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Energy-absorption behavior of a metallic double-sine-wave beam under axial crushing
ARTICLE IN PRESS Thin-Walled Structures 47 (2009) 1168–1176
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Energy-absorption behavior of a metallic double-sine-wave beam under axial crushing W. Jiang, J.L. Yang The Solid Mechanics Research Centre, Beijing University of Aeronautics and Astronautics, Beijing 100191, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 September 2008 Received in revised form 13 April 2009 Accepted 24 April 2009 Available online 17 May 2009
Energy-absorbing behavior of a metallic double-sine-wave beam under axial crushing is studied in this paper. The aims of the study are to improve the energy-absorbing capability of traditional corrugated beams with sine-wave at single direction and reduce the initial peak force by forcing the beam plasticly deformed at the predetermined intervals along the axial direction. The theoretical approach based on a rigid, perfectly plastic model is adopted to predict the mean crushing force under axial crush loading. Besides, the numerical analysis of energy-absorption behavior of aluminium double-sine-wave beams is conducted by using nonlinear finite element code MSC.DYTRAN. The numerical results are compared well with theoretical predictions and show that although the crushing mean force slightly decreases, the double-sine-wave beam produces better uniform load displacement relationship and lower initial peak force. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Metallic double-sine-wave beam Energy-absorption capability Axial crushing Rigid perfectly plastic Concertina mode
1. Introduction Sine-wave beams are widely used as energy absorbers for aircraft subfloor beam since they are weight efficient [1]. In most cases, although sine-wave beams are not designed to bear inservice loads under normal condition, they play an important role in improving the fuselage crashworthiness which must be considered in primary stage of designing aircraft according to crashworthiness requirement forced by FAA in the US. In last decades, many researches have been conducted experimentally and theoretically in both metallic and composite sine-wave beams. Some methods of predicting the energy-absorption capability of sine-wave beams based on empirical data of circular-tube specimens have been developed [2,3]. Currently, the crushing behavior of sine-wave beams can be numerically investigated by some commercial codes, such as MSC.DYTRAN, PAM-CRASH, ABAQUS Explicit and LS-DYNA [4,5]. One target of the work of these numerical simulations is to examine whether or not the code can efficiently predict the individual crushing failure mode under large structural deformation. Although the traditional-corrugated beam with single-directional sine-wave (referred to as a straight sine-wave beam in the following sections), depending upon specimen geometry and material parameters, shows high energy-absorption potential, there are still some disadvantages which limit its further applications. Experiments have shown that axial crushing of a straight sine-wave beam produces a load displacement relationship characterized by high initial peak
Corresponding author. Tel.: +86 1082317528; fax: +86 1082315606.
E-mail address: [email protected] (J.L. Yang). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.04.006
force [5]. This is due to the fact that the perfect straight sine-wave beam can hardly develop into the formation of the initial buckle. Although in the design of the composite sine-wave beam, precrush triggers may be introduced, some experiments showed that the trigger mechanism implemented on the sine-wave beam to control the start of crushing did not function and hence, the traditional straight sine-wave beam did not give full play to its energy-absorption capability [5]. Besides, in most cases, the traditional straight sine-wave beam hardly follows prescribed deformation mode which is available for conducting the energyabsorption capability of the structure to be maximum. To eliminate the disadvantages of the traditional straight sinewave beam mentioned above, a double-sine-wave beam with corrugation (see Fig. 1(a)) is proposed. Compared with the traditional straight sine-wave beam, the double-sine-wave beam has another curvature along the Z-direction. An illustrative figure is shown in Fig. 1(b). The surface of the double-sine-wave beam is generated by the CURVE 2 sweeping along the CURVE 1. The CURVE 1 on the X–Y plane and the CURVE 2 on the Y–Z plane are illustrated in Fig. 1(c) and Fig. 1(d), respectively. Note that R0 ¼ 23 mm, f ¼ 1261, N refers to the amplitude of the sinusoid, and 4H is the cycle length of the sinusoid. The purpose of this improvement is to induce the sine-wave beam to produce so-called ‘‘concertina’’ mode, and a load displacement relationship with desirable uniformity may be obtained. The corrugations introduced along the axial direction cause large plastic bending moment at some interval in the beam and therefore plastic hinges may form in the beam at the same interval. Since the double-sine-wave beam exhibits the energyabsorption behavior as comparable to that of a circular tube, the theoretical analysis of the circular tube in ‘‘concertina’’ mode can
ARTICLE IN PRESS W. Jiang, J.L. Yang / Thin-Walled Structures 47 (2009) 1168–1176
Nomenclature
f N H h L n m R0 Eb EIb EIIb M0 Em EIm EIIm w
radian of the arc sinusoid amplitude half folding length double-sine-wave beam wall thickness height of the double-sine-wave beam amplitude factor eccentricity factor radius of the arc total bending energy bending energy of the first phase bending energy of the second phase fully plastic bending moment per unit circumferential length total membrane energy membrane energy of the first phase membrane energy of the second phase radial displacement of the shell element
be used on the study of double-sine-wave beam crushing with some improvement. In the early analysis of an axially compressed circular tube by Alexander (1960), the material was assumed to be rigid perfectly plastic, and the tube was assumed to undergo axisymmetric folding [6]. The model of progressive crushing of circular tubes was developed by many subsequent researchers such as Amdahl and Soreide (1981), Andronicou and Walker (1981), Abramowicz and Jones (1984, 1986), Wierzbick and Bhat (1986) and Grzebieta (1990), in which a more realistic radially outward folding mechanism was formulated [7–12]. The more recent study of the problem was carried out by Wierzbicki (1992), Singace (1995), and Singace and EI-Sobky (1996), where a new approach to the representation of the concertina collapse mode of tubes was introduced [13–15]. The authors selected structural deformation fields based on three stationary plastic hinge mechanisms. As the fold develops, the deformation fields allow both inward and outward radial displacement of the tube to develop. In this paper, by taking the amplitude factor n into account, the stationary plastic hinge mechanisms which captured several
N0 P¯ S
1169
fully plastic membrane force per unit length mean crushing load wave-number in Y-direction for the double-sine-wave beam
Greek letters
a b
a0 b0 d
s0
angle between the leg of the inward fold and the horizontal axis angle between the leg of the outward fold and the horizontal axis critical angle corresponding to an established inward fold critical angle corresponding to an established outward fold shortening of the column yield stress
key features of the crushing process with great accuracy are developed to derive the mean crushing force of the doublesine-wave beam. Moreover, numerical analysis of energyabsorption behavior of the beam is conducted by using nonlinear finite element code MSC.DYTRAN. The results of the numerical simulation are compared with those of the theoretical predictions.
