean crushing force versus displacement curves.
loading [30]. This quasi-static axial crushing test of the tube was conducted on an INSTRON 5984 (150kN) testing machine, as shown in Fig. 6. Fig. 7 provides a comparison of force-displacement curves obtained from the experiment test and FE modeling. Obviously the simulation was well correlated to the experimental test. Furthermore, the deformed tube shapes or collapse modes in the experiment and simulation were compared at two displacement points as presented in Fig. 7. Clearly, the collapse mode from the experiment agrees fairly well with that from the FE simulation. For the energy absorption, the experimental and numerical results are 1406 J and 1387 J, respectively, which are also in reasonable agreement. In order to further validate the accuracy of the FE model, the theoretical solutions available in literature were also utilized to
4. Crashworthiness analysis 4.1. Collapse modes under axial quasi static loading According to Table 2, a total of 118 design cases with different A and W were simulated. As shown in Fig. 9, the collapse of SCTs has mainly displayed in two modes: namely ring mode and mixed mode (i.e. more than one mode developed during the entire crashing process, which will be discussed later in more detail), and these corresponding 118 collapse modes are plotted collectively in Fig. 10a. When amplitude A r0.4 mm, the collapses are shown in a mixed mode regardless of wavelengths. However, the collapse mode begins to transform to a ring mode in general, when amplitude AZ0.5 mm. As shown in Fig. 10a, the black dots indicate the area of ring mode, and clearly the area increases with the amplitude. On the other hand, When wavelength 4.0 r Wr6.0 mm and 25r Wr30 mm, the mixed mode dominates. Besides, the collapse mode transfers from a mixed mode to a ring mode gradually. For the further study, Fig. 10b plots the experimental force-displacement curve, the nominal average wavelength is 15.75 mm, which can be obtained by calculating the average of the three data. W¼ 15 is close to the nominal average wavelength, which means that W¼ 15 can account for easy formation of the ring mode. Table 3 shows the transition process of collapse modes of SCTs with the amplitude of A ¼1.8 mm and different wavelengths ranging from 0 to 30 mm. From which, it is easily found that the collapse follows a mixed mode when wavelength is quite small, such as W ¼0 or W¼6 mm. When 8 rWr25 mm, the deformation was transferred from the mixed mode to the stable ring mode. However, collapse mode was transformed to the mixed mode again when W is long enough, such as W¼ 30 mm. Table 4 compares the collapse modes of straight tube with that of the SCT (A ¼1.6 mm and W ¼20 mm) at different displacements (0, 30, 60, 90, and 120 mm). Obviously the collapse of straight tube follows a ring mode in the first plastic folding, but it transforms to the diamond mode in the subsequent progressive deformation,
Fig. 9. The two representative collapse modes of SCTs: (a) Ring mode (A ¼0.8 mm, W ¼15 mm); (b) Mixed mode (A ¼ 0.8 mm, W ¼25 mm).
S. Wu et al. / Thin-Walled Structures 105 (2016) 121–134
127
50 35
Straight circular tube
Ring mode Mixed mode
40
25
Force (kN)
Wavelength (mm)
30
20 15 10
30 15.70
15.43
16.12
20 10
5
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
20
Amplitude (mm)
40
60
80
Displacement (mm)
Fig. 10. Collapse modes and force-displacement curve: (a)The distribution of collapse modes of 118 specimens; (b) The experimental force-displacement curve.
Table 3 The collapse modes of different wavelengths at the same amplitude of 1.8 mm.
whilst the sinusoidal corrugation tube always keeps the stable ring mode until the end of deformation. 4.2. Influence on the initial peak force The initial peak forces of SCTs with the ring modes are listed in Table 5. It shows that the initial peak force of SCTs can be reduced significantly by adopting the tubal corrugations compared with conventional straight tube. For example, IPCF of the circular straight tube is 43.69 kN, whereas IPCF (W ¼15 mm, A¼0.5 mm) reduces to 23.15 kN. Overall, around 47.01–84.80% reduction of IPCF can be achieved in the corrugating tubes.
Furthermore, Fig. 11 (a) shows the trend of IPCF with the amplitude of A ¼1.8 mm vs. different wavelengths, where the force gradually increases with the wavelength and the maximal initial peak force reaches 13.96 kN, which is however far lower than that of straight circular tube (43.69 kN). Fig. 11 (b) shows the trend of IPCF of sinusoidal corrugation tube with the wavelength of W¼12 mm vs. amplitude. Clearly, the IPCF increases with the reduction of amplitude, but it is still much smaller than that of the straight counterpart. Overall, by introducing sinusoidal corrugations to a straight tube, the crashing peak force can be reduced significantly. Specifically, the longer the wavelength and the smaller the amplitude,
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Table 4 Comparison of collapse modes between straight tube and SCT (W¼ 20 mm, A ¼1.6 mm) at different displacements.
the higher the initial peak force; and this tendency can be more intuitively observed from Fig. 12. 4.3. Influence on undulation of load-carrying capacity The undulation of load-carrying capacity (ULC) can be obtained by Eq. (4), which evaluates the smoothness of energy absorption. If the force variation and force range between peak and valley in collapse process are too large, the energy-displacement curves will be less smooth, which should be avoided in an energy absorption device. Table 5 shows the ULC of SCTs with the ring modes, the ULC of straight tube is 0.222, which is higher than most of other ring modes except at A¼ 0.8, W ¼20 mm and A¼ 2.0, W¼ 25 mm. The minimum ULC value is 0.049, which corresponds to a 77.9% reduction from the straight tube. From a unilateral consideration, the smaller the ULC, the smoother the energy absorption. Fig. 13 compares the force vs displacement curves between the straight tube and SCT (W ¼15 mm and A ¼1.6 mm). It is observed that the straight tube has a larger deviation between the peak and trough forces. However, the red curve in Fig. 13 displays a relatively smoother loading process, so SCTs are of certain advantages on reducing the undulation of load-carrying capacity and keeping the steady state of energy absorption process.
