Abstract
To mitigate shock forces in collision events, thinwalled members are used as energy absorber. In this article, crashworthiness of singlecell and multicell Sshaped members with various crosssections including triangular, square, hexagonal, decagon and circular were investigated under axial dynamic loading using finite element code LSDYNA. Furthermore, crashworthiness of the Srails with the same outer tubes and different inner ones was studied as well. The multicell members employed in this task were doublewalled tubes with several ribs connecting the inner and outer tubes together. Modified multi criteria decision making method known as complex proportional assessment (COPRAS) was used to rank the members using three conflicting crashworthiness criteria namely specific energy absorber (SEA), peak crash force (F_{max}) and crash force efficiency (CFE). Moreover, the multicell Sshaped members were found to perform better than singlecell ones in terms of crashworthiness. In addition, the multicell Srail with decagonal crosssection was found as the best energy absorber, and also the Srail having the same inner and outer tube with decagonal crosssection displayed desirable crashworthiness performance. Optimum geometry of this Srail was eventually obtained from the parametric study.
Keywords:
Multicell members; Srails; axial dynamic loading; crashworthiness; COPRAS
1 INTRODUCTION
The front end part in the vehicles is desired to be folded progressively during car frontal crash so as to absorb more energy and provide safety for the passengers. In order to develop more progressive folding wrinkles, at the early stage of frontal crash, and to absorb more energy in buckling at the late stage of crash, it is important to improve the crashworthiness of the front end member. Figure 1 shows an example for the possible use of members as energy absorbers in the longitudinal frames of automobiles (Marsolek and Reimerdes, 2004Marsolek, J., Reimerdes, H.G. (2004). Energy absorption of metallic cylindrical shells with induced nonaxisymmetric folding patterns. International Journal of Impact Engineering 30: 120923.). Designing automotive body structure to achieve maximum safety, as significant vehicle attribute, is an important research field that has received considerable attention (Atahan et al., 2014Atahan, A.O., Yucel, A.O., Erdem, M. (2014). Crash testing and evaluation of a new generation L1 containment level guardrail. Engineering Failure Analysis 38: 25−37.; Belingardi et al., 2013Belingardi, G., Boria, S., Obradovic, J. (2013). Energy absorbing sacrificial structures made of composite materials for vehicle crash design. Dynamic Failure of Composite and Sandwich Structures. 192: 577−609.; Pawlus et al., 2011Pawlus, W., Karimi, H.R., Robbersmyr, K.G. (2011). Mathematical modeling of a vehicle crash test based on elastoplastic unloading scenarios of springmass models. The International Journal of Advanced Manufacturing Technology 55: 369−378.).
Curved tubes used in the vehicle structures have significant effects on prolonging deformation of the frame (Han and Yamazaki, 2003Han, J., Yamazaki, K. (2003) Crashworthiness Optimization of Sshape Square Tubes. International Journal of Vehicle Design 31: 7285.; Cheon and Meguid, 2004Cheon, S.S., Meguid, S.A. (2004). Crush Behavior of Metallic Foams for Passenger Car Design. International Journal of Automotive Technology 5: 4753.) in the crash events. There are some experimental and numerical studies on the collapse behavior of Sshaped beams which have been done by Ohkami et al. (1990Ohkami, Y., Takada, K., Motomura, K., Shimamura, M. (1990). Tomizawa H, Usuda M. Collapse of thinwalled curved beam with closedhat sectionpart 1: study on collapse characteristics. SAE Technical Paper. 10, 900460.), Abe et al. (1990Abe, K., Nishigaki, K., Ishivama, S., Ohta, M., Takagi, M., Matsukawa, F. (1990). Collapse of thin walled curved beam with closedhat sectionpart 2: Simulation by plane plastic hinge model. SAE Technical Paper 10, 900461.) and Zhang (2005Zhang, C. (2005). Study of Crash Behavior of a 3D SShape Space Frame Using Finite Element Method. M.S. Thesis, Tufts University, Medford.). They reinforced these beams to promote their energy absorption capacity.