2. Theoretical analysis The transformation of the double-sine-wave beam under the axial load is similar to one of the circular tubes into ‘‘concertina’’ mode. Such collapse of beams can be beautifully modeled by stationary plastic hinge mechanisms based on the rigid perfectly plastic idealization for the beam material. In any given cycle of deformation representing a complete fold, the sum of the plastic bending and stretching energies in the beam is equal to the work done by the applied force, and an expression for the mean crushing load is obtained.
Fig. 1. Sketch of the double-sine-wave beam.
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In the following analysis, it is assumed that crushing progress by virtue of instantaneous formation of three stationery plastic hinges leads to a fold comprised of two elements of equal lengths. Fig. 2 illustrates the crushing of the two-element model in this representation, where Element 1 begins at A and ends at B; Element 2 begins at B and ends at C. The length of each Element is 2H. The value of H is 7.5 mm in all cases in this paper, which is pffiffiffiffiffiffiffiffiffiffiffi referred to the approximate equation H 0:95 2R0 h=2 6 mm (the value of h is given in Section 3.1) derived by Alexander [6], and furthermore, present numerical simulation shows that the selection of H ¼ 7.5 mm is better than that of H ¼ 6 mm.
2.1. Geometry It is convenient to take an inclination angle of the first element (AB) a0 as a reference configuration such that the point B and point C of the next element (BC) are positioned on the two dashed lines (see Fig. 3). The distance between the dashed line and the real line is N, which is the depth of the corrugation, where N ¼ n 2H. Parameter m is known as eccentricity factor, which defines the outward portion over whole length 2H and is related to angle a0 and amplitude factor n by cos a0 ¼ n þ m
and its rate obeys the equation
d_ ¼ 2H½a_ cos a þ b_ cos b
2.2. Bending energy The rate of bending energy in a sine-wave beam, E_ b , is given by E_ b ¼ 2fR0 M 0 ðja_ j þ jb_ jÞ 2
(1)
(2)
and their rates are related by sin aa_
b_ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(3)
½1 ðcos a þ n mÞ2
The shortening of the column d, measured from the reference position, can be obtained as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ 2H½ 1 4n2 þ sin a0 sin a sin b (4)
(6)
where M0 ¼ s0h /4 is the fully plastic bending moment per unit circumferential length, s0 is yield stress and h is the doublesine-wave beam thickness, respectively. Using Eqs. (2) and (3), Eq. (6) can be integrated over the duration of the complete folding cycle and the bending energy, therefore, is obtained. For the first phase of the folding cycle, a varies from a0 to 0 while b changes from p/2
As the folding process goes on, point C moves along the dashed line. The inclination of the first deforming element, AB, with respect to the horizontal line is denoted by a. The second element BC is inclined by an angle of b. The compatible relationship between a and b can be obtained by cos b ¼ cos a þ n m
(5)
Fig. 3. A transition zone consisting of two superfolding elements.
Fig. 2. Subsequent deformation stages of the two-element model.
ARTICLE IN PRESS W. Jiang, J.L. Yang / Thin-Walled Structures 47 (2009) 1168–1176
2.5. The load displacement behavior
to b0.Thus, EIb ¼ 2fR0 M 0 ½sin1 ð1 þ n mÞ þ cos1 ðn þ mÞ sin1 ð2nÞ
(7)
The instantaneous equilibrium of the double-sine-wave beam in the transition zone can be represented by the principle of virtual velocity
(8)
Pd_ ¼ SðE_ b þ E_ m Þ
During the next half cycle m becomes 1m. Therefore, EIIb ¼ 2fR0 M 0 ½sin1 ðn þ mÞ þ cos1 ðn þ 1 mÞ sin1 ð2nÞ
The total bending energy Eb for the full fold is found by summing Eqs. (7) and (8). Hence, Eb ¼ EIb þ EIIb ¼ 2fR0 M0 ½p 2 sin1 ð2nÞ
(9)
(17)
Using Eqs. (5), (6), (10) and (15), the normalized instantaneous crushing force in the first half of the cycle can be recast in the form PðaÞ ¼ f 1 ða; m; nÞ þ f 2 ða; m; nÞ P¯
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hR0 1 þ ðsin a= ½1 ðcos a þ n mÞ2 f1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f8H2 ð1 2nÞ þ R0 h½p 2 sin1 ð2nÞg cos a þ sin aðcos a þ n mÞ= ½1 ðcos a þ n mÞ2 f2 ¼
1171
16H2 sin a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f8H2 ð1 2nÞ þ R0 h½p 2 sin1 ð2nÞg cos a þ sin aðcos a þ n mÞ= ½1 ðcos a þ n mÞ2
(18)
(19)
(20)
2.3. Membrane energy The membrane energy results from compression or extension of a shell element in the hoop direction, it gives Z wðsÞ _ ds (10) E_ m ¼ fR0 N0 ABC R0 where N0 ¼ s0h is the fully plastic membrane force per unit _ length and w(s) is the radial velocity of material point ) wðsÞ ¼ s cos a 2Hm on AB _ wðsÞ ¼ s sin aa_ wðsÞ ¼ s cos b 2Hn _ wðsÞ ¼ s sin aa_
) on BC
(11)
For the first phase, the membrane energy is EIm
¼ 4fN0 H2 ð1 m nÞ
(12)
For the second phase, it is EIIm
¼ 4fN0 H2 ðm nÞ
In the second half of the cycle the same formulation would apply with m replaced by (1m). According to the Wierzbicki’s theoretical analysis, the eccentricity factor m is arbitrary and indeterminate. Although Singace [14] derived the value of m by using the Newton–Raphson numerical scheme, such value of m can be only used for the problem of circular tube without corrugation. For the doublesine-wave beam, as shown in Fig. 6(f), the folds extend the same distance on both the internal and external sides. Besides, the load displacement curves show that the peaks of each fold are almost the same. Due to these two points, setting m as 0.5 is reasonable for the theoretical analysis of the double-sine-wave beam. The load displacement curves for different values of n are shown in Fig. 4. It is observed that with the increasing value of n, both the peak crushing force and crushing distance of each fold is decreased: the curves exhibit the tendency to approach the load uniformity.
