Abstract
This paper aims to explore crashworthiness performance of new designed foamfilled tapered structures under axial and oblique impact load conditions. The structures consisted of inner and outer tapered tubes connected together by four stiffening plates. They also included square, hexagonal, octagonal, decagonal and circular crosssectional shapes with different ratios of a/b (ratio of the inner tube side length to the outer tube one). The ratio of a/b was assumed to be 0, 0.25, 0.5, 0.75 and 1. Crashworthiness criteria of specific energy absorption (SEA), peak collapse force (F_{max}) and crash force efficiency (CFE) were obtained for these structures using the experimentally validated model in LSDYNA. Based on the numerical results, semifoam filled structures (i.e. those with the ratio of a/b=0.5) demonstrated greater crashworthiness performance than foam filled singlecell ones. Furthermore, according to the TOPSIS calculations, cross sectional shape of decagonal was found to be an outstanding potential profile for the automobile structural components.
Keywords
Crashworthiness; Foamfilled tapered structure; Axial and oblique loads; TOPSIS
1 INTRODUCTION
Application of aluminum in automobile body and components has been increased in recent decades due to its low cost and high weightstiffness efficiency. In addition, automobile parts made of aluminum can be produced to almost any shape via extrusion processes ( Hou et al., 2007 Hou, S. Li, Q., Long, S., Yang, X., Li, W., (2007). Design optimization of regular hexagonal thinwalled columns with crashworthiness criteria, Finite Elements in Analysis and Design 43(6) (2007) 555565. ). Because of the mentioned manufacturing benefits, new interesting thinwalled structures made of aluminum have been introduced by the researchers to further improve the crashworthiness capacity and automobile safety. For example, HosseiniTehrani and Pirmohammad (2007 HosseiniTehrani, P., Pirmohammad, S., (2007). Collapse study of thinwalled polygonal section columns subjected to oblique loads, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 221(7) 801810. , ^{2010} HosseiniTehrani, P., Pirmohammad, S., (2010). Study on Crashworthiness Characteristics of Several Concentric Thin Wall Tubes, ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis, American Society of Mechanical Engineers, pp. 18. , ^{2011} HosseiniTehrani, P., Pirmohammad, S., (2011). Collapse Study of a Pair ThinWalled Prismatic Column Subjected to Oblique Loads, Int. J. Auto. Eng1 267279. ) studied crushing behavior of aluminum extruded tubes with different crosssectional shapes. In another study, EsmaeiliMarzdashti et al. (2017) EsmaeiliMarzdashti, S. Sebaey, T.A., Pirmohammad, S., EsmaeiliMarzdashti, S., (2017). Crash Analysis And Design of MultiCell Octagonal SShape Members Under Axial And Oblique Impacts. investigated crashworthiness of the Sshaped octagonal tubes with different crosssections. Liu and Day (2008) Liu, Y. Day, M.L., (2008). Bending collapse of thinwalled circular tubes and computational application, ThinWalled Structures 46(4) 442450 studied bending collapse of thinwalled circular tubes made of aluminum. Tran et al. (2014) Tran, T. Hou, S., Han, X., Nguyen, N., Chau, M., (2014). Theoretical prediction and crashworthiness optimization of multicell square tubes under oblique impact loading, International Journal of Mechanical Sciences 89 177193. investigated the collapse behavior of aluminum extruded square tubes with different profiles.
Many investigations have been also made on the tapered tubes. For example, Ghamarian et al. (2011) Ghamarian, A. Zarei, H.R. Abadi, M.T., (2011). Experimental and numerical crashworthiness investigation of empty and foamfilled endcapped conical tubes, ThinWalled Structures 49(10) 13121319. investigated crashworthiness performance of circular endcapped tapered columns experimentally and numerically. Qi et al. (2012) Qi, C. Yang, S. Dong, F., (2012). Crushing analysis and multiobjective crashworthiness optimization of tapered square tubes under oblique impact loading, ThinWalled Structures 59 (2012) 103119. and HosseiniTehrani et al. (2008) HosseiniTehrani, P. Pirmohammad, S. Golmohammadi, M., (2008). Study on the collapse of tapered tubes subjected to oblique loads, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 222(11) 20252039. investigated crashworthiness of tapered square tubes. Nagel and Thambiratnam (2005 Nagel, G. Thambiratnam, D., (2005). Computer simulation and energy absorption of tapered thinwalled rectangular tubes, Thinwalled structures 43(8) 12251242. , ^{2006} Nagel, G.M. Thambiratnam, D.P., (2006). Dynamic simulation and energy absorption of tapered thinwalled tubes under oblique impact loading, International Journal of Impact Engineering 32(10) 15951620. ) compared the energy absorption response of straight and tapered thinwalled rectangular tubes under quasistatic axial loading. They showed that the tapered tubes are desirable energy absorbers due to their relatively stable crush load and deformation response compared with straight tubes.