In addition to the energy absorption capacity of the vehicle front end structure, weight is considered to be an important issue attempted to be minimized. Kim and Wierzbicki (2000Kim, H.S., Wierzbicki, T. (2000). Effect of the crosssectional shape on crash behavior of a 3D space frame [R]. Impact and Crashworthiness Laboratory Report No. 34, MIT.) have carried out a study by investigating different methods to ameliorate structural crashworthiness of internal members. In another study, Ohkami et al. (1990Ohkami, Y., Takada, K., Motomura, K., Shimamura, M. (1990). Tomizawa H, Usuda M. Collapse of thinwalled curved beam with closedhat sectionpart 1: study on collapse characteristics. SAE Technical Paper. 10, 900460.) experimentally investigated collapse behavior of the thinwalled curved tube with closedhat section subjected to static and dynamic loading.
HosseiniTehrani and Nikahd (2006HosseiniTehrani, P., Nikahd, M. (2006). Two materials Sframe representation for improving crashworthiness and lightening. ThinWalled Structures 44: 40714.) employed various arrangements of straight and sidling ribs within the Sshaped tubes so as to achieve the members possessing better crashworthiness performance and higher weight efficiency. Although structural modification has a significant effect on crashworthiness performance and produces light weight tubes, recent studies have revealed that the crashworthiness capacity and weight efficiency can be further improved by applying some materials as well (HosseiniTehrani and Nikahd, 2006HosseiniTehrani, P., Nikahd, M. (2006). Two materials Sframe representation for improving crashworthiness and lightening. ThinWalled Structures 44: 40714.). In order to increase the weight efficiency of the Sshaped tubes, Kim et al. (2002Kim, H.S., Chen, W., Wierzbicki, T. (2002). Weight and crash optimization of foamfilled threedimensional S frame. Computational Mechanics. 28: 41724.) have applied aluminum foam filler. Finding an appropriate contact between the wall and foam filler was the advantage of their study.
Several works have been done in recent years on the effect of foam filler on the energy absorption of structures (Reyes et al., 2004Reyes, A., Hopperstad, O.S., Langseth, M. (2004). Aluminum foamfilled extrusions subjected to oblique loading: experimental and numerical study. International Journal of Solids and Structures 41: 16451675.; Chen, 2001Chen, W., Wierzbicki, T. (2001). Relative merits of singlecell, multicell and foam filled thinwalled structures in energy absorption. ThinWalled Structures 39(4):287306.; Hong et al., 2005Hong, H.W., Fan, Z.J., Yu, G., Wang, Q.Ch, Tobota, A. (2005). Partition energy absorption of axially crushed aluminum foamfilled hat sections International Journal of Solids and Structures 42: 25752600.; Li et al., 200Li, Q.M., Mines, R.A.W., Birch, R.S. (2000). The crush behaviour of Rohacell51WF structural foam. International Journal of Solids and Structures 37: 63216341.). Kim and Wierzbicki (2004Kim, H.S., Wierzbicki, T. (2004). Closedform solution for crushing response of threedimensional thinwalled S frames with rectangular section. International Journal of Impact Engineering 30: 87−112.) studied crushing behavior of Srails with rectangular crosssection. Their research indicated that the critical aspect ratio of the rectangular crosssection was 1.366. Also, according to their investigation, analytically derived crushing force gave excellent correlation with the finite element results. Khalkhali et al. (2011Khalkhali, A., Darvizeh, A., Masoumi, A., NarimanZadeh, N. (2011). Experimental and numerical investigation into the quasistatic crushing behaviour of the Sshape square tube International Journal of Mechanical Sciences 27: 585−596.) made an experimental and numerical investigation on the singlecell Sshaped tubes with square crosssection under quasistatic axial loading. In another work carried out by khalkhali et al. (2013Khalkhali, A., Agha Hossinali Shirazi, V., Mohseni Kabir, M. (2013). Closedform solution for peak crushing force of the Srails [J]. International Journal of Automotive Technology 3: 446−456.) they derived a closed form solution for calculating crushing force of the Srails. More recently, Elmarakbi et al. (2011Elmarakbi, A., Fielding, N., Hadavinia, H. (2011). Finite element simulation of the axial crush of thinwalled tubes with different crosssections: vehicle/ pole impact application. International Journal of Vehicle Structures & Systems 3:15460.) studied energy absorption capacity of the simple Sshaped members with different crosssection shapes and several inner ribs. Reviewing the literature, as mentioned above, indicates that many investigations have been performed on the Sshaped members. However, some new designs of sectional configurations are presented in the current work.