(13)
By summing Eqs. (12) and (13), the total membrane energy term is given by Em ¼ EIm þ EIIm ¼ 4fN0 H2 ð1 2nÞ
(14)
2.4. Expression for P¯ The total work done by the mean crushing load P¯ acting over the two phases of compression is P¯ 4H. The mean crushing load is calculated from the balance of global energy such that P¯ 4H ¼ SðEb þ Em Þ
(15)
where S is the wave-number in Y-direction for the double-sinewave beam. The normalized mean crushing load can be written in the form P¯ 4fHð1 2nÞ fR0 ½p 2 sin1 ð2nÞ ¼S þ 2H M0 h
(16) Fig. 4. A load displacement diagram of different values of n.
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3. Numerical simulation 3.1. Finite element models The material of double-sine-wave beam employed here is made of aluminium alloy with elastic perfectly plastic constitution, and the mechanical properties are that Young’s modulus E ¼ 70 GPa, yield stress s0 ¼ 284 MPa, and Poisson’s ratio n ¼ 0.35. The finite element model of the double-sine-wave beam is shown in Fig. 5. The bottom of the beam is clamped, while the other end is free. The beam is crushed on the top in the axial direction by a rigid wall with prescribed velocity 1 m/s. The beam is modeled with Belytschko–Tsay 4-node shell element with 3 mm thickness. The aircraft structures with 3 mm thickness could serve good function of energy absorption without adding too much weight to the aircraft. Two types of contact elements are employed in the simulation of crushing. For the sine-wave beam itself an automatic single-surface contact is adopted to account for the contact between lobes during crushing process. Between the rigid body and the sine-wave beam, an automatic node-tosurface contact is defined. 3.2. Pre-crushed trigger Compared with the straight sine-wave beam, the initial peak force of the double-sine-wave beam is largely reduced due to the pre-crushed bending. However, the initial peak forces are still higher than the average forces. To obtain a desirable load uniformity, the pre-crushed trigger could be introduced by a variety of methods. Among them the indentation trigger [16,17] is used most commonly in thin-walled structures. Other methods such as triggers based on elastic buckling mode shapes [18] are also used as an alternative. In this section, the measure of reducing the thickness of the top part of the double-sine-wave beam is used as the pre-crushed trigger. This measure is found to be convenient and effective in eliminating the initial peak force. According to the numerical simulation, a better result could be obtained by reducing the thickness near the top of the beam from 3 to 2 mm. 3.3. Numerical results Using MSC.DYTRAN for the simulation, the numerical results of the double-sine-wave beam under axial crushing can be obtained and some typical collapse modes are shown in Fig. 6. It is shown that the collapse modes of the double-sine-wave beam with different value of N are similar, and the plastic deformation of the double-sinewave beam occurs at the predetermined intervals along the axial direction. In this part, the collapse modes of the double-sine-wave
beam with N ¼ 2.5 mm are shown in Fig. 6(a–e) and as a comparison, the collapse mode of the straight sine-wave beam is shown in Fig. 7. Fig. 6(f) demonstrated the complete one-fold crushing of the doublesine-wave beam. It is easy to see that the double-sine-wave beam deforms in axisymmetric or ‘‘concertina’’ mode when axially crushed. The straight sine-wave beam, however, exhibits non-symmetric mode when subjected to axial crushing load. The comparison between them indicates that the deformation mode of a doublesine-wave beam during crushing could be controlled, making its energy-absorption capability fully utilized; but for a straight sinewave beam, usually it is hard to predict its deformation mode during crushing and its energy-absorption capability is hard to be evaluated accurately. The load displacement curves of both the straight sine-wave beam and the double-sine-wave beam with N ¼ 1.25 mm are shown in Fig. 8. It is shown that the axial force of the straight sine-wave beam reaches a significant high initial peak force, followed by a steep drop. The initial peak force of the doublesine-wave beam, however, is far lower than that of the straight sine-wave beam. Furthermore, for the double-sine-wave beam, the load displacement behavior exhibits a repeated pattern. Each pair of the peaks is associated with the development of one full wrinkle or buckle. The 12 fluctuations on the load displacement curve correspond to 12 corrugations of the double-sine-wave beam. Thus, the double-sine-wave beam produces a better load displacement response and uniform collapse load than the straight one. As it can be seen in Fig. 8, the introduction of the corrugations in the aluminium double-sine-wave beam reduces the mean collapse load when compared with that of the straight sine-wave beam. This is due to the fact that the introduction of the corrugations reduces the amount of external work required for a complete crush. The load displacement curves of the double-sine-wave beam with different values of N are shown in Fig. 9. With the increasing value of N, the mean crushing force is decreased. This is due to the fact that the deeper corrugation results in more decrease of the amount of external work required for a complete crush. As expected, the more shallow the corrugations, the higher the tendency for the double-sine-wave beam to approach the mean crushing force of the straight sine-wave beam. In addition, the sine-wave beam with the deeper corrugation produces a better load displacement response with smaller fluctuations of the collapse load. And this behavior, known by the load uniformity ratio, is favorable for an energy-absorption device. 3.4. Comparison between theoretical predictions and numerical results The theoretical predictions and the numerical simulation results of the energy absorption of the straight and double-sine-wave
Fig. 5. Finite element model of the double-sine-wave beam.
