- Email: [email protected]
Effect of eccentric loading on energy absorbing circular cap and open end frusta tube structures
Vacuum xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Effect of eccentric loading on energy absorbing circular cap and open end frusta tube structures Vivek Patel∗, Gaurav Tiwari, Ravikumar Dumpala Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, 440010, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Crashworthiness Energy absorption Quasi-static eccentric loading Cap end frusta Open end frusta LS-DYNA
The response of cap end (C) and open end (O) frusta tube made of aluminium alloy AA-1080 were explored against eccentric loading by carrying out numerical simulation. The thickness of both type of frusta tube was varied as 1.15, 1.2 and 1.3 mm whereas height of open end and cap end frusta tube were considered as 91.6 and 103 mm respectively. The semi-apical angle was varied as 5.71°and 7.59° which resulted base diameter 60.8 and 67.2 mm for both type of tubes respectively. The variation in eccentricity of loading from the axis of the tubes was kept as 5, 8 and 12 mm. The numerical simulations were carried out by using commercial finite element code LS-DYNA. The performance of structures was decided based on crashworthiness parameters like Peak force, Mean crushing force, Crash load efficiency and Energy absorption capability. For validating the numerical results, experiments were performed on compression testing machine wherein both cap end and open end specimens with the thickness of 1.15 mm were compressed with eccentricity 5 mm. The actual results were found close enough with predicted results. Cap end structure showed higher Crash load efficiency and energy absorption compared to the open end frusta tube.
1. Introduction The structures with light weight, stronger and tougher material show superior performance in automobile, naval, aerospace and locomotive industries where these are very often subjected to impact or shock loading during their service life. In this regard thin walled structure in the form of tubular structure, honeycomb structure, metal and non-metal foams have drawn the major attention of most of the researchers due to their superior strength to weight ratio. Even though the structures having honeycomb and metal foam confirmed their superiority compare to tubular structure, the tubular structures have secured its own application in various industrial applications due to vast availability and low cost. Plenty of studies are available wherein the energy absorption of the tubular structure with varying cross-section like circular, rectangular, square, conical, triangular, pyramidal and hexagonal have been reported. In spite of good energy absorber against longitudinal or lateral compressive loading, the circular tubes failed in global buckling mode particularly when subjected to eccentric or oblique crushing load. Recently graded structures with the combination of functionally graded material were more concern to increase energy absorbing capability [1,2]. Gradient structures mainly corrugated tube [3–7] and tapered or frusta tube having a good load uniformity that enhance the ∗
capability of structures to absorb maximum part of impact energy. Performance of these structures were good comparatively to the uniform structures like circular, rectangular, square, triangular, pyramidal and hexagonal. Graded structures having gradient properties like thickness, diameter and width considered as main dominating factors when designing an energy absorber. Frusta tube structures are one of the type of graded structure that shows the gradient property like graded diameter (change in diameter throughout the height) [8] and graded thickness (non-uniformity in thickness throughout the length of tube) [9–11] that help to sustain both axial and oblique impact. Tapered tubes is employed for improving the uniformity in load as it overcame the tendency of Euler – buckling mode of deformation which rises due to the effect of inertia [12] at the time of loading. Mamalis et al. [13] investigated the axial compression behaviour of aluminium conical frusta of constant large end diameter with varying thickness from 0.28 to 1.27 mm for two semi-apical angles, 5° and 10°. Specimens were deformed with the dominant lobe deformation mechanism of in extensional two, three and four lobe formation. In the similar context Alghamdi et al. [14] explored the collapse mode formation of frusta tube with varying semi apical angle (15°–60°) and thickness (1–3 mm). Obtained deformation modes were classified in five modes which are: (1) outward inversion, (2) limited inward
Corresponding author. E-mail address: [email protected] (V. Patel).
https://doi.org/10.1016/j.vacuum.2018.10.056 Received 31 July 2018; Received in revised form 14 October 2018; Accepted 20 October 2018 0042-207X/ © 2018 Published by Elsevier Ltd.
Please cite this article as: Patel, V., Vacuum, https://doi.org/10.1016/j.vacuum.2018.10.056
Vacuum xxx (xxxx) xxx–xxx
V. Patel et al.
θ e t Dl Ds L
Nomenclature Peak Force PF Mean Crushing Force MCF Crash Load Efficiency CLE Energy Absorption Capacity EAC
Semi –apical angle Eccentricity Thickness Larger end diameter Smaller end diameter Length of tube
fabricated with a constant semi apical angle, thickness and lower end diameter as 5.71°, 1.15 mm and 60.8 mm respectively. Upper end diameter for open end taken as 42.8 mm while 41.6 mm for cap end frusta tube. Fig. 2 shows the fabricated samples of both type of frusta tube that used for experimental analysis.
