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A new bi-tubular conical–circular structure for improving crushing behavior under axial and oblique impacts
International Journal of Mechanical Sciences 105 (2016) 253–265
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A new bi-tubular conical–circular structure for improving crushing behavior under axial and oblique impacts Mohammadbagher B. Azimi, Masoud Asgari n Faculty of Mechanical Engineering, K. N. Toosi University of Technology, P. O. Box: 19395-1999, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 17 August 2015 Received in revised form 25 October 2015 Accepted 6 November 2015 Available online 14 November 2015
In this study single and double wall structures for crashworthiness are investigated to introduce a novel system with better energy absorption and crushing characteristics under both axial and oblique loading. The new developed double wall structure is constructed from inner conical and outer circular tubes, incorporated with and without foam filler. Foam filled bi-tubular structure subject to direct axial and oblique impact loadings are simulated using nonlinear finite element analysis software package LSDYNA. Numerical simulations obtained via non-linear explicit dynamic FEM are firstly validated using theoretical and experimental solutions. Next, effectiveness of the new developed bi-tubular structure has been shown by comparing with similar common thin-walled structures. Different types of structures namely bi-tubular empty and foam filled new design, empty and filled frusta as well as empty and foam filled circular tubes were considered in order to make a more insightful of capabilities of the proposed structure. A parametric study including the effect of geometrical and material properties of the structure on the crashworthiness has been carried out. Results show the possibility of amending the peak crushing load along with keeping other energy absorption characteristics unchanged or even improved. Also an improvement in absorbed energy under oblique loading by using the new developed structure is observed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Double-walled tube Foam filled Oblique impact Energy absorption Crushing Crashworthiness
1. Introduction The use of thin-walled structures as energy absorbent has been recently proven to be an efficient and applicable solution as they offer high values of absorbed energy in spite of their light weight in the automotive, high speed railway and aerospace industries.These crush boxes are made in different geometries, along which circular [1–3], squared [4–6] and conical tubes [7–8] with a variety of materials such as aluminum, steel, compositesas well as use of foam-filler [8–9] has gained a notable interest. To make an inference of energy absorption capacity and measure the effectiveness of these structures, energy absorption per unit mass called specific energy absorption (SEA) and mean crush forceas well as maximum induced load during crushing called peak crushing force (PCF) has been utilized by researchers. Alexander [1] firstly introduced axial crushing of thin walled circular tubes as a mechanism for energy absorption. Furthermore he was able to derive a theoretical solution by assuming perfect plastic material behavior. Ever since numerous attempts have been made by researchers to improve his solution [10–12]. Singace et al. [13] indicated that frusta have a more stable structure compared to cylinders. n
Corresponding author. Tel.: þ 98 21 8406 3209; fax: þ 98 21 8867 7274. E-mail address: [email protected] (M. Asgari).
http://dx.doi.org/10.1016/j.ijmecsci.2015.11.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
They also observed that constraining the end of frusta leads to more energy absorption [13]. Improving energy absorption capacity and reducing crushing force and weight of structures is the main object considered by researchers in studying of energy absorbers. Numerous researches have been conducted theoretically, experimentally and numerically in thin-walled structures as energy absorbers. In this regard, Seitzberger et al. [14] studied steel tube with aluminum foam fillers. Hanssen et al. [15] confirmed that by using aluminum foam fillers considerable weight savings are achievable. Foam filling would affect the buckling shape causing tubes to go under more plastic deformation. Hanssen et al. [16] also observed a shift in deformation pattern from diamond to concertina mode for a critical value of foam density. Hanefi et al. [17] tested reinforced compound metal/composite wall and derived an analytical model for it based on Alexander's theory. Xue et al. [18] analyzed flat-topped conical cells made of textile composite and found them in good agreement with both theoretical and experimental solution. Kim [19] proposed new multi-cell profiles base on the idea of adding more square elements would cause in more energy absorption and also investigated effect of different types of triggering. By performing optimization they alsowere able to increase the SEA up to 200% [19]. Zhang et al. [20] introduced patterns to conventional surface of squared tube cross section which led to a new octagonal collapse mode as well as an enhancement of the absorbed energy. Ahmad et al. [21] performed a parametric study on foam-filled
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Nomenclature Af CFE Df Dt dm E EA Fx F mean F initialpeak F davg F savg hf ht k; C kD
Ν
cross section area of the foam Crush force efficiency frusta's diameter circular tube's diameter mean diameter Young's modulus energy absorption resultant impact force mean load initial peak force average dynamic crush force average static crush force frusta'slength circular tube's length material parameters dimensionless constant Poisson's ratio
conical tubes and concluded that foam-filled conical tubes collapse in a stable manner while having more lobes in comparison to empty tubes. Hosseini et al. [22] developed a mathematical model by considering the change in thickness of the frusta and tubes during collapse progressing as well as assuming different values for compressive and tensile strength of the materials and found the results more justifiable with experiments. Ghasemnejad et al. [23] investigated the effect of delamination crack growth of hybrid composite tubes and concluded the ones with higher fracture toughness will absorb more energy in crushing process. As an innovative way to absorb energy, Morris et al. [24,25] loaded circular tube segments transversally. This allows the possibility to reduce the occupied space without compromising the crashworthiness characteristics. Further investigation by Baroutaji et al. [26] showed that exposing these tubes to external constraints will lead to more deformed volume as well as specific energy absorption. Li et al. [27] made a comparison between functionally graded thickness (FGT) tubes and straight and tapered tubes with constant thickness found them in advantage to previous models under oblique loading. Lu et al. [28] introduce double functionally graded tubes by filling FGT tube with functionally graded foams (FGF) and found them to improve energy absorption characteristics without raising the initial peak load. Recently a great notice has been focused on using multiwall and multicell structures as they have offered a significant step up on absorbed energy [29–31]. Tang et al. [32] investigated multi-cell cylindrical and square columns and inferred that cylindrical columns offer better energy absorption properties. Jusuf et al. [33] performed a comparison between single-walled and double-walled square tubes and improved the double-walled tubes energy absorption efficiency by introducing internal ribs to them. Fang et al. [34] investigated rectangular multi-cell tubes under axial and oblique loading. They concluded that tubes with fewer cells perform better under oblique loading with large angle. Qiu et al. [35] studied multi-cell hexagonal tubes and found that number of corners plays a significant part in improving energy absorption. Borvik et al. [36] investigated empty and foam filled circular tubes under oblique loading for load angles of β ¼ 0; 5; 15; 30 and found that dramatic reduction in energy absorption occur even for small impact angles. Fang et al. [37] investigated multi-cell square tubes under oblique loading and found that despite their good response under axial loading, global bending occurs for impact angle of 201. Reyes et al. [38] investigated the crushing behavior of aluminum squared section extrusions under oblique loading both experimentally and
Ρ
density specific energy absorption wall thickness velocity plastic coefficient contraction yield stress α semi-epical angle δ maximum deflection of the crushed structure ε^ equivalent strain ε_ plastic strain rate εp plastic strain α2 , γ , εD and β material parameters for foam σf plateau stress σy yield stress σp plateau stress of the foam σe effective Von Mises stress σm mean stress σt true stress ρf 0 density of the foam base material
SEA t V Vp Y
numerically, and concluded that energy absorption drops drastically in a small impact angle of 51 and will continue to drop by increasing the impact angle. While there have been several studies on multi-cell tubes, few studies have investigated the use of bi-tubular structures on crashworthiness parameters. In those studies multi-wall structure with identical thin walls mainly investigated for direct axial loadings [29–31]. So, there exist wide extents of research to optimize the energy absorber structures for different practical applications especially including oblique loadings. Based on this fact, this paper focuses on combining the existing structures for crashworthiness to introduce a novel system with better energy absorption characteristics under both axial and oblique loading. A double walled structure including a frustum inside of a circular tube, incorporated with and without foam filler. Empty and foam filled bi-tubular structure subject to direct and oblique axial impacts are simulated using explicit dynamic finite element analysis software LS-DYNA. The numerical simulations were validated using experimental and theoretical solutions presented in appropriate publications. In order to make a more insightful of capabilities of the proposed structure, a comparison between it and previously existed models has been done. In this regard, different types of structures namely bi-tubular foam Filled (BTF), bi-tubular (BTE), empty and filled frusta (SFE, SFF) and empty and filled circular tubes (ACE, ACF) were considered under oblique impact. In addition a parametric study of new design including effects of geometrical parameters on the structural crashworthiness has been carried out. Obtained results clearly show that the crash ability of the new developed double wall structure is better than the other structures as it achieve a more absorbed energy while the peak crushing force decreases significantly. It is also observed that the empty bitubularcircular-conical structure improves the crushing performance considerably in oblique impacts.
2. Description of the new developed bi-tubular structure The study consists of a new innovative model proposed to offer better energy absorption and comparisons to simpler existing models. Improving crushing behaviors in axial impacts as well as oblique ones has been the motivation to combine circular and conical tubes. This double walled model includes a single frustuminside of a
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3. Finite element modeling 3.1. Mesh convergence and boundary conditions The finite element simulation is executed using non-linear explicit dynamic LS-DYNA solver. Tubes are modeled using shell elements based on Belytschko–Tsay formulation with five integration points through shell thickness to obtain proper accuracy [39]. Aluminum foam filler is modeled by constant stress solid elements. Hourglass control based on the Belytschko hypothesis as well as Bindeman assumed strain stiffness form are applied to avoid both artificial zero energy deformation modes and volumetric locking [41].