Because of some outstanding reasons, such as having higher energy absorption capacity and lower weight, foamfilled thinwalled structures have been widely used in automobile, railway and aerospace industries. Many investigations have been performed experimentally ( Meguid et al., 2004 Meguid, S. Attia, M. Monfort, A., (2004). On the crush behaviour of ultralight foamfilled structures, Materials & design 25(3) 183189. ; Reyes et al., 2004 Reyes, A. Hopperstad, O. Langseth, M., (2004). Aluminum foamfilled extrusions subjected to oblique loading: experimental and numerical study, International journal of solids and structures 41(5) 16451675. ; Mamalis et al., 2008 Mamalis, A. Manolakos, D., Ioannidis, M., Spentzas, K., Koutroubakis, S., (2008). Static axial collapse of foamfilled steel thinwalled rectangular tubes: experimental and numerical simulation, International Journal of Crashworthiness 13(2) 117126. ; Seitzberger et al., 1997 Seitzberger, M. Rammerstorfer, F.G., Degischer, H.P., Gradinger, R., (1997). Crushing of axially compressed steel tubes filled with aluminium foam, Acta Mechanica 125(1) 93105. ) analytically ( Wang et al., 2007 Wang, Q. Fan, Z. Gui, L., (2007). Theoretical analysis for axial crushing behaviour of aluminium foamfilled hat sections, International Journal of Mechanical Sciences 49(4) 515521. ; Niknejad et al., 2011 Niknejad, A. Liaghat,G., Naeini, H.M., Behravesh, A., (2011). Theoretical and experimental studies of the instantaneous folding force of the polyurethane foamfilled square honeycombs, Materials & Design 32(1) 6975. ; Gupta and Velmurugan, 1999 Gupta, N., Velmurugan, R., (1999). Axial compression of empty and foam filled composite conical shells, Journal of composite materials 33(6) 567591. ) and numerically ( Ahmad and Thambiratnam, 2009a Ahmad, Z. Thambiratnam, D.P., (2009a). Application of foamfilled conical tubes in enhancing the crashworthiness performance of vehicle protective structures, International journal of crashworthiness 14(4) 349363. ; ZiaeiRad et al., 2008 ZiaeiRad, S. Salimi, M. Mirfendereski, L., (2008). Finite Element Modelling of FoamFilled Tapered ThinWalled Rectangular Tubes under Oblique Impact Loading, Steel Research 317324. ; Santosa et al., 2001 Santosa, S. Banhart, J. Wierzbicki, T., (2001). Experimental and numerical analyses of bending of foamfilled sections, Acta Mechanica 148(1) 199213. ; Fang et al., 2016 Fang, J. Gao, Y., An, X., Sun, G., Chen, J., Li, Q., (2016). Design of transverselygraded foam and wall thickness structures for crashworthiness criteria, Composites Part B: Engineering 92 338349. ) on the foamfilled thinwalled structures. Reid et al. (1986) Reid, S. Reddy, T. Gray, M., (1986). Static and dynamic axial crushing of foamfilled sheet metal tubes, International Journal of Mechanical Sciences 28(5) 295322. studied quasistatic and dynamic axial crushing of empty and polyurethane foamfilled square and rectangular tubes manufactured from sheet metal. They discussed on both the stability and the energy absorbing characteristics of these tubes. Reddy and Wall (1988) Reddy, T. Wall, R., (1988). Axial compression of foamfilled thinwalled circular tubes, International Journal of Impact Engineering 7(2) 151166. performed some experiments on the circular tubes filled with polyurethane foam. They observed that the foam filler not only stabilizes the buckling mechanism of the empty tubes but also improves the energy absorption capacity. Abramowicz and Wierzbicki (1988) Abramowicz, W. Wierzbicki, T., (1988). Axial crushing of foamfilled columns, International Journal of Mechanical Sciences 30(34) 263271. developed a new method of treating progressive folding of foam filled columns. They compared the results with experimental data. Kavi et al. (2006) Kavi, H. Toksoy, A.K. Guden, M., (2006). Predicting energy absorption in a foamfilled thinwalled aluminum tube based on experimentally determined strengthening coefficient, Materials & design 27(4) 263269. investigated the energy absorption capacity of foamfilled circular tube. They found that the foam filler changed the deformation mode of tube from diamond into concertina. In another study, the experimental