Microsoft Word - 1 D.H. Chen, K. Ushijima--Estimation of maximum compressive load for circular tubes under axial impact.doc Advances in Systems Science and Applications (2011), Vol.11, No.3-4 203-213 ISSN 1078-6236 International Institute for General Systems Studies, Inc. Estimation of Maximum Compressive Load for Circular Tubes under Axial Impact D.H. Chen1 and K. Ushijima2 1Department of Mechanical Engineering, Faculty of Engineering, Tokyo University of Science, Tokyo, 1628601, Japan 2Department of Mechanical Engineering, Faculty of Engineering, Kyushu Sangyo University, Fukuoka, 8138503,Japan Abstract The influence of impact velocity on the crushing behaviour of cylindrical shells subjected to an axial impact was investigated using a finite element analysis. The effects of the material properties, tube geometries and impact velocity V0 on the initial peak stressσ1 are explored. In this study, the applied material is assumed to be insensitive to the strain rate, and the effect of impact velocity is discussed as an inertia effect. It is shown that the initial peak stressσ1 during dynamic loading increases with increase of the impact velocity V0, which is due to the fact that the displacement in radial direction is delayed as the velocity V0 increases. Also, based on our numerical simulations, the peak stressσ1 can be regarded as a function of the ratio of tube thickness to radius t/R, hardening modulus to Young's modulus Eh/E and impact velocity to elastic stress wave speed V0/c. Moreover, an approximate equation to evaluate the peak stress is proposed and in good agreement with the FEM results and other researcher's results under a relatively low impact velocity (V0<40m/s). Keywords Elastic-plastic cylindrical shells, Axial impact, Energy absorption, FEM 1.Introduction Thin-walled structures such as circular and square tubes have been widely used in automotive and aerospace engineering as impact energy absorbing devices. A large number of studies concerning the static and dynamic response of thin-walled tubes subjected to axial load have been conducted by many researchers, and investigated some important parameters such as crushing distance, the peak load and the buckling shape in crashworthiness design [1-13]. Figure 1 shows a schematic of compressive axial stressσx and displacement Ux for a cylindrical tube subjected to axial loading. The purpose of this study is to explore the effect of impact velocity on the peak load for circular tubes, and to propose an empirical equation to estimate the peak load by numerical simulation. 2.Method of Analysis In this study, the dynamic numerical simulation of the impact crush test was carried out using the non-linear FE commercial code, MSC.Dytran. The geometry of the FE model and its boundary condition is shown in Fig.2. The model is struck from the upper edge by a rigid mass M having an initial kinetic energy T0=MV0 2/2=99 kJ. In the FE model, 4-node Key-Hoff shell elements(QUAD4, PSHELL) with three integration points are used to evaluate domain integrals, and the whole model is divided into 1440 elements. Here, parameters L, t and R are the tube length, thickness and mean radius, respectively. Also, the lower end of a tube is fixed to another rigid body, and a contact condition between the tube and the striker, and a self-contact condition at the inner and the outer surface of the tube are defined with the dynamic and static frictional coefficients of 0.2 and 0.3, respectively. 204 Chen: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending σ1 Axial displacement , Ux Ax ia l s tre ss , σ x Fig.1. Schematic of axial compressive stress-displacement relationship for a tube subjected to axial impact Fig.2. Shell geometry and loading condition The analyzed FE model with a densityρ=2685 kg/m3 is assumed to be isotropic, and to obey the Mises yield criterion with strain hardening, and a strain rate insensitive bilinear relationship between the uniaxial stress and strain as: ( ) ( ) ( )⎩ ⎨ ⎧ >−+ ≤ = EEE EE yyhy y σεσεσ σεε σ (1) Here, the Young's modulus, E, yielding stress, σy and hardening coefficient, Eh are assumed to be 72.4 GPa, 72.4 MPa and 3.62 GPa, respectively. All models in our calculation have the same tube length L=150 mm, mean radius R=25 mm and thickness t=1 mm, unless otherwise mentioned. 