To get accurate results, a mesh convergence analysis on a bitubular foam filled structure with semi-epical angle of α ¼ 51, kg with thickness of t¼1.5 mm and foam density of ρf ¼ 220 m 3 various different mesh sizes has been done. As mentioned in some other related references 4 mm mesh size for solid elements is able to capture the deformation of foam filler and leads to accurate results [21,42]. Shell element of different sizes are used to represent finer, fine, medium and coarse mesh effects as represented in Table 1 and Fig. 2. The mesh size of 4 mm produced acceptable results on internal energy and mean force while it caused the inaccurate peak force which is an important parameter and therefore adequate precision is essential to solution. Also a further analysis on foam filled hybrid structures revealed the necessity of applying a finer mesh. Therefore all the simulations are done by 2 mm mesh size for shell tubes that is ensuring sufficient mesh density and will accurately capture the deformation process. An approximate number of 27,000 and 40,000 were produced for 2 mm shell and 4 mm solid elements respectively. “Automatic Surface To Surface” algorithm [43] with static coefficient friction of 0.3 and dynamic coefficient friction of 0.2 was used to account for 160 155 150 Force(KN)
circular tube, incorporated with and without foam filler as shown in Fig. 1a. This bi-tubular structure consists of a frustum with height of hf ¼ 200 mm and mean diameter, dmf ¼56 mm with some specified values of semi-epical angle, α and thickness, t as variables in parametric study. A triggered circular tube with diameterof Dt ¼ 98mm and heightof ht ¼ 230mm is used as outer wall. Fig.1b illustrates the proposed model geometry. The outside geometry was kept the same in parametric study to keep the focus on effects of changes in conical tube and foam filler. Geometries undergo axial and oblique impact loading. Columns are fixed at base to a completely constrained rigid plate. On the other end a 200 kg rigid mass block impacts with 20 m/s initial velocity with only allowable movement in the initial direction.
255
145 140
Peak force
135
Mean crush force
130 125 120 6mm
4mm
2mm
1mm
mesh size Fig. 2. Mesh sensitivity to initial peak and mean force.
Fig. 1. Geometry model of the new design bi-tubular crush box.
Table 1 Mesh convergence study results. Shell element size (mm)
Peak force Internal (kN) energy (kJ)
6
128.4
4 2 1
147.19 156.1 154.83
– 18.37 19.69 19.32
Mean crush force (kN) 122.25 130.98 128.85
Approximate time (min) Termination due to error 32 40 160
Fig. 3. Section view of finite element setup for simulation.
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contacts between the rigid bodies and structures components as well as for interactions between the components themselves. “Tied Node to Surface offset” algorithm [43] was employed to constraint the base of crush box. Also to capture the selfinteraction of shell components “Automatic Single Surface” contact type was utilized. Fig. 3 illustrates the developed model for finite element simulation.
fracture such as choosing critical value for maximum principal stress in elements and energy based fracture criterion developed by Reyes et al. [44], in the following simulations element erosion only occurs when reaching a positive value of 0.02 in volumetric strain. The eroded element mass is retained in solution. The yield criterion is as [44]: ∅ ¼ σ^ Y r U
ð1Þ
where
3.2. Material properties 3.2.1. Material of thin walled tubes Circular tubes are modeled as elasto-plastic materials without considering the strain rate properties since previous studies showed that aluminumcan be considered as strain rate independent [40]. “Mat Piecewise Linear Plasticity” has the ability to model the tube with mentioned configurations. Also arbitrary stress versus strain curve can be defined with this material which is later used to model the carbon steel material [43]. Aluminum 6061 T6 with mechanical properties of density ρal ¼2700 kg/m3, Young's modulus Eal ¼ 68.2 GPa, Poisson'sratio ν ¼0.3 and yield stress of Yal ¼276 Mpa was chosen for study. The aluminum was considered to have a perfect plastic behavior. For steel tubes, on the other hand, strain hardening effects should be considered, therefore to get more accurate results truestress versusplastic strain curve is also specified as described in Table 2. The conical tubes were all made of carbon steel with mechanical properties of density of ρs ¼7809 kg/m3, Young modulus, Es ¼ 200 GPa, Poisson's ratio,ν ¼ 0.3 and yield stress, Ys ¼ 401.4 MPa. Material properties were derived from standard tensile test. In Table 2, εp and σ t denote the plastic strainand the true stress respectively. 3.2.2. Material of foam filler Since volume changes due to collapsing of foam cells under compression, yield criterion should contain a hydrostatic stress term. Deshpande–Fleck [43] material was chosen to model the aluminum closed cell foam filler. This material can simulate the fracture by eroding the elements that have reached a critical value for volumetric strain. Although other criteria exist for modeling
2 1 2 2 α3 i σ e þ α σ m 1þ 3
σ^ 2 ¼ h
ð2Þ
where σ m is the mean stress and σ e shows the effective Von Mises stress. α is a function of plastic coefficient contraction Vp and describes the shape of yield surface. 9 1 2V p ð3Þ α2 ¼ 2 1þVp V p is assumed to be zero for aluminumfoams [36,45] which will lead to α ¼ 2:12. The yield stress Y is defined as " # ε^ 1 Y ¼ σ p þ γ þ α2 ln ð4Þ εD 1 ðεε^D Þβ In which σ p (plateau stress of the foam), α2 ,γ ,εD and β are material parameters and equivalent strain is defined as ε^ . Densification strain, εD is derived from the following equation ! ρf 9 þ α2 εD ¼ ln ð5Þ ρf 0 3α 2 In which ρf and ρf 0 are the foam density and the density of the foam base material which equals to 2700 kg/m3 respectively. The materialproperties of foam are modeled based on experimental data obtained from uniaxial compression test and are presented in Table 3.
4. Numerical results and discussion 4.1. Energy absorption and crushing characteristics
Table 2 True stress versus plastic strain pairs for carbon steel material [21]. εp
0.000 401.40
σ t N=mm2
0.021 473.04
0.046 521.10
0.086 552.94
0.106 563.94
0.141 576.99
Table 3 Material properties for aluminum foam filler [21]. ρf
kg m3
220 534
σp
N mm2
2.14 12.56
α
α2
2.12 2.12
N mm2
169 1544
β 2.942 3.680
γ
N mm2
2.45 1.00
εD 2.5074 1.6206
Let firstly introduce the parameters used for study of theenergy absorption and crushing behavior. Effectiveness of crushable tube is measured by energy absorption characteristics to make a meaningful comparison between tubes. One of the most prevalent criterions in crashworthiness isabsorbed energy (EA). This parameter is often used in validating the FE simulation results and is defined as below: Z δ EA ¼ F x dx ð6Þ 0
where F x is the resultant impact force and δ stands for the maximum deflection of the crushed structure. The EA efficiency is then evaluated by subtracting the absorbed energy to total mass ðMÞ of
Table 4 Theoretical solution and FE simulation of foam filled conical tube. Mass block velocity
Geometry properties
Material properties
Mean dynamic force (kN)
V m=s
thickness t ðmmÞ
Diameter dðmmÞ
Foam density kg ρf m 3
Plateau stress N σ p mm 2
Yield stress tube σ y ðMPaÞ
Theory [16]
FE simulation
Error (%)
20 20 30
1.43 1.43 1.4
78.5 78.5 78.5
0 220 534
0 2.14 12.56
109 401.1 401.1
14.4 69.75 130.17
13.26 73.35 126.75
5.6 5.16 2.9
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Table 5 Theoretical solution and FE simulation of foam filled bi-tubular structure. Mass block velocity
Geometry properties
Material properties
Thickness of tubes
Mean diameter of circular tube
Mean diameter of conical tube
Semi-epical angle
Foam density
V m=s
t ðmmÞ
dmc ðmmÞ
dmf ðmmÞ
α
ρf
20 20 20 20
1.5 1.5 1.5 1.5
97.25 97.25 97.25 97.25
55.25 55.25 55.25 55.25
5 5 7.5 7.5
0 220 0 220
kg m3
Plateau stress σp
N mm2
0 2.14 0 2.14
Mean dynamic force (kN) Yield stress tube
Developed theory
FE simulation
93.687 133.224 93.687 133.224
88.11 130.98 86.87 136.09
Conical tube Circular tube σ y ðMPaÞ 401.1 401.1 401.1 401.1
276 276 276 276
180 160
Laod (kN)
140 120 100
Peak Force
80
Mean Force
60 40 20 Fig. 4. Numerical result and experimental crushing test.