and numerical crushing responses of polystyrene foamfilled thinwalled aluminum tubes were investigated by Aktay et al. (2006) Aktay, L. Toksoy, A.K. Güden, M., (2006). Quasistatic axial crushing of extruded polystyrene foamfilled thinwalled aluminum tubes: experimental and numerical analysis, Materials & design 27(7) 556565. . They observed that the energy absorption capacity of foamfilled tubes were higher than the sum of the energy absorption capacities of empty tube and foamfiller. Ghamarian et al. (2013) Ghamarian, A. Zarei, H., Farsi, M., Ariaeifar, N., (2013). Experimental and Numerical Crashworthiness Investigation of the Empty and Foam‐Filled Conical Tube with Shallow Spherical Caps, Strain 49(3) 199211. studied crashworthiness of the foamfilled conical tubes with shallow spherical caps. Ahmad et al. (2010) Ahmad, Z. Thambiratnam, D. Tan, A., (2010). Dynamic energy absorption characteristics of foamfilled conical tubes under oblique impact loading, International Journal of Impact Engineering 37(5) 475488. investigated collapse response of foamfilled conical tubes. Ahmad and Thambiratnam (2009b) Ahmad, Z. Thambiratnam, D., (2009b). Crushing response of foamfilled conical tubes under quasistatic axial loading, Materials & design 30(7) 23932403 made an investigation on the crushing behavior of circular tapered tubes under quasistatic and dynamic axial loading. Hou et al. (2011) Hou, S. Han, X., Sun, G., Long, S., Li, W., Yang, X., Li, Q., (2011). Multiobjective optimization for tapered circular tubes, ThinWalled Structures 49(7) 855863. investigated the collapse behavior of tapered structures of hollow single, foam filled single and collinear double tubes. Song et al. (2013) Song, X. Sun, G., Li, G., Gao, W., Li, Q., (2013). Crashworthiness optimization of foamfilled tapered thinwalled structure using multiple surrogate models, Structural and Multidisciplinary Optimization 47(2) 221231. optimized geometry of the foam filled tapered columns using several models.
In addition, some researchers used foamfilled doublecell tubes to improve the energy absorption capabilities of thinwalled tubes. For example, Yuen et al. (2008) Yuen, S.C.K. Nurick, G. Starke, R., (2008). The energy absorption characteristics of doublecell tubular profiles, Latin American Journal of Solids and Structures 5(4) 289317. mad an experimental work on the crushing characteristics of doublecell square and circular tubes filled with foam. According to their results, the foamfilled doublecell profiles made with circular tubes had higher energy absorption capabilities than those made with square tubes. Fang et al. (2014) Fang, J. Gao, Y., Sun, G., Zhang, Y., Li, Q., (2014). Crashworthiness design of foamfilled bitubal structures with uncertainty, International Journal of NonLinear Mechanics 67 120132. studied crashworthiness of foamfilled bitubal structures with circular cross section. In another study, Zhang et al. (2012) Zhang, Y. Sun, G., Li, G., Luo, Z., Q. Li, Q., (2012). Optimization of foamfilled bitubal structures for crashworthiness criteria, Materials & Design 38 99109. optimized dimensions of foamfilled bitubal tubes with square cross section. Zheng et al. (2014) Zheng, G. Wu, S., Sun, G., Li, G., Li, Q., (2014). Crushing analysis of foamfilled single and bitubal polygonal thinwalled tubes, International Journal of Mechanical Sciences 87 226240. compared the energy absorption capabilities of foamfilled single and bitubal polygonal tubes under axial loading. Their results showed that the foamfilled circular bitubal column was the most promising configuration.
Previous studies reveal that combination of tapered tubes and foam filler perform very well as the energy absorption devices. Therefore, new designs of foamfilled tapered structures were investigated in the present study. Crushing behavior of these structures was simulated in LSDYNA. In order to compare the structures, the crashworthiness indicators, SEA (specific energy absorption), F_{max} (peak collapse force) and CFE (crash force efficiency) were calculated from the numerical analyses. TOPSIS method was finally used to select the best structure in terms of energy absorption capability.