3.Results and Discussion 3.1 Effect of Impact Velocity V0 on the initial peak stressσ1 0 0.01 0.02 0.03 0 50 100 :V0=5 (km/h) Ax ia l F or ce (K N ) Axial Displacement (m) B A C D :V0=180 (km/h) :V0=360 (km/h) Fig.3. Comparison of axial force and displacement behaviour for a tube under different impact velocity V0 Figure 3 shows comparisons of axial compressive force and displacement diagram for a tube under the impact velocity V0=5, 180 and 360 km/h. Advances in Systems Science and Applications (2011), Vol.11, No.3-4 205 ISSN 1078-6236 International Institute for General Systems Studies, Inc. Also, the deformed shapes at the initiation of the initial peak stress and soon after the stress for the case of V0=5 km/h and 360 km/h are shown in Fig.4. It is evident from Fig.4 that the initial peak stress is associated with the initiation of local buckling deformation which occurs near the upper and lower ends of a tube. From the relationship between the axial load and displacement for each V0 which is shown in Fig.3, the initial peak stress can be calculated and summarized in Fig.5. Also in Fig.5, quasi-static buckling stress for the tube obtained by implicit finite element code, MSC.Marc, is shown by a dashed line. It is found from this figure that the initial peak stressσ1 for a lower impact velocity is almost equal to the value of the quasi-static result, and the stress value becomes higher as the impact velocity V0 increases. Fig.4. Comparisons of deformed shape at points 'A', 'B', 'C' and 'D' in Fig.3. 'A': buckling point for V0=5(km/h); 'B': point just after buckling for V0=5(km/h); 'C': buckling point for V0=360(km/h); 'D': point just after buckling for V0=360 (km/h) 0 100 200 300 400 0 250 500 In iti al P ea k S tre ss σ 1 (M P a) Impact Velocity V0 (km/h) Quasi−Static buckling stress Fig.5. Variation of peak stress σ1 with impact velocity V0 0 0.08 0.16 0 200 400 A xi al S tre ss σ x (M Pa ) Position x (m) Impacted endV0=5 (km/h) t=6.2 (μsec) 4.16 3.16 2.16 1.16 0.16 0.08 0.05 0.03 0.02 0.01 :Yield stress :Quasi−Static buckling stress (a) V0=5(km/h) Figure 6 illustrates propagations of the stress wave in the axial direction for V0=5 km/h (Fig.6(a)) and 360 km/h (Fig.6(b)) until the initial peak stress occurs. Also in Figure 6, values of the yield stressσy and the quasi-static buckling stress are shown by dotted and dashed lines, respectively. It is evident from Fig.6(a) that for the case of a lower impact velocity(V0=5 km/h), 206 Chen: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending the amplitude of the stress wave in axial direction at the initiation of impact is relatively small, and the stress wave travels and reflects many times along the tube until the local buckling deformation arises near the fixed end. In the end, the axial stress distribution developing over the tube becomes almost uniformly at t=0.16μs, and its amplitude increases stably to the quasi-static buckling stress. On the other hand, for the case of a higher impact velocity (Fig.6(b)), the development of axial stress distribution apparently differs from that for a lower impact velocity(Fig.6(a)). 0 0.08 0.16 0 400 800 A xi al S tre ss σ x (M P a) Position x (m) Impacted endV0=360 (km/h) t=0.14 (μsec) 0.12 0.1 0.03 0.02 0.01 :Yield stress :Quasi−Static buckling stress (b) V0=360(km/h) Fig.6. Axial stress distribution until the initiation of the peak stressσ1 In Fig.6(b), the stress concentration can be found at x=0.156 m. Such a stress concentration occurs by the existence of the flange at the impacted end, and does not affect the overall buckling behaviour of a circular tube. It is evident from Fig.6(b) that the amplitude of the stress wave bigger than the value of quasi-static buckling stress develops near the impacted end at the beginning. However, such the high stress field is relatively narrow, and no buckling behaviours seem to be observed even if the amplitude of stress wave is bigger than that of the quasi-static result. Moreover, even though the axial stress developing over the tube becomes larger than the quasi-static result at t=0.12μs, no buckling can be observed. Finally, the local buckling can be observed at t=0.14μs when the amplitude of the axial stress is almost twice larger than the quasi-static buckling stress. The mechanism of the buckling for circular tubes under higher impact velocities will be discussed as follows. Fig.7. Comparisons of deformed shape near the fixed end at the initiation of the peak stressσ1 Based on the previous work concerning the quasi-static axial compressive behaviour for circular tube, it is evident that the initial