inversion followed by outward inversion, (3) full inward inversion followed by outward inversion, (4) limited extensible crumpling followed by outward inversion, and (5) full extensible crumpling. In another study of Alghamdi [15] focused on higher range of semi-apical angle (65°–80°). These structure were deformed with the formation of maximum plastic hinges and formed a new mode called as folding crumpling mode. Taper angle in the range of 75° having better choice as it absorbed maximum energy. Frusta tubes were also studied by constraining its radial movement. Sobky et al. [16] worked on such type of frusta where they constrained the radial movement and prepared four type of frusta which named as (i) Top constrained frusta (ii) Base constrained frusta (iii) Fully constrained frusta & (iv) Non-constrained frusta. Top end constrained frusta exhibited the highest increase in specific energy compared with non-constrained frusta tubes. In spite of constrained radial motion, frusta tubes having square or rectangular cross-section were studied with the effect of tapered faces. Nagel et al. [3] explored the behaviour of uniform and tapered tubular structure with the variation in thickness, taper angle and the number of tapered faces. They were concluded that taper face have dominant effect along with the taper angle on the energy absorption as it more when compared to the uniform tubes. Most of the available studies have been addressed the response of frusta tube against longitudinal and lateral load and the studies pertaining the effect of eccentric or oblique load are very limited in spite of its severity on energy absorption capability of the structure. Moreover the effect of cap end effect has also not been studied much. Therefore in depth study is required to explore the behaviour of frusta tubes against eccentric loading with cap and open end. In the present study the combined effect of top end constrained (open and cap), frusta wall thickness (1.15, 1.2 and 1.3 mm), semi-apical angle (5.71°and 7.59°) and eccentricity of the loading (5, 8 and 12 mm) from the axis of the structure on the crashworthiness behaviour of the frusta tube has been studied.
3.2. Experimental setup The quasi-static eccentric test was performed using the compression-testing machine (CTM) having a capacity of 20 Tonne. To incorporate eccentricity, line contact was provided at corresponding offset radial distance from the axis of frusta tube. A round bar of 4 mm radius and 65 mm length was attached with upper jaw of the machine. Fig. 3 shows the complete setup arrangement for experimental and numerical analysis. The bottom plate was fixed while the upper plate moved towards downward to compress the specimen, which achieved by using a displacement-controlled mode in CTM. Compression test provides information related to the deformation behaviour from the obtained load-displacement curve. PF and MF are the main parameters which are found with the help of load-displacement curve. 4. Numerical description Low-velocity impact response of aluminum frusta tube was analysed using a commercial finite element code LS-DYNA [17]. FEM model was used for analysis which consists of four parts: the circular frusta tube, a roller, a bottom fixed rigid plate and a movable top rigid plate. The frusta tube was modelled with a four-node shell element known as Belytschko–Tsay shell element. The specimen mesh size was defined as per the run time of the simulation which was, in this case, discretized a mesh of 1.2 mm. To incorporate the large deformation behaviour of the material, MAT24 PIECEWISE LINEAR PLASTICITY material model was used. AUTOMATIC SINGLE SURFACE contact was used to provide a self-contact and contact between the specimen and both rigid plate was simulated with AUTOMATIC SURFACE TO SURFACE contact. All contact surfaces was simulated with 0.3 and 0.4 static and dynamic coefficient of friction respectively. The top and bottom plate was considered as rigid plate. Test samples having thickness 1.15, 1.2 and 1.3 mm with angle 5.71° and 7.59° were model for both capped end frusta and open end frusta.
2. Material properties The material chosen for analysis of frusta tube under eccentric loading is commercially available aluminium alloy AA-1080, which chemical composition has been shown in Table 1. The true stress-strain curve of the aluminium AA-1080 was obtained by conducting uni-axial tension test for that the specimens of the standard ASTM E 8M-98 was cut from the frusta tubes, see Fig. 1.
4.1. Coding of specimen 3. Experimental details For identification of samples, each was coded to a combination of alphabets and number having a specific meaning. Each samples have 4 parts showing different notations. First denote the structure of sample (either capped (C) or open end (O)), second shows the semi apical angle (θ1 = 5.71° and θ2 = 7.59°), third for thickness (t1 = 1.15 mm, t2 = 1.2 mm and t3 = 1.2 mm) and fourth part shows eccentricity factor
3.1. Sample preparation The circular frusta tubes were fabricated through spinning process of an aluminium sheet having thickness and diameter 1.2 mm and 140 mm respectively. Cap end (C) and Open-end (O) frusta were Table 1 Chemical composition of aluminium alloy. %
Si
Fe
Cu
Mn
Zn
Cr
Ti
Other
Aluminium
0.043
0.145
0.0017
0.0013
< 0.0017
0.0013
0.025
0.0182
99.77
2
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Fig. 1. True stress – strain plot from tensile test of AA-1080 specimen.
Fig. 2. Fabricated thin walled a) Open end (O) and b) Cap end (C) frusta tube samples.
ring transformed to diamond mode on further application of load. Deformation pattern observed were similar in both experimental and numerical simulations. Although pattern of load-deformation curve were similar but numerical results of peak force, mean load and CLE increased by 9.11%, 20.3% and 9% respectively. This difference occurred in both analysis due to the non-linearity (thickness varied in the range of 1–1.15) of thickness arises in the fabrication process, which was considered as uniform in case of simulation. Considering same material model series of simulation test were performed at different eccentric position for both cap and open end frusta.
(e1 = 5 mm, e2 = 8 mm, and e3 = 12 mm). Total 36 simulations were performed with variation in thickness, taper angle and eccentricity. Specifications of all samples are listed in Table 2. 4.2. Validation For validating the results of the numerical analysis, one of the samples (O_θ1_t1_e1) was tested experimentally on compression testing machine (CTM) with 20 Tonne capacity under quasi-static loading with a rate of 1.2 mm/min. Boundary and loading condition used in the experimental test was similar to that adopted in simulation analysis. Mechanical properties extracted from the stress-strain graph used for simulation which provided similar characteristics of experimental observation. Collapse properties of experimental and simulation results are shown in Table 3 and the variation of force during the deformation is shown in Fig. 4. Deformation pattern also considered for validating the result that found similar modes of deformation during compression of the structure. Fig. 5 shows the comparison of obtain deformation pattern. Initially line loading at pre-defined eccentricity happens that initiate the deformation with inside folding of structure. Inner folded concentric
5. Results and discussion Two tube structures cap and open end were tested with the different eccentric position under quasi-static loading. The height difference of cap and open end was 11.4 mm which was negligible in crash analysis. The main dominant factors were the semi-apical angle, eccentric position and thickness analysed with reference to the obtained deformation behaviour. The performance of cap and open end frusta tubes were compared at their respective height.