2 GEOMETRY OF STIFFENED FOAM FILLED TAPERED STRUCTURES AND FINITE ELEMENT MODELING
Fig. 1 a displays the structures studied in the present research. According to this figure, the structures consist of two inner and outer tapered tubes, which have been connected together by four stiffening plates. These structures include square, hexagonal, octagonal, decagonal and circular crosssectional shapes with different ratios of a/b (ratio of the inner tube side length to the outer tube one). Meanwhile, polyurethane foam was employed to fill the spaces between the inner and outer tubes. The ratio of a/b was assumed to be 0, 0.25, 0.5, 0.75 and 1 for all the cross sectional shapes illustrated in Fig. 1 a. It is noticed that the ratios of a/b=0 and a/b=1 refer to the simple stiffened foamfilled tapered structure (without the inner tube) and singlecell tapered structure (without the inner tube and foam), respectively; whereas, the other ratios of a/b refer to the stiffened foamfilled structures with two concentric tapered tubes connected together by four plates. According to Fig. 1 a, the structures studied in the present work were designated as Sr, Hr, Or, Dr, and Cr for convenience, which S, H, O, D, C represent the structures with square, hexagonal, octagonal, decagonal and circular crosssectional shapes, respectively, and r represents the value of R_{1}/R_{2} (for circular structure) or a/b (for other structures).
a) Stiffened foamfilled tapered structures with different cross sectional shapes, b) boundary and impact conditions
All the tapered structures studied in the current work had the same thickness of 2mm and total length of 120mm. Perimeter of the outer tube at the large end was also taken the same (equal with 160mm) for all the studied structures. In addition, the tapered angle was taken α=5° for all the structures (see Fig. 1 b).
All the structures had the same boundary and impact conditions. According to Fig. 1 b, the large end of the tapered structures was constrained in all directions to simulate the clamping support, and the small end was assumed free. A moving rigidwall with a speed of 15m/s and mass of 600kg impacted on the small end of the tapered structures at different axial (θ=0°) and oblique (θ=10°, 20° and 30°) angles. In order to measure the crashworthiness capacity, all the structures were crushed 90mm (75% of the total length).
Material of the structures was assumed aluminum alloy AA6060 with the mechanical properties of: Young’s modulus E=68GPa, Yield stress σ_{y} =80MPa, Ultimate stress σ_{u}=173MPa, Density ρ=2700 kg/m3 and Poisson’s ratio υ=0.33. Fig. 2 a displays the plastic behavior of AA6060. Furthermore, mechanical properties of the polyurethane foam were taken as Young’s modulus E=8.5GPa, Yield stress σ _{y}=1.24MPa, Density ρ=174kg/m^{3} ( Rezvani and Jahan, 2015 Rezvani, M.J. Jahan, A., (2015). Effect of initiator, design, and material on crashworthiness performance of thinwalled cylindrical tubes: A primary multicriteria analysis in lightweight design, ThinWalled Structures 96 169182. ). Meanwhile, stressstrain curve of the polyurethane foam, obtained from the uniaxial compression test according to the ASTMD 162194 standard, has been given in Fig. 2 b. Based on this figure, three different regions can be seen in the stressstrain curve of the polymeric foams: linear elastic, plateau and densification region ( Azarakhsh et al., 2015 Azarakhsh, S. Rahi, A. Ghamarian, A. Motamedi, H., (2015) Axial crushing analysis of empty and foamfilled brass bitubular cylinder tubes, ThinWalled Structures 95 60–72. , Ghamarian et al., 2011 Ghamarian, A. Zarei, H.R. Abadi, M.T., (2011). Experimental and numerical crashworthiness investigation of empty and foamfilled endcapped conical tubes, ThinWalled Structures 49(10) 13121319. , Yan et al., 2014 Yan, L. Chouw, N. Jayaraman, N., (2014) Effect of triggering and polyurethane foamfiller on axial crushing of natural flax/epoxy composite tubes, Materials and Design 56 528–541. , Rezvani and Jahan, 2015 Rezvani, M.J. Jahan, A., (2015). Effect of initiator, design, and material on crashworthiness performance of thinwalled cylindrical tubes: A primary multicriteria analysis in lightweight design, ThinWalled Structures 96 169182. ). Polyurethane foam behaves in linear elastic at the smaller strains, witch slope of the curve is equal with the Young's modulus (AB). This region is followed by a plateau region in which the stress nearly remains unchanged (BC). Finally, a densification region occurs where the stress enhances rapidly by further increase of the strain (CD).