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Fig. 6. Collapse modes of the double-sine-wave beam: (a) d ¼ 0.03 m; (b) d ¼ 0.06 m; (c) d ¼ 0.09 m; (d) d ¼ 0.12 m; (e) d ¼ 0.15 m; and (f) formation of a complete fold.
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Fig. 7. Collapse modes of the straight sine-wave beam.
the double-sine-wave beam also plays an important role in the deformation mode. Usually, the double-sine-wave beam with short length would prefer to deform in concertina mode, and the one with long length might result in Euler collapse mode. Finally, it is still worth discussing the issues with manufacturability of such a double-sine-wave beam. Although it has some difficulties in the manufactures, two current methods could be considered: one is using CNC milling machine and the other is to employ the stamp press. In the future, the advanced manufacturing technology could also be employed for the beam manufacture.
4.2. Conclusion remark
Fig. 8. The load displacement characteristics of the straight and double-sine-wave beam.
beam are shown in Table 1 in which the crushing distance is 120 mm. It should be noted that the theoretical value for the straight sine-wave beam is approximately deemed as the value for the double-sine-wave beam with n equating to 0. It is shown that the numerical results of the mean crushing force of different values of N are compared well with theoretical predictions, with error less than 12%. In addition, all the theoretical predictions are slightly overestimated.
4. Discussion and conclusion 4.1. Discussion The present investigation was conducted on the case of R0 ¼ 23 mm, L ¼ 180 mm, f ¼ 1261 and h ¼ 3 mm. The different energy-absorption characteristics can be obtained by a combination of different values of R0, L, f and h. For tubes subjected to axial crushing load, relative thick-walled tubes deform in concertina mode, however, thin-walled tubes exhibit diamond mode, which is a non-symmetrical mode [19]. Similar phenomenon happens for the double-sine-wave beam, i.e. the beams with thick wall would prefer to deform in concertina mode. Besides, the quasi-static behavior of axially compressed ductile tubes indicates the influence of tube height on collapse mode [20]. The height of
The energy-absorption behavior of the double-sine-wave beam was investigated. In the theoretical analysis, the rigid perfectly plastic material model was adopted and the stationary plastic hinge mechanisms were employed. The introduction of the amplitude factor n in the analysis, successfully lead to a quantitative expression of the energy-absorbing capability of the double-sine-wave beam with different depth of the corrugation. Numerical simulations were conducted by using nonlinear explicit finite element codes MSC.DYTRAN. The mean crushing forces of double-sine-wave beam obtained by numerical simulation were in good agreement with those of theoretical analysis. Compared with the traditional straight sine-wave beam, the double-sine-wave beam has shown favorable characteristics as an energy absorber in terms of load uniformity and low initial peak force. Furthermore, it is found that the energy absorbed by the axial crushing of the double-sine-wave beam could be controlled by the introduction of corrugations with suitable depth along the axial direction. This makes its energy-absorption capability fully utilized. It has been shown that with increase of the depth of corrugation, the amount of external work required for a complete crush is decreased, and hence the mean crushing force. Meanwhile, the double-sine-wave beam of the deeper corrugation produces an acceptable load displacement curve with smaller fluctuations of the mean crushing force. Although the classification of the axial collapse modes of cylindrical tubes under quasi-static loading has been discussed by Andrews [20], the expressions for energy absorption were mainly based on the experimental data. Compared with the tubes, the deformation mode of the double-sine-wave beam is more complicated due to its axis asymmetry. Thus, the collapse mode of the double-sine-wave beam is suggested for future study. In addition, more attention should be dedicated to the investigation of quantifying the parameter effects of the double-sine-wave beam, such as height, diameter and thickness, on collapse mode.
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Fig. 9. The load displacement characteristics of the double-sine-wave beam with different values of N: (a) N ¼ 1 mm; (b) N ¼ 1.5 mm; (c) N ¼ 2 mm; (d) N ¼ 2.5 mm; (e) N ¼ 3 mm; and (f) N ¼ 4 mm.
Table 1 Numerical results and theoretical predictions for energy absorption (R0 ¼ 23 mm, L ¼ 180 mm, h ¼ 3 mm, d(crushing distance) ¼ 120 mm). N (mm)
Numerical results (kJ)
Theoretical predictions (kJ)
Error (%)
0 1 1.25 1.5 2 2.5 3 4
33.971 29.907 28.065 26.557 25.354 24.336 20.301 17.377
34.957 30.864 29.833 28.801 26.730 24.650 22.558 18.317
2.90 3.20 6.30 8.45 5.43 1.29 11.11 5.41
Acknowledgement The work described in this paper is financially supported by the National Natural Science Foundation of China under Grant number 10532020. The authors would like to gratefully acknowledge this support.
References [1] Carden HD. Energy-absorbing-beam design for composite aircraft subfloors. AIAA 1993;1339:378–88. [2] Farley GL. A method of predicting the energy-absorption capability of composite subfloor beams. NASA 1987;800:428–37. [3] Farley GL. Crash energy-absorbing composite subfloor structure. AIAA 1986:0944. [4] McCarthy MA, Wiggenraad JFM. Numerical investigation of a crash test of a composite helicopter subfloor structure. Compos Struct 2001;51:345–59. [5] Mahe M, Ribet H, LePage F. Composite fuselage crash FE modeling dedicated to enhance the design in correlation with full scale drop test. Mec. Ind. 2001;2:5–17. [6] Alexander JM. An approximate analysis of the collapse of thin cylindrical shell under axial loading. Q J Mech Appl Math 1960;13(1):5–10. [7] Amdahl J, Soreide TH. Energy absorption in axially compressed cylindrical shells with special reference to bulbous bows in collision. Norw Marit Res 1981;4:2–11. [8] Andronicous A, Walker AC. A plastic collapse mechanism for cylinders under uniaxial end compression. J Constr Steel Res 1981;1:23–34. [9] Abramowicz W, Jones N. Dynamic axial crushing of circular tubes. Int J Impact Eng 1984;2:263–81. [10] Abramowicz W, Jones N. Dynamic progressive buckling of circular and square tubes. Int J Impact Eng 1986;4:243–70.