3
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Fig. 3. Eccentric loading setup: a) Finite element arrangement and b) experimental setup.
Table 2 Sample specification.
Table 3 Comparison of experimental and numerical results.
Sample
t (mm)
e (mm)
Ds (mm)
Dl (mm)
Experimental
Numerical
O_θ1_t1_e1 O_θ1_t1_e2 O_θ1_t1_e3 O_θ1_t2_e1 O_θ1_t2_e2 O_θ1_t2_e3 O_θ1_t3_e1 O_θ1_t3_e2 O_θ1_t3_e3 O_θ2_t1_e1 O_θ2_t1_e2 O_θ2_t1_e3 O_θ2_t2_e1 O_θ2_t2_e2 O_θ2_t2_e3 O_θ2_t3_e1 O_θ2_t3_e2 O_θ2_t3_e3 C_θ1_t1_e1 C_θ1_t1_e2 C_θ1_t1_e3 C_θ1_t2_e1 C_θ1_t2_e2 C_θ1_t2_e3 C_θ1_t3_e1 C_θ1_t3_e2 C_θ1_t3_e3 C_θ2_t1_e1 C_θ2_t1_e2 C_θ2_t1_e3 C_θ2_t2_e1 C_θ2_t2_e2 C_θ2_t2_e3 C_θ2_t3_e1 C_θ2_t3_e2 C_θ2_t3_e3
1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3 1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3 1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3 1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3
5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12
42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6
60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2
PF (KN)
MF (KN)
CLE
PF (KN)
MF (KN)
CLE
7.68
5.06
0.66
8.38
6.09
0.72
contact was present other than surface contact. Plastic hinges formed initially, initiate the folding of the collapsible structure. Due to line loading the first plastic hinge was formed at the 6.2 mm from upper end. Initially concentric mode of deformation was observed which subsequently transformed to diamond mode of deformation. Experimental and numerical analysis showed same mode of deformation. Fold formation were started with the internal folding that stretched the structure toward the axis and it was reflected as an elliptical shape cross-section. 5.2. Effect of eccentricity In real condition impact might be at some offset distance or some oblique condition (angle to the horizontal plane). Considering this state of loading condition, three offsets or eccentricity distance (5, 8 and 12 mm) was adopted and the behaviour of the structure was analysed. According to literature, non-uniformity in eccentricity [18] were chosen to explore the maximum region with the minimum set of variation in eccentric distance. As the upper end diameter kept constant, selection of eccentricity distance is such that it covers the maximum distance of half part of the upper end diameter. Therefore, 5, 8 and 12 mm non uniform eccentricity were chosen. Eccentric loading effect was tested on both type of structure and found that with the increase in eccentricity PF decreases up to 5% for cap end frusta having semi apical angle 5.71° and up to 9% for taper angle 7.59° while it was up to 12% for open end frusta tube for both taper angle. Less difference between PF and MCF is always beneficial for a better energy absorber system [19].
5.1. Deformation mode At the time of impact line load applied that helped to initiate the deformation at lower initial maximum peak which is a desirable condition for the absorber. In eccentric loading condition at beginning of deformation line
5.3. Effect of thickness The behaviour of the structure under impact affected mainly by the thickness of the shell that reflects how much capable the structure to 4
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Fig. 4. Comparison of force – displacement plot from experimentally and numerically compressive testing of open end frusta tube.
end. In case of open end PF and MF varied up to 31% and 38% respectively for angle 5.71° and for taper angle 7.59° PF increased up to 32% and MF increased up to 38%. With the constrained motion of upper end, cap end frusta absorbed slightly more energy compared to the open end. It is due to the formation of plastic hinge in case of open end at very early stage of impact while for cap end it not formed as motion is constrained.
resist the initial impact force. Three wall thickness 1.15, 1.2 and 1.3 mm was considered as wall thickness of frusta tube for simulation. Fig. 6 shows the load-displacement plot for both cap and open end structure. From curve, it reflected that for the open end frusta, the load increased drastically with initial increase in compression, after reaching the initial maximum peak, load sudden drop to half of the initial load while in case of cap end frusta load not drop suddenly while it progressed certain distance after it dropped down and then continuously formed stable deformation. In cap end frusta plastic hinges did not form as early as formed in open end. This happened due to the constraint motion of cap end tube which avoids the elliptical shape of deformation. In case of open end structure, top surface expanded at the time of impact which resemble as the elliptical cross section during first peak. It was not evident in cap end due to restriction offered by the cap surface. After getting initial peak the load became approximately constant till the formation of plastic hinge in case of cap end frusta tube due to the constraint motion while sudden drop was found in case of open end. From Fig. 7 the variation in PF and MCF for samples of both cap and open end structure can be easily observed. With the increase in thickness PF increased up to 33% and MF up to 34% for taper angle 5.71° while for angle 7.59°, PF and MF increased up to 31% and 36% respectively for the case of cap
5.4. Crash load efficiency The maximum force was considered as critical parameter during the crash event that shown in Fig. 1 as initial peak force. Along with that focus on the mean force as it help to getting response of structure during the deformation phase rather than finding instantaneous force. Both parameter decided the crashing efficiency which defined as the ratio of MCF to the PF and it should be maximum [20] for good energy absorber. Fig. 8 shows the comparison in CLE of tested samples which was maximum for the case of cap end tubular frusta. Cap end restrict the radial movement due to the constraint at upper diameter end. Constraint motion provide stability in load that help structure to collapse with approximately constant variation in load. It helped to maintain the mean force maximum which was not achieved in
Fig. 5. Comparison of deformation pattern of open end frusta tube, tested: a) experimentally and b) numerically. 5
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Fig. 6. Load-displacement plot of a) cap end and b) open end frusta tube. 6
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Fig. 7. Comparison of peak force and mean force of a) cap end and b) open end frusta tube.