peak stress is associated with the sufficient local bending deformation which is observed near the tube end. That is, during the axial compression, the tube seems to expand in radial direction, but near the tube end, such a movement is restricted by the existence of the fixed boundary condition. As a result, the local bending deformation can be observed near the tube end, and the local bending deformation is necessary for initiating the buckling behaviour. Figure 7 shows comparisons of deformed shape near the fixed end when the initial peak stress is observed for some cases of impact velocity V0=5 km/h, 180 km/h and 360 Advances in Systems Science and Applications (2011), Vol.11, No.3-4 207 ISSN 1078-6236 International Institute for General Systems Studies, Inc. km/h. It is evident that for the case of a higher impact velocity, the amount of the local bending deformation is smaller than that under a slower impact velocity. In order to initiate the buckling behaviour where the smaller amount of the local bending deformation occurs under higher impact velocity, that is, in order to increase the radial displacement, a large amount of axial stress should be needed. Therefore, it could be understood that the reason why the initial peak stress increases as increase of the impact velocity V0 is due to the fact that the impact for a higher impact velocity causes a smaller radial displacement than that for a lower impact velocity. Moreover, the reason why the bending deformation decreases as V0 increases can be explained by the radial inertia effect. That is, the more faster the impact velocity becomes, the more rapidly the axial stress increases, but the expansion in radial direction would be delayed by the radial inertia effect. Such a mechanism can be observed in Fig.8. 0 200 400 0 0.8 1.6 0 400 800 Ra di al D is pl ac em en t U r ( m m ) Impact Velocity V0 (km/h) :Ur :σx A xi al S tre ss σ x ( M P a) Ur σx Fig.8.Variation of radial displacement Ur and axial stress σx at the apex of wrinkle with impact velocity V0 0 1 2 0 200 400 600 A xi al S tre ss σ x ( M P a) Radial Displacement Ur (mm) :Peak Stress :σx−Ur curve V0=300 (km/h) V0=360 (km/h) V0=180 (km/h) V0=5 (km/h) Fig.9. Variation of axial stressσx with radial displacement Ur at the apex of wrinkle for V0=5, 180, 300 and 360 (km/h) Figure 8 presents the relationship between the radial displacement Ur and the impact velocity V0 under almost the same amount of the axial stressσx. It is evident that the displacement Ur decreases as increase of V0 even if the same amount ofσx occurs. Moreover, the relationship between the axial stressσx and the radial displacement Ur for four cases of V0= 5, 180, 300 and 360 km/h are chased and the initial peak stress for every V0 is summarized in Fig.9 by marks ●. From the figure, the reason why the higher velocity causes the larger initial peak stress can be explained as follows. While the axial stress increases by propagating the stress waves over the tube, the radial displacement Ur develops by the axial compression, but the rate of Ur relatively decreases as increase of V0. As a result, the axial stress increases by the stress wave reflecting many times until the sufficient radial displacement can be reached for initiating the local buckling. In order to discuss the influence of V0 on the peak stress quantitatively, an effective parameter considering the effect of mechanical properties on the inertia effect is needed. Figure 10 shows the relationship between the normalized axial stress σx/E and displacement Ux/L for 208 Chen: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending three cases of a tube problem having different elastic modulus E, tube density ρ and the impact velocity V0. In the figure, parameter c represents the elastic wave speed for a one-dimensional rod, and can be shown as (E/ρ)1/2. Here, all models have different values of E, ρ and V0, but keep the same ratio V0/c. It is evident that all models behave the same response of the axial stressσx/E and Ux/L diagram. Strictly speaking, the elastic and the plastic wave speed for a circular tube are distinct from the elastic wave speed c for a one-dimensional rod. However, based on the fact that the relationship between the normalized axial stressσx/E and displacement Ux/L is the same under the same V0/c, the non-dimensional parameter V0/c can be used for evaluating the effect of impact velocity on the initial peak stressσ1. Fig.10. Normalized axial compressive stress and displacement behaviour for tubes having the same ratio of V0/c 3.2 Effects of tube geometries and material properties on the initial peak stress As the other effective parameters for the dynamic initial peak stressσ1, tube geometries (such as mean radius, R and thickness, t) and material properties (such as Young's modulus, E, hardening coefficient, Eh and yield stress,σy) can be given, and the effects of these parameters onσ1 are discussed as follows. 