Therefore, crashworthiness of new designed multicell Sshaped members together with the singlecell Srails was studied under axial impact loading in the present work. These members have included several sectional configurations namely triangular, square, hexagonal, decagon and circular crosssections. Finite element code LSDYNA was used to simulate the collapse behavior of these members. In addition, modified multi criteria decision making method namely complex proportional assessment (COPRAS) was used to rank the considered members from the crashworthiness point of view.
2 SRAIL GEOMETRY AND MATERIAL
This study focuses on the crashworthiness of different singlecell and multicell Sshaped tubes. Figure 2 shows the various crosssection configurations namely triangular, square, hexagonal, decagon and circular assumed for the Sshaped members. The singlewalled and doublewalled Sshaped tubes have been designated as SCTS, SCSS, SCHS, SCDS, SCCS (see Figure 2a) and MCTS, MCSS, MCHS, MCDS, MCCS (see Figure 2b), respectively. The ribs in the doublewalled members connected middle of the outer and inner tube sides together. Geometrical model of the Srail has been depicted in Figure 3, where L, R, ϑ, and t denote the length, radius of curvature, curve angle and wall thickness, respectively. The outer perimeter of all crosssections was chosen the same and equal with 534 mm. In addition, ratio of the inner tube to the outer one of doublewalled members was assumed to be 0.5 as well as the wall thickness of all the tubes shown in Figure 2 was selected 3mm.
The material of tubes was assumed Aluminum alloy AA6060T4 with Young's modulus E=68.2Gpa, Poisson's ratio v=0.3, mass density ϱ = 2700 kg/m^{3}, yield strength σ_{y} = 80MPa and Ultimate strength σ_{u} =173MPa. The stressstrain curve obtained from the tensile test has been shown in Figure 4. Since, Aluminum alloy is insensitive to the strain rate (Langseth and Hopperstad, 1996Langseth, M., Hopperstad, O.S. (1996). Static and dynamic axial crushing of square thinwalled aluminium extrusions. International Journal of Impact Engineering 18(78): 949968.); therefore, its effect was ignored in the finite element analyses.
3 CRASHWORTHINESS CRITERIA
Three important criteria namely specific energy absorption (SEA), peak crash force (F_{max}) and crash force efficiency (CFE) have been used in this paper to assess crashworthiness of the Sshaped members. Equation (1) indicates formula of the SEA which is typically defined as the total strain energy absorbed during the plastic deformation. It is calculated by dividing the energy absorption capacity (EA) during the crushing process to the total mass of member. Thus, an appropriate energy absorber must have higher value of SEA.
Where F(x) denotes variations of the instantaneous crushing load and δ is the effective stroke length (Chen and Wierzbicki, 2001Chen, W., Wierzbicki, T. (2001). Relative merits of singlecell, multicell and foam filled thinwalled structures in energy absorption. ThinWalled Structures 39(4):287306.), which is taken as 0.4L in this study, and L is the total length of the energy absorbing device.