SCTs must sacrifice some energy absorption capacity. Fig. 14(a) displays the variation in the EA of SCTs with different amplitudes and wavelengths. It can be easily found that the EA decreases when wavelength starts increasing. From Table 3, when the wavelength is quite small, the collapse mode has more ringmoded plastic deformation, which is the reason for a higher EA. However, when wavelength increases, SCT tends to be more straight, which enhances the resistant capacity of structure, so the total energy absorption increases. For example, when the amplitude is 1.8 mm, EA of SCTs decreases from 1.28 to 1.00 kJ, but increases to 1.11 kJ in the last. On the other hand, when the wavelength is constant, the smaller the amplitude, the higher the EA. This phenomenon can be observed in a range of wavelengths in Fig. 14(a), indicating that the larger the amplitude, the lower the ability of loading resistance. Fig. 14 (b) plots the SEA vs. wavelength and amplitude. Obviously, SEA has a fairly similar tendency to EA. From Table 5, the structural mass decrease with the wavelength. SEA was calculated by Eq. (2), the denominator ‘m’ has the same decreasing tendency, thus SEA is similar tendency to EA. In this regard, the SEA of SCTs does not exhibit advantage. Therefore, a tradeoff needs to be made among the collapse mode, initial peak force, ULC, EA and SEA. 4.5. Influence on diameter and thickness on collapse mode
4.4. Absorbed energy and specific energy absorption The energy absorption (EA) and specific energy absorption (SEA) for different amplitudes and wavelengths in the ring mode are listed in Table 5. The EA and SEA of straight tube are 1.79 kJ and 17.27 kJ/kg, respectively. The highest EA and SEA of SCT are 1.71 KJ and 16.29 KJ/Kg. It shows that straight tube is more advantageous in terms of EA and SEA. This implies that to keep a stable ring mode, reduce the initial peak force and achieve a small ULC value,
The aforementioned study has focused on the influence of wavelength and amplitude of SCTs with constant diameter D, thickness t and length L on crashworthiness. However, for conventional straight circular tube, these parameters are also important design variables for energy absorber; so their influence on collapse mode and energy absorption characteristics should be also investigated. By introducing the thickness to diameter ratio t* ¼t / D and diameter to length ratio D* ¼D/L, Table 6 exhibits
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129
Table 5 Crashworthiness performance criteria of straight tube and the SCTS with ring modes. Number
A (mm)
W (mm)
Mass (Kg)
IPCF (kN)
Pm (kN)
ULC
EA (kJ)
SEA (kJ/kg)
1
0.5
15
0.1051
23.15
14.08
0.178
1.71
16.26
2 3 4
0.8
15 20 15
0.1066 0.1059 0.1085
18.84 20.05 16.53
12.30 11.38 11.57
0.168 0.227 0.086
1.50 1.33 1.40
14.07.0 12.55 12.90
5 6
1.2
12 15
0.1136 0.1103
14.04 14.45
12.00 10.52
0.109 0.077
1.43 1.27
12.58 11.51
7 8 9 10
1.4
8 10 12 15
0.1304 0.12150 0.1167 0.1125
9.22 12.14 12.42 12.81
13.09 12.24 11.08 9.80
0.109 0.102 0.094 0.077
1.53 1.48 1.33 1.19
11.77 12.18 11.39 10.57
11 12 13
1.5
12 15 20
0.1184 0.1138 0.1094
11.84 12.15 13.59
10.74 9.64 9.21
0.113 0.076 0.183
1.29 1.16 1.09
10.89 10.19 9.96
14 15 16 17
1.6
10 12 15 20
0.1268 0.1206 0.1143 0.1199
10.78 11.13 11.47 12.96
10.94 10.06 9.24 8.94
0.082 0.069 0.063 0.185
1.30 1.21 1.11 1.05
10.27 10.03 9.71 9.55
18 19 20 21 22 23
1.8
8 10 12 15 20 25
0.1443 0.1318 0.1236 0.1173 0.1042 0.1023
8.86 9.69 10.09 10.39 11.89 13.96
10.96 10.00 9.47 8.82 8.44 9.18
0.075 0.078 0.088 0.083 0.176 0.126
1.28 1.21 1.14 1.08 1.00 1.11
8.87 9.18 9.22 9.20 9.59 10.13
24 25 26 27 28 29
2.0
8 10 12 15 25 Straight tube
0.1515 0.1375 0.1284 0.1200 0.1106 0.1043
6.64 8.59 8.95 9.48 12.89 43.69
9.75 9.14 8.64 8.22 8.67 14.76
0.123 0.061 0.049 0.059 0.231 0.222
1.15 1.10 1.03 0.99 1.07 1.79
7.59 8.00 8.02 8.25 9.67 17.27
1.0
60
60
50
50 43.69
43.69
40 IPCF (kN)
IPCF (kN)
40 30 20 10
8.86
9.69
8
10
10.09
10.39
11.89
30 20 14.03
13.96
12.42
11.84
10
11.13
10.09
1.6
1.8
0
0 12 15 20 Wavelength (mm)
25 straight tube
(a)
Straight tube
1.2
1.4 1.5 Amplitude (mm)
(b)
Fig. 11. The tread of IPCF vs. wavelength and amplitude: (a) IPCF vs. wavelength (A¼ 1.8 mm); (b) IPCF vs. amplitude (W¼ 12 mm).
three groups of collapse modes for the SCTs with different diameters to length ratio D* for D¼ 40, 50, 60 and 70 mm. When W¼10 mm, A¼1.2 mm, t¼ 1.4 mm, the smallest diameter (D ¼40 mm and 50 mm) increases the tubal slimness, making it
less stable under axial compression. So the collapse follows an Euler global buckling mode. Nevertheless, the collapse remains the ring mode when D ¼60 mm and D¼ 70 mm. When W¼ 15 mm, A¼1.4 mm, t ¼1.4 mm, all attained the ring mode. Table 7 shows
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mode to the ring mode when increasing the wall thickness, but if the thickness increases further, the ring mode will be transferred to the mixed mode. The larger the thickness is, the more difficult the ring mode is taken a shape.
5. Multiobjective optimization for sinusoidal corrugation tubes 5.1. Design methodology
Fig. 12. The initial peak force in different amplitudes and wavelengths.
50 Straight tube W=15mm, A=1.6mm
Force (kN)
40 30
As an energy absorber, the sinusoidal corrugation tubes are expected to absorb as much crushing energy per unit mass as possible. Thus, the SEA should be chosen as one of objectives herein to be maximized. Furthermore, the IPCF is another critical criterion to protect occupants from severe injury or death. From vehicle safety perspective, a smaller IPCF would lead to a lower deceleration, thus increasing the safety; so IPCF should be minimized. Meanwhile, to keep the stability of energy absorption, the ULC needs to be limited to a certain value. The straight tube of ULC is 0.222, which is not an expected value, so this design optimization will attempt to achieve a value no higher than 0.2. According to the area of most black diamond dots in Fig. 10, such a multiobjective optimization problem is formulated mathematically as:
⎧ Min {−SEA ( W, A, t, r),IPCF ( W, A, t, r) } ⎪ ⎪ s . t . ULC (W, A, t, r) ≤ 0. 2 ⎪ ⎪ 10 ≤ W ≤ 20 ⎨ ⎪ 0. 8 ≤ A≤2. 0 ⎪ 1. 0 ≤ t≤2. 0 ⎪ ⎪ ⎩ 40 ≤ D≤70
(7)
To compare with sinusoidal corrugation tubes, the straight circular tubes with different diameters are also considered in design optimization as follows:
20
⎧ Min {−SEA ( t, r),IPCF ( t, r) } ⎪ ⎪ s . t .ULC (t, D) ≤ 0. 2 ⎨ ⎪ 1. 0 ≤ t≤2. 0 ⎪ ⎩ 40 ≤ D≤70
10 0 0
20
40 60 Displacement (mm)
(8)
80
Fig. 13. Comparison of force vs displacement curves between straight tube and SCT with W¼ 15 mm and A ¼1.6 mm.
the detailed development of the ring modes when displacement is 90 mm. Obviously, the larger the diameter is, the more easily the ring mode could be generated. Also it can be shown that increasing amplitude A and decreasing wavelength W will also promote formation of a ring mode. In order to analyze the effect of thickness on the collapse mode, three groups of representative collapse modes of SCTs were considered with different wall thicknesses, as shown in Table 8. First, with increase in the wall thickness, the left group (W¼ 15 mm, A¼ 1.0 mm, D ¼60 mm as Tabel 1 column (a)) shows transformation from the mixed mode to ring mode, and then from the ring mode to mixed mode. Second all the middle group (W ¼15 mm, A¼ 1.4 mm, D¼ 60 mm as Tabel 1 column (b)) generates the ring mode. Table 9 exhibits the deformation details with different thicknesses when displacement reached 90 mm. It was found that the ring mode may not be easy to form when increasing the wall thickness. Third, the right group (W¼20 mm, A¼1.6 mm, D¼ 60 mm as Tabel 1 column (c)) shows a similar tendency to the left group somehow. The collapse mode transfers from the mixed
5.2. Surrogate model A major challenge to solve the abovementioned multiobjective optimization problems is significant difficulty of sensitivity analysis and considerably high computational cost. To address such issues, surrogated modeling techniques have proven fairly effective [32]. It has been shown that the radial basic function (RBF) method is applicable for highly nonlinear problems [21,22]. In this paper, therefore, the RBF model was chosen to approximate the crashworthiness responses. For establishing the surrogate models, a number of sample points were first generated in the design space. In order to construct an accurate RBF model with low computational cost, the optimal Latin hypercube design was utilized to generate 80 sample points in the design domain for sinusoidal corrugation tubes and straight tubes, respectively. These corresponding FE runs were performed to construct the RBF surrogates. To verify the fitting accuracy of the RBF surrogate models, extra 10 validation points were generated using the optimal Latin hypercube design and were then evaluated by relative error (RE) metric, given as:
RE =
f f ( x) − fk ( x) f f ( x)
(9)
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131
1.6
12
1.5
A=1.4 (mm) A=1.5 (mm)
11
1.4
A=1.6 (mm) A=1.8 (mm)
1.3 1.2 1.1
A=1.5 (mm) A=1.6 (mm) A=1.8 (mm) A=2.0 (mm)
SEA (kJ/kg)
EA (kJ)
A=2.0 (mm)
A=1.4 (mm)
10 9 8
1.0
7 8
10 12 14 16 18 20 22 24 26
8
10 12 14 16 18 20 22 24 26
Wavelength (mm)
Wavelength (mm)
(a)
(b)
Fig. 14. EA and SEA vs wavelength at different A: (a) EA vs A and W in ring mode collapse, (b) SEA vs A and W in ring mode collapse.