0 ACE
ACF
SFE
SFF
BTE
BTF
the absorber and is described as specific energy absorption (SEA). ð7Þ
Crush force efficiency ðCFEÞ is another indicator that is defined as the ratio of mean load to peak load. Since in foam filled conical tubes, peak force tends to increase as the crush distance progresses, initial peak force was employed to calculate the CFE, as done by previous researchers [21,42]. A higher value of CFE is desired for protection of vehicle's passengers. CFE ¼ F mean =F initialpeak
ð8Þ
25 20 SEA,EA and CFE
SEA ¼ EA=M
15
SEA(KJ/Kg) EA(KJ)
10
CFE*10 5
4.2. Validation of numerical results FE model is validated by relevant experimental test results and theoretical solution to establish whether it is sufficiently accurate in this section. 4.2.1. Validation via theoretical solution In order to validate the numerical simulation results, single foam filled circular tube has been modeled and compared to theoretical solution found in literature [16]. The average crush force of empty tubes can be derived from F savg ¼ kD σ y dm t 5=3 1=3
ð9Þ
where σ y , dm and t are the yield stress, mean diameter and wall thickness of the tube, respectively and kD is dimensionless constant and equals to 17 in this case [16]. To consider the effect of plastic strain rate ε_ on yield stress, under the dynamic impact Abramowicz and Jones [49] suggest the following equation: 1=k ε_ ð10Þ F davg ¼ F savg þ F savg C where k ¼ 3:91 and C ¼ 6844s 1 are material parameters for steel tubes applied in previous studies [45,46].
0 ACE
ACF
SFE
SFF
BTE
BTF
Fig. 5. Comparison between previously developed crush boxes and the proposed bi-tubular model in (a) peak crushing force and mean crushing force, (b) specific energy absorbed, energy absorption and crushing force efficiency.
The resistance of the foam inside under uni-axial compression can be expressed as: F foam ¼ Af σ f
ð11Þ
where σ f and Af are plateau stress and cross section area of the foam. Altogether the mean force of foam filled circular tube then can be expressed as: F davgðf Þ ¼ F davg þC avg dm t
pffiffiffiffiffiffiffiffiffiffi
σ f σ y þ Af σ f
ð12Þ
Last term indicates the interaction effect between extrusion and foam filler. C avg is the interaction constant and equals to 2.68 for 50% crush capacity. Table 4 shows a comparison between FEM and theoretical solution based on the average force of impact. It can be seen that simulations provide a satisfactory results.
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Table 6 Comparison between FE simulation and experimental results. m
Test no.
Thickness (mm)
Diameter (mm)
Length (mm)
M ðkgÞ
V
1 2
1 2
40 40
180 180
104.5 104.5
4.3 6.6
s
EAExp ðJÞ[47]
P Exp m ðkNÞ[47]
EAFE ðJÞ
P FE m ðkNÞ
998 2326
13.03 45.6
954.3 2530.3
11.47 42.88
ACE
ACF
Fig.7. Force–displacement curve for a bi-tubular structure's components.
SFE 200
Initial Peak Force(KN)
180
SFF
BTE
8.7
160 140
86.1
120 100 80
154.8
60 86.8
40 20
0 circular tube
BTF
frusta
foams
proposed structure
Fig. 8. Effect of using new structure on initial peak force. Fig. 6. Deformed shapes of different crush boxes.
4.2.2. Further validation of bi-tubular with theoretical solution In this part, a theoretical solution is presented based on the previous section and works of Hanssen et al. [16] to validate the bi-tubular conical–circular thin walled model. To do so, these assumptions are taken into account to simplify the solution. 1. Conical tubes with length almost equal to their height are assumed to be identical to circular tubes with same diameter as mean diameter of conical tubes; in other words cos α Z0:99.
2. The foam filler is assumed to have the same volume as ACF model. In other words, the volume of conical tube is not subtracted from the foam filler. 3. The foam must not reach the densification area. Mean force of the inside foam filled tube can be expressed as: in in F in avg ðf Þ ¼ F avg þ 2 C avg dmf t f
pffiffiffiffiffiffiffiffiffiffi
σf σy þ σf
R
Af in dx R dx
! ð13Þ
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259
Fig. 9. Effect of thickness of energy absorption and crushing characteristics.
where dmf and t f are the mean diameter and thickness of frusta. The interaction term is multiplied by 2 because of the interactions effect in both sides of inner tube. Mean force for the outer tube and middle foam filler is found by the following expression R out ! Af dx pffiffiffiffiffiffiffiffiffiffi out out R F avgðf Þ ¼ F avg þ C avg dmc t c σ f σ y þ σ f ð14Þ dx dmc and t c stand for mean diameter and thickness of circular outer tube. Based on the second assumption mean force of the foam filler will be simplified to R in R out ! Af dx Af dx R F foam ¼ σ f ð15Þ þ R ffi Af σ f dx dx Altogether the mean force of the foam filled bi-tubular structures is calculated from in out F avg ðbi tube Þh¼ F avgðf Þ þ F avgðf Þ pffiffiffiffiffiffiffiffiffiffii h out pffiffiffiffiffiffiffiffiffiffii ¼ Af σ f þ F in þ 2 C σ f σ y þ F avg þC avg dmc t c σ f σ y avg dmf t f avg Table 5 shows the comparison between numerical simulations and theoretical solution for bi-tubular structure. 4.2.3. Validation with experimental results A validation of empty circular tube has also been done to assure the deformability patterns are in agreement with experiments. Tube is modeled with isotropic viscoplastic aluminum, with σ y ¼ 231N=
mm2 and isotropic hardening parameters are defined as in reference [47]. As it can be seen clearly in Fig. 4 deformation patterns obtained from finite element simulation show good concurrence with experimental result. Numerical simulations predict also the absorbed energy and mean load with acceptable accuracy for different tests. Table 6 shows the comparison between numerical simulations and experimental tests. 4.3. Comparison between new and previous structures To make an inference of new structure capabilities, a comparison between it and previously developed structures for crashworthiness purposes has been made. The study includes investigating the crushing of new developed bi-tubular empty (BTE) and foam filled (BTF) structures, aluminum circular tube without (ACE) and with foam core (ACF) and also empty steel frusta (SFE) as well as foam filled steel frustum (SFF) under axial impact loading. Loading conditions include a 200 kg rigid mass block with a 20 m/s initial directional velocity. Maximum deflection of 65% for foam filled tubes and 70% for empty circular tubes' length is used to calculate the energy absorption characteristics. Shell thickness of tubes are 1.5 mm for all tubes while the diameter is 98 mm for circular and mean diameter of conical tubes is 56 mm respectively as it described in finite element modeling. The aluminum foam of 220 kg/m3 density is used as core filler. All other geometrical and material properties are also kept the same for new model as discussed previously. To get a better understanding of results,
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Fig. 10. Deformed shapes of selected geometries of new bi-tubular structure, (a) t ¼0.5 mm, empty, (b) t ¼0.5 mm, foam filled, (c) t¼ 2 mm, empty, (d) t ¼2 mm, foam filled.
column charts of them is depicted in Fig. 5. Final deformed state of the crushed structures along with their mid-plane viewis also illustrated in Fig. 6. It is worth noting that foam filling the structures will not always alter the crashworthiness for better result from SEA point of view as in case of conical filled tube below. By using a bi-tubular mechanism, a possibility for amending the peak load along with keeping other energy absorption characteristics unchanged, becomes tangible. Fig. 7 illustrates the force–displacement curve for each components of the bi-tubular foam filled tube as well as whole structure. It is evident that peak loads of one tube coincide with the lowest load of the other on several points. This has led to 14.7% decrease in Initial peak load as it can be seen in Fig. 8. This figure shows the effects of bi-tubular mechanism on the crushing force characteristics. It should be noted that this is achievable for any multiwalled structure by choosing right geometric parameters and material properties which would result in finding the hinge points of tubes. 4.4. Parametric study In order to investigate the effects of geometry variations on energy absorption, a parametric study has been carried out. Geometrical properties are as described in Section 4.3. In the study mean radius and height of the conical tubes are kept unchanged. While the semiepical angle, wall thickness and foam density are the input parameters.
4.4.1. Effects of wall thickness Fig. 9 shows the effect of wall thickness in energy absorption characteristics. The semi-epical angle used for cones is α ¼ 51 and
foam density of ρf ¼ 0; 220
kg m3
is chosen for study. Wall thickness
has rather a direct effect on energy absorption. Tubes with thicker walls tend to have higher peak load. It is evident that mean load and SEA also increases by using thicker tubes. Although higher values are achieved in energy absorption characteristics, by increasing the thickness, CFE would decrease substantially in foam filled tubes and in thicker empty tubes. That said it is evident that a reasonable thickness should be adopted for tubes. Deformation shapes of some selected crush boxes are presented in Fig. 10. Different crushing modes of the structure affected by wall thickness and foam filling can be clearly seen in this figure. 4.4.2. Effects of semi-epical angle The effects of semi-epical angle in energy absorption characteristics are depicted in Fig. 11. All tubes have the similar thickness of t¼1.5 mm. Cones have the semi-epical angle of α ¼ 5; 7:5; 101 and kg foam density of ρf ¼ 0; 220 m is chosen to perform the study. 3 Generally speaking increasing this angle would improve the energy absorption characteristics. Nevertheless it should be noted that characteristics only enhance slightly by increasing the angle while keeping
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261
Fig.11. Effect of change in semi-epical angle.