(a) Plastic behavior of Aluminum alloy AA6060, (b) true stressstrain for the polyurethane foam ( Rezvani and Jahan, 2015 Rezvani, M.J. Jahan, A., (2015). Effect of initiator, design, and material on crashworthiness performance of thinwalled cylindrical tubes: A primary multicriteria analysis in lightweight design, ThinWalled Structures 96 169182. )
Crashworthiness performance of the foamfilled tapered structures was studied using the nonlinear finite element code LSDYNA. Fournode Belytschko–Tsay shell elements with five integration points through the thickness were used in the present study, which suitably model the crushing behavior of the structures. In addition, eightnode solid elements with hourglass control (to avoid spurious zero energy collapse modes) and reduced integration techniques (to avoid volumetric locking) were used for modeling the foam ( Mamalis et al., 2008 Mamalis, A. Manolakos, D., Ioannidis, M., Spentzas, K., Koutroubakis, S., (2008). Static axial collapse of foamfilled steel thinwalled rectangular tubes: experimental and numerical simulation, International Journal of Crashworthiness 13(2) 117126. ). Mesh convergence analyses were carried out in LSDYNA, and the element sizes of 2mm×2mm for the shell elements and 2mm×2mm×2mm for the solid elements were finally found suitable.
A suitable contact algorithm for modeling the interaction between the structure walls and foam filler was needed. Thereby, AUTOMATICSURFACETOSURFACE algorithm was defined for the contact interfaces of the foam and structure walls. This contact algorithm was also used for the contact between the rigidwall and the structure walls. Furthermore, AUTOMATICSINGLESURFACE algorithm was employed for not penetrating structure walls into each other. MODIFIEDPIECEWISELINEARPLASTICITY code was employed to simulate the aluminum properties, and crushable foam material model (MAT_CRUSHABLE_FOAM code) was used to model the foam properties in LSDYNA. Also, the moving rigidwall was modelled by MATRIGID code in LSDYNA. The rigidwall was constrained to move only along the vertical axis, while a tied constraint was defined for the large end of the tapered structures.
The friction coefficient related to the contact between the foam filler and structure walls influences the crushing force remarkably, and it is very hard to characterize this parameter from the experiments. This was regarded as a variable in the numerical analysis to achieve correlation with experimental results. Therefore, the friction coefficient of 0.2 was found a suitable value. In addition, the contact between the structure walls and rigidwall was assumed frictionless.
3 MODEL VALIDATION
At the first phase of this research, finite element model generated in LSDYNA was validated by performing impact experiments to ascertain if it was sufficiently accurate. For this purpose, some experiments were carried out. Two experiments were conducted on bitubal tubes with circular and square cross sectional shapes under axial impact, which the spaces between the tubes had been filled with polyurethane foam. The experiments upon these tubes were performed using a free fall drop hammer shown in Fig. 3 a. The bitubal circular specimens had the inner and outer radii, length and thickness of 15mm, 30mm, 150mm and 1.8mm, respectively (see Fig. 3 a). As such, the bitubal square specimens had the inner and outer side widths, length and thickness of 30mm, 60mm, 150mm and 1.8mm, respectively (see Fig. 3 b). Meanwhile, materials of the tubes and foam were those described in Section 2. Procedure of the impact tests was such that the specimens were placed upon a fixture, and then the 250kg hammer dropped on the specimens from 3.27m height to crush them. It is noticed that velocity of the hammer was 8m/s at the moment of impact. Variations of force versus displacement were recorded by load cell and data logger. Collapse behavior of the foamfilled bitubal circular and square tubes was then simulated in LSDYNA based on the conditions regarded for the impact tests. Fig. 3 displays the test and LSDYNA results including the deformation patterns and forcedisplacement responses for the mentioned foamfilled bitubal specimens. According to Fig. 3 , good agreements were achieved between the impact tests and numerical simulations in terms of the deformation patterns and the force–displacement responses.
