Bending collapse of thin-walled circular tubes and computational application PDF

Preview

Summary

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel...

Description

Accelerat ing t he world's research.

Bending collapse of thin-walled circular tubes and computational application Yucheng Liu

Cite this paper

Downloaded from Academia.edu 

Get the citation in MLA, APA, or Chicago styles

Related papers

Download a PDF Pack of t he best relat ed papers 

Mult i-axis bending of channel sect ion beam and modeling Yucheng Liu

Simplified modeling of t hin-walled box sect ion Yucheng Liu Opt imisat ion of t he crushing performance of t ubular st ruct ures Yucheng Liu

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

Author's personal copy ARTICLE IN PRESS

Thin-Walled Structures 46 (2008) 442–450 www.elsevier.com/locate/tws

Bending collapse of thin-walled circular tubes and computational application Yucheng Liu, Michael L. Day Department of Mechanical Engineering, University of Louisville, Louisville, KY 40292, USA Received 6 July 2006; received in revised form 27 June 2007; accepted 4 July 2007 Available online 5 November 2007

Abstract This paper focuses on describing the bending collapse behavior of thin-walled circular tubes. In this paper, global energy equilibrium theory is applied to derive the relationship between the applied moment and the bending angle of circular tubes. A general bending collapse mode of circular tubes is referenced for the derivation, and it is assumed that during bending crush, all impact energy is absorbed and distributed along the hinge lines. After obtaining the relationship, it is compared to a published theory of tubular structure’s bending resistance, which was obtained from analytical and experimental studies. The derived bending resistance is then applied to generate simplified circular tube models, which have different cross-sections and are made of different materials. Crashworthiness analyses are performed on these simplified models as well as detailed tube models, and the crash results are compared to verify the efficiency of the generated simplified model and the accuracy of the derived tube’s bending resistance. All the problems involved in this paper are solved by means of LS-DYNA. r 2007 Elsevier Ltd. All rights reserved. Keywords: Thin-walled circular tube; Bending collapse; Crash analysis; Simplified model

1. Introduction Thin-walled structures are widely used in automotive industry and other engineering industry for the purpose of increasing energy absorption efficiency and safety as well as reliability. The crash behavior of the thin-walled structures during axial collapse and bending collapse have aroused a lot of interest, including the relationship between the axial loading and the axial shortening during axial compression as well as the relationship between the applied moment and the rotation angle during bending crushing. Previous researchers have developed various mathematical equations to describe or predict the relationships of certain types of thin-walled structures. These equations are very important in correctly estimating and predicting the crash behavior of such structures. Moreover, the published equations can also be used to develop simplified crash models for thin-walled structures, which can be used to Corresponding author.

E-mail address: [email protected] (Y. Liu). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.07.014

replace the detailed models in crashworthiness analysis and save computer time and resources. Liu and Day [1] developed the bending characteristics of thin-walled channel section beams. Also Wierzbicki and Abramowicz [2,3,5,6], and Kecman [4] developed mathematical equations to predict the axial and bending resistances of thinwalled box section beams. Besides the above thin-walled structures, thin-walled circular tubes are also important structures widely used as auto-body structures in lightweight vehicle design. The axial crushing of the thin-walled circular tubes has been thoroughly investigated and well described experimentally, analytically, and theoretically. Abramowicz and other researchers [7–10] studied the crushing behavior of circular tubes when subjected to an axial impact, including the influence of impact velocity and the energy-absorption characteristics of such tubes. In their work, important equations were derived based on numerical methods and verified through experiments, and proposed to predict and describe the crush behavior of thin-walled tubular columns during axial compression.