4.4.3. Effects of foam density The effect of foam density was investigated by comparing the results of structures with similar angleα ¼ 51, thickness, t¼ 1.5 mm
and different foam densities of ρ ¼ 0; 0:22; 0:534
kg m3
. Increasing
the foam density will result in a considerable improvement in the mean load and SEA. On the other hand a large amount has been added to initial peak load along with higher peak load during the entire deflection. From Fig. 12 it can be inferred that a higher foam density is not desirable for crashworthiness purposes at all as the structure absorbs the energy in an impractical manner. A high foam density can also cause a major drop in maximum crush distance [16]. This indicates the importance of choosing the right foam density as a determinative parameter for structure. The effect of foam density is illustrated in Fig. 13.
5. New structure under oblique loading Fig. 12. Force–displacement curves for different foam densities.
5.1. Comparison between new and previously developed structures
the mean radius unchanged. Some researchers kept the base diameter of frusta constant instead of mean radius when changing the semiepical angle, but by keeping the mean radius constant, it can be inferred that conical tubes with small semi-epical angle can be treated as circular tubes with similar mean radius to them.
It is important to consider the possibility of oblique loading when designing crush boxes as this type of impact is very common in vehicle crash. If the crush box is subjected to oblique loading, it is probable to experience global bending without creation of numerous local plastic hinges, which will lead to considerable loss
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Fig.13. Effect of foam density on crashworthiness.
14 12 10 8 SEA(Kj/Kg) 6
EA(Kj)
related studies [34,36–38]. All the other loading conditions such as boundaries, impact velocity are similar to Section 4.3. As shown in Fig. 14 bi-tubular structures can absorb more energy than previously suggested models, particularly using aluminum made BTE results in higher SEA value than others. Fig. 15 represents the deformed shapes of tubes after 8.5 ms. It is evident that by foam filling the all the structures, SEA value has dropped and foam filled structures tend to go under Euler buckling sooner than empty ones, resulting in less plastic work on shell components. The amount of absorbed energy versus time is illustrated in Fig. 16.
4
5.2. Investigation of bi-tubular structure under oblique loading with different impact angle
2 0 ACE
ACF
SFE
SFF
BTE
BTF
Fig. 14. Comparison between previously developed crush boxes and the proposed bi-tubular model under oblique loading with impact angle of β ¼ 301 after 8.5 ms.
in energy absorption. The study includes investigating the crushing response of previously mentioned structures in Section 4.3 under oblique loading. The mass block is a rigid wall with only allowable movement in structures axis of revolution. The impact angle of β ¼ 301 is chosen for this study as it was chosen as the maximum investigated angle under oblique loading by most
It is important to find the critical angle in which structure would start to go under global buckling. To perform the study, thealuminum made structure with a 51 semi-epical angle, thick ness of t ¼ 1:5mm and foam densities of ρ ¼ 0; 220 kg=m3 Þ are subjected to 10°, 20° and 30° constrained rigid wall. The results are evaluated for constant time of 8.5 ms after collision. As it can be seen in Fig. 17 under 10° and 20° oblique impact, the crushed structures show similar behavior to axial loading, while others [27,36] have announced a drastic decrease in absorbed energy or found that for impact angle of β ¼ 201 the deformation mode will change to Euler buckling [48]. By increasing the angle to 30°, foam filled structure's folding pattern transforms from concentric to global bending as the momentum caused by oblique load, exceeds.
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263
ACE
ACF Fig. 16. Energy absorption time history previously developed crush boxes and the proposed bi-tubular model under oblique loading with impact angle of β ¼ 301.
BTE
translates the force harmonically to conical tubes. As for the empty bi-tubular structure, the absence of foam filler lets the conical tube to go under more plastic deformation. Fig. 17 shows deformation modes offoam filled and empty bi-tubular crush box under different impact angles. From the results provided in Table7 it is evident that drastic drop in energy absorption due to global buckling starts around 30° impact angle. Inconclusion more stable deformation and energy absorption can be expected from the bitubular structure in oblique loading and using BTE model is suggested for crashworthiness purposes.
6. Conclusion
SFF
BTE
This study was performed to analyze the behavior of a newly proposed bi-tubular structure under axial and oblique loading. The effect of concentric conical and circular tubes configurations, geometrical parameters of inner and outer wall and foam filled in comparison with empty tube as well as material properties have been investigated. Behavior of bi-tubular structure is compared with single empty and foam filled common structures in terms of crushing force and energy absorptions. The crashworthiness capability of the new design in oblique loading has also been considered. Based on the obtained results the following points can be concluded.
The study showed that foam filling structures such as single
BTF Fig. 15. Deformation shapes of different crush boxes under oblique loading with impact angle of β ¼ 301 after 8.5 ms.
It is notable that outside tube has a more profound part in energy absorption than conical tube when using foam filler. This could be justifiedby saying that because the length of the conical tube is shorter than circular tube, it involves in the later stages of crash. On the other hand conical structures have a semiepical angle, they tend to slide and get parallel to rigid body since foam filler
frusta or the bi-tubular model may not always result in better crashworthiness characteristics and may cause higher initial peak load as well as decreased value in SEA. Based on the obtained results it can be concluded that by implanting the new developed double wall structure considerable decrease in initial peak load is been made possible, while maintaining other crashworthiness characteristics unaffected. A theoretical solution, based on the work of Hanssen et al. [16] is suggested to predict the mean crushing force of circular– conical foam filled structure. Obtained results show good agreement with FE simulation. It is inferred from the parametric study that by keeping the mean radius of frusta unchanged, changing semi-epical angle will have negligible effect on energy absorption characteristics. Although foam density can be used as a parameter to control the amplification of crushing force and absorbed energy, it is
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Fig. 17. Deformed shapes of foam filled and empty new absorbers under oblique loading after 8.5 ms, (a) β ¼ 10, (b) β ¼ 20, (c) β ¼ 30. Table 7 simulation results of bi-tubular structure under oblique loading with different impact angle.
Structure
Impact-angle
EA (kJ)
SEA (kJ/kg)
BTE BTF BTE BTF BTE BTF
10 10 20 20 30 30
10.96 19.1 8.19 14.29 5.567 6.19
25.61 25.94 19.31 19.52 13.01 8.456
essential to note that usage higher foam densities is impractical for crash protection purposes. This is because the foam filler will have a smaller plateau region and reaches the densification area shortly after impact. The bi-tubular structure can be effectively used to improve crushing and energy absorption characteristics under oblique loading. Under low impact angles it has a similar performance to axial loading; and a higher specific energy absorption value is reached under higher impact angle.
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[4] Langseth M, Hopperstad OS, Berstad T. Crashworthiness of aluminium extrusions: validation of numerical simulation, effect of mass ratio and impact velocity. Int J Impact Eng 1999;22:829–54. [5] Wierzbicki T, Abramowicz W. On the crushing mechanics of thin-walled structures. J Appl Mech 1983;50(4):727–34. [6] Abramowicz W, Wierzbicki T. Axial crushing of multicorner sheet metal columns. J Appl Mech 1989;56(1):113–20. [7] Alghamdi AAA, Aljawi AAN, Abu-Mansour TM. Modes of axial collapse of unconstrained capped frusta. Int J Mech Sci 2002;44:1145–61. [8] Li Z, Yu J, Guo L. Deformation and energy absorption of aluminum foam-filled tubes subjected to oblique loading. Int J Mech Sci 2012;54:48–56. [9] Liu H, Yao G, Cao Z. Energy absorption of aluminium foam-filled tubes under quasi-static loading. Light Met 2012:517–20. [10] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. Int J Impact Eng 1984;2(3):263–81. [11] Wierzbicki T, Bhat SU, Abramowicz W, Brodkin D. Alexander revisited-a two folding elements model of progressive crushing of tubes. Int J Solids Struct 1992;29:3269–88. [12] Hanefi E, Bloch JA, Cesari D. Static and dynamic crushing of externally reinforced tubes. In: Proceedings of JST AMAC-GAMAC, Lyon; 10 June 1993. [13] Singace AA, Sobky HE, Petsios M. Influence of end constraints on the collapse of axially impacted frusta. Thin Walled Struct 2001;39:415–28. [14] Seitzberger M, Rammerstorfer FG, Degischer HP, Gradinger R. Crushing of axially compressed steel tubes filled with aluminium foam. Acta Mech 1997;125:93–105. [15] Hanssen AG, Langseth M, Hopperstad OS. Optimum design for energy absorption of square aluminium extrusions with aluminium foam filler. Int. J. Mech. Sci. 2001;43:153–76. [16] Hanssen AG, Langseth M, Hopperstad OS. Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler. Int J Mech Sci 2000;24:475–507. [17] Hanefi E, Bloch JA, Cesari D. Static and dynamic crushing of externally reinforced tubes. In: Proceedings of JST AMAC-GAMAC, Lyon; 10 June 1993. [18] Xue P, Yua TX, Tao XM. Flat-topped conical shell under axial compression. Int J Mech Sci 2001;43:2125–45. [19] Kim HS. New extruded multi-cell aluminum profile for maximum crash energy absorption and weight efficiency. Thin-Walled Struct 2002;40:311–27.