Fig. 8. Comparison of crash load efficiency: a) cap end and b) open end frusta tube. 7
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case of open end. The minimum difference between peak and mean force were obtained for cap end, hence the CLE was higher for cap end.
[2] M. Merklein, M. Wieland, M. Lechner, S. Bruschi, A. Ghiotti, Hot stamping of boron steel sheets with tailored properties: a review, J. Mater. Process. Technol. 228 (2016) 11–24. [3] D. Chen, S. Ozaki, Numerical study of axially crushed cylindrical tubes with corrugated surface, Thin-Walled Struct. 47 (11) (2009) 1387–1396. [4] G. Daneshi, S. Hosseinipour, Elastic–plastic theory for initial buckling load of thinwalled grooved tubes under axial compression, J. Mater. Process. Technol. 125 (2002) 826–832. [5] A. Eyvazian, M.K. Habibi, A.M. Hamouda, R. Hedayati, Axial crushing behavior and energy absorption efficiency of corrugated tubes, Mater. Des. (1980-2015) 54 (2014) 1028–1038. [6] S. Salehghaffari, M. Tajdari, M. Panahi, F. Mokhtarnezhad, Attempts to improve energy absorption characteristics of circular metal tubes subjected to axial loading, Thin-Walled Struct. 48 (6) (2010) 379–390. [7] A. Taştan, E. Acar, M. Güler, Ü. Kılınçkaya, Optimum crashworthiness design of tapered thin-walled tubes with lateral circular cutouts, Thin-Walled Struct. 107 (2016) 543–553. [8] F. Xu, X. Zhang, H. Zhang, A review on functionally graded structures and materials for energy absorption, Eng. Struct. 171 (2018) 309–325. [9] X. Zhang, H. Zhang, Relative merits of conical tubes with graded thickness subjected to oblique impact loads, Int. J. Mech. Sci. 98 (2015) 111–125. [10] P. Gupta, A study on mode of collapse of varying wall thickness metallic frusta subjected to axial compression, Thin-Walled Struct. 46 (5) (2008) 561–571. [11] H. Zhang, X. Zhang, Crashworthiness performance of conical tubes with nonlinear thickness distribution, Thin-Walled Struct. 99 (2016) 35–44. [12] G. Nagel, D. Thambiratnam, A numerical study on the impact response and energy absorption of tapered thin-walled tubes, Int. J. Mech. Sci. 46 (2) (2004) 201–216. [13] A. Mamalis, D. Manolakos, M. Ioannidis, P. Kostazos, Numerical simulation of thinwalled metallic circular frusta subjected to axial loading, Int. J. Crashworthiness 10 (5) (2005) 505–513. [14] A. Alghamdi, A. Aljawi, T.-N. Abu-Mansour, Modes of axial collapse of unconstrained capped frusta, Int. J. Mech. Sci. 44 (6) (2002) 1145–1161. [15] A. Alghamdi, Folding-crumpling of thin-walled aluminium frusta, Int. J. Crashworthiness 7 (1) (2002) 67–78. [16] H. El-Sobky, A. Singace, M. Petsios, Mode of collapse and energy absorption characteristics of constrained frusta under axial impact loading, Int. J. Mech. Sci. 43 (3) (2001) 743–757. [17] J. Hallquist, L. D.K.U.s, Manual, Livermore Software Technology Corporation, 2007, Google Scholar, 2006. [18] A. Rashedi, I. Sridhar, K. Tseng, N. Srikanth, Minimum mass design of thin tubular structures under eccentric compressive loading, Thin-Walled Struct. 90 (2015) 191–201. [19] A.A. Nia, J.H. Hamedani, Comparative analysis of energy absorption and deformations of thin walled tubes with various section geometries, Thin-Walled Struct. 48 (12) (2010) 946–954. [20] G. Nagel, D. Thambiratnam, Computer simulation and energy absorption of tapered thin-walled rectangular tubes, Thin-Walled Struct. 43 (8) (2005) 1225–1242.
6. Conclusion The crushing performance of cap and open end frusta tube structure against eccentric loading were investigated numerically with varying wall thickness and semi-epical angle with reference to the EAC, PF, MF and CLE.
• For the open end frusta tube, during deformation the load suddenly • • • •
dropped after reaching a peak while it takes some time for the case of cap end frusta tube that reflected maximum energy absorption capacity of cap end frusta tube. This happened due to the constrained motion at initial impact. The eccentric loading acted as an initial trigger that started a deformation at an early stage of impact with low initial maximum PF. The early stage of deformation led to the stability of the load-deformation curve that helped to control the MF during the crash event. With the increase in eccentricity, the EAC of the frusta tube structures was reduced which enhanced the probability of Euler-buckling mode of deformation. Increase in taper angle slightly increased the PF which caused the Euler buckling mode deformation. Crash load efficiency (CLF) was maximum for the cap end compared to open end frusta tube.
Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.vacuum.2018.10.056. References [1] M. Merklein, M. Johannes, M. Lechner, A. Kuppert, A review on tailored blanks—production, applications and evaluation, J. Mater. Process. Technol. 214 (2) (2014) 151–164.