Comparison of the impact test and LSDYNA results, a) bitubal circular tube, b) bitubal square tuve
In order to further validate the finite element model, two more experiments were performed on foamfilled singlecell circular and square tubes using the universal test machine shown in Fig. 4 a under an oblique loading of 8°. The foamfilled singlecell circular tube had the radius of 15mm, length of 90mm and thickness of 1.8mm. As such, the foamfilled singlecell square tube had the cross sectional dimensions of 60mm×60mm, the thickness of 1.8mm and the length of 100mm. As seen in Fig. 4 , these tubes were constrained on the lower part of the fixture; while, the upper part of this fixture crushed over 80% of the total length of the tubes with a constant velocity of 10mm/min. Forcedisplacement curves were then recorded by load cell and data logger. These tubes were finally simulated in LSDYNA similar to the experimental conditions explained above, and the results have been given in Fig. 4 for comparing with the experimental ones. As seen in this figure, there are satisfied agreement between the two sets of results in terms of deformation patterns and force–displacement responses for both the circular and square tubes. Therefore, the finite element model constructed in LSDYNA had capability of simulating the collapse response of the foam filled tubes.
Comparison of the universal test and LSDYNA results, a) singlecell circular tube, b) singlecell square tube
4 CRASHWORTHINESS INDICATORS
Specific energy absorption (SEA) is considered as a very important crashworthiness indicator, which is defined as follows:
where, EA is the energy absorbed by a structure, which can be calculated as the area under the forcedisplacement curve. In addition, δ_{c} is the maximum displacement that rigidwall collapses a structure, δ is the rigidwall displacement, and F(δ) is variations of force versus displacement. Meanwhile, m is the total mass of structure.
SEA is frequently used for indicating energy absorption capacity of structures. A suitable thinwalled structure must have higher value of SEA.
Another important crashworthiness criterion used by vehicle engineers is the peak collapse force which is defined as below:
Hence, the maximum force available in the forcedisplacement curve is defined as the F _{max}, which is attempted to reduce for the sake of vehicle safety. Therefore, a suitable thinwalled structure must have less value of F_{max}.
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:
Mean collapse force (F_{mean}) is the ratio of the total energy absorbed by a structure to the effective stroke length δ. Similar to the SEA, the greater is the value of CFE the better is the crashworthiness of thinwalled structure.
5 NUMERICAL RESULTS AND DISCUSSIONS
Crushing behavior of the structures displayed in Fig. 1 was analyzed in LSDYNA based on the explanations given in Section 2. Fig. 5 shows deformation patterns of the foamfilled tapered structures for the case of a/b=0.5. It is noticed that for the sake of brevity, the deformation patterns have been given in Fig. 5 only for the case of a/b=0.5. It is also reminded that the rigidwall identically crushed 75% of the total length of the structures (i.e. 90mm from the total length of 120mm). As seen in Fig. 5 , the square structure showed progressive and concertina deformation pattern; while, the structures with other cross sectional shapes progressively collapsed in diamond mode under axial impact.
Deformation patterns observed for the foamfilled structures under axial and oblique impacts
When the structures were loaded under the oblique impact of θ=10°, all the structures showed mixed mode of progressive diamond and global bending. Meanwhile, the structures tended to slide sideways a little because of the oblique loading. However, effective progressive deformation pattern was still observed for θ=10° similar to the axial impact. For the larger oblique impacts of θ=20° and θ=30°, all the structures collapsed in the less effective global bending mode. Therefore, all the structures absorbed the rigidwall energy by bending at the regions near to the incidental and distal ends. As is well known, structures are desired to be loaded axially due to dissipating remarkable collision energy; nevertheless, the oblique impact is inevitable in the real vehicle collisions, so the structures must have ability of absorbing higher collision energy under oblique impacts.
Fig. 6 displays samples of the forcedisplacement curves extracted from the numerical simulations for the foamfilled square tapered structures. Other structures studied in this research have the similar forcédisplacement curves. Based on Fig. 6 , for the case of axial impact, the force initially enhanced to reach its peak value to form the first wrinkle, and then oscillated several times corresponding to the formation of the subsequent wrinkles. It is noteworthy that magnitude of the first peak forces is nearly equal with the subsequent peak forces due to the tapered geometry of the structures; while, there is a remarkable difference between the mentioned peak forces in the straight structures. For the case of the oblique impact of θ=10°, the trends of force variations were similar to the case of axial impact with two differences: 1) the force initially increased with less steep compared to the axial impact, 2) the peak values of force were less than those for the case of axial impact. This behavior is imputed to this fact that the rigidwall collapsed only a partial area of cross section of the structures at the beginning of the oblique impact. For the cases of larger oblique impacts (θ=20° and θ=30°), the force increased up to a maximum value followed by a rapid reduction due to the formation of global bending occurred at the distal end of the structures.