Author's personal copy ARTICLE IN PRESS Y. Liu, M.L. Day / Thin-Walled Structures 46 (2008) 442–450

Nomenclature Eext Eint H M(y) M0

total external energy total internal energy half length of plastic folding wave bending moment fully plastic moment per unit length of section wall

In addition to the axial crushing behavior of the thinwalled circular tubes, Zheng and Wierzbicki [11] obtained a moment–angle response for the tubular structures via experimental and analytical studies. Nevertheless, the bending behavior of the thin-walled circular tubes during impact analyses still needs to be determined. In this paper, the thin-walled tube is considered as an efficient energy absorber and its bending collapse mechanism is studied. Based on its bending collapse mechanism, the tube’s moment–angle response is derived using the global energy equilibrium method. The findings in this paper are compared with Zheng and Wierzbicki’s results for validation. One of the significant applications of the derived moment–angle response is simplified modeling. Simplified modeling is a critical modeling technique which was extensively used in early design stages for evaluation, crashworthiness analyses, and computer simulation. The simplified computer model is a finite-element model that is composed of beam and spring elements. Compared to detailed model, it requires less labor, time and effort, and is regarded as a very promising solution for quick and efficient computer simulation. In simplified thin-walled beam models, the moment–angle relationships are used to define nonlinear spring elements, which simulate the crash behavior of plastic hinges during the crashes. Kim [12], Drazetic [13], and Liu [14] have made considerable achievements in that field. In this paper, the derived thinwalled tube bending resistance is applied to develop simplified circular tube models. The generated simplified model is then used for crashworthiness analyses and the results are compared to those from the detailed model. Good agreement is achieved through the comparisons between the detailed model and the simplified model and therefore the derived circular tube moment–angle response is validated.

P t a d y s0 su R

instantaneous crushing force wall thickness folding angle axial shortening rotation angle energy equivalent flow stress ultimate stress mean radius of circular beam

cylindrical tube subjected to three-point bending collapse (Fig. 2), and verified through bending experiments. The thin-walled circular tubes used in those experiments and analyses were made of low-carbon steel and whose axial length L, outside diameter D, and wall-thickness t were 200, 31.6, and 1 mm, respectively. Even though the collapse mode was obtained from a three-point bending collapse model, it can still be adopted as the mode resulting in a bending crash. This assumption is also verified by comparing the deformed configurations of a thin-walled circular tube model after a crash analysis and a three-point static bending test (Fig. 3). During the derivation, it is

Fig. 1. Collapse mechanisms of circular tube: (a) general view; (b) plan view; (c) instantaneous force P, and resultant moment M(y).

2. Bending collapse mechanism In this section, the moment–angle relationship of thinwalled circular tube is derived by applying the global energy equilibrium method [1]. A common collapse mode of such tubes is shown in Fig. 1, which has also been used in previous research [15]. According to [15], the mode of bending collapse shown in Fig. 1 is predicted from the numerical simulation results of a simply supported

443

Fig. 2. Configuration of three-point bending model.

Author's personal copy ARTICLE IN PRESS Y. Liu, M.L. Day / Thin-Walled Structures 46 (2008) 442–450

444

From Fig. 1, it can be seen that during the crash, the internal energy is dissipated along the six hinge lines 1–6. Also, due to its symmetry mechanism, the internal energy dissipating along the hinge lines 3–6 are equal. Thus, the rate of total absorbed energy equals E_ int ¼

Fig. 3. Bending collapse mode from computer simulation: (a) from a crash analysis and (b) from a three-point static bending test.

assumed that all the work done by external forces is absorbed by the structure and transformed into internal energy. Also, the internal energy is assumed to be distributed along the six moving hinge lines (1–6 in Fig. 1), which equals the limit bending moment per unit length M0 times the relative rotational angle, multiplied by the length of each hinge line. Based on the above assumptions, the moment M(y) versus the angle of rotation y relationship can be derived following the routines. From the geometrical relationship showed in Fig. 1, we have d ¼ 2Hð1  cos aÞ,

(1)

where d is the axial shortening of the plastic fold, H is its half length, and a is the folding angle. Based on the equilibrium of the energy during the crash, the rate of the internal energy absorbed by the collapsed mechanism should equal the rate of work of the external forces. Therefore E ext ¼ E int

and

E_ ext ¼ E_ int .