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[20] Zhang Xiong, Cheng Gengdong, You Zong, Zang Hui. Energy absorption of axially compressed thin-walled square tubes with patterns. Thin-Walled Struct 2007;45:737–46. [21] Ahmad Z, Thambiratnam DP. Dynamic computer simulation and energy absorption of foam-filled conical tubes under axial impact loading. Comput Struct 2009;87:186–97. [22] Hosseini M, Abbas H, Gupta NK. Change in thickness in straight fold models for axial crushing of thin-walled frusta and tubes. Thin-Walled Struct 2009;47:98–108. [23] Ghasemnejad H, Hadavinia H, Aboutorabi A. Effect of delamination failure in crashworthiness analysis of hybrid composite box structures. Mater Des 2010;31:1105–16. [24] Morris E, Olabi AG, Hashmi MSJ. Analysis of nested tube type energy absorberswith different indenters and exterior constraints. Thin-Walled Struct 2006;44:872–85. [25] Morris E, Olabi AG, Hashmi MSJ. Lateral crushing of circular and non-circular tube systems under quasi-static conditions. J Mater Process Technol 2007;191:132–5. [26] Baroutaji A, Morris E, Olabi AG. Quasi-static response and multi-objective crashworthiness optimization of oblong tube under lateral loading. ThinWalled Struct 2014;82:262–77. [27] Li Guangyao, et al. A comparative study on thin-walled structures with functionally graded thickness (FGT) and tapered tubes withstanding oblique impact loading. Int J Impact Eng 2015;77:68–83. [28] Lu Minghao, et al. On functionally graded composite structures for crashworthiness. Compos Struct 2015;132:393–405. http://dx.doi.org/10.1016/j. compstruct.2015.05.034. [29] Goel Manmohan Dass. Deformation, energy absorption and crushing behavior of single-, double- and multi-wall foam filled square and circular tubes. ThinWalled Struct 2015;90:1–11. [30] Guo LW, Yu JL. Bending behavior of aluminum foam-filled double cylindrical tubes. ActaMech 2011;222:233–44. [31] Guo L, Yu J. Dynamic bending response of double cylindrical tubes filled with aluminum foam. Int J Impact Eng 2011;38:85–94. [32] Tang Zhiliang, Liu Shutian, Zhang Zonghua. Analysis of energy absorption characteristics of cylindrical multi-cell columns. Thin-Walled Struct 2013;62:75–84. [33] Jusuf Annisa, Dirgantara Tatacipta, Gunawan Leonardo, Putra Ichsan Setya. Crashworthiness analysis of multi-cell prismatic structures. Int J Impact Eng 2015;78:34–50. [34] Fang Jianguang, Gao Yunkai, Sun Guangyong, Qiu Na, Li Qing. On design of multi-cell tubes under axial and oblique impact loads. Thin-Walled Struct 2015;95:115–26.
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Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
A new bi-tubular conical–circular structure for improving crushing behavior under axial and oblique impacts Mohammadbagher B. Azimi, Masoud Asgari n Faculty of Mechanical Engineering, K. N. Toosi University of Technology, P. O. Box: 19395-1999, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 17 August 2015 Received in revised form 25 October 2015 Accepted 6 November 2015 Available online 14 November 2015
In this study single and double wall structures for crashworthiness are investigated to introduce a novel system with better energy absorption and crushing characteristics under both axial and oblique loading. The new developed double wall structure is constructed from inner conical and outer circular tubes, incorporated with and without foam filler. Foam filled bi-tubular structure subject to direct axial and oblique impact loadings are simulated using nonlinear finite element analysis software package LSDYNA. Numerical simulations obtained via non-linear explicit dynamic FEM are firstly validated using theoretical and experimental solutions. Next, effectiveness of the new developed bi-tubular structure has been shown by comparing with similar common thin-walled structures. Different types of structures namely bi-tubular empty and foam filled new design, empty and filled frusta as well as empty and foam filled circular tubes were considered in order to make a more insightful of capabilities of the proposed structure. A parametric study including the effect of geometrical and material properties of the structure on the crashworthiness has been carried out. Results show the possibility of amending the peak crushing load along with keeping other energy absorption characteristics unchanged or even improved. Also an improvement in absorbed energy under oblique loading by using the new developed structure is observed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Double-walled tube Foam filled Oblique impact Energy absorption Crushing Crashworthiness
1. Introduction The use of thin-walled structures as energy absorbent has been recently proven to be an efficient and applicable solution as they offer high values of absorbed energy in spite of their light weight in the automotive, high speed railway and aerospace industries.These crush boxes are made in different geometries, along which circular [1–3], squared [4–6] and conical tubes [7–8] with a variety of materials such as aluminum, steel, compositesas well as use of foam-filler [8–9] has gained a notable interest. To make an inference of energy absorption capacity and measure the effectiveness of these structures, energy absorption per unit mass called specific energy absorption (SEA) and mean crush forceas well as maximum induced load during crushing called peak crushing force (PCF) has been utilized by researchers. Alexander [1] firstly introduced axial crushing of thin walled circular tubes as a mechanism for energy absorption. Furthermore he was able to derive a theoretical solution by assuming perfect plastic material behavior. Ever since numerous attempts have been made by researchers to improve his solution [10–12]. Singace et al. [13] indicated that frusta have a more stable structure compared to cylinders. n
Corresponding author. Tel.: þ 98 21 8406 3209; fax: þ 98 21 8867 7274. E-mail address: [email protected] (M. Asgari).
http://dx.doi.org/10.1016/j.ijmecsci.2015.11.012 0020-7403/& 2015 Elsevier Ltd. All rights reserved.
They also observed that constraining the end of frusta leads to more energy absorption [13]. Improving energy absorption capacity and reducing crushing force and weight of structures is the main object considered by researchers in studying of energy absorbers. Numerous researches have been conducted theoretically, experimentally and numerically in thin-walled structures as energy absorbers. In this regard, Seitzberger et al. [14] studied steel tube with aluminum foam fillers. Hanssen et al. [15] confirmed that by using aluminum foam fillers considerable weight savings are achievable. Foam filling would affect the buckling shape causing tubes to go under more plastic deformation. Hanssen et al. [16] also observed a shift in deformation pattern from diamond to concertina mode for a critical value of foam density. Hanefi et al. [17] tested reinforced compound metal/composite wall and derived an analytical model for it based on Alexander's theory. Xue et al. [18] analyzed flat-topped conical cells made of textile composite and found them in good agreement with both theoretical and experimental solution. Kim [19] proposed new multi-cell profiles base on the idea of adding more square elements would cause in more energy absorption and also investigated effect of different types of triggering. By performing optimization they alsowere able to increase the SEA up to 200% [19]. Zhang et al. [20] introduced patterns to conventional surface of squared tube cross section which led to a new octagonal collapse mode as well as an enhancement of the absorbed energy. Ahmad et al. [21] performed a parametric study on foam-filled
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Nomenclature Af CFE Df Dt dm E EA Fx F mean F initialpeak F davg F savg hf ht k; C kD
Ν
cross section area of the foam Crush force efficiency frusta's diameter circular tube's diameter mean diameter Young's modulus energy absorption resultant impact force mean load initial peak force average dynamic crush force average static crush force frusta'slength circular tube's length material parameters dimensionless constant Poisson's ratio
conical tubes and concluded that foam-filled conical tubes collapse in a stable manner while having more lobes in comparison to empty tubes. Hosseini et al. [22] developed a mathematical model by considering the change in thickness of the frusta and tubes during collapse progressing as well as assuming different values for compressive and tensile strength of the materials and found the results more justifiable with experiments. Ghasemnejad et al. [23] investigated the effect of delamination crack growth of hybrid composite tubes and concluded the ones with higher fracture toughness will absorb more energy in crushing process. As an innovative way to absorb energy, Morris et al. [24,25] loaded circular tube segments transversally. This allows the possibility to reduce the occupied space without compromising the crashworthiness characteristics. Further investigation by Baroutaji et al. [26] showed that exposing these tubes to external constraints will lead to more deformed volume as well as specific energy absorption. Li et al. [27] made a comparison between functionally graded thickness (FGT) tubes and straight and tapered tubes with constant thickness found them in advantage to previous models under oblique loading. Lu et al. [28] introduce double functionally graded tubes by filling FGT tube with functionally graded foams (FGF) and found them to improve energy absorption characteristics without raising the initial peak load. Recently a great notice has been focused on using multiwall and multicell structures as they have offered a significant step up on absorbed energy [29–31]. Tang et al. [32] investigated multi-cell cylindrical and square columns and inferred that cylindrical columns offer better energy absorption properties. Jusuf et al. [33] performed a comparison between single-walled and double-walled square tubes and improved the double-walled tubes energy absorption efficiency by introducing internal ribs to them. Fang et al. [34] investigated rectangular multi-cell tubes under axial and oblique loading. They concluded that tubes with fewer cells perform better under oblique loading with large angle. Qiu et al. [35] studied multi-cell hexagonal tubes and found that number of corners plays a significant part in improving energy absorption. Borvik et al. [36] investigated empty and foam filled circular tubes under oblique loading for load angles of β ¼ 0; 5; 15; 30 and found that dramatic reduction in energy absorption occur even for small impact angles. Fang et al. [37] investigated multi-cell square tubes under oblique loading and found that despite their good response under axial loading, global bending occurs for impact angle of 201. Reyes et al. [38] investigated the crushing behavior of aluminum squared section extrusions under oblique loading both experimentally and
Ρ
density specific energy absorption wall thickness velocity plastic coefficient contraction yield stress α semi-epical angle δ maximum deflection of the crushed structure ε^ equivalent strain ε_ plastic strain rate εp plastic strain α2 , γ , εD and β material parameters for foam σf plateau stress σy yield stress σp plateau stress of the foam σe effective Von Mises stress σm mean stress σt true stress ρf 0 density of the foam base material
SEA t V Vp Y
numerically, and concluded that energy absorption drops drastically in a small impact angle of 51 and will continue to drop by increasing the impact angle. While there have been several studies on multi-cell tubes, few studies have investigated the use of bi-tubular structures on crashworthiness parameters. In those studies multi-wall structure with identical thin walls mainly investigated for direct axial loadings [29–31]. So, there exist wide extents of research to optimize the energy absorber structures for different practical applications especially including oblique loadings. Based on this fact, this paper focuses on combining the existing structures for crashworthiness to introduce a novel system with better energy absorption characteristics under both axial and oblique loading. A double walled structure including a frustum inside of a circular tube, incorporated with and without foam filler. Empty and foam filled bi-tubular structure subject to direct and oblique axial impacts are simulated using explicit dynamic finite element analysis software LS-DYNA. The numerical simulations were validated using experimental and theoretical solutions presented in appropriate publications. In order to make a more insightful of capabilities of the proposed structure, a comparison between it and previously existed models has been done. In this regard, different types of structures namely bi-tubular foam Filled (BTF), bi-tubular (BTE), empty and filled frusta (SFE, SFF) and empty and filled circular tubes (ACE, ACF) were considered under oblique impact. In addition a parametric study of new design including effects of geometrical parameters on the structural crashworthiness has been carried out. Obtained results clearly show that the crash ability of the new developed double wall structure is better than the other structures as it achieve a more absorbed energy while the peak crushing force decreases significantly. It is also observed that the empty bitubularcircular-conical structure improves the crushing performance considerably in oblique impacts.