Table 6 Three groups collapse modes of sinusoidal corrugation tubes in different diameters.
where f f (x ) is the FE value, and fk (x ) is the corresponding approximate value obtained from the established RBF surrogate model. Fig. 15(a) and (b) show the surrogate accuracy of the RBF models for the sinusoidal corrugation and straight circular tubes, respectively. It can be seen that all the REs are less than 3%. Therefore, these RBF surrogate models were considered sufficiently accurate and were used to conduct the subsequent design optimization.
5.3. Optimization results To obtain the optimal results of the sinusoidal corrugation tube and straight circular tubs, the well-known non-domain sorting genetic algorithm II (NSGA-II) [16] was used herein, which has proven rather effective for solving the crashworthiness optimal problems [33,34]. Fig. 16 plots the Pareto fronts for the sinusoidal corrugation tube and straight circular tube, respectively.
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Table 7 The detailed collapse with ring modes at displacement of 90 mm (W ¼ 15 mm, A ¼ 1.4 mm, t¼ 1.4 mm).
Table 8 Three groups collapse modes of sinusoidal corrugation tubes in different thicknesses.
Obviously, these two objective SEA and IPCF are found to conflict with each other. Also the Pareto curve of optimal sinusoidal corrugation tube is overall lower than that of the optimal straight circular tube, indicating the crashworthiness benefits of SCT. These two Pareto frontiers provide designers with a reference to select a best compromised design. Fig. 16 can be divided into three areas as per SEA: namely ‘Area
one’, ‘Area two’ and ‘Area three’, respectively. If designers pay more attention to SEA, circular straight tubes have certain advantages than the sinusoidal corrugation tube as displayed in ‘Area one’, where higher SEA can be attained. If the designers focus on the lower IPCF for safety concerns, the sinusoidal tubes is superior to the straight counterpart as indicated in ‘Area three’. If the designers would like to compromise SEA with IPCF, the sinusoidal
S. Wu et al. / Thin-Walled Structures 105 (2016) 121–134
133
Table 9 The deformation details of different thicknesses when displacement reached 90 mm.
5
5 IPCF SEA ULC
4
4 3
RE(%)
3 RE(%)
IPCF SEA ULC
2
2 1
1
0
0 1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
Validation point number
Validation point number
(a)
(b)
9
10
Fig. 15. The results of accuracy assessment: (a) the RBF models for SCTs, (b) the RBF models for straight circular tubes.
50 45
IPCF (kN)
40
Area one
Area two
Table 10 Ideal optimums of the two single objective functions for the SCT columns and corresponding FE results.
SCT Circle straight tubes Area three
Single objective
Design variable
SEA kJ/kg Error
IPCF kN Error
35
A
30
Ideal Max RBF 0.80 10.00 1.80 SEA FEA 0.80 10.00 1.80
41.09 19.54 41.09 19.23
1.58%
17.63 17.80
0.96%
Ideal Min IPCF
40.26 40.26
3.79%
2.84 2.79
1.76%
W
t
D
25 20
RBF 1.74 FEA 1.74
10.02 1.15 10.02 1.15
8.62 8.96
15 10 5 0 -24
-22
-20
-18 -16 -14 -SEA (kJ/kg)
-12
-10
-8
Fig. 16. The Pareto fronts of the sinusoidal corrugation tube and straight circular tube.
corrugation tube can be still advantageous in ‘Area two’, where SCT has a lower IPCF for the same level of SEA. For the Pareto frontiers of the SCT and straight tubes, there are two optimal points which lie in the end of the Pareto curves over the design space. In ‘Area one’, the value of SEA ranges from 19.933 to 23.450 kJ/kg, while the corresponding IPCF value ranges from 30.745 to 44.939 kN, where SCT has no Pareto points. In ‘Area two’, there are two Pareto fronts curves for SCT and straight tube. Clearly, the red curve (straight tube) is higher than the black curve (SCT), indicating
that the straight tube is inferior to the SCT overall. Finally, in ‘Area three’, SEA ranges from 8.624 to 14.218 kJ/kg and IPCF ranges from 2.884 to 9.315 kN, only sinusoidal corrugation tube extended the Pareto to this area, which yields a substantially lower IPCF. In addition, Table 10 summarizes the comparison between the RBF optima and corresponding FE results. It can know that the optimal result have a fairly good accuracy, comparing RBF modeling values with the corresponding FE result. Hence, the optimal results can be considered as the best SCT columns for design of an energy absorber.
6. Conclusions In this paper, the crashing behaviors of SCTs with different wavelengths and amplitudes have been explored. A range of crashworthiness characteristics, such as collapse modes, initial peak force, mean crushing force, undulation of load-carrying capacity, energy absorption, specific energy absorption, has been analyzed. The study also investigated the influence of tubal
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diameter and wall thickness on collapse modes and crushing performance. Finally, the multiobjective design optimization was conducted for maximizing specific energy absorption (SEA), minimizing peak crushing force (IPCF) and limiting the undulation of load-carrying capacity (ULC) for both SCT and straight tubes. Within the limitation of the study, the following conclusions can be drawn: (1) The crashworthiness of sinusoidal corrugation tube is determined by the wavelength and amplitude, and their effects on collapse modes were studied. (2) Around 47.01–84.80% reduction in the initial peak force was observed in SCTs over the straight counterpart. There is a tendency that the initial peak force decreases when the wavelength decreases and amplitude increases. (3) The sinusoidal corrugation tube exhibits advantages on the ULC with the maximum reduction of 77.90%, which indicates that the sinusoidal corrugation tube has a much better uniformity of the load-displacement curve. (4) For sinusoidal corrugation tube, the increase in mean diameter can promote formation of the ring collapse mode. Within the range studied, increase in the wall thickness of SCTs will transfer collapse from a mixed mode to a ring mode in a lower thickness range, but can further transfer back to the mixed mode in a higher thickness range. (5) The multiobjective optimization was conducted within a limit of the ULC, through which the Pareto frontiers can be divided into three areas: namely ‘Area one’, ‘Area two’ and ‘Area three’ respectively. ‘Area one’ attains higher SEA with straight tube only; ‘Area two’ shows the advantage of sinusoidal corrugation tubes over a straight tube on a balanced IPCF and SEA; and ‘Area three’ achieves a lower IPCF in SCT only. These different areas provide designers with a good reference for design selection.
Acknowledgments This work is supported by National Natural Science Foundation of China (61232014) and (51575172), the Hunan Provincial Science Foundation of China (13JJ4036), the Doctoral Fund of Ministry of Education of China (20120161120005).