The average value of F(x), called as the mean collapse load (F_{mean}), is calculated as the ratio of the total energy absorbed by tube to the effective stroke length δ. This parameter is defined as:
Another crashworthiness indicator is the crash force efficiency (CFE) which is calculated by the division of the mean collapse load (F_{mean}) to the F_{max} as follows:
4 NUMERICAL ANALYSIS
4.1 Finite Element Modeling
Nonlinear finite element code LSDYNA was employed to study energy absorption capacity and crushing behavior of the Srails under axial dynamic loading. Geometry of these Srails was designed in CATIA software, and they were then exported to the postprocessor LSPREPOST in LSDYANA to analyze the impact problem so that acquire afore mentioned crash worthiness criteria. Figure 5 shows schematic of the finite element analysis setup. A rigid striker with initial velocity of 10 m/s and added mass of 500kg impacted on the Sshaped members axially. This process was modeled using the RIGID WALL_PLANNAR MOVING FORCE command in LSDYNA. To avoid any movement under the crash, the stationary boundary condition was employed at the end of tubes. Besides, the members were modeled using the quadrilateral fournode shell elements with five integration points through the thickness. The decreased hourglass and integration technique were applied in the analyses to avoid volumetric locking and spurious zero energy deformation cases. Mesh convergence analysis was performed, and the optimal element size was finally found to be 5mm×5mm; while, the element size at the corners was taken finer (about 1mm×1mm). AUTOMATICNODETOSURFACE algorithm was used to model the contact between the striker and members. In addition, AUTOMATICSINGLESURFACE algorithm was used to consider the contact between the Srail walls to avoid penetrating the walls into together. The friction coefficient of 0.15 was adopted for all the contact conditions (Ahmed et al., 2013Ahmed, E., Yee, X.L., John, M. (2013). Crash analysis and energy absorption characteristics of Sshape longitudinal members. ThinWalled Structures 68: 6574.). It is also noticed that material of the tubes was modeled by MAT024 (namely modifiedpiecewiseLinearplasticity) in LSDYNA.
4.2 Validation of FE Simulations
To validate the finite element simulations of the crash problem, experiments were performed on the square tubes using the universal test machine under the longitudinal loading, and they were then simulated in LSDYNA. The tubes with square crosssection of 40×40mm, the thickness of 2 mm and the length of 90 mm were constrained on the lower part of the fixture; while, the upper part of this fixture crushed 40% of the tube length with a constant displacement rate of 10 mm/min (see Figure 6). These tubes were also simulated in LSDYNA similar to the experimental conditions mentioned above, and the results have been given in Figure 6 for comparing with the experimental results. As is evident in this figure, there is a satisfied agreement between the two sets of results in terms of deformation modes, forcedisplacement responses, and crashworthiness indicators. The relative error (R_{e}) between the three crashworthiness criteria for the numerical (_{fNum.} ) and experimental (_{fExp.} ) results was calculated by the Equation (4).
Comparison of the experimental and numerical results: (a) deformation modes, (b) force versus displacement curves, (c) crashworthiness indicators.
4.3 Finite Element Results for the Designed SRails
Collapse behavior of the Sshaped members with different crosssections illustrated in Figure 2 was analyzed in LSDYNA according to the notes mentioned in the section 4.1. Deformation modes of these Srails have been shown in Figure 7. It is reminded that the striker identically crushed 40 percent of the total length of the Srails. As is clear from Fig. 7, the plastic hinges (formed at the curved zones of the Srails due to the global bending mode) absorbed the main kinematic energy of the striker. Forcedisplacement curves for the studied Srails have been plotted in Figure 8. From this figure, the force initially increased to reach its maximum value, and then suddenly decreased with a sharp steep due to global bending deformation mode.
The SEA calculated from the Figure 8 has been plotted in Figure 9a. From which it is clear that the decagonal and triangular crosssections had the highest and the lowest values of SEA, respectively compared to the other crosssections. However, the ranking in terms of the SEA value is: SCDS> SCHS> SCSS> SCCS> SCTS. The amount of energy absorption for the singlecell decagonal (SCDS) and hexagonal tubes (SCHS) were 0.855kJ/kg and 0.81kJ/kg, respectively which were found to be the best energy absorbing devices. This result exhibits that the energy absorption generally goes up by increasing sides of the cross section.
Figure 9b presents values of the F_{max} for the considered Srails. As is evident from this figure, the singlecell Srails were ranked as follows: SCHS> SCSS> SCDS> SCCS> SCTS. Therefore, the singlecell hexagonal (SCHS) and square (SCSS) members possessed higher peak crash force values which could be detrimental to the vehicle safety. Results for the crash force efficiency (CFE) have been given in Figure 9c. By comparing values of the CFE, it is seen that the SCTS, SCCS and SCHS members have greater values among the singlecell structures.