2. Description of the new developed bi-tubular structure The study consists of a new innovative model proposed to offer better energy absorption and comparisons to simpler existing models. Improving crushing behaviors in axial impacts as well as oblique ones has been the motivation to combine circular and conical tubes. This double walled model includes a single frustuminside of a
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3. Finite element modeling 3.1. Mesh convergence and boundary conditions The finite element simulation is executed using non-linear explicit dynamic LS-DYNA solver. Tubes are modeled using shell elements based on Belytschko–Tsay formulation with five integration points through shell thickness to obtain proper accuracy [39]. Aluminum foam filler is modeled by constant stress solid elements. Hourglass control based on the Belytschko hypothesis as well as Bindeman assumed strain stiffness form are applied to avoid both artificial zero energy deformation modes and volumetric locking [41].
To get accurate results, a mesh convergence analysis on a bitubular foam filled structure with semi-epical angle of α ¼ 51, kg with thickness of t¼1.5 mm and foam density of ρf ¼ 220 m 3 various different mesh sizes has been done. As mentioned in some other related references 4 mm mesh size for solid elements is able to capture the deformation of foam filler and leads to accurate results [21,42]. Shell element of different sizes are used to represent finer, fine, medium and coarse mesh effects as represented in Table 1 and Fig. 2. The mesh size of 4 mm produced acceptable results on internal energy and mean force while it caused the inaccurate peak force which is an important parameter and therefore adequate precision is essential to solution. Also a further analysis on foam filled hybrid structures revealed the necessity of applying a finer mesh. Therefore all the simulations are done by 2 mm mesh size for shell tubes that is ensuring sufficient mesh density and will accurately capture the deformation process. An approximate number of 27,000 and 40,000 were produced for 2 mm shell and 4 mm solid elements respectively. “Automatic Surface To Surface” algorithm [43] with static coefficient friction of 0.3 and dynamic coefficient friction of 0.2 was used to account for 160 155 150 Force(KN)
circular tube, incorporated with and without foam filler as shown in Fig. 1a. This bi-tubular structure consists of a frustum with height of hf ¼ 200 mm and mean diameter, dmf ¼56 mm with some specified values of semi-epical angle, α and thickness, t as variables in parametric study. A triggered circular tube with diameterof Dt ¼ 98mm and heightof ht ¼ 230mm is used as outer wall. Fig.1b illustrates the proposed model geometry. The outside geometry was kept the same in parametric study to keep the focus on effects of changes in conical tube and foam filler. Geometries undergo axial and oblique impact loading. Columns are fixed at base to a completely constrained rigid plate. On the other end a 200 kg rigid mass block impacts with 20 m/s initial velocity with only allowable movement in the initial direction.
255
145 140
Peak force
135
Mean crush force
130 125 120 6mm
4mm
2mm
1mm
mesh size Fig. 2. Mesh sensitivity to initial peak and mean force.
Fig. 1. Geometry model of the new design bi-tubular crush box.
Table 1 Mesh convergence study results. Shell element size (mm)
Peak force Internal (kN) energy (kJ)
6
128.4
4 2 1
147.19 156.1 154.83
– 18.37 19.69 19.32
Mean crush force (kN) 122.25 130.98 128.85
Approximate time (min) Termination due to error 32 40 160
Fig. 3. Section view of finite element setup for simulation.
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contacts between the rigid bodies and structures components as well as for interactions between the components themselves. “Tied Node to Surface offset” algorithm [43] was employed to constraint the base of crush box. Also to capture the selfinteraction of shell components “Automatic Single Surface” contact type was utilized. Fig. 3 illustrates the developed model for finite element simulation.
fracture such as choosing critical value for maximum principal stress in elements and energy based fracture criterion developed by Reyes et al. [44], in the following simulations element erosion only occurs when reaching a positive value of 0.02 in volumetric strain. The eroded element mass is retained in solution. The yield criterion is as [44]: ∅ ¼ σ^ Y r U
ð1Þ
where
3.2. Material properties 3.2.1. Material of thin walled tubes Circular tubes are modeled as elasto-plastic materials without considering the strain rate properties since previous studies showed that aluminumcan be considered as strain rate independent [40]. “Mat Piecewise Linear Plasticity” has the ability to model the tube with mentioned configurations. Also arbitrary stress versus strain curve can be defined with this material which is later used to model the carbon steel material [43]. Aluminum 6061 T6 with mechanical properties of density ρal ¼2700 kg/m3, Young's modulus Eal ¼ 68.2 GPa, Poisson'sratio ν ¼0.3 and yield stress of Yal ¼276 Mpa was chosen for study. The aluminum was considered to have a perfect plastic behavior. For steel tubes, on the other hand, strain hardening effects should be considered, therefore to get more accurate results truestress versusplastic strain curve is also specified as described in Table 2. The conical tubes were all made of carbon steel with mechanical properties of density of ρs ¼7809 kg/m3, Young modulus, Es ¼ 200 GPa, Poisson's ratio,ν ¼ 0.3 and yield stress, Ys ¼ 401.4 MPa. Material properties were derived from standard tensile test. In Table 2, εp and σ t denote the plastic strainand the true stress respectively. 3.2.2. Material of foam filler Since volume changes due to collapsing of foam cells under compression, yield criterion should contain a hydrostatic stress term. Deshpande–Fleck [43] material was chosen to model the aluminum closed cell foam filler. This material can simulate the fracture by eroding the elements that have reached a critical value for volumetric strain. Although other criteria exist for modeling
2 1 2 2 α3 i σ e þ α σ m 1þ 3
σ^ 2 ¼ h
ð2Þ
where σ m is the mean stress and σ e shows the effective Von Mises stress. α is a function of plastic coefficient contraction Vp and describes the shape of yield surface. 9 1 2V p ð3Þ α2 ¼ 2 1þVp V p is assumed to be zero for aluminumfoams [36,45] which will lead to α ¼ 2:12. The yield stress Y is defined as " # ε^ 1 Y ¼ σ p þ γ þ α2 ln ð4Þ εD 1 ðεε^D Þβ In which σ p (plateau stress of the foam), α2 ,γ ,εD and β are material parameters and equivalent strain is defined as ε^ . Densification strain, εD is derived from the following equation ! ρf 9 þ α2 εD ¼ ln ð5Þ ρf 0 3α 2 In which ρf and ρf 0 are the foam density and the density of the foam base material which equals to 2700 kg/m3 respectively. The materialproperties of foam are modeled based on experimental data obtained from uniaxial compression test and are presented in Table 3.