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[11] Wierzbicki T, Bhat SU. A moving hinge solution for axisymmetric crushing of tubes. Int J Mech Sci 1986;28:135–51. [12] Grzebieta RH. An alternative method for determining the behavior of round stocky tubes subjected to an axial crush load. Thin-Walled Struct 1990;9: 61–89. [13] 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. [14] Singace AA, El-Sobky H, Reddy TY. On the eccentricity factor in the progressive crushing of tubes. Int J Solids Struct 1995;32(24):3589–602. [15] Singace AA, El-Sobky H. Further experimental investigation on the eccentricity factor in the progressive crushing of tubes. Int J Solids Struct 1996; 33:3517–38.
[16] Langseth M, Hopperstad OS, Hanssen AG. Crash behavior of thin-walled aluminium members. Thin-Walled Struct 1998;32(1–3):127–50. [17] Hanssen AG, Langseth M, Hopperstad OS. Static and dynamic crushing of square aluminium extrusions with aluminium foam filler. Int J Impact Eng 2000;24(4):347–83. [18] Kim HS. New extruded multi-cell aluminium profile for maximum crash energy absorption and weight efficiency. Thin-Walled Struct 2002;40(4): 311–27. [19] Singace AA, EI-Sobky H. Behavior of axially crushed corrugated tubes. Int. J Mech Sci 1997;39(3):249–68. [20] Andrews KRF, England GL, Ghani E. Classification of the axial collapse of cylindrical tubes under quasi-static loading. Int J Mech Sci 1983;25(9–10): 687–96.
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
Energy-absorption behavior of a metallic double-sine-wave beam under axial crushing W. Jiang, J.L. Yang The Solid Mechanics Research Centre, Beijing University of Aeronautics and Astronautics, Beijing 100191, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 September 2008 Received in revised form 13 April 2009 Accepted 24 April 2009 Available online 17 May 2009
Energy-absorbing behavior of a metallic double-sine-wave beam under axial crushing is studied in this paper. The aims of the study are to improve the energy-absorbing capability of traditional corrugated beams with sine-wave at single direction and reduce the initial peak force by forcing the beam plasticly deformed at the predetermined intervals along the axial direction. The theoretical approach based on a rigid, perfectly plastic model is adopted to predict the mean crushing force under axial crush loading. Besides, the numerical analysis of energy-absorption behavior of aluminium double-sine-wave beams is conducted by using nonlinear finite element code MSC.DYTRAN. The numerical results are compared well with theoretical predictions and show that although the crushing mean force slightly decreases, the double-sine-wave beam produces better uniform load displacement relationship and lower initial peak force. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Metallic double-sine-wave beam Energy-absorption capability Axial crushing Rigid perfectly plastic Concertina mode
1. Introduction Sine-wave beams are widely used as energy absorbers for aircraft subfloor beam since they are weight efficient [1]. In most cases, although sine-wave beams are not designed to bear inservice loads under normal condition, they play an important role in improving the fuselage crashworthiness which must be considered in primary stage of designing aircraft according to crashworthiness requirement forced by FAA in the US. In last decades, many researches have been conducted experimentally and theoretically in both metallic and composite sine-wave beams. Some methods of predicting the energy-absorption capability of sine-wave beams based on empirical data of circular-tube specimens have been developed [2,3]. Currently, the crushing behavior of sine-wave beams can be numerically investigated by some commercial codes, such as MSC.DYTRAN, PAM-CRASH, ABAQUS Explicit and LS-DYNA [4,5]. One target of the work of these numerical simulations is to examine whether or not the code can efficiently predict the individual crushing failure mode under large structural deformation. Although the traditional-corrugated beam with single-directional sine-wave (referred to as a straight sine-wave beam in the following sections), depending upon specimen geometry and material parameters, shows high energy-absorption potential, there are still some disadvantages which limit its further applications. Experiments have shown that axial crushing of a straight sine-wave beam produces a load displacement relationship characterized by high initial peak
Corresponding author. Tel.: +86 1082317528; fax: +86 1082315606.
E-mail address: [email protected] (J.L. Yang). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.04.006
force [5]. This is due to the fact that the perfect straight sine-wave beam can hardly develop into the formation of the initial buckle. Although in the design of the composite sine-wave beam, precrush triggers may be introduced, some experiments showed that the trigger mechanism implemented on the sine-wave beam to control the start of crushing did not function and hence, the traditional straight sine-wave beam did not give full play to its energy-absorption capability [5]. Besides, in most cases, the traditional straight sine-wave beam hardly follows prescribed deformation mode which is available for conducting the energyabsorption capability of the structure to be maximum. To eliminate the disadvantages of the traditional straight sinewave beam mentioned above, a double-sine-wave beam with corrugation (see Fig. 1(a)) is proposed. Compared with the traditional straight sine-wave beam, the double-sine-wave beam has another curvature along the Z-direction. An illustrative figure is shown in Fig. 1(b). The surface of the double-sine-wave beam is generated by the CURVE 2 sweeping along the CURVE 1. The CURVE 1 on the X–Y plane and the CURVE 2 on the Y–Z plane are illustrated in Fig. 1(c) and Fig. 1(d), respectively. Note that R0 ¼ 23 mm, f ¼ 1261, N refers to the amplitude of the sinusoid, and 4H is the cycle length of the sinusoid. The purpose of this improvement is to induce the sine-wave beam to produce so-called ‘‘concertina’’ mode, and a load displacement relationship with desirable uniformity may be obtained. The corrugations introduced along the axial direction cause large plastic bending moment at some interval in the beam and therefore plastic hinges may form in the beam at the same interval. Since the double-sine-wave beam exhibits the energyabsorption behavior as comparable to that of a circular tube, the theoretical analysis of the circular tube in ‘‘concertina’’ mode can