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Full length article
Crashworthiness analysis and optimization of sinusoidal corrugation tube Shengyin Wu a, Guangyao Li a, Guangyong Sun a,n, Xin Wu b, Qing Li c a
State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha 410082, China Mechanical Engineering, Wayne State University, Detroit, MI 48202, USA c School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia b
art ic l e i nf o
a b s t r a c t
Article history: Received 23 November 2015 Received in revised form 22 March 2016 Accepted 28 March 2016 Available online 14 April 2016
Thin-walled structures are widely used as energy absorbing devices for their proven advantages on lightweight and crashworthiness. However, conventional thin-walled structures often exhibit unstable collapse modes and excessive initial peak crushing force (IPCF) followed by undesirable fluctuation in force-displacement curves under impact loading. This paper introduces a novel tubal configuration, namely sinusoidal corrugation tube (SCT), to control the collapse mode, and minimize the IPCF and fluctuations. Through validating the finite element (FE) models established, the effects of wavelength, amplitude, thickness and diameter of SCTs on collapse mode and energy absorption were investigated. The results showed that SCTs can make the deformation mode more controllable and predictable, which can be transformed from a mixed mode to a ring mode by simply changing the wavelength and amplitude. Compared with the traditional straight circular tube, the IPCF is reduced appreciably. Furthermore, SCTs have lower fluctuation in the force–displacement curves than traditional straight circular tubes. Finally, a multiobjective optimization is conducted to obtain the optimized SCT configuration for maximizing specific energy absorption (SEA), minimizing IPCF under the constraint of fluctuation criterion. The optimal SCTs are of even more superior crashworthiness and great potential as an energy absorber. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Sinusoidal corrugation tube Collapse mode Energy absorption Multiobjective optimization
1. Introduction Thin-walled tubes are frequently used as energy absorbing devices in the vehicular frontal structures, attributable to their excellent energy absorption efficiency, lightweight and low cost for manufacturing. To better understand energy absorption of thin-walled tubes and fully explore their potential as energy absorbers, a large body of studies has been reported by utilizing analytical, experimental and numerical methods for various crosssectional configurations, such as circular, square, hexagonal, octagon, top-hat, etc. For example, Alexander [1] first derived an approximate formula for predicting the mean crushing force of circular tubes under axial loading; and then Abramowicz et al. [2] developed a more realistic theoretical model by introducing an effective plastic hinge length. Huh et al. [3] analyzed the collapse modes of square tubes for sequential crushing loads as well as deformation modes, numerically and experimentally. Tarlochan et al. [4] explored the crashworthiness of thin-walled tubes with n
Corresponding author. E-mail address: [email protected] (G. Sun).
http://dx.doi.org/10.1016/j.tws.2016.03.029 0263-8231/& 2016 Elsevier Ltd. All rights reserved.
several different cross-sectional shapes (e.g. circular, rectangle, square, hexagonal, octagonal and elliptical cross-sections) under dynamic loading. The previous studies demonstrated that crosssectional configurations have critical influences in energy absorption (EA) capacity of thin-walled columns. Further studies showed that circular tubes can more easily generate a stable progressive folding mode [5–8], which is often considered as an effectie way of energy absorption. For this reason, the study on crashworthiness of circular thin-walled tubes had drawn considerable attentions. Generally speaking, the collapse mode of circular tubes depends on both material properties and geometric dimensions, typically leading to three different modes: symmetric (concertina or ring) mode, non-symmetric (diamond or mixed) mode and Euler mode (global buckling), respectively [8]. Although circular tubes have great advantages on energy absorption and weight reduction, their collapse modes are easily affected by geometric dimension and manufacturing imperfections. Moreover, they are often associated with an excessive peak crushing force. In many cases, the collapse process of an ideal energy absorber is expected to be stable to dissipate the maximum kinetic energy. Since the high peak crushing force can lead to a
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large deceleration, which increases probability of damage and passenger’s injury, severe initial peak force should be avoided. To reduce the initial peak force and enhance the stability of the crushing process, there have been many exploratory studies. For example, Arameh et al. [9] investigated the effect of corrugations on the crushing behaviors such as energy absorption and failure mode of circular aluminum tubes experimentally. The results showed that the corrugated tubes can make the collapse mode more predictable and controllable. From this point, corrugated tube was considered a good candidate for a controllable energy absorption device. Singace et al. [10] also studied the energy absorption characteristics of corrugated tubes experimentally, in which the corrugations are introduced to force the plastic deformation to occur at predetermined intervals along the tube generator for improving the uniformity of the load-displacement behavior under axial loading. Chen and Ozaki [11] used the finite element method (FEM) to investigate the nonlinear elastoplastic behavior of circular tubes with corrugated profiles. Their results showed that the corrugations are one of the dominant factors to influence the deformation mode and fluctuations in the load-displacement curves of circular tubes. Song et al. [12] introduced the origami patterns to thin-walled tubes to minimize the initial peak force and subsequent fluctuations. Their numerical results showed that the patterned tubes exhibit a lower initial peak force and more uniform crushing load. To facilitate the axisymmetric collapse mode in frusta tube, Rezvani and Damghani [13] presented a new design by cutting the circumferential grooves from the outer and inner surfaces of the frusta at different positions in the tube. A series of experimental tests showed that the grooves can stabilize the collapse mode and control energy absorption of the frusta tubes. Daneshi and Hosseinipour [14,15] investigated the buckling mode and energy absorption capacity of grooved tubes and further presented a theoretical formulation for predicting the energy absorption and mean crushing load of these tubes under axial loading. The results showed that the load-displacement curve and energy absorption could be controlled by introducing the grooves with different distances. It was found that these grooves can stabilize the deformation behaviors. In summary, these corrugated tubes have a great potential to reduce the initial peak force and improve the deformation stability, which has been attracting more and more attention. To the authors’ best knowledge, however, there is rather limited research to systematically quantify the influence of the wavelength, amplitude, thickness and diameter of SCTs on the collapse mode, initial peak force and energy absorption. To explore the energy absorption mechanism of SCTs and provide some insights into the design of the SCTs, this study aimed to investigate the parametric effects of wavelength, amplitude, thickness and basic diameter of SCTs on the deformation mechanisms, distribution area of different collapse modes, initial peak crushing force (IPCF), mean crushing force, undulation of load-carrying capacity (ULC), absorbed energy (EA) and specific energy absorption (SEA). It will show that SCT parameters can significantly affect the uniformity of the load-displacement curve, as well as the controllability and predictability of collapse modes. To obtain the optimal configuration of SCTs, the Non-domain Sorting Genetic Algorithm II (NSGA-II) [16] that is integrated with surrogate models was utilized to seek the optimal wavelength, amplitude, thickness and basic diameter of SCTs. Of some common surrogates, such as radial basis function (RBF) [17], response surface method (RSM) [18], Kriging (KRG) [19] and support vector machine (SVM) [20], the radial basis function (RBF) surrogate models have proven effective for such highly nonlinear problems as crashworthiness [21,22]. Thus, the RBF surrogate modeling technique is adopted in this study. In the optimization process, the IPCF is minimized and SEA is maximized simultaneously, the load-
carrying capacity is constrained within a level of 0.2. The results demonstrated that the SCTs can provide a better crashworthiness than corresponding straight circular tubes.
2. Crashworthiness criteria and material properties 2.1. Crashworthiness criteria Crashing criteria are essential to evaluate the crashworthiness performance. In literature, different criteria have been adopted in different structures [23,24]. In this paper, the following criteria are utilized to evaluate the crashworthiness characteristics of SCT structures.