4. Numerical results and discussion 4.1. Energy absorption and crushing characteristics
Table 2 True stress versus plastic strain pairs for carbon steel material [21]. εp
0.000 401.40
σ t N=mm2
0.021 473.04
0.046 521.10
0.086 552.94
0.106 563.94
0.141 576.99
Table 3 Material properties for aluminum foam filler [21]. ρf
kg m3
220 534
σp
N mm2
2.14 12.56
α
α2
2.12 2.12
N mm2
169 1544
β 2.942 3.680
γ
N mm2
2.45 1.00
εD 2.5074 1.6206
Let firstly introduce the parameters used for study of theenergy absorption and crushing behavior. Effectiveness of crushable tube is measured by energy absorption characteristics to make a meaningful comparison between tubes. One of the most prevalent criterions in crashworthiness isabsorbed energy (EA). This parameter is often used in validating the FE simulation results and is defined as below: Z δ EA ¼ F x dx ð6Þ 0
where F x is the resultant impact force and δ stands for the maximum deflection of the crushed structure. The EA efficiency is then evaluated by subtracting the absorbed energy to total mass ðMÞ of
Table 4 Theoretical solution and FE simulation of foam filled conical tube. Mass block velocity
Geometry properties
Material properties
Mean dynamic force (kN)
V m=s
thickness t ðmmÞ
Diameter dðmmÞ
Foam density kg ρf m 3
Plateau stress N σ p mm 2
Yield stress tube σ y ðMPaÞ
Theory [16]
FE simulation
Error (%)
20 20 30
1.43 1.43 1.4
78.5 78.5 78.5
0 220 534
0 2.14 12.56
109 401.1 401.1
14.4 69.75 130.17
13.26 73.35 126.75
5.6 5.16 2.9
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257
Table 5 Theoretical solution and FE simulation of foam filled bi-tubular structure. Mass block velocity
Geometry properties
Material properties
Thickness of tubes
Mean diameter of circular tube
Mean diameter of conical tube
Semi-epical angle
Foam density
V m=s
t ðmmÞ
dmc ðmmÞ
dmf ðmmÞ
α
ρf
20 20 20 20
1.5 1.5 1.5 1.5
97.25 97.25 97.25 97.25
55.25 55.25 55.25 55.25
5 5 7.5 7.5
0 220 0 220
kg m3
Plateau stress σp
N mm2
0 2.14 0 2.14
Mean dynamic force (kN) Yield stress tube
Developed theory
FE simulation
93.687 133.224 93.687 133.224
88.11 130.98 86.87 136.09
Conical tube Circular tube σ y ðMPaÞ 401.1 401.1 401.1 401.1
276 276 276 276
180 160
Laod (kN)
140 120 100
Peak Force
80
Mean Force
60 40 20 Fig. 4. Numerical result and experimental crushing test.
0 ACE
ACF
SFE
SFF
BTE
BTF
the absorber and is described as specific energy absorption (SEA). ð7Þ
Crush force efficiency ðCFEÞ is another indicator that is defined as the ratio of mean load to peak load. Since in foam filled conical tubes, peak force tends to increase as the crush distance progresses, initial peak force was employed to calculate the CFE, as done by previous researchers [21,42]. A higher value of CFE is desired for protection of vehicle's passengers. CFE ¼ F mean =F initialpeak
ð8Þ
25 20 SEA,EA and CFE
SEA ¼ EA=M
15
SEA(KJ/Kg) EA(KJ)
10
CFE*10 5
4.2. Validation of numerical results FE model is validated by relevant experimental test results and theoretical solution to establish whether it is sufficiently accurate in this section. 4.2.1. Validation via theoretical solution In order to validate the numerical simulation results, single foam filled circular tube has been modeled and compared to theoretical solution found in literature [16]. The average crush force of empty tubes can be derived from F savg ¼ kD σ y dm t 5=3 1=3
ð9Þ
where σ y , dm and t are the yield stress, mean diameter and wall thickness of the tube, respectively and kD is dimensionless constant and equals to 17 in this case [16]. To consider the effect of plastic strain rate ε_ on yield stress, under the dynamic impact Abramowicz and Jones [49] suggest the following equation: 1=k ε_ ð10Þ F davg ¼ F savg þ F savg C where k ¼ 3:91 and C ¼ 6844s 1 are material parameters for steel tubes applied in previous studies [45,46].
0 ACE
ACF
SFE
SFF
BTE
BTF
Fig. 5. Comparison between previously developed crush boxes and the proposed bi-tubular model in (a) peak crushing force and mean crushing force, (b) specific energy absorbed, energy absorption and crushing force efficiency.
The resistance of the foam inside under uni-axial compression can be expressed as: F foam ¼ Af σ f
ð11Þ
where σ f and Af are plateau stress and cross section area of the foam. Altogether the mean force of foam filled circular tube then can be expressed as: F davgðf Þ ¼ F davg þC avg dm t
pffiffiffiffiffiffiffiffiffiffi
σ f σ y þ Af σ f
ð12Þ
Last term indicates the interaction effect between extrusion and foam filler. C avg is the interaction constant and equals to 2.68 for 50% crush capacity. Table 4 shows a comparison between FEM and theoretical solution based on the average force of impact. It can be seen that simulations provide a satisfactory results.
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Table 6 Comparison between FE simulation and experimental results. m
Test no.
Thickness (mm)
Diameter (mm)
Length (mm)
M ðkgÞ
V
1 2
1 2
40 40
180 180
104.5 104.5
4.3 6.6
s
EAExp ðJÞ[47]
P Exp m ðkNÞ[47]
EAFE ðJÞ
P FE m ðkNÞ
998 2326
13.03 45.6
954.3 2530.3
11.47 42.88
ACE
ACF
Fig.7. Force–displacement curve for a bi-tubular structure's components.
SFE 200
Initial Peak Force(KN)
180
SFF
BTE
8.7
160 140
86.1
120 100 80
154.8
60 86.8
40 20
0 circular tube
BTF
frusta
foams
proposed structure
Fig. 8. Effect of using new structure on initial peak force. Fig. 6. Deformed shapes of different crush boxes.
4.2.2. Further validation of bi-tubular with theoretical solution In this part, a theoretical solution is presented based on the previous section and works of Hanssen et al. [16] to validate the bi-tubular conical–circular thin walled model. To do so, these assumptions are taken into account to simplify the solution. 1. Conical tubes with length almost equal to their height are assumed to be identical to circular tubes with same diameter as mean diameter of conical tubes; in other words cos α Z0:99.
2. The foam filler is assumed to have the same volume as ACF model. In other words, the volume of conical tube is not subtracted from the foam filler. 3. The foam must not reach the densification area. Mean force of the inside foam filled tube can be expressed as: in in F in avg ðf Þ ¼ F avg þ 2 C avg dmf t f
pffiffiffiffiffiffiffiffiffiffi
σf σy þ σf
R
Af in dx R dx
! ð13Þ
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259
Fig. 9. Effect of thickness of energy absorption and crushing characteristics.
where dmf and t f are the mean diameter and thickness of frusta. The interaction term is multiplied by 2 because of the interactions effect in both sides of inner tube. Mean force for the outer tube and middle foam filler is found by the following expression R out ! Af dx pffiffiffiffiffiffiffiffiffiffi out out R F avgðf Þ ¼ F avg þ C avg dmc t c σ f σ y þ σ f ð14Þ dx dmc and t c stand for mean diameter and thickness of circular outer tube. Based on the second assumption mean force of the foam filler will be simplified to R in R out ! Af dx Af dx R F foam ¼ σ f ð15Þ þ R ffi Af σ f dx dx Altogether the mean force of the foam filled bi-tubular structures is calculated from in out F avg ðbi tube Þh¼ F avgðf Þ þ F avgðf Þ pffiffiffiffiffiffiffiffiffiffii h out pffiffiffiffiffiffiffiffiffiffii ¼ Af σ f þ F in þ 2 C σ f σ y þ F avg þC avg dmc t c σ f σ y avg dmf t f avg Table 5 shows the comparison between numerical simulations and theoretical solution for bi-tubular structure. 4.2.3. Validation with experimental results A validation of empty circular tube has also been done to assure the deformability patterns are in agreement with experiments. Tube is modeled with isotropic viscoplastic aluminum, with σ y ¼ 231N=
mm2 and isotropic hardening parameters are defined as in reference [47]. As it can be seen clearly in Fig. 4 deformation patterns obtained from finite element simulation show good concurrence with experimental result. Numerical simulations predict also the absorbed energy and mean load with acceptable accuracy for different tests. Table 6 shows the comparison between numerical simulations and experimental tests. 4.3. Comparison between new and previous structures To make an inference of new structure capabilities, a comparison between it and previously developed structures for crashworthiness purposes has been made. The study includes investigating the crushing of new developed bi-tubular empty (BTE) and foam filled (BTF) structures, aluminum circular tube without (ACE) and with foam core (ACF) and also empty steel frusta (SFE) as well as foam filled steel frustum (SFF) under axial impact loading. Loading conditions include a 200 kg rigid mass block with a 20 m/s initial directional velocity. Maximum deflection of 65% for foam filled tubes and 70% for empty circular tubes' length is used to calculate the energy absorption characteristics. Shell thickness of tubes are 1.5 mm for all tubes while the diameter is 98 mm for circular and mean diameter of conical tubes is 56 mm respectively as it described in finite element modeling. The aluminum foam of 220 kg/m3 density is used as core filler. All other geometrical and material properties are also kept the same for new model as discussed previously. To get a better understanding of results,
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Fig. 10. Deformed shapes of selected geometries of new bi-tubular structure, (a) t ¼0.5 mm, empty, (b) t ¼0.5 mm, foam filled, (c) t¼ 2 mm, empty, (d) t ¼2 mm, foam filled.
column charts of them is depicted in Fig. 5. Final deformed state of the crushed structures along with their mid-plane viewis also illustrated in Fig. 6. It is worth noting that foam filling the structures will not always alter the crashworthiness for better result from SEA point of view as in case of conical filled tube below. By using a bi-tubular mechanism, a possibility for amending the peak load along with keeping other energy absorption characteristics unchanged, becomes tangible. Fig. 7 illustrates the force–displacement curve for each components of the bi-tubular foam filled tube as well as whole structure. It is evident that peak loads of one tube coincide with the lowest load of the other on several points. This has led to 14.7% decrease in Initial peak load as it can be seen in Fig. 8. This figure shows the effects of bi-tubular mechanism on the crushing force characteristics. It should be noted that this is achievable for any multiwalled structure by choosing right geometric parameters and material properties which would result in finding the hinge points of tubes. 4.4. Parametric study In order to investigate the effects of geometry variations on energy absorption, a parametric study has been carried out. Geometrical properties are as described in Section 4.3. In the study mean radius and height of the conical tubes are kept unchanged. While the semiepical angle, wall thickness and foam density are the input parameters.