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Nomenclature
f N H h L n m R0 Eb EIb EIIb M0 Em EIm EIIm w
radian of the arc sinusoid amplitude half folding length double-sine-wave beam wall thickness height of the double-sine-wave beam amplitude factor eccentricity factor radius of the arc total bending energy bending energy of the first phase bending energy of the second phase fully plastic bending moment per unit circumferential length total membrane energy membrane energy of the first phase membrane energy of the second phase radial displacement of the shell element
be used on the study of double-sine-wave beam crushing with some improvement. In the early analysis of an axially compressed circular tube by Alexander (1960), the material was assumed to be rigid perfectly plastic, and the tube was assumed to undergo axisymmetric folding [6]. The model of progressive crushing of circular tubes was developed by many subsequent researchers such as Amdahl and Soreide (1981), Andronicou and Walker (1981), Abramowicz and Jones (1984, 1986), Wierzbick and Bhat (1986) and Grzebieta (1990), in which a more realistic radially outward folding mechanism was formulated [7–12]. The more recent study of the problem was carried out by Wierzbicki (1992), Singace (1995), and Singace and EI-Sobky (1996), where a new approach to the representation of the concertina collapse mode of tubes was introduced [13–15]. The authors selected structural deformation fields based on three stationary plastic hinge mechanisms. As the fold develops, the deformation fields allow both inward and outward radial displacement of the tube to develop. In this paper, by taking the amplitude factor n into account, the stationary plastic hinge mechanisms which captured several
N0 P¯ S
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fully plastic membrane force per unit length mean crushing load wave-number in Y-direction for the double-sine-wave beam
Greek letters
a b
a0 b0 d
s0
angle between the leg of the inward fold and the horizontal axis angle between the leg of the outward fold and the horizontal axis critical angle corresponding to an established inward fold critical angle corresponding to an established outward fold shortening of the column yield stress
key features of the crushing process with great accuracy are developed to derive the mean crushing force of the doublesine-wave beam. Moreover, numerical analysis of energyabsorption behavior of the beam is conducted by using nonlinear finite element code MSC.DYTRAN. The results of the numerical simulation are compared with those of the theoretical predictions.
2. Theoretical analysis The transformation of the double-sine-wave beam under the axial load is similar to one of the circular tubes into ‘‘concertina’’ mode. Such collapse of beams can be beautifully modeled by stationary plastic hinge mechanisms based on the rigid perfectly plastic idealization for the beam material. In any given cycle of deformation representing a complete fold, the sum of the plastic bending and stretching energies in the beam is equal to the work done by the applied force, and an expression for the mean crushing load is obtained.
Fig. 1. Sketch of the double-sine-wave beam.
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In the following analysis, it is assumed that crushing progress by virtue of instantaneous formation of three stationery plastic hinges leads to a fold comprised of two elements of equal lengths. Fig. 2 illustrates the crushing of the two-element model in this representation, where Element 1 begins at A and ends at B; Element 2 begins at B and ends at C. The length of each Element is 2H. The value of H is 7.5 mm in all cases in this paper, which is pffiffiffiffiffiffiffiffiffiffiffi referred to the approximate equation H 0:95 2R0 h=2 6 mm (the value of h is given in Section 3.1) derived by Alexander [6], and furthermore, present numerical simulation shows that the selection of H ¼ 7.5 mm is better than that of H ¼ 6 mm.
2.1. Geometry It is convenient to take an inclination angle of the first element (AB) a0 as a reference configuration such that the point B and point C of the next element (BC) are positioned on the two dashed lines (see Fig. 3). The distance between the dashed line and the real line is N, which is the depth of the corrugation, where N ¼ n 2H. Parameter m is known as eccentricity factor, which defines the outward portion over whole length 2H and is related to angle a0 and amplitude factor n by cos a0 ¼ n þ m
and its rate obeys the equation
d_ ¼ 2H½a_ cos a þ b_ cos b
2.2. Bending energy The rate of bending energy in a sine-wave beam, E_ b , is given by E_ b ¼ 2fR0 M 0 ðja_ j þ jb_ jÞ 2
(1)
(2)
and their rates are related by sin aa_
b_ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(3)
½1 ðcos a þ n mÞ2
The shortening of the column d, measured from the reference position, can be obtained as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ 2H½ 1 4n2 þ sin a0 sin a sin b (4)
(6)
where M0 ¼ s0h /4 is the fully plastic bending moment per unit circumferential length, s0 is yield stress and h is the doublesine-wave beam thickness, respectively. Using Eqs. (2) and (3), Eq. (6) can be integrated over the duration of the complete folding cycle and the bending energy, therefore, is obtained. For the first phase of the folding cycle, a varies from a0 to 0 while b changes from p/2
As the folding process goes on, point C moves along the dashed line. The inclination of the first deforming element, AB, with respect to the horizontal line is denoted by a. The second element BC is inclined by an angle of b. The compatible relationship between a and b can be obtained by cos b ¼ cos a þ n m
(5)
Fig. 3. A transition zone consisting of two superfolding elements.
Fig. 2. Subsequent deformation stages of the two-element model.