0 0.1 0.2 0 100 200 300 σ x (M Pa ) Ux / L :t=0.5(mm) , R=12.5(mm) V0=40 (km/h) L=160(mm) :t=1.0(mm) , R=25.0(mm) :t=1.5(mm) , R=37.5(mm) Fig.11. Comparison of the compressive stress and displacement behaviour for tubes having the same ratio of tube thickness to radius t/R Figure 11 summarizes effects of mean radius R and thickness t on the axial stress and displacement behaviour for tubes having different tube radius R and thickness t, but the same ratio of t/R. It is evident from Fig.11 that if the ratio of t/R has the same value, the initial peak stressσ1 has the same value. Advances in Systems Science and Applications (2011), Vol.11, No.3-4 209 ISSN 1078-6236 International Institute for General Systems Studies, Inc. 0 0.1 0.2 0 0.002 0.004 σ x / E Ux / L E ρ V0=40 (km/h) σy=E/1000 E0 2E0 ρ0 2ρ0 c E0=72.4 (GPa) ρ0=2685 (kg/m3) Eh E0/20 E0/10 E0 /ρ0 Fig.12. Normalized axial compressive stressσx/E and displacement Ux/L behaviour for tubes having the same ratios of Eh/E and V0/c Figure 12 shows the comparison of the relationship between two tubes having the different E and Eh but the same ratio of Eh/E. In the figure, the stress value is normalized by E. It can be observed from Fig.12 that the normalized initial peak stressσ1/E can be arranged as a function of the ratio, Eh/E. Moreover, the yield stressσy for a tube also affects the initial peak stress, but if the valueσy is quite smaller than the peak stressσ1, the effect seems to be negligible. From the above results and discussions, the normalized initial peak stressσ1/E for a tube obeying a bilinear stress and strain relationship can be expressed by a function composed of four parameters, V0/c, t/R, Eh/E andσy /E as: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = EE E R t c Vf E yh σσ ,,,0 1 1 (2) Figure 13 shows the relationship between the normalized impact velocity V0/c and the peak stressσ1/E for some combinations of ratios, t/R, Eh/E andσ1/E. It is clear from the figure that the stressσ1/E increases linearly with the parameter (V0/c)2. Also, the slope and intersect of the relationship depend on these ratios, Eh/E and t/R. 0 20 40 0 0.004 0.008 0.012 σ 1 / E (V0 / c)2 10−5 t/R σy=E/1000 σy=3E/1000 Eh/E 0.01 0.05 0.04 0.08 0.01 0.05 Fig.13. Normalized impact velocity V0/c and the peak stressσ1/E for some combinations of t/R, Eh/E andσy/E Based on these charateristics, the normalized initial peak stressσ1/E can be written by the following type of equation as: 2 2 0 1 1 C c VC E +⎟ ⎠ ⎞ ⎜ ⎝ ⎛= σ (3) where, 210 Chen: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending .,,,,,,, 3221 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = EE E R tfC EE E R tfC yhyh σσ Equation (3) means that the initial stress can be expressed by the sum of the inertial term including the parameter V0/c and the quasi-static term for V0=0. Here, the quasi-static term C2 has already been discussed in the previous study[11], and proposed an approximate equation as follows: ( ) ( ) ( ) ( ) ( ) . 1 2 13 0 17.0 02 1 2 ⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ >−⎟ ⎠ ⎞ ⎜ ⎝ ⎛⋅+ ≤ − = = − xRte E E R t E xRt R t E C k EE hy static hσ ν σ (4) where, ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = − = 0 2 0 22, 13 x R t E EEk E x y hy σ νσ Figure 14 shows comparisons of the parameter C2 obtained by FEM and its approximation obtained by Eq.(4). In the figure, results forσy/E=1/1000 and 3/1000 correspond to solid and dotted lines, respectively. It is clear that both results coincide with each other, and the approximate equation which is shown as Eq.(4) can be used to estimate the quasi-static peak stress with a good accuracy. 0 0.05 0.1 0 0.005 0.01 0.015 C 2 t / R σy/E=0.001 σy/E=0.003 Eh/E 0.01 0.05 0.1 Line :equation (4) :σy/E=0.001 :σy/E=0.003 Fig.14. Comparisons of quasi-static term C2 in Eq.