Results of the three mentioned criteria for the multicell Srails have been given in Figure 10. The ranking in order of the SEA values were obtained as: MCDS> MCSS> MCHS> MCTS> MCCS according to Figure 10a. Comparing the results of singlecell tubes with the multicell ones indicates that addition of the ribs to the doublewalled Srails contributed to the improvement of SEA values due to developing the plastic hinges at the corners. The multicell tube with decagonal crosssection (MCDS) has greater ability to absorb energy than MCHS tubes and MCSS tubes.
Initial peak force causes severe injury or damage to the people or automobile structures so it is significantly undesirable in crash performance. The results for this negative crashworthiness indicator have been given in Figure 10b for the multicell Srails. Results for the crash force efficiency (CFE) have been given in Figure 10c. By comparing values of the CFE, it is clear that the SCDS had the highest value among the considered multicell structures.
4.4 Crashworthiness Comparison of the SRails Using the COPRAS Method
The multi criteria decision making method is applied in various situations where a number of alternatives need to be chosen. In this research, a multicriteria decision making (MCDM) process known as the complex proportional assessment (COPRAS) developed by Zavadskas et al. (2007) and Zavadskas et al. (2008Zavadskas, E.K., Turskis, Z., Tamošaitiene, J., Marina, V. (2008). Multi criteria selection of project managers by applying grey criteria. Technology and Economic Development 14: 46277.) was employed to select the better energy absorbers investigated in this research regarding the crashworthiness indicators calculated in the section 4.3. Actually, this is the ranking method which by considering conflicting criteria, the optimum alternative is selected. The steps to apply this method for the abovementioned results are explained below in detail.
Step 1: Construct the initial decisionmaking matrix, X as:
Where Xij is the performance value of i^{th} criterion on j^{th} alternative. m and n state the number of criteria and alternatives, respectively.
Step 2: calculate the normalized initial decisionmaking matrix using:
The purpose of normalizing the decision matrix is to find dimensionless values of different criteria for comparing all of them.
Step 3: Find the weighted normalized decision matrix, D as below:
Where r_{ij} is the normalized performance value of i^{th} criterion on j^{th} alternative. W_{i} is the weight of i^{th} criterion. The sum of dimensionless weighted normalized values of each criterion is always equal to the weight for that criterion:
In other words, the weight, w_{i} of the investigated criterion is proportionally distributed among all the alternatives according to their weighted normalized value, y_{ij}.
Step 4: The sums of weighted normalized values are calculated for both the beneficial criteria and nonbeneficial criteria. The lower is the value of a nonbeneficial criterion like F_{max} the better is the crashworthiness of the members. On the other hand, the greater is the value of a beneficial criterion like the SEA or CFE the better is the crashworthiness of the members. These sums are obtained as:
Where y_{+ij} and y_{ij} are the weighted normalized values for the beneficial and nonbeneficial criteria, respectively.
The greater the value of S_{+j}, the better is the alternative, and the lower the value of S_{j}, the better is the alternative. The values of S_{+j} and S_{j} state the degree of goals attained by each alternative. In any case, the sums of S_{+j} and S_{j} of the alternatives are always respectively equal to the sums of weights for the beneficial and nonbeneficial criteria as written by the following equations:
These equations can be used for verifying the calculations.
Step 5: characterize the significances of the alternatives based on defining the positive alternatives S_{+j} and negative alternatives S_{j} characteristics.
Step 6: Determine the priorities of the alternatives. The priorities of the candidate alternatives are calculated based on Q_{j} written as:
where S_{min} is the minimum value of S_{j}.
The greater the value of Q_{j}, the higher is the priority of the alternative. The relative significance value of an alternative states the degree of satisfaction achieved by that alternative. The alternative with the highest relative significance value (Q_{max}) is the best choice among the all alternatives.
Step 7: Compute the quantitative utility (U_{j}) for j^{th} alternative. The degree of an alternative’s utility is directly associated with its relative significance value (Q_{j}). The degree of an alternative’s utility, leading to a complete ranking of the candidate alternatives, is determined by comparing the priorities of all the alternatives with the most efficient one and can be denoted as below (Mandal and Sarkar, 2012Mandal, U.K., Sarkar, B. A. (2012). Exploratory analysis of intelligent manufacturing system (Ims) under fuzzy utopian environment. IOSR J of Eng 2: 129140.):
Where Q_{max} is the maximum relative significance value. These utility values of the candidate alternatives range from 0% to 100%. Thus, this approach allows for evaluating the direct and proportional dependence of significance and utility degree of the considered alternatives in a decisionmaking problem involving multiple criteria, their weights and performance values of the alternatives with respect to all the criteria.