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2.5. The load displacement behavior
to b0.Thus, EIb ¼ 2fR0 M 0 ½sin1 ð1 þ n mÞ þ cos1 ðn þ mÞ sin1 ð2nÞ
(7)
The instantaneous equilibrium of the double-sine-wave beam in the transition zone can be represented by the principle of virtual velocity
(8)
Pd_ ¼ SðE_ b þ E_ m Þ
During the next half cycle m becomes 1m. Therefore, EIIb ¼ 2fR0 M 0 ½sin1 ðn þ mÞ þ cos1 ðn þ 1 mÞ sin1 ð2nÞ
The total bending energy Eb for the full fold is found by summing Eqs. (7) and (8). Hence, Eb ¼ EIb þ EIIb ¼ 2fR0 M0 ½p 2 sin1 ð2nÞ
(9)
(17)
Using Eqs. (5), (6), (10) and (15), the normalized instantaneous crushing force in the first half of the cycle can be recast in the form PðaÞ ¼ f 1 ða; m; nÞ þ f 2 ða; m; nÞ P¯
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hR0 1 þ ðsin a= ½1 ðcos a þ n mÞ2 f1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f8H2 ð1 2nÞ þ R0 h½p 2 sin1 ð2nÞg cos a þ sin aðcos a þ n mÞ= ½1 ðcos a þ n mÞ2 f2 ¼
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16H2 sin a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f8H2 ð1 2nÞ þ R0 h½p 2 sin1 ð2nÞg cos a þ sin aðcos a þ n mÞ= ½1 ðcos a þ n mÞ2
(18)
(19)
(20)
2.3. Membrane energy The membrane energy results from compression or extension of a shell element in the hoop direction, it gives Z wðsÞ _ ds (10) E_ m ¼ fR0 N0 ABC R0 where N0 ¼ s0h is the fully plastic membrane force per unit _ length and w(s) is the radial velocity of material point ) wðsÞ ¼ s cos a 2Hm on AB _ wðsÞ ¼ s sin aa_ wðsÞ ¼ s cos b 2Hn _ wðsÞ ¼ s sin aa_
) on BC
(11)
For the first phase, the membrane energy is EIm
¼ 4fN0 H2 ð1 m nÞ
(12)
For the second phase, it is EIIm
¼ 4fN0 H2 ðm nÞ
In the second half of the cycle the same formulation would apply with m replaced by (1m). According to the Wierzbicki’s theoretical analysis, the eccentricity factor m is arbitrary and indeterminate. Although Singace [14] derived the value of m by using the Newton–Raphson numerical scheme, such value of m can be only used for the problem of circular tube without corrugation. For the doublesine-wave beam, as shown in Fig. 6(f), the folds extend the same distance on both the internal and external sides. Besides, the load displacement curves show that the peaks of each fold are almost the same. Due to these two points, setting m as 0.5 is reasonable for the theoretical analysis of the double-sine-wave beam. The load displacement curves for different values of n are shown in Fig. 4. It is observed that with the increasing value of n, both the peak crushing force and crushing distance of each fold is decreased: the curves exhibit the tendency to approach the load uniformity.
(13)
By summing Eqs. (12) and (13), the total membrane energy term is given by Em ¼ EIm þ EIIm ¼ 4fN0 H2 ð1 2nÞ
(14)
2.4. Expression for P¯ The total work done by the mean crushing load P¯ acting over the two phases of compression is P¯ 4H. The mean crushing load is calculated from the balance of global energy such that P¯ 4H ¼ SðEb þ Em Þ
(15)
where S is the wave-number in Y-direction for the double-sinewave beam. The normalized mean crushing load can be written in the form P¯ 4fHð1 2nÞ fR0 ½p 2 sin1 ð2nÞ ¼S þ 2H M0 h
(16) Fig. 4. A load displacement diagram of different values of n.
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3. Numerical simulation 3.1. Finite element models The material of double-sine-wave beam employed here is made of aluminium alloy with elastic perfectly plastic constitution, and the mechanical properties are that Young’s modulus E ¼ 70 GPa, yield stress s0 ¼ 284 MPa, and Poisson’s ratio n ¼ 0.35. The finite element model of the double-sine-wave beam is shown in Fig. 5. The bottom of the beam is clamped, while the other end is free. The beam is crushed on the top in the axial direction by a rigid wall with prescribed velocity 1 m/s. The beam is modeled with Belytschko–Tsay 4-node shell element with 3 mm thickness. The aircraft structures with 3 mm thickness could serve good function of energy absorption without adding too much weight to the aircraft. Two types of contact elements are employed in the simulation of crushing. For the sine-wave beam itself an automatic single-surface contact is adopted to account for the contact between lobes during crushing process. Between the rigid body and the sine-wave beam, an automatic node-tosurface contact is defined. 3.2. Pre-crushed trigger Compared with the straight sine-wave beam, the initial peak force of the double-sine-wave beam is largely reduced due to the pre-crushed bending. However, the initial peak forces are still higher than the average forces. To obtain a desirable load uniformity, the pre-crushed trigger could be introduced by a variety of methods. Among them the indentation trigger [16,17] is used most commonly in thin-walled structures. Other methods such as triggers based on elastic buckling mode shapes [18] are also used as an alternative. In this section, the measure of reducing the thickness of the top part of the double-sine-wave beam is used as the pre-crushed trigger. This measure is found to be convenient and effective in eliminating the initial peak force. According to the numerical simulation, a better result could be obtained by reducing the thickness near the top of the beam from 3 to 2 mm. 3.3. Numerical results Using MSC.DYTRAN for the simulation, the numerical results of the double-sine-wave beam under axial crushing can be obtained and some typical collapse modes are shown in Fig. 6. It is shown that the collapse modes of the double-sine-wave beam with different value of N are similar, and the plastic deformation of the double-sinewave beam occurs at the predetermined intervals along the axial direction. In this part, the collapse modes of the double-sine-wave
beam with N ¼ 2.5 mm are shown in Fig. 6(a–e) and as a comparison, the collapse mode of the straight sine-wave beam is shown in Fig. 7. Fig. 6(f) demonstrated the complete one-fold crushing of the doublesine-wave beam. It is easy to see that the double-sine-wave beam deforms in axisymmetric or ‘‘concertina’’ mode when axially crushed. The straight sine-wave beam, however, exhibits non-symmetric mode when subjected to axial crushing load. The comparison between them indicates that the deformation mode of a doublesine-wave beam during crushing could be controlled, making its energy-absorption capability fully utilized; but for a straight sinewave beam, usually it is hard to predict its deformation mode during crushing and its energy-absorption capability is hard to be evaluated accurately. The load displacement curves of both the straight sine-wave beam and the double-sine-wave beam with N ¼ 1.25 mm are shown in Fig. 8. It is shown that the axial force of the straight sine-wave beam reaches a significant high initial peak force, followed by a steep drop. The initial peak force of the doublesine-wave beam, however, is far lower than that of the straight sine-wave beam. Furthermore, for the double-sine-wave beam, the load displacement behavior exhibits a repeated pattern. Each pair of the peaks is associated with the development of one full wrinkle or buckle. The 12 fluctuations on the load displacement curve correspond to 12 corrugations of the double-sine-wave beam. Thus, the double-sine-wave beam produces a better load displacement response and uniform collapse load than the straight one. As it can be seen in Fig. 8, the introduction of the corrugations in the aluminium double-sine-wave beam reduces the mean collapse load when compared with that of the straight sine-wave beam. This is due to the fact that the introduction of the corrugations reduces the amount of external work required for a complete crush. The load displacement curves of the double-sine-wave beam with different values of N are shown in Fig. 9. With the increasing value of N, the mean crushing force is decreased. This is due to the fact that the deeper corrugation results in more decrease of the amount of external work required for a complete crush. As expected, the more shallow the corrugations, the higher the tendency for the double-sine-wave beam to approach the mean crushing force of the straight sine-wave beam. In addition, the sine-wave beam with the deeper corrugation produces a better load displacement response with smaller fluctuations of the collapse load. And this behavior, known by the load uniformity ratio, is favorable for an energy-absorption device. 3.4. Comparison between theoretical predictions and numerical results The theoretical predictions and the numerical simulation results of the energy absorption of the straight and double-sine-wave
Fig. 5. Finite element model of the double-sine-wave beam.