(3) for tubes having some combinations of t/R,σy/E and Eh/E In Figure 15, results of the parameter C1 for some ratios, Eh/E,σy/E and t/R are summarized. It is found from the figure that the parameter C1 increases with increase of Eh/E, and its value becomes larger as the ratio of tube thickness to radius t/R decreases. Also, the value C1 depends on the yield stress σy. For example, C1 for t/R=0.02 and 0.08 under the same ratios, Eh/E=0.1 andσy/E=0.001 are almost equal to 13.0 and 8.0, respectively, so that the difference between them is almost 5.0. However, by concerning the term (V0/c)2, for example, if the impact velocity V0 is 100km/h, the order of the parameter (V0/c)2 is about 10-5, and the scale of the difference is one digit smaller than the parameter C2. Based on the fact, the parameter C1 can be written by the following equation as: 7.0 1 50 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛= E EC h (5) and shown in Fig.15 as a solid line. Advances in Systems Science and Applications (2011), Vol.11, No.3-4 211 ISSN 1078-6236 International Institute for General Systems Studies, Inc. 0 0.04 0.08 0.12 0 5 10 15 C 1 Eh / E t/R 0.04 :equation (5) σy/E=0.001 0.08 σy/E=0.003 Fig.15. Comparisons of quasi-static term C1 in Eq.(3) for tubes having some combinations of t/R,σy/E and Eh/E Consequently, an approximate equation for the non-dimensional initial peak stress for a cylindrical tube subjected to axial impact load is proposed in this paper as follows: static h Ec V E E E 1 2 0 7.0 1 50 σσ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛= (6) 0 200 400 0 500 1000 σ 1 (M P a) V0 (km/h) :Eh/E=0.01 :Eh/E=0.05 :Eh/E=0.1 t/R=0.04 :equation (6) (a) for the case of t/R=0.04 0 200 400 0 500 1000 1500 σ 1 (M Pa ) V0 (km/h) :Eh/E=0.01 :Eh/E=0.05 :Eh/E=0.1 t/R=0.08 :equation (6) (b) for the case of t/R=0.08 Fig.16. Estimation of the peak stress σ1 under some cases of impact velocity V0 Figure 16 shows the relationship between the initial peak stress and the impact velocity for some cases of Eh/E for the ratio of t/R=0.04 (Fig.16(a)) and 0.08 (Fig.16(b)). In these figures, solid lines correspond to the approximate results obtained by Eq.(6), and the symbols show the numerical results obtained by FEM. It is clear from these figures that the predictedσ1/E agrees well with the numerical results for a wide range of impact velocity V0. 4. Validation of the proposed prediction for σ1 212 Chen: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending In order to check the validity of the proposed prediction forσ1 as shown in Eq.(6), the same impact problem studied by Karagiozova and Jones[5] is examined and compared the prediction with their results in Fig. 17. In the figure, the solid line shows the approximation by Eq.(6), and dotted line and solid circles are the predictions and FEM results which can be found in Karagiozova and Jones' paper[5]. Here, the model is intended for a circular tube made of aluminium alloy, and the material and geometrical parameters of the model are shown in Fig. 17. It is evident that for relatively low impact velocity (V0<40m/s), the proposed approximation gives a good agreement with numerical results obtained by Karagiozova and Jones[5], which means that the proposed approximation can be effectively used for estimating the initial peak stressσ1 under a relatively low impact velocity. 0 40 80 200 400 600 V0 (m/sec) σ 1 (M Pa ) : FEM : prediction Karagiozova and Jones(2001) : Eq.(6) E =72.4(GPa), Eh =542.6(MPa), σy=295(MPa) ρ =2685(kg/m3), t =1.65(mm), R =11.875(mm) Fig.17. Comparison of estimating the peak stressσ1 under some cases of impact velocity V0 5. Conclusion In this paper, large displacement numerical simulation based on FEM is undertaken to explore the relationship between the impact velocity V0 and the initial peak stressσ1 for circular tubes when subjected to an axial impact. Based on our numerical results, the following points have been revealed. (1) The initial peak stressσ1 becomes higher with increases of the impact velocity V0. That is because the local displacement Ur in radial direction decreases as V0 increases, under the same axial stressσx. (2) The initial peak stressσ1 can be expressed by the sum of a term including V0/c and a term obtained by quasi-static numerical simulation. (3) The proposed approximate equation forσ1 can be effectively used under low impact velocity (V0<40m/s). References [1] Karagiozova D. and Alves M., Jones N., Inertia effects in axisymmetrically deformed cylindrical shells under axial impact. Int. J. Impact Engng. 24(10)(2000) 1083-1115. 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