Forcedisplacement curves for the square structures subjected to the impact angles a) Ɵ=0°, b) Ɵ=10°, c) Ɵ=20°, d) Ɵ=30°
Fig. 7 a depicts values of the SEA calculated for the foamfilled square tapered structures with different ratios of a/b under different impact angles. As seen in Fig. 7 a, by increasing the ratio of a/b, the value of SEA increased to reach its maximum value at the ratio of a/b=0.5, and then decreased. Meanwhile, this trend of SEA variations was observed for all the structures with different cross sectional shapes under both the axial and oblique impacts (see Tables 1 a5 a). Furthermore, as seen in Fig. 7 a, by increasing the impact angle, values of the SEA reduced due to changing the collapse mode from progressive folding into the global bending. As is well known, the bending mode dissipates less energy than the progressive folding mode.
Crashworthiness indicators of a) SEA, b) F_{max} and c) CFE for the square structures under different loading angles
TOPSIS calculations for the foamfilled square tapered structures, a) decision making matrix, b) normalized decision making matrix, c) ranking results
Fig. 7 b shows values of the F_{max} for the square structures. Similar to the results observed for the SEA, by increasing the ratio of a/b, value of the F_{max} increased to reach its peak value at the ratio of a/b=0.5, and then reduced. In addition, by increasing the impact angle, values of the F_{max} decreased. Because, the collapse mode was changed from progressive folding into the global bending as mentioned above; therefore, the level of force values for the bending collapses was considerably less than that for the folding one. It is worthmentioning that this trend of F_{max} variations was also observed for all the structures studied in this research (see Tables 1 a5 a).
Fig. 7 c displays the CFE values calculated for the square structures. Variations of this parameter against the ratio of a/b and impact angle is similar to the results observed for the SEA and F _{max}. In other words, by increasing the ratio of a/b, the CFE enhanced to reach its máximum value at the ratio of a/b=0.5, and then decreased. Furthermore, the CFE reduced as the impact angle decreased. Meanwhile, this trend of CFE variations was observed for all the structures with different cross sectional shapes under both the axial and oblique impacts (see Tables 1 a5 a).
6 SELECTION OF THE SUITABLE RATIO OF a/b AND THE BEST CROSS SECTIONAL SHAPE
In order to select the suitable ratio of a/b and the best cross sectional shape for the structures studied in the present work, TOPSIS (technique for ordering preferences by similarity to ideal solution) method was employed. Indeed, TOPSIS is a decision making method when more than one criterion exists. Since there were three crashworthiness indicators (i.e. SEA, F_{max} and CFE) for stating the crashworthiness capacity of the structures, so a decision making method was needed for selecting the suitable structure. In the TOPSIS, two artificial alternatives are considered i.e. ideal alternative (which has the best level for the crashworthiness indicators) and negative ideal alternative (which has the worst values of crashworthiness indicators). TOPSIS selects the alternative closest to the ideal alternative and farthest from the negative ideal one. More details on the formulation and theory of the TOPSIS can be found in ( Pirmohammad and Nikkhah, 2017 Pirmohammad, S. Nikkhah, H., (2017). Crashworthiness investigation of bitubal columns reinforced with several inside ribs under axial and oblique impact loads, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 0954407017702986 ).
The TOPSIS method was implemented on the numerical results. It is noticed that there were twelve criteria (4 impact angles× 3 crashworthiness indicators, SEA, F_{max} and CFE) in the current work. Therefore, eight criteria were taken into account for the selection of appropriate ratio of a/b and cross sectional shape. As mentioned above, an appropriate energy absorber must have greater SEA and CFE, and smaller F_{max}. Table 1 presents the TOPSIS calculations including the decision matrix, normalized decision matrix and the ranking results for the foamfilled tapered square structures. For the sake of brevity, only the ranking results for the structures with other cross sectional shapes (hexagon, octagon, decagon and circle) have been given in Tables 2  5 . According to Tables 1  5 , ranking of the alternatives was finally obtained as: S0.5>S0.75>S0.25>S0>S1, H0.5>H0.75>H0.25>H0>H1, O0.5>O0.75>O0.25>O0>O1, D0.5>D0.75>D0.25>D0>D1 and C0.5>C0.75>C0.25 >C0>C1. As a result, foamfilled tapered structures with the ratio of a/b=0.5 demonstrated more suitable energy absorbing devices compared with other ratios of a/b due to having more and useful interactions between the structure walls and foam. This result signifies that the structures fully filled with foam (i.e. S0, H0, O0, D0 and C0) were not the efficient structure in terms of the crashworthiness capacity. Interestingly, semifoam filled structures (i.e. S0.5, H0.5, O0.5, D0.5 and C0.5) exhibited greater crashworthiness capability. Therefore, by using the stiffened bitubal structures with the ratio of 0.5, we can save in foam consumption, and weight of the structures can be significantly reduced as well. On the other hand, the singlecell structures (without the inner tube and foam) showed the worst energy absorption capacity. Consequently, the foamfilled structures generally demonstrated better crashworthiness performance than singlecell structures without the foam filler. Indeed, the foam available within the structures resisted against the inward penetration of the structure walls and stiffening plates during the collapsing process. These useful interactions between the foam and structure walls improved the crashworthiness capacity of the foamfilled structures.