8
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Effect of eccentric loading on energy absorbing circular cap and open end frusta tube structures Vivek Patel∗, Gaurav Tiwari, Ravikumar Dumpala Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, 440010, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Crashworthiness Energy absorption Quasi-static eccentric loading Cap end frusta Open end frusta LS-DYNA
The response of cap end (C) and open end (O) frusta tube made of aluminium alloy AA-1080 were explored against eccentric loading by carrying out numerical simulation. The thickness of both type of frusta tube was varied as 1.15, 1.2 and 1.3 mm whereas height of open end and cap end frusta tube were considered as 91.6 and 103 mm respectively. The semi-apical angle was varied as 5.71°and 7.59° which resulted base diameter 60.8 and 67.2 mm for both type of tubes respectively. The variation in eccentricity of loading from the axis of the tubes was kept as 5, 8 and 12 mm. The numerical simulations were carried out by using commercial finite element code LS-DYNA. The performance of structures was decided based on crashworthiness parameters like Peak force, Mean crushing force, Crash load efficiency and Energy absorption capability. For validating the numerical results, experiments were performed on compression testing machine wherein both cap end and open end specimens with the thickness of 1.15 mm were compressed with eccentricity 5 mm. The actual results were found close enough with predicted results. Cap end structure showed higher Crash load efficiency and energy absorption compared to the open end frusta tube.
1. Introduction The structures with light weight, stronger and tougher material show superior performance in automobile, naval, aerospace and locomotive industries where these are very often subjected to impact or shock loading during their service life. In this regard thin walled structure in the form of tubular structure, honeycomb structure, metal and non-metal foams have drawn the major attention of most of the researchers due to their superior strength to weight ratio. Even though the structures having honeycomb and metal foam confirmed their superiority compare to tubular structure, the tubular structures have secured its own application in various industrial applications due to vast availability and low cost. Plenty of studies are available wherein the energy absorption of the tubular structure with varying cross-section like circular, rectangular, square, conical, triangular, pyramidal and hexagonal have been reported. In spite of good energy absorber against longitudinal or lateral compressive loading, the circular tubes failed in global buckling mode particularly when subjected to eccentric or oblique crushing load. Recently graded structures with the combination of functionally graded material were more concern to increase energy absorbing capability [1,2]. Gradient structures mainly corrugated tube [3–7] and tapered or frusta tube having a good load uniformity that enhance the ∗
capability of structures to absorb maximum part of impact energy. Performance of these structures were good comparatively to the uniform structures like circular, rectangular, square, triangular, pyramidal and hexagonal. Graded structures having gradient properties like thickness, diameter and width considered as main dominating factors when designing an energy absorber. Frusta tube structures are one of the type of graded structure that shows the gradient property like graded diameter (change in diameter throughout the height) [8] and graded thickness (non-uniformity in thickness throughout the length of tube) [9–11] that help to sustain both axial and oblique impact. Tapered tubes is employed for improving the uniformity in load as it overcame the tendency of Euler – buckling mode of deformation which rises due to the effect of inertia [12] at the time of loading. Mamalis et al. [13] investigated the axial compression behaviour of aluminium conical frusta of constant large end diameter with varying thickness from 0.28 to 1.27 mm for two semi-apical angles, 5° and 10°. Specimens were deformed with the dominant lobe deformation mechanism of in extensional two, three and four lobe formation. In the similar context Alghamdi et al. [14] explored the collapse mode formation of frusta tube with varying semi apical angle (15°–60°) and thickness (1–3 mm). Obtained deformation modes were classified in five modes which are: (1) outward inversion, (2) limited inward
Corresponding author. E-mail address: [email protected] (V. Patel).
https://doi.org/10.1016/j.vacuum.2018.10.056 Received 31 July 2018; Received in revised form 14 October 2018; Accepted 20 October 2018 0042-207X/ © 2018 Published by Elsevier Ltd.
Please cite this article as: Patel, V., Vacuum, https://doi.org/10.1016/j.vacuum.2018.10.056
Vacuum xxx (xxxx) xxx–xxx
V. Patel et al.
θ e t Dl Ds L
Nomenclature Peak Force PF Mean Crushing Force MCF Crash Load Efficiency CLE Energy Absorption Capacity EAC
Semi –apical angle Eccentricity Thickness Larger end diameter Smaller end diameter Length of tube
fabricated with a constant semi apical angle, thickness and lower end diameter as 5.71°, 1.15 mm and 60.8 mm respectively. Upper end diameter for open end taken as 42.8 mm while 41.6 mm for cap end frusta tube. Fig. 2 shows the fabricated samples of both type of frusta tube that used for experimental analysis.
inversion followed by outward inversion, (3) full inward inversion followed by outward inversion, (4) limited extensible crumpling followed by outward inversion, and (5) full extensible crumpling. In another study of Alghamdi [15] focused on higher range of semi-apical angle (65°–80°). These structure were deformed with the formation of maximum plastic hinges and formed a new mode called as folding crumpling mode. Taper angle in the range of 75° having better choice as it absorbed maximum energy. Frusta tubes were also studied by constraining its radial movement. Sobky et al. [16] worked on such type of frusta where they constrained the radial movement and prepared four type of frusta which named as (i) Top constrained frusta (ii) Base constrained frusta (iii) Fully constrained frusta & (iv) Non-constrained frusta. Top end constrained frusta exhibited the highest increase in specific energy compared with non-constrained frusta tubes. In spite of constrained radial motion, frusta tubes having square or rectangular cross-section were studied with the effect of tapered faces. Nagel et al. [3] explored the behaviour of uniform and tapered tubular structure with the variation in thickness, taper angle and the number of tapered faces. They were concluded that taper face have dominant effect along with the taper angle on the energy absorption as it more when compared to the uniform tubes. Most of the available studies have been addressed the response of frusta tube against longitudinal and lateral load and the studies pertaining the effect of eccentric or oblique load are very limited in spite of its severity on energy absorption capability of the structure. Moreover the effect of cap end effect has also not been studied much. Therefore in depth study is required to explore the behaviour of frusta tubes against eccentric loading with cap and open end. In the present study the combined effect of top end constrained (open and cap), frusta wall thickness (1.15, 1.2 and 1.3 mm), semi-apical angle (5.71°and 7.59°) and eccentricity of the loading (5, 8 and 12 mm) from the axis of the structure on the crashworthiness behaviour of the frusta tube has been studied.