In order to select the suitable tube (among the tubes with different crosssections illustrated in Fig. 2) in crashworthiness point of view, the complex proportional assessment (COPRAS) method was adopted. A good energy absorber must have greater SEA and CFE as well as less F_{max} to prevent the vehicle passengers from the severe damage during a crash. Therefore, all of the three criteria (i.e. SEA, F_{max} and CFE) were required to be considered as the design criteria for comparing the Srails.
COPRAS method was separately applied on the numerical results of both singlecell and multicell Srails according to the procedures explained above. The decision matrix, normalized decision matrix, beneficial (S_{i}) and nonbeneficial(S_{+i}) attributes, relative significance or priority (Q) and quantitative utility (U) were computed as have been shown in Tables 12.
COPRAS results for the singlecell Sshaped members, (a) decision making matrix, (b) normalized decision making matrix, (c) ranking
COPRAS results for the multicell Sshaped members, (a) decision making matrix, (b) normalized decision making matrix, (c) ranking
From the COPRAS results presented in Tables 12, final ranking for the singlecell Srails was obtained as SCDS, SCTS, SCCS, SCHS and SCSS; while the multicell Sshaped tubes were ranked as: MCDS, MCSS, MCHS, MCTS and MCCS. Therefore, the decagonal crosssection for both the singlecell and multicell Srails was found the best crosssection shape to improve crashworthiness capability of the Srails. It is worth noting that the mentioned ranking of the Srails in terms of the crashworthiness capacity is only for the specified crosssections illustrated in Figure 2. For example, the increase in the number of stiffener can result in increasing the F_{max}, the m (mass of the structure) and EA. Hence, SEA may not be increased because of increasing the m (or in the case of increasing the SEA, this increase may not compensate the increase of F_{max}). Because of this reason, complex conditions are created by increasing the number of stiffeners, and so any change in the crosssectional configurations must be individually studied.
5 CRASH ANALYSIS OF SRAILS WITH THE SAME OUTER TUBES AND DIFFERENT INNER ONES
According to the previous section, the multicell Srail with decagon crosssection (MCDS) was found to perform as the best energy absorber. Thus, further analysis was carried out on this member by assuming the same outer crosssection (decagon) and changing the inner one to triangular, square, hexagonal, decagon and circular (see Figure 11). Number of the ribs was assumed as N=5 similar to the MCDS. Ratio of the inner tube perimeter to the outer one was also assumed equal with 0.5 like MCDS. Figure 12 exhibits deformation modes of these tubes. The corresponding forcedisplacement curves were also given in Figure 13. In addition, the crashworthiness indicators including SEA, F_{max} and CFE, calculated from the LSDYNA, were presented in Figure 14.
COPRAS method was implemented on the results of these five tubes as is seen in Table 3. Based on the COPRAS calculations, the ranking was obtained as type 4>type 3>type 2>type 1>type 5. This signifies that the Srail with the same outer and inner crosssection shape performed as the best energy absorbing device.
COPRAS results for the Srails with the same outer crosssection and different inner one, (a) decision making matrix, (b) normalized decision making matrix, (c) ranking.
6 PARAMETRIC STUDY
According to the previous section, the multicell Srail with decagonal crosssection and five ribs (namely MCDS member) was identified as the best alternative for dissipating collision energy. A parametric study was performed to evaluate effects of the geometrical parameters on the crashworthiness capacity of the selected Srail (MCDS). These parameters consisted of the wall thickness t, curve angle ϑ, and the perimeter ratio S (see Table 4). These parameters namely t, ϑ and S were varied to be 2, 2.5, 3 and 3.5 mm, 30°, 40°, 50° and 60°, 0.3, 0.45, 0.6 and 0.7, respectively. The Srails with these geometries were modeled and analyzed in LSDYNA (according to the notes and values mentioned in the section 4.1), and the corresponding results for the SEA, F_{max} and CFE were computed as are seen in Table 5. The material properties were similar to the ones mentioned in the section 2. Variations of the SEA, F_{max} and CFE against the parameters of S and ϑ for the wall thickness of t=2, 2.5, 3, 3.5 have been plotted in Figure 1517, respectively. As is evident from these figures, by increasing the wall thickness (while S and ϑ were considered to be constant), the values of SEA and F_{max} increased.