Initial Peak Crushing Force (IPCF)
Typically, thin-walled tubes hit a peak force in an early stage under crushing. The force at this point is known as initial peak force. As a device for energy adsorption in real life, it may cause severe injury or damage when the initial peak force is overly high [25]. For this reason, Initial Peak Crushing Force (IPCF) should be minimized or constrained to a safe level. Energy Absorption (EA) Energy absorption has been widely used to evaluate the crashing characteristic of structure, which indicates the amount of absorbed energy during crashing process; it can be calculated mathematically as follows:
∫0
EA (d) =
F (x) dx
(1)
where F (x ) is the axial impact force as a function of displacement x during crashing and d denotes the effective deformation distance, which was set to be 110 mm over the initial tube length of 150 mm (so that the final length of crushed tube is 40 mm) in this study. Specific Energy Absorption (SEA) To compare the energy absorbed capability of different materials and structures, the specific energy absorption (SEA) is often used to evaluate the capability of energy absorption by taking into account the mass factor, defined by:
SEA ( d)=
d
EA (d) m
(2)
where m is the mass of energy absorber. As a potential device for energy absorption, the larger the SEA, the higher the mass efficiency of energy absorber [26]. Mean Crushing Force ( Pm ) Mean crush force is the average value of the compressive force of the energy absorber through total effective deformation, given as: d
Pm =
∫ F (x) dx EA (d) = 0 d d
(3)
Undulation of Load-carrying Capacity (ULC) In order to estimate the stability of energy absorption mathematically, Xiang et al. [27] introduced the undulation of load-carrying capacity (ULC) to evaluate the stability of load, as: d
ULC =
∫0 | F (x) − Pm | dx d
∫0 F (x) dx
(4)
which indicates that when the deviation between peak and trough is too great, ULC is high as shown in Fig. 1; in which case the forcedisplacement curves is considered less effective for energy
S. Wu et al. / Thin-Walled Structures 105 (2016) 121–134
as the amplitude A approaches to zero, as shown in Fig. 3(b). Therefore, the straight tube can be regarded as a special case of SCTs.
50
2.3. Material properties
Force (kN)
40
The tube material is aluminum alloy 6063-T6, whose chemical composition (in weight percent) is Si-0.20–0.60, Fe-0.35, Cu-0.10, Mn-0.10, Mg-0.45–0.90, Cr-0.10, Zn-0.10, Ti-0.10 and the remaining major composition is Al. The stress-strain curve of the material was obtained by using the tensile tests with specimen dimensions as specified in ASTM standard B557M. The tensile experiment was performed in the MTS 5457 material testing machine with computer control and data acquisition system under a constant speed of 2 mm/ min. The specimen geometry with the unit of mm and the corresponding stress-strain curve of Al6063-T6 are shown in Fig. 4, and the obtained material properties are summarized in Table 1. The real strain rate in crashing scenarios would be two orders of magnitude higher than this testing condition, and based on the reported strain rate sensitivity of aluminum alloy [28], the flow stress is slightly higher; thus, the FE models neglected the rate-dependency. To model the tubes made of such an aluminum alloy, the piece-wise linear plasticity material (material model 24 in LS-DYNA) was utilized.
30
Peak
20
Mean crushing force
10
Trough 0 0
20
40
60
80
Displacement (mm) Fig. 1. Typical force-displacement curve.
absorption. On contrary, if ULC is small, the force-displacement curves become a rectangular shape and the energy-displacement curve is approaching to a straight line approximatively, indicating better energy absorption efficiency. 2.2. Geometric configuration In this study, the configuration of a sinusoidal corrugation tube (SCT) is shown in Fig. 2. The following function was utilized to describe the corrugation profile of the tube walls:
y=
123
⎛ 2π ⎞ D + A sin ⎜ x⎟ ⎝W ⎠ 2
(5)
where A is the amplitude, W is the wavelength (the vertical distance), D is the mean diameter. This structure is used for energy absorption in the front side rail of a passenger car, and the length of energy absorbing box is often 100–200 mm, so this paper chose L ¼150 mm as the total length of this tube. When the amplitude keeps a constant value A¼ 1.0 mm, the profile will tend to be a straight line with increasing wavelength W, as shown in Fig. 3(a). Similarly, the profile also becomes straight line
3. Finite element modeling and validation 3.1. Finite element model A representative 3D experimental schematic diagram for SCT is displayed in Fig. 5a. The tube specimen was placed in between two flat steel plates in the testing machine and it was compressed at a constant speed (1000 mm/s). During the test, the top plate of the machine moved downward to compress the specimen vertically by the displacement control. The real-time loading and compressive displacement were acquired automatically. When the compressive displacement reached 110 mm, the specimen was unloaded and the experiment stopped. The explicit finite element code LS-DYNA was used to simulate the experimental process. The tube wall was meshed using the 4-node quadrilateral Belytschko-Lin-Tsay shell elements with reduced integration. Five integration points were used across the thickness to capture the local bending behavior, and one integration point was used in the plane of the element. The stiffness-type hourglass control was employed to eliminate spurious zero energy
Fig. 2. The profile of sinusoidal corrugation.
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W=6
W=8
W=12
W=15
W=20
W=30
W=∞
A=0
A=0.2
A=0.4
A=0.8
A=1.2
A=1.5
A=2.0
Fig. 3. The profiles of SCTs with various wavelengths W or amplitudes A at W ¼ 1 and/or A ¼ 0 the profile reduces to a straight tube: (a) A ¼ 1 mm; (b) W ¼10 mm.
Fig. 4. Axial tensile test set-up and specimen: (a) MTS 5457material testing machine; (b) geometry dimension of specimen;(c) a specimen after failure; (d) engineering stress versus engineering strain curve.
Table 1 Material properties of the aluminum alloy 6063-T6. Density
Young's modulus
2700 kg/m^3 65,000 MPa
Poisson’s ratio
Yield stress Ultimate tensile stress
0.33
145 MPa
161 MPa
models commonly arising when such reduced integration elements are used. As the steel plate did not deform much during compression process, thus it was treated as a rigid body using the 8 node brick elements. The interface between the specimen and the two steel plates was modeled using a ‘node-to-surface’ contact. The ‘automatic single surface’ contact was applied to the column wall to avoid interpenetration of folding generated during axial collapse. The dynamic and static coefficients of friction of
0.2 and 0.3 were respectively used as suggested in [29]. Several simulations were first run to conduct a convergence test and finally the element size of 0.8 mm 0.8 mm was determined for its acceptable accuracy and computational cost. Following the above description, the final FE model is shown in Fig. 5b. To investigate the effect of the wavelength and amplitude of SCTs on the collapse mode and energy absorption characteristics, a series of SCTs with different wavelengths and amplitudes were considered as summarized in Table 2, in which the amplitude ranges from 0.1 to 2.0 mm, and the wavelength ranges from 4 to 30 mm, while the tube wall thickness t¼1.4 mm and length L¼150 mm remain unchanged. 3.2. Validation of the FE model From Eq. (5), it can be easily found that the SCTs will become a simple straight circular tube when amplitude A¼0 mm, so the
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Fig. 5. Model of SCT: (a) A representative 3D experimental schematic diagram for SCT; (b) the final FE model for SCT. Table 2 Wavelengths and amplitudes of various models.
2st
1st
4st
3st
5st
7st
6 st
8st
9 st
11st
10 st
12st
13st
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
A
W
0.1
4 5 6 8 10 12 15 20 25 30
0.2
4 5 6 8 10 12 15 20 25 30
0.3
4 5 6 8 10 12 15 20 25 30
0.4
4 5 6 8 10 12 15 20 25 30
0.5
4 5 6 8 10 12 15 20 25 30
0.8
4 5 6 8 10 12 15 20 25 30
1.0
4 5 6 8 10 12 15 20 25 30
1.2
/ / 6 8 10 12 15 20 25 30
1.4
/ / 6 8 10 12 15 20 25 30
1.5
/ / 6 8 10 12 15 20 25 30
1.6
/ / 6 8 10 12 15 20 25 30
1.8
/ / 6 8 10 12 15 20 25 30
2.0
/ / 6 8 10 12 15 20 25 30
Fig. 6. Quasi-static axial compressive test for the straight column.
straight tube can be regarded as a special SCT. The FE model was thus validated using a simple straight circular tube with the diameter of 60 mm, height of 150 mm and thickness of 1.4 mm under
axial compression at a constant velocity of 2 mm/min. To improve the computing efficiency, the tube is compressed by a rigid plane with the velocity of 1000 mm/s in the FE model for quasi-static
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compare with the numerical results. From the literature [31], the mean crushing force of circular tubes can be expressed as follows:
⎛ D ⎞0.32 Fm = 72. 3⎜ ⎟ ⎝t⎠ MP
(6)
where Fm is the mean crushing force which can be calculated by Eq. (3). D is the diameter of circular tube, t is the wall thickness, MP 1 is the full plastic moment, calculated by MP = 4 σ0 t 2 (where σ0 is a characteristic stress of material). The comparison of the mean crushing force versus displacement between FE simulation and the analytical solution is plotted in Fig. 8. Obviously, the numerical result well agrees with the theoretical result. Therefore, the FE model is of an acceptable accuracy and can be used to analyze the crashing characteristics of sinusoidal corrugation tubes under quasi-static axial loading.