3.2. Experimental setup The quasi-static eccentric test was performed using the compression-testing machine (CTM) having a capacity of 20 Tonne. To incorporate eccentricity, line contact was provided at corresponding offset radial distance from the axis of frusta tube. A round bar of 4 mm radius and 65 mm length was attached with upper jaw of the machine. Fig. 3 shows the complete setup arrangement for experimental and numerical analysis. The bottom plate was fixed while the upper plate moved towards downward to compress the specimen, which achieved by using a displacement-controlled mode in CTM. Compression test provides information related to the deformation behaviour from the obtained load-displacement curve. PF and MF are the main parameters which are found with the help of load-displacement curve. 4. Numerical description Low-velocity impact response of aluminum frusta tube was analysed using a commercial finite element code LS-DYNA [17]. FEM model was used for analysis which consists of four parts: the circular frusta tube, a roller, a bottom fixed rigid plate and a movable top rigid plate. The frusta tube was modelled with a four-node shell element known as Belytschko–Tsay shell element. The specimen mesh size was defined as per the run time of the simulation which was, in this case, discretized a mesh of 1.2 mm. To incorporate the large deformation behaviour of the material, MAT24 PIECEWISE LINEAR PLASTICITY material model was used. AUTOMATIC SINGLE SURFACE contact was used to provide a self-contact and contact between the specimen and both rigid plate was simulated with AUTOMATIC SURFACE TO SURFACE contact. All contact surfaces was simulated with 0.3 and 0.4 static and dynamic coefficient of friction respectively. The top and bottom plate was considered as rigid plate. Test samples having thickness 1.15, 1.2 and 1.3 mm with angle 5.71° and 7.59° were model for both capped end frusta and open end frusta.
2. Material properties The material chosen for analysis of frusta tube under eccentric loading is commercially available aluminium alloy AA-1080, which chemical composition has been shown in Table 1. The true stress-strain curve of the aluminium AA-1080 was obtained by conducting uni-axial tension test for that the specimens of the standard ASTM E 8M-98 was cut from the frusta tubes, see Fig. 1.
4.1. Coding of specimen 3. Experimental details For identification of samples, each was coded to a combination of alphabets and number having a specific meaning. Each samples have 4 parts showing different notations. First denote the structure of sample (either capped (C) or open end (O)), second shows the semi apical angle (θ1 = 5.71° and θ2 = 7.59°), third for thickness (t1 = 1.15 mm, t2 = 1.2 mm and t3 = 1.2 mm) and fourth part shows eccentricity factor
3.1. Sample preparation The circular frusta tubes were fabricated through spinning process of an aluminium sheet having thickness and diameter 1.2 mm and 140 mm respectively. Cap end (C) and Open-end (O) frusta were Table 1 Chemical composition of aluminium alloy. %
Si
Fe
Cu
Mn
Zn
Cr
Ti
Other
Aluminium
0.043
0.145
0.0017
0.0013
< 0.0017
0.0013
0.025
0.0182
99.77
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Fig. 1. True stress – strain plot from tensile test of AA-1080 specimen.
Fig. 2. Fabricated thin walled a) Open end (O) and b) Cap end (C) frusta tube samples.
ring transformed to diamond mode on further application of load. Deformation pattern observed were similar in both experimental and numerical simulations. Although pattern of load-deformation curve were similar but numerical results of peak force, mean load and CLE increased by 9.11%, 20.3% and 9% respectively. This difference occurred in both analysis due to the non-linearity (thickness varied in the range of 1–1.15) of thickness arises in the fabrication process, which was considered as uniform in case of simulation. Considering same material model series of simulation test were performed at different eccentric position for both cap and open end frusta.
(e1 = 5 mm, e2 = 8 mm, and e3 = 12 mm). Total 36 simulations were performed with variation in thickness, taper angle and eccentricity. Specifications of all samples are listed in Table 2. 4.2. Validation For validating the results of the numerical analysis, one of the samples (O_θ1_t1_e1) was tested experimentally on compression testing machine (CTM) with 20 Tonne capacity under quasi-static loading with a rate of 1.2 mm/min. Boundary and loading condition used in the experimental test was similar to that adopted in simulation analysis. Mechanical properties extracted from the stress-strain graph used for simulation which provided similar characteristics of experimental observation. Collapse properties of experimental and simulation results are shown in Table 3 and the variation of force during the deformation is shown in Fig. 4. Deformation pattern also considered for validating the result that found similar modes of deformation during compression of the structure. Fig. 5 shows the comparison of obtain deformation pattern. Initially line loading at pre-defined eccentricity happens that initiate the deformation with inside folding of structure. Inner folded concentric
5. Results and discussion Two tube structures cap and open end were tested with the different eccentric position under quasi-static loading. The height difference of cap and open end was 11.4 mm which was negligible in crash analysis. The main dominant factors were the semi-apical angle, eccentric position and thickness analysed with reference to the obtained deformation behaviour. The performance of cap and open end frusta tubes were compared at their respective height.