Geometrical parameters, range and step size for the parametric study of the multicell Srails with decagonal crosssection.
Crashworthiness indicators of the multicell Srails with decagonal crosssection at the design points.
Variations of SEA against ϑ & S for the MCDS Srails with different thicknesses of (a) t=2, (b) t=2.5, (c) t=3, (d) t=3.5.
Variations of SEA against ϑ & S for the MCDS Srails with different thicknesses of (a) t=2, (b) t=2.5, (c) t=3, (d) t=3.5.
Variations of CFE against ϑ & S for the MCDS Srails with different thicknesses of (a) t=2, (b) t=2.5, (c) t=3, (d) t=3.5.
It is obvious in Figure 15 that as the perimeter ratio S increased, SEA initially reduced up to S=0.6 and then increased. Furthermore, SEA had the maximum value at S=0.3. On the other hand, by increasing the perimeter ratio S, the peak force F_{max} enhanced and CFE decreased (see Figure 1617). It is also clear from these figures that as the curve angle θ increased, SEA initially reduced and then remained approximately unchanged. This behavior of the Srails corresponds to changing progressive folding collapse mode to the global bending. Moreover, by increasing the curve angle θ, generally both the peak force F_{max} and CFE decreased.
The COPRAS method was finally implemented on the results presented in Table 5 considering three crashworthiness criteria namely SEA, F_{max} and CFE. This calculations lead to find the optimum geometrical parameters of Srail as: ϑ =60°, S=0.3 and t=2 mm. The results reveal that, the optimum values have been taken place at the upper or lower bounds. These values are indeed the usual values taken in the literatura (khalkhali, 2015; Elmarakbi et al., 2013Elmarakbi, Ahmed, Long, Y.X., MacIntyre, J. (2013). Crash analysis and energy absorption characteristics of Sshaped longitudinal members. ThinWalled Structures 68: 6574.; Han and Yamazaki, 2001Han, J., Yamazaki, K. (2001) A study on the crashworthiness of Sshape square tubes, Transactions on the Built Environment 52, ISSN 17433509.; Kim et al., 2002Kim, H.S., Chen, W., Wierzbicki, T. (2002). Weight and crash optimization of foamfilled threedimensional S frame. Computational Mechanics. 28: 41724.). It is evident that changing the lower and upper bounds would probabaly give different results.
7 CONCLUSIONS
In this research, crashworthiness performance of Sshaped members with different crosssections was numerically studied under axial crash. The following important results were drawn:

Singlecell and multicell Srails with triangular, square, hexagonal, decagon and circular crosssections were analyzed in LSDYNA, and three main crashworthiness indicators i.e. SEA, F_{max} and CFE were computed. In order to select the suitable tube among these Srails, the COPRAS method was applied on the numerical results. The proposed multicell Srail with decagonal crosssection was finally found to be the best energy absorbing device.

Doublewalled Srails with the same outer crosssection (decagonal) and different inner one (i.e. triangular, square, hexagonal, decagon and circular) were also investigated from the crashworthiness point of view. The results demonstrated that the Srail with the same outer and inner crosssection (decagonal) had the best crashworthiness performance using the COPRAS method.

Parametric study was performed on the afore mentioned selected Srail to evaluate effects of the geometrical parameters including the wall thickness t, curve angle ϑ, and the perimeter ratio S (each one at four levels) on the crashworthiness capacity. COPRAS calculations lead to find the optimum geometrical parameters of Srail as: ϑ = 60°, S = 0.3 and t = 2 mm.
References
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Publication Dates

Publication in this collection
June 2017
History

Received
13 Oct 2016 
Reviewed
18 Feb 2017 
Accepted
21 Feb 2017