Fig. 7. Comparison of force-displacement curves in the experiment and FE simulation.
Fig. 8. Mean crushing force versus displacement curves.
loading [30]. This quasi-static axial crushing test of the tube was conducted on an INSTRON 5984 (150kN) testing machine, as shown in Fig. 6. Fig. 7 provides a comparison of force-displacement curves obtained from the experiment test and FE modeling. Obviously the simulation was well correlated to the experimental test. Furthermore, the deformed tube shapes or collapse modes in the experiment and simulation were compared at two displacement points as presented in Fig. 7. Clearly, the collapse mode from the experiment agrees fairly well with that from the FE simulation. For the energy absorption, the experimental and numerical results are 1406 J and 1387 J, respectively, which are also in reasonable agreement. In order to further validate the accuracy of the FE model, the theoretical solutions available in literature were also utilized to
4. Crashworthiness analysis 4.1. Collapse modes under axial quasi static loading According to Table 2, a total of 118 design cases with different A and W were simulated. As shown in Fig. 9, the collapse of SCTs has mainly displayed in two modes: namely ring mode and mixed mode (i.e. more than one mode developed during the entire crashing process, which will be discussed later in more detail), and these corresponding 118 collapse modes are plotted collectively in Fig. 10a. When amplitude A r0.4 mm, the collapses are shown in a mixed mode regardless of wavelengths. However, the collapse mode begins to transform to a ring mode in general, when amplitude AZ0.5 mm. As shown in Fig. 10a, the black dots indicate the area of ring mode, and clearly the area increases with the amplitude. On the other hand, When wavelength 4.0 r Wr6.0 mm and 25r Wr30 mm, the mixed mode dominates. Besides, the collapse mode transfers from a mixed mode to a ring mode gradually. For the further study, Fig. 10b plots the experimental force-displacement curve, the nominal average wavelength is 15.75 mm, which can be obtained by calculating the average of the three data. W¼ 15 is close to the nominal average wavelength, which means that W¼ 15 can account for easy formation of the ring mode. Table 3 shows the transition process of collapse modes of SCTs with the amplitude of A ¼1.8 mm and different wavelengths ranging from 0 to 30 mm. From which, it is easily found that the collapse follows a mixed mode when wavelength is quite small, such as W ¼0 or W¼6 mm. When 8 rWr25 mm, the deformation was transferred from the mixed mode to the stable ring mode. However, collapse mode was transformed to the mixed mode again when W is long enough, such as W¼ 30 mm. Table 4 compares the collapse modes of straight tube with that of the SCT (A ¼1.6 mm and W ¼20 mm) at different displacements (0, 30, 60, 90, and 120 mm). Obviously the collapse of straight tube follows a ring mode in the first plastic folding, but it transforms to the diamond mode in the subsequent progressive deformation,
Fig. 9. The two representative collapse modes of SCTs: (a) Ring mode (A ¼0.8 mm, W ¼15 mm); (b) Mixed mode (A ¼ 0.8 mm, W ¼25 mm).
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50 35
Straight circular tube
Ring mode Mixed mode
40
25
Force (kN)
Wavelength (mm)
30
20 15 10
30 15.70
15.43
16.12
20 10
5
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0
20
Amplitude (mm)
40
60
80
Displacement (mm)
Fig. 10. Collapse modes and force-displacement curve: (a)The distribution of collapse modes of 118 specimens; (b) The experimental force-displacement curve.
Table 3 The collapse modes of different wavelengths at the same amplitude of 1.8 mm.
whilst the sinusoidal corrugation tube always keeps the stable ring mode until the end of deformation. 4.2. Influence on the initial peak force The initial peak forces of SCTs with the ring modes are listed in Table 5. It shows that the initial peak force of SCTs can be reduced significantly by adopting the tubal corrugations compared with conventional straight tube. For example, IPCF of the circular straight tube is 43.69 kN, whereas IPCF (W ¼15 mm, A¼0.5 mm) reduces to 23.15 kN. Overall, around 47.01–84.80% reduction of IPCF can be achieved in the corrugating tubes.
Furthermore, Fig. 11 (a) shows the trend of IPCF with the amplitude of A ¼1.8 mm vs. different wavelengths, where the force gradually increases with the wavelength and the maximal initial peak force reaches 13.96 kN, which is however far lower than that of straight circular tube (43.69 kN). Fig. 11 (b) shows the trend of IPCF of sinusoidal corrugation tube with the wavelength of W¼12 mm vs. amplitude. Clearly, the IPCF increases with the reduction of amplitude, but it is still much smaller than that of the straight counterpart. Overall, by introducing sinusoidal corrugations to a straight tube, the crashing peak force can be reduced significantly. Specifically, the longer the wavelength and the smaller the amplitude,
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Table 4 Comparison of collapse modes between straight tube and SCT (W¼ 20 mm, A ¼1.6 mm) at different displacements.
the higher the initial peak force; and this tendency can be more intuitively observed from Fig. 12. 4.3. Influence on undulation of load-carrying capacity The undulation of load-carrying capacity (ULC) can be obtained by Eq. (4), which evaluates the smoothness of energy absorption. If the force variation and force range between peak and valley in collapse process are too large, the energy-displacement curves will be less smooth, which should be avoided in an energy absorption device. Table 5 shows the ULC of SCTs with the ring modes, the ULC of straight tube is 0.222, which is higher than most of other ring modes except at A¼ 0.8, W ¼20 mm and A¼ 2.0, W¼ 25 mm. The minimum ULC value is 0.049, which corresponds to a 77.9% reduction from the straight tube. From a unilateral consideration, the smaller the ULC, the smoother the energy absorption. Fig. 13 compares the force vs displacement curves between the straight tube and SCT (W ¼15 mm and A ¼1.6 mm). It is observed that the straight tube has a larger deviation between the peak and trough forces. However, the red curve in Fig. 13 displays a relatively smoother loading process, so SCTs are of certain advantages on reducing the undulation of load-carrying capacity and keeping the steady state of energy absorption process.