4.4.1. Effects of wall thickness Fig. 9 shows the effect of wall thickness in energy absorption characteristics. The semi-epical angle used for cones is α ¼ 51 and
foam density of ρf ¼ 0; 220
kg m3
is chosen for study. Wall thickness
has rather a direct effect on energy absorption. Tubes with thicker walls tend to have higher peak load. It is evident that mean load and SEA also increases by using thicker tubes. Although higher values are achieved in energy absorption characteristics, by increasing the thickness, CFE would decrease substantially in foam filled tubes and in thicker empty tubes. That said it is evident that a reasonable thickness should be adopted for tubes. Deformation shapes of some selected crush boxes are presented in Fig. 10. Different crushing modes of the structure affected by wall thickness and foam filling can be clearly seen in this figure. 4.4.2. Effects of semi-epical angle The effects of semi-epical angle in energy absorption characteristics are depicted in Fig. 11. All tubes have the similar thickness of t¼1.5 mm. Cones have the semi-epical angle of α ¼ 5; 7:5; 101 and kg foam density of ρf ¼ 0; 220 m is chosen to perform the study. 3 Generally speaking increasing this angle would improve the energy absorption characteristics. Nevertheless it should be noted that characteristics only enhance slightly by increasing the angle while keeping
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261
Fig.11. Effect of change in semi-epical angle.
4.4.3. Effects of foam density The effect of foam density was investigated by comparing the results of structures with similar angleα ¼ 51, thickness, t¼ 1.5 mm
and different foam densities of ρ ¼ 0; 0:22; 0:534
kg m3
. Increasing
the foam density will result in a considerable improvement in the mean load and SEA. On the other hand a large amount has been added to initial peak load along with higher peak load during the entire deflection. From Fig. 12 it can be inferred that a higher foam density is not desirable for crashworthiness purposes at all as the structure absorbs the energy in an impractical manner. A high foam density can also cause a major drop in maximum crush distance [16]. This indicates the importance of choosing the right foam density as a determinative parameter for structure. The effect of foam density is illustrated in Fig. 13.
5. New structure under oblique loading Fig. 12. Force–displacement curves for different foam densities.
5.1. Comparison between new and previously developed structures
the mean radius unchanged. Some researchers kept the base diameter of frusta constant instead of mean radius when changing the semiepical angle, but by keeping the mean radius constant, it can be inferred that conical tubes with small semi-epical angle can be treated as circular tubes with similar mean radius to them.
It is important to consider the possibility of oblique loading when designing crush boxes as this type of impact is very common in vehicle crash. If the crush box is subjected to oblique loading, it is probable to experience global bending without creation of numerous local plastic hinges, which will lead to considerable loss
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Fig.13. Effect of foam density on crashworthiness.
14 12 10 8 SEA(Kj/Kg) 6
EA(Kj)
related studies [34,36–38]. All the other loading conditions such as boundaries, impact velocity are similar to Section 4.3. As shown in Fig. 14 bi-tubular structures can absorb more energy than previously suggested models, particularly using aluminum made BTE results in higher SEA value than others. Fig. 15 represents the deformed shapes of tubes after 8.5 ms. It is evident that by foam filling the all the structures, SEA value has dropped and foam filled structures tend to go under Euler buckling sooner than empty ones, resulting in less plastic work on shell components. The amount of absorbed energy versus time is illustrated in Fig. 16.
4
5.2. Investigation of bi-tubular structure under oblique loading with different impact angle
2 0 ACE
ACF
SFE
SFF
BTE
BTF
Fig. 14. Comparison between previously developed crush boxes and the proposed bi-tubular model under oblique loading with impact angle of β ¼ 301 after 8.5 ms.
in energy absorption. The study includes investigating the crushing response of previously mentioned structures in Section 4.3 under oblique loading. The mass block is a rigid wall with only allowable movement in structures axis of revolution. The impact angle of β ¼ 301 is chosen for this study as it was chosen as the maximum investigated angle under oblique loading by most
It is important to find the critical angle in which structure would start to go under global buckling. To perform the study, thealuminum made structure with a 51 semi-epical angle, thick ness of t ¼ 1:5mm and foam densities of ρ ¼ 0; 220 kg=m3 Þ are subjected to 10°, 20° and 30° constrained rigid wall. The results are evaluated for constant time of 8.5 ms after collision. As it can be seen in Fig. 17 under 10° and 20° oblique impact, the crushed structures show similar behavior to axial loading, while others [27,36] have announced a drastic decrease in absorbed energy or found that for impact angle of β ¼ 201 the deformation mode will change to Euler buckling [48]. By increasing the angle to 30°, foam filled structure's folding pattern transforms from concentric to global bending as the momentum caused by oblique load, exceeds.
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263
ACE
ACF Fig. 16. Energy absorption time history previously developed crush boxes and the proposed bi-tubular model under oblique loading with impact angle of β ¼ 301.
BTE
translates the force harmonically to conical tubes. As for the empty bi-tubular structure, the absence of foam filler lets the conical tube to go under more plastic deformation. Fig. 17 shows deformation modes offoam filled and empty bi-tubular crush box under different impact angles. From the results provided in Table7 it is evident that drastic drop in energy absorption due to global buckling starts around 30° impact angle. Inconclusion more stable deformation and energy absorption can be expected from the bitubular structure in oblique loading and using BTE model is suggested for crashworthiness purposes.
6. Conclusion
SFF
BTE
This study was performed to analyze the behavior of a newly proposed bi-tubular structure under axial and oblique loading. The effect of concentric conical and circular tubes configurations, geometrical parameters of inner and outer wall and foam filled in comparison with empty tube as well as material properties have been investigated. Behavior of bi-tubular structure is compared with single empty and foam filled common structures in terms of crushing force and energy absorptions. The crashworthiness capability of the new design in oblique loading has also been considered. Based on the obtained results the following points can be concluded.
The study showed that foam filling structures such as single
BTF Fig. 15. Deformation shapes of different crush boxes under oblique loading with impact angle of β ¼ 301 after 8.5 ms.
It is notable that outside tube has a more profound part in energy absorption than conical tube when using foam filler. This could be justifiedby saying that because the length of the conical tube is shorter than circular tube, it involves in the later stages of crash. On the other hand conical structures have a semiepical angle, they tend to slide and get parallel to rigid body since foam filler
frusta or the bi-tubular model may not always result in better crashworthiness characteristics and may cause higher initial peak load as well as decreased value in SEA. Based on the obtained results it can be concluded that by implanting the new developed double wall structure considerable decrease in initial peak load is been made possible, while maintaining other crashworthiness characteristics unaffected. A theoretical solution, based on the work of Hanssen et al. [16] is suggested to predict the mean crushing force of circular– conical foam filled structure. Obtained results show good agreement with FE simulation. It is inferred from the parametric study that by keeping the mean radius of frusta unchanged, changing semi-epical angle will have negligible effect on energy absorption characteristics. Although foam density can be used as a parameter to control the amplification of crushing force and absorbed energy, it is
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Fig. 17. Deformed shapes of foam filled and empty new absorbers under oblique loading after 8.5 ms, (a) β ¼ 10, (b) β ¼ 20, (c) β ¼ 30. Table 7 simulation results of bi-tubular structure under oblique loading with different impact angle.
Structure
Impact-angle
EA (kJ)
SEA (kJ/kg)
BTE BTF BTE BTF BTE BTF
10 10 20 20 30 30
10.96 19.1 8.19 14.29 5.567 6.19
25.61 25.94 19.31 19.52 13.01 8.456
essential to note that usage higher foam densities is impractical for crash protection purposes. This is because the foam filler will have a smaller plateau region and reaches the densification area shortly after impact. The bi-tubular structure can be effectively used to improve crushing and energy absorption characteristics under oblique loading. Under low impact angles it has a similar performance to axial loading; and a higher specific energy absorption value is reached under higher impact angle.
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