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Fig. 3. Eccentric loading setup: a) Finite element arrangement and b) experimental setup.
Table 2 Sample specification.
Table 3 Comparison of experimental and numerical results.
Sample
t (mm)
e (mm)
Ds (mm)
Dl (mm)
Experimental
Numerical
O_θ1_t1_e1 O_θ1_t1_e2 O_θ1_t1_e3 O_θ1_t2_e1 O_θ1_t2_e2 O_θ1_t2_e3 O_θ1_t3_e1 O_θ1_t3_e2 O_θ1_t3_e3 O_θ2_t1_e1 O_θ2_t1_e2 O_θ2_t1_e3 O_θ2_t2_e1 O_θ2_t2_e2 O_θ2_t2_e3 O_θ2_t3_e1 O_θ2_t3_e2 O_θ2_t3_e3 C_θ1_t1_e1 C_θ1_t1_e2 C_θ1_t1_e3 C_θ1_t2_e1 C_θ1_t2_e2 C_θ1_t2_e3 C_θ1_t3_e1 C_θ1_t3_e2 C_θ1_t3_e3 C_θ2_t1_e1 C_θ2_t1_e2 C_θ2_t1_e3 C_θ2_t2_e1 C_θ2_t2_e2 C_θ2_t2_e3 C_θ2_t3_e1 C_θ2_t3_e2 C_θ2_t3_e3
1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3 1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3 1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3 1.15 1.15 1.15 1.2 1.2 1.2 1.3 1.3 1.3
5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12 5 8 12
42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 42.8 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6
60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 60.8 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2
PF (KN)
MF (KN)
CLE
PF (KN)
MF (KN)
CLE
7.68
5.06
0.66
8.38
6.09
0.72
contact was present other than surface contact. Plastic hinges formed initially, initiate the folding of the collapsible structure. Due to line loading the first plastic hinge was formed at the 6.2 mm from upper end. Initially concentric mode of deformation was observed which subsequently transformed to diamond mode of deformation. Experimental and numerical analysis showed same mode of deformation. Fold formation were started with the internal folding that stretched the structure toward the axis and it was reflected as an elliptical shape cross-section. 5.2. Effect of eccentricity In real condition impact might be at some offset distance or some oblique condition (angle to the horizontal plane). Considering this state of loading condition, three offsets or eccentricity distance (5, 8 and 12 mm) was adopted and the behaviour of the structure was analysed. According to literature, non-uniformity in eccentricity [18] were chosen to explore the maximum region with the minimum set of variation in eccentric distance. As the upper end diameter kept constant, selection of eccentricity distance is such that it covers the maximum distance of half part of the upper end diameter. Therefore, 5, 8 and 12 mm non uniform eccentricity were chosen. Eccentric loading effect was tested on both type of structure and found that with the increase in eccentricity PF decreases up to 5% for cap end frusta having semi apical angle 5.71° and up to 9% for taper angle 7.59° while it was up to 12% for open end frusta tube for both taper angle. Less difference between PF and MCF is always beneficial for a better energy absorber system [19].
5.1. Deformation mode At the time of impact line load applied that helped to initiate the deformation at lower initial maximum peak which is a desirable condition for the absorber. In eccentric loading condition at beginning of deformation line
5.3. Effect of thickness The behaviour of the structure under impact affected mainly by the thickness of the shell that reflects how much capable the structure to 4
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Fig. 4. Comparison of force – displacement plot from experimentally and numerically compressive testing of open end frusta tube.
end. In case of open end PF and MF varied up to 31% and 38% respectively for angle 5.71° and for taper angle 7.59° PF increased up to 32% and MF increased up to 38%. With the constrained motion of upper end, cap end frusta absorbed slightly more energy compared to the open end. It is due to the formation of plastic hinge in case of open end at very early stage of impact while for cap end it not formed as motion is constrained.
resist the initial impact force. Three wall thickness 1.15, 1.2 and 1.3 mm was considered as wall thickness of frusta tube for simulation. Fig. 6 shows the load-displacement plot for both cap and open end structure. From curve, it reflected that for the open end frusta, the load increased drastically with initial increase in compression, after reaching the initial maximum peak, load sudden drop to half of the initial load while in case of cap end frusta load not drop suddenly while it progressed certain distance after it dropped down and then continuously formed stable deformation. In cap end frusta plastic hinges did not form as early as formed in open end. This happened due to the constraint motion of cap end tube which avoids the elliptical shape of deformation. In case of open end structure, top surface expanded at the time of impact which resemble as the elliptical cross section during first peak. It was not evident in cap end due to restriction offered by the cap surface. After getting initial peak the load became approximately constant till the formation of plastic hinge in case of cap end frusta tube due to the constraint motion while sudden drop was found in case of open end. From Fig. 7 the variation in PF and MCF for samples of both cap and open end structure can be easily observed. With the increase in thickness PF increased up to 33% and MF up to 34% for taper angle 5.71° while for angle 7.59°, PF and MF increased up to 31% and 36% respectively for the case of cap
5.4. Crash load efficiency The maximum force was considered as critical parameter during the crash event that shown in Fig. 1 as initial peak force. Along with that focus on the mean force as it help to getting response of structure during the deformation phase rather than finding instantaneous force. Both parameter decided the crashing efficiency which defined as the ratio of MCF to the PF and it should be maximum [20] for good energy absorber. Fig. 8 shows the comparison in CLE of tested samples which was maximum for the case of cap end tubular frusta. Cap end restrict the radial movement due to the constraint at upper diameter end. Constraint motion provide stability in load that help structure to collapse with approximately constant variation in load. It helped to maintain the mean force maximum which was not achieved in
Fig. 5. Comparison of deformation pattern of open end frusta tube, tested: a) experimentally and b) numerically. 5
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Fig. 6. Load-displacement plot of a) cap end and b) open end frusta tube. 6
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Fig. 7. Comparison of peak force and mean force of a) cap end and b) open end frusta tube.