TOPSIS calculations for the foamfilled hexagonal tapered structures, a) decision making matrix, b) normalized decision making matrix, c) ranking results
TOPSIS calculations for the foamfilled octagonal tapered structures, a) decision making matrix, b) normalized decision making matrix, c) ranking results
TOPSIS calculations for the foamfilled decagonal tapered structures, a) decision making matrix, b) normalized decision making matrix, c) ranking results
TOPSIS calculations for the foamfilled circular tapered structures, a) decision making matrix, b) normalized decision making matrix, c) ranking results
In the next step, TOPSIS calculations were made on the foamfilled tapered structures with the ratio of a/b=0.5 to find the best cross sectional shape. Table 6 presents the TOPSIS results for these structures. As is evident from this Table, ranking of the structures are: D0.5>C0.5>O0.5>H0.5>S0.5. Therefore, foamfilled decagonal structure with the ratio of a/b=0.5 was found as the best energy absorber, which can be imputed to having more corners compared with other cross sectional shapes.
TOPSIS calculations for the foamfilled tapered structures with the ratio of a/b=0.5, a) decision making matrix, b) normalized decision making matrix, c) ranking results
7 CONCLUSIONS
This paper explored the crashworthiness responses of several new designed foamfilled tapered thinwalled structures with different ratios of a/b under axial and oblique impact load conditions. The crashworthiness criteria of specific energy absorption (SEA), peak collapse force (F _{max}) and CFE (crash force efficiency) were taken into account to compare these structures. The following important notes can be presented as outcomes of this research study:

Generally, all the foamfilled tapered structures showed progressive and concertina deformation pattern under axial impact. When they were loaded under the oblique impact of θ=10°, all the structures showed mixed mode of progressive diamond and global bending. For the larger oblique impacts of θ=20° and θ=30°, all the structures were collapsed in the less effective global bending mode.

As the ratio of a/b increased, values of the SEA and CFE increased to reach their maximum value at the ratio of a/b=0.5, and then decreased for all the structures with different cross sectional shapes under the axial and oblique impacts. Furthermore, by increasing the impact angle, values of the SEA and CFE reduced due to changing the collapse mode from progressive folding into global bending.

By increasing the ratio of a/b, value of the F_{max} increased to reach its peak value at the ratio of a/b=0.5, and then reduced. In addition, by increasing the impact angle, value of the F_{max} decreased.

According to the TOPSIS results, the foamfilled tapered structures with the ratio of a/b=0.5 demonstrated more suitable energy absorbing devices than those with other ratios of a/b for the structures with any cross sectional shape. This result signifies that the semifoam filled structures (i.e. S0.5, H0.5, O0.5, D0.5 and C0.5) offered greater crashworthiness performance. It is noticed that by using these structures, we can save in foam consumption as well as weight of the structures can be significantly reduced.

From the TOPSIS calculations, the decagonal foamfilled tapered structure was finally found as the best design of cross sectional shape in terms of crashworthiness performance. Therefore, this foamfilled tapered structure can be an outstanding potential profile for the automobile structural components.
ACKNOWLEDGMENTS
This research project has been financially supported by the Office of Vice Chancellor for Research under a research grant of Project No.: 136596/7/22 from University of Mohaghegh Ardabili. This support is gratefully acknowledged.
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Publication Dates

Publication in this collection
28 June 2018 
Date of issue
2018
History

Received
17 Oct 2017 
Reviewed
21 Dec 2017 
Accepted
26 Jan 2018