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Fig. 6. Collapse modes of the double-sine-wave beam: (a) d ¼ 0.03 m; (b) d ¼ 0.06 m; (c) d ¼ 0.09 m; (d) d ¼ 0.12 m; (e) d ¼ 0.15 m; and (f) formation of a complete fold.
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Fig. 7. Collapse modes of the straight sine-wave beam.
the double-sine-wave beam also plays an important role in the deformation mode. Usually, the double-sine-wave beam with short length would prefer to deform in concertina mode, and the one with long length might result in Euler collapse mode. Finally, it is still worth discussing the issues with manufacturability of such a double-sine-wave beam. Although it has some difficulties in the manufactures, two current methods could be considered: one is using CNC milling machine and the other is to employ the stamp press. In the future, the advanced manufacturing technology could also be employed for the beam manufacture.
4.2. Conclusion remark
Fig. 8. The load displacement characteristics of the straight and double-sine-wave beam.
beam are shown in Table 1 in which the crushing distance is 120 mm. It should be noted that the theoretical value for the straight sine-wave beam is approximately deemed as the value for the double-sine-wave beam with n equating to 0. It is shown that the numerical results of the mean crushing force of different values of N are compared well with theoretical predictions, with error less than 12%. In addition, all the theoretical predictions are slightly overestimated.
4. Discussion and conclusion 4.1. Discussion The present investigation was conducted on the case of R0 ¼ 23 mm, L ¼ 180 mm, f ¼ 1261 and h ¼ 3 mm. The different energy-absorption characteristics can be obtained by a combination of different values of R0, L, f and h. For tubes subjected to axial crushing load, relative thick-walled tubes deform in concertina mode, however, thin-walled tubes exhibit diamond mode, which is a non-symmetrical mode [19]. Similar phenomenon happens for the double-sine-wave beam, i.e. the beams with thick wall would prefer to deform in concertina mode. Besides, the quasi-static behavior of axially compressed ductile tubes indicates the influence of tube height on collapse mode [20]. The height of
The energy-absorption behavior of the double-sine-wave beam was investigated. In the theoretical analysis, the rigid perfectly plastic material model was adopted and the stationary plastic hinge mechanisms were employed. The introduction of the amplitude factor n in the analysis, successfully lead to a quantitative expression of the energy-absorbing capability of the double-sine-wave beam with different depth of the corrugation. Numerical simulations were conducted by using nonlinear explicit finite element codes MSC.DYTRAN. The mean crushing forces of double-sine-wave beam obtained by numerical simulation were in good agreement with those of theoretical analysis. Compared with the traditional straight sine-wave beam, the double-sine-wave beam has shown favorable characteristics as an energy absorber in terms of load uniformity and low initial peak force. Furthermore, it is found that the energy absorbed by the axial crushing of the double-sine-wave beam could be controlled by the introduction of corrugations with suitable depth along the axial direction. This makes its energy-absorption capability fully utilized. It has been shown that with increase of the depth of corrugation, the amount of external work required for a complete crush is decreased, and hence the mean crushing force. Meanwhile, the double-sine-wave beam of the deeper corrugation produces an acceptable load displacement curve with smaller fluctuations of the mean crushing force. Although the classification of the axial collapse modes of cylindrical tubes under quasi-static loading has been discussed by Andrews [20], the expressions for energy absorption were mainly based on the experimental data. Compared with the tubes, the deformation mode of the double-sine-wave beam is more complicated due to its axis asymmetry. Thus, the collapse mode of the double-sine-wave beam is suggested for future study. In addition, more attention should be dedicated to the investigation of quantifying the parameter effects of the double-sine-wave beam, such as height, diameter and thickness, on collapse mode.
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Fig. 9. The load displacement characteristics of the double-sine-wave beam with different values of N: (a) N ¼ 1 mm; (b) N ¼ 1.5 mm; (c) N ¼ 2 mm; (d) N ¼ 2.5 mm; (e) N ¼ 3 mm; and (f) N ¼ 4 mm.
Table 1 Numerical results and theoretical predictions for energy absorption (R0 ¼ 23 mm, L ¼ 180 mm, h ¼ 3 mm, d(crushing distance) ¼ 120 mm). N (mm)
Numerical results (kJ)
Theoretical predictions (kJ)
Error (%)
0 1 1.25 1.5 2 2.5 3 4
33.971 29.907 28.065 26.557 25.354 24.336 20.301 17.377
34.957 30.864 29.833 28.801 26.730 24.650 22.558 18.317
2.90 3.20 6.30 8.45 5.43 1.29 11.11 5.41
Acknowledgement The work described in this paper is financially supported by the National Natural Science Foundation of China under Grant number 10532020. The authors would like to gratefully acknowledge this support.
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