Fig. 8. Comparison of crash load efficiency: a) cap end and b) open end frusta tube. 7
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case of open end. The minimum difference between peak and mean force were obtained for cap end, hence the CLE was higher for cap end.
[2] M. Merklein, M. Wieland, M. Lechner, S. Bruschi, A. Ghiotti, Hot stamping of boron steel sheets with tailored properties: a review, J. Mater. Process. Technol. 228 (2016) 11–24. [3] D. Chen, S. Ozaki, Numerical study of axially crushed cylindrical tubes with corrugated surface, Thin-Walled Struct. 47 (11) (2009) 1387–1396. [4] G. Daneshi, S. Hosseinipour, Elastic–plastic theory for initial buckling load of thinwalled grooved tubes under axial compression, J. Mater. Process. Technol. 125 (2002) 826–832. [5] A. Eyvazian, M.K. Habibi, A.M. Hamouda, R. Hedayati, Axial crushing behavior and energy absorption efficiency of corrugated tubes, Mater. Des. (1980-2015) 54 (2014) 1028–1038. [6] S. Salehghaffari, M. Tajdari, M. Panahi, F. Mokhtarnezhad, Attempts to improve energy absorption characteristics of circular metal tubes subjected to axial loading, Thin-Walled Struct. 48 (6) (2010) 379–390. [7] A. Taştan, E. Acar, M. Güler, Ü. Kılınçkaya, Optimum crashworthiness design of tapered thin-walled tubes with lateral circular cutouts, Thin-Walled Struct. 107 (2016) 543–553. [8] F. Xu, X. Zhang, H. Zhang, A review on functionally graded structures and materials for energy absorption, Eng. Struct. 171 (2018) 309–325. [9] X. Zhang, H. Zhang, Relative merits of conical tubes with graded thickness subjected to oblique impact loads, Int. J. Mech. Sci. 98 (2015) 111–125. [10] P. Gupta, A study on mode of collapse of varying wall thickness metallic frusta subjected to axial compression, Thin-Walled Struct. 46 (5) (2008) 561–571. [11] H. Zhang, X. Zhang, Crashworthiness performance of conical tubes with nonlinear thickness distribution, Thin-Walled Struct. 99 (2016) 35–44. [12] G. Nagel, D. Thambiratnam, A numerical study on the impact response and energy absorption of tapered thin-walled tubes, Int. J. Mech. Sci. 46 (2) (2004) 201–216. [13] A. Mamalis, D. Manolakos, M. Ioannidis, P. Kostazos, Numerical simulation of thinwalled metallic circular frusta subjected to axial loading, Int. J. Crashworthiness 10 (5) (2005) 505–513. [14] A. Alghamdi, A. Aljawi, T.-N. Abu-Mansour, Modes of axial collapse of unconstrained capped frusta, Int. J. Mech. Sci. 44 (6) (2002) 1145–1161. [15] A. Alghamdi, Folding-crumpling of thin-walled aluminium frusta, Int. J. Crashworthiness 7 (1) (2002) 67–78. [16] H. El-Sobky, A. Singace, M. Petsios, Mode of collapse and energy absorption characteristics of constrained frusta under axial impact loading, Int. J. Mech. Sci. 43 (3) (2001) 743–757. [17] J. Hallquist, L. D.K.U.s, Manual, Livermore Software Technology Corporation, 2007, Google Scholar, 2006. [18] A. Rashedi, I. Sridhar, K. Tseng, N. Srikanth, Minimum mass design of thin tubular structures under eccentric compressive loading, Thin-Walled Struct. 90 (2015) 191–201. [19] A.A. Nia, J.H. Hamedani, Comparative analysis of energy absorption and deformations of thin walled tubes with various section geometries, Thin-Walled Struct. 48 (12) (2010) 946–954. [20] G. Nagel, D. Thambiratnam, Computer simulation and energy absorption of tapered thin-walled rectangular tubes, Thin-Walled Struct. 43 (8) (2005) 1225–1242.
6. Conclusion The crushing performance of cap and open end frusta tube structure against eccentric loading were investigated numerically with varying wall thickness and semi-epical angle with reference to the EAC, PF, MF and CLE.
• For the open end frusta tube, during deformation the load suddenly • • • •
dropped after reaching a peak while it takes some time for the case of cap end frusta tube that reflected maximum energy absorption capacity of cap end frusta tube. This happened due to the constrained motion at initial impact. The eccentric loading acted as an initial trigger that started a deformation at an early stage of impact with low initial maximum PF. The early stage of deformation led to the stability of the load-deformation curve that helped to control the MF during the crash event. With the increase in eccentricity, the EAC of the frusta tube structures was reduced which enhanced the probability of Euler-buckling mode of deformation. Increase in taper angle slightly increased the PF which caused the Euler buckling mode deformation. Crash load efficiency (CLF) was maximum for the cap end compared to open end frusta tube.
Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.vacuum.2018.10.056. References [1] M. Merklein, M. Johannes, M. Lechner, A. Kuppert, A review on tailored blanks—production, applications and evaluation, J. Mater. Process. Technol. 214 (2) (2014) 151–164.
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