SCTs must sacrifice some energy absorption capacity. Fig. 14(a) displays the variation in the EA of SCTs with different amplitudes and wavelengths. It can be easily found that the EA decreases when wavelength starts increasing. From Table 3, when the wavelength is quite small, the collapse mode has more ringmoded plastic deformation, which is the reason for a higher EA. However, when wavelength increases, SCT tends to be more straight, which enhances the resistant capacity of structure, so the total energy absorption increases. For example, when the amplitude is 1.8 mm, EA of SCTs decreases from 1.28 to 1.00 kJ, but increases to 1.11 kJ in the last. On the other hand, when the wavelength is constant, the smaller the amplitude, the higher the EA. This phenomenon can be observed in a range of wavelengths in Fig. 14(a), indicating that the larger the amplitude, the lower the ability of loading resistance. Fig. 14 (b) plots the SEA vs. wavelength and amplitude. Obviously, SEA has a fairly similar tendency to EA. From Table 5, the structural mass decrease with the wavelength. SEA was calculated by Eq. (2), the denominator ‘m’ has the same decreasing tendency, thus SEA is similar tendency to EA. In this regard, the SEA of SCTs does not exhibit advantage. Therefore, a tradeoff needs to be made among the collapse mode, initial peak force, ULC, EA and SEA. 4.5. Influence on diameter and thickness on collapse mode
4.4. Absorbed energy and specific energy absorption The energy absorption (EA) and specific energy absorption (SEA) for different amplitudes and wavelengths in the ring mode are listed in Table 5. The EA and SEA of straight tube are 1.79 kJ and 17.27 kJ/kg, respectively. The highest EA and SEA of SCT are 1.71 KJ and 16.29 KJ/Kg. It shows that straight tube is more advantageous in terms of EA and SEA. This implies that to keep a stable ring mode, reduce the initial peak force and achieve a small ULC value,
The aforementioned study has focused on the influence of wavelength and amplitude of SCTs with constant diameter D, thickness t and length L on crashworthiness. However, for conventional straight circular tube, these parameters are also important design variables for energy absorber; so their influence on collapse mode and energy absorption characteristics should be also investigated. By introducing the thickness to diameter ratio t* ¼t / D and diameter to length ratio D* ¼D/L, Table 6 exhibits
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Table 5 Crashworthiness performance criteria of straight tube and the SCTS with ring modes. Number
A (mm)
W (mm)
Mass (Kg)
IPCF (kN)
Pm (kN)
ULC
EA (kJ)
SEA (kJ/kg)
1
0.5
15
0.1051
23.15
14.08
0.178
1.71
16.26
2 3 4
0.8
15 20 15
0.1066 0.1059 0.1085
18.84 20.05 16.53
12.30 11.38 11.57
0.168 0.227 0.086
1.50 1.33 1.40
14.07.0 12.55 12.90
5 6
1.2
12 15
0.1136 0.1103
14.04 14.45
12.00 10.52
0.109 0.077
1.43 1.27
12.58 11.51
7 8 9 10
1.4
8 10 12 15
0.1304 0.12150 0.1167 0.1125
9.22 12.14 12.42 12.81
13.09 12.24 11.08 9.80
0.109 0.102 0.094 0.077
1.53 1.48 1.33 1.19
11.77 12.18 11.39 10.57
11 12 13
1.5
12 15 20
0.1184 0.1138 0.1094
11.84 12.15 13.59
10.74 9.64 9.21
0.113 0.076 0.183
1.29 1.16 1.09
10.89 10.19 9.96
14 15 16 17
1.6
10 12 15 20
0.1268 0.1206 0.1143 0.1199
10.78 11.13 11.47 12.96
10.94 10.06 9.24 8.94
0.082 0.069 0.063 0.185
1.30 1.21 1.11 1.05
10.27 10.03 9.71 9.55
18 19 20 21 22 23
1.8
8 10 12 15 20 25
0.1443 0.1318 0.1236 0.1173 0.1042 0.1023
8.86 9.69 10.09 10.39 11.89 13.96
10.96 10.00 9.47 8.82 8.44 9.18
0.075 0.078 0.088 0.083 0.176 0.126
1.28 1.21 1.14 1.08 1.00 1.11
8.87 9.18 9.22 9.20 9.59 10.13
24 25 26 27 28 29
2.0
8 10 12 15 25 Straight tube
0.1515 0.1375 0.1284 0.1200 0.1106 0.1043
6.64 8.59 8.95 9.48 12.89 43.69
9.75 9.14 8.64 8.22 8.67 14.76
0.123 0.061 0.049 0.059 0.231 0.222
1.15 1.10 1.03 0.99 1.07 1.79
7.59 8.00 8.02 8.25 9.67 17.27
1.0
60
60
50
50 43.69
43.69
40 IPCF (kN)
IPCF (kN)
40 30 20 10
8.86
9.69
8
10
10.09
10.39
11.89
30 20 14.03
13.96
12.42
11.84
10
11.13
10.09
1.6
1.8
0
0 12 15 20 Wavelength (mm)
25 straight tube
(a)
Straight tube
1.2
1.4 1.5 Amplitude (mm)
(b)
Fig. 11. The tread of IPCF vs. wavelength and amplitude: (a) IPCF vs. wavelength (A¼ 1.8 mm); (b) IPCF vs. amplitude (W¼ 12 mm).
three groups of collapse modes for the SCTs with different diameters to length ratio D* for D¼ 40, 50, 60 and 70 mm. When W¼10 mm, A¼1.2 mm, t¼ 1.4 mm, the smallest diameter (D ¼40 mm and 50 mm) increases the tubal slimness, making it
less stable under axial compression. So the collapse follows an Euler global buckling mode. Nevertheless, the collapse remains the ring mode when D ¼60 mm and D¼ 70 mm. When W¼ 15 mm, A¼1.4 mm, t ¼1.4 mm, all attained the ring mode. Table 7 shows
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mode to the ring mode when increasing the wall thickness, but if the thickness increases further, the ring mode will be transferred to the mixed mode. The larger the thickness is, the more difficult the ring mode is taken a shape.
5. Multiobjective optimization for sinusoidal corrugation tubes 5.1. Design methodology
Fig. 12. The initial peak force in different amplitudes and wavelengths.
50 Straight tube W=15mm, A=1.6mm
Force (kN)
40 30
As an energy absorber, the sinusoidal corrugation tubes are expected to absorb as much crushing energy per unit mass as possible. Thus, the SEA should be chosen as one of objectives herein to be maximized. Furthermore, the IPCF is another critical criterion to protect occupants from severe injury or death. From vehicle safety perspective, a smaller IPCF would lead to a lower deceleration, thus increasing the safety; so IPCF should be minimized. Meanwhile, to keep the stability of energy absorption, the ULC needs to be limited to a certain value. The straight tube of ULC is 0.222, which is not an expected value, so this design optimization will attempt to achieve a value no higher than 0.2. According to the area of most black diamond dots in Fig. 10, such a multiobjective optimization problem is formulated mathematically as:
⎧ Min {−SEA ( W, A, t, r),IPCF ( W, A, t, r) } ⎪ ⎪ s . t . ULC (W, A, t, r) ≤ 0. 2 ⎪ ⎪ 10 ≤ W ≤ 20 ⎨ ⎪ 0. 8 ≤ A≤2. 0 ⎪ 1. 0 ≤ t≤2. 0 ⎪ ⎪ ⎩ 40 ≤ D≤70
(7)
To compare with sinusoidal corrugation tubes, the straight circular tubes with different diameters are also considered in design optimization as follows:
20
⎧ Min {−SEA ( t, r),IPCF ( t, r) } ⎪ ⎪ s . t .ULC (t, D) ≤ 0. 2 ⎨ ⎪ 1. 0 ≤ t≤2. 0 ⎪ ⎩ 40 ≤ D≤70
10 0 0
20
40 60 Displacement (mm)
(8)
80
Fig. 13. Comparison of force vs displacement curves between straight tube and SCT with W¼ 15 mm and A ¼1.6 mm.
the detailed development of the ring modes when displacement is 90 mm. Obviously, the larger the diameter is, the more easily the ring mode could be generated. Also it can be shown that increasing amplitude A and decreasing wavelength W will also promote formation of a ring mode. In order to analyze the effect of thickness on the collapse mode, three groups of representative collapse modes of SCTs were considered with different wall thicknesses, as shown in Table 8. First, with increase in the wall thickness, the left group (W¼ 15 mm, A¼ 1.0 mm, D ¼60 mm as Tabel 1 column (a)) shows transformation from the mixed mode to ring mode, and then from the ring mode to mixed mode. Second all the middle group (W ¼15 mm, A¼ 1.4 mm, D¼ 60 mm as Tabel 1 column (b)) generates the ring mode. Table 9 exhibits the deformation details with different thicknesses when displacement reached 90 mm. It was found that the ring mode may not be easy to form when increasing the wall thickness. Third, the right group (W¼20 mm, A¼1.6 mm, D¼ 60 mm as Tabel 1 column (c)) shows a similar tendency to the left group somehow. The collapse mode transfers from the mixed
5.2. Surrogate model A major challenge to solve the abovementioned multiobjective optimization problems is significant difficulty of sensitivity analysis and considerably high computational cost. To address such issues, surrogated modeling techniques have proven fairly effective [32]. It has been shown that the radial basic function (RBF) method is applicable for highly nonlinear problems [21,22]. In this paper, therefore, the RBF model was chosen to approximate the crashworthiness responses. For establishing the surrogate models, a number of sample points were first generated in the design space. In order to construct an accurate RBF model with low computational cost, the optimal Latin hypercube design was utilized to generate 80 sample points in the design domain for sinusoidal corrugation tubes and straight tubes, respectively. These corresponding FE runs were performed to construct the RBF surrogates. To verify the fitting accuracy of the RBF surrogate models, extra 10 validation points were generated using the optimal Latin hypercube design and were then evaluated by relative error (RE) metric, given as:
RE =
f f ( x) − fk ( x) f f ( x)
(9)
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1.6
12
1.5
A=1.4 (mm) A=1.5 (mm)
11
1.4
A=1.6 (mm) A=1.8 (mm)
1.3 1.2 1.1
A=1.5 (mm) A=1.6 (mm) A=1.8 (mm) A=2.0 (mm)
SEA (kJ/kg)
EA (kJ)
A=2.0 (mm)
A=1.4 (mm)
10 9 8
1.0
7 8
10 12 14 16 18 20 22 24 26
8
10 12 14 16 18 20 22 24 26
Wavelength (mm)
Wavelength (mm)
(a)
(b)
Fig. 14. EA and SEA vs wavelength at different A: (a) EA vs A and W in ring mode collapse, (b) SEA vs A and W in ring mode collapse.
Table 6 Three groups collapse modes of sinusoidal corrugation tubes in different diameters.
where f f (x ) is the FE value, and fk (x ) is the corresponding approximate value obtained from the established RBF surrogate model. Fig. 15(a) and (b) show the surrogate accuracy of the RBF models for the sinusoidal corrugation and straight circular tubes, respectively. It can be seen that all the REs are less than 3%. Therefore, these RBF surrogate models were considered sufficiently accurate and were used to conduct the subsequent design optimization.
5.3. Optimization results To obtain the optimal results of the sinusoidal corrugation tube and straight circular tubs, the well-known non-domain sorting genetic algorithm II (NSGA-II) [16] was used herein, which has proven rather effective for solving the crashworthiness optimal problems [33,34]. Fig. 16 plots the Pareto fronts for the sinusoidal corrugation tube and straight circular tube, respectively.
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Table 7 The detailed collapse with ring modes at displacement of 90 mm (W ¼ 15 mm, A ¼ 1.4 mm, t¼ 1.4 mm).
Table 8 Three groups collapse modes of sinusoidal corrugation tubes in different thicknesses.
Obviously, these two objective SEA and IPCF are found to conflict with each other. Also the Pareto curve of optimal sinusoidal corrugation tube is overall lower than that of the optimal straight circular tube, indicating the crashworthiness benefits of SCT. These two Pareto frontiers provide designers with a reference to select a best compromised design. Fig. 16 can be divided into three areas as per SEA: namely ‘Area
one’, ‘Area two’ and ‘Area three’, respectively. If designers pay more attention to SEA, circular straight tubes have certain advantages than the sinusoidal corrugation tube as displayed in ‘Area one’, where higher SEA can be attained. If the designers focus on the lower IPCF for safety concerns, the sinusoidal tubes is superior to the straight counterpart as indicated in ‘Area three’. If the designers would like to compromise SEA with IPCF, the sinusoidal
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Table 9 The deformation details of different thicknesses when displacement reached 90 mm.
5
5 IPCF SEA ULC
4
4 3
RE(%)
3 RE(%)
IPCF SEA ULC
2
2 1
1
0
0 1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
Validation point number
Validation point number
(a)
(b)
9
10
Fig. 15. The results of accuracy assessment: (a) the RBF models for SCTs, (b) the RBF models for straight circular tubes.
50 45
IPCF (kN)
40
Area one
Area two
Table 10 Ideal optimums of the two single objective functions for the SCT columns and corresponding FE results.
SCT Circle straight tubes Area three
Single objective
Design variable
SEA kJ/kg Error
IPCF kN Error
35
A
30
Ideal Max RBF 0.80 10.00 1.80 SEA FEA 0.80 10.00 1.80
41.09 19.54 41.09 19.23
1.58%
17.63 17.80
0.96%
Ideal Min IPCF
40.26 40.26
3.79%
2.84 2.79
1.76%
W
t
D
25 20
RBF 1.74 FEA 1.74
10.02 1.15 10.02 1.15
8.62 8.96
15 10 5 0 -24
-22
-20
-18 -16 -14 -SEA (kJ/kg)
-12
-10
-8
Fig. 16. The Pareto fronts of the sinusoidal corrugation tube and straight circular tube.
corrugation tube can be still advantageous in ‘Area two’, where SCT has a lower IPCF for the same level of SEA. For the Pareto frontiers of the SCT and straight tubes, there are two optimal points which lie in the end of the Pareto curves over the design space. In ‘Area one’, the value of SEA ranges from 19.933 to 23.450 kJ/kg, while the corresponding IPCF value ranges from 30.745 to 44.939 kN, where SCT has no Pareto points. In ‘Area two’, there are two Pareto fronts curves for SCT and straight tube. Clearly, the red curve (straight tube) is higher than the black curve (SCT), indicating
that the straight tube is inferior to the SCT overall. Finally, in ‘Area three’, SEA ranges from 8.624 to 14.218 kJ/kg and IPCF ranges from 2.884 to 9.315 kN, only sinusoidal corrugation tube extended the Pareto to this area, which yields a substantially lower IPCF. In addition, Table 10 summarizes the comparison between the RBF optima and corresponding FE results. It can know that the optimal result have a fairly good accuracy, comparing RBF modeling values with the corresponding FE result. Hence, the optimal results can be considered as the best SCT columns for design of an energy absorber.
6. Conclusions In this paper, the crashing behaviors of SCTs with different wavelengths and amplitudes have been explored. A range of crashworthiness characteristics, such as collapse modes, initial peak force, mean crushing force, undulation of load-carrying capacity, energy absorption, specific energy absorption, has been analyzed. The study also investigated the influence of tubal
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diameter and wall thickness on collapse modes and crushing performance. Finally, the multiobjective design optimization was conducted for maximizing specific energy absorption (SEA), minimizing peak crushing force (IPCF) and limiting the undulation of load-carrying capacity (ULC) for both SCT and straight tubes. Within the limitation of the study, the following conclusions can be drawn: (1) The crashworthiness of sinusoidal corrugation tube is determined by the wavelength and amplitude, and their effects on collapse modes were studied. (2) Around 47.01–84.80% reduction in the initial peak force was observed in SCTs over the straight counterpart. There is a tendency that the initial peak force decreases when the wavelength decreases and amplitude increases. (3) The sinusoidal corrugation tube exhibits advantages on the ULC with the maximum reduction of 77.90%, which indicates that the sinusoidal corrugation tube has a much better uniformity of the load-displacement curve. (4) For sinusoidal corrugation tube, the increase in mean diameter can promote formation of the ring collapse mode. Within the range studied, increase in the wall thickness of SCTs will transfer collapse from a mixed mode to a ring mode in a lower thickness range, but can further transfer back to the mixed mode in a higher thickness range. (5) The multiobjective optimization was conducted within a limit of the ULC, through which the Pareto frontiers can be divided into three areas: namely ‘Area one’, ‘Area two’ and ‘Area three’ respectively. ‘Area one’ attains higher SEA with straight tube only; ‘Area two’ shows the advantage of sinusoidal corrugation tubes over a straight tube on a balanced IPCF and SEA; and ‘Area three’ achieves a lower IPCF in SCT only. These different areas provide designers with a good reference for design selection.
Acknowledgments This work is supported by National Natural Science Foundation of China (61232014) and (51575172), the Hunan Provincial Science Foundation of China (13JJ4036), the Doctoral Fund of Ministry of Education of China (20120161120005).
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