(2)

On the basis of the principle of virtual velocities [16], the rate of work of the external force can be expressed as the product of instantaneous force P and the generalized relative velocity of the shortened folding mechanism. Then we can have E_ ext ¼ Pd_ ¼ 2PH sin a_a,

(3)

where P, as presented in Fig. 1, is the external instantaneous crushing force generated as the thin-walled tube impacts the rigid wall [3,6], and will later be represented in terms of the tube’s mean radius R, its wall thickness t, and the instantaneous bending angle y.

6 X i¼1

E_ i ¼ E_ 1 þ E_ 2 þ 4E_ 3 .

(4)

As shown in the figure, lengths of hinge lines1 and 2 equal half of circumference of the circular cross-section, pR, and the lengths of hinge line 3–6 are calculated using qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 the Pythagorean Theorem as ðpRÞ 4 þ H (Fig. 1). Therefore, the rate of rotation of hinge line 1 equals the rate of the bending of H, which is a_ . Also, it is deduced from the _ For same figure that the rate of rotation of hinge line 2 is y. hinge lines 3–6, it is assumed that during the bending collapse, they rotated by angle g, and the rate of rotation is g_ (as shown in Fig. 4). Table 1 lists the rotational rates and corresponding hinge lengths of the six hinge lines shown in Fig. 1. From Table 1, the rates of energy distributed along each hinge line are E_ 1 ¼ M 0 pR_a,

(5)

_ E_ 2 ¼ M 0 pRy, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 R2 E_ 3 ¼ M 0 þ H 2 g_ . 4

(6)

(7) 



 From Fig. 5, it is inferred that tgðy=2Þ ¼ ðd=2Þ=a ¼ ðHð1  cos aÞÞ=a , where a is width of the equivalent square cross-section for a circular cross-section. From previous research [11] based on the thin-walled structures, it was found that the characteristics of the solutions for circular and square cross-sections were very similar. Therefore, any circular tube can be approximated using an equivalent square tube. According to [11], the width of the equivalent square tube was calculated using analytical methods as 1.63R and it was verified that the circular crosssection and its equivalent square cross-section provided the same first moment of inertia. For small angle values of a and y, this relationship can be rewritten as     1=2 a2 y H 1 1 2 H ay ) y ¼ a2 ) a ¼ ¼ . (8) a 2 a H

α1

γ

α2 α2

Fig. 4. Rotation angle g of hinge lines 3–6.

Author's personal copy ARTICLE IN PRESS Y. Liu, M.L. Day / Thin-Walled Structures 46 (2008) 442–450 Table 1 Rotational rates and corresponding hinge lengths in collapse thin-walled circular tube model Hinge lines

Rotational rates

Lengths

Line 1 Line 2 Lines 3–6

a_ y_ g_

pR pR qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpRÞ2 2 4 þH

/2

a

/2

 Fig. 5. Relationship between a and y.

the relationship between the instantaneous force P and the angle y is obtained:  1=4    R 1 pffiffiffi þ 1:93 P ¼ M 0 1:08 t y    pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1=4 1 . ð13Þ þ ð0:75R þ 0:51tÞ 3=4 y R

From (11) and (13), it can be seen that for a given thinwalled circular beam, the instantaneous crushing force P during the bending collapse is a function of its bending angle y. Fig. 6 shows the variation of P as the bending angle y changes from 0 to p. Finally, M(y), the bending moment caused by the instantaneous crushing force P, can be easily determined following the relationship moment equals force times its arm:   5=4   R 1 pffiffiffi þ 3:15 MðyÞ ¼ Pa ¼ Pð1:63RÞ ¼ M 0 1:76 1=4 t y   pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð2R þ 1:36tÞðRtÞ1=4 . ð14Þ y

 Angle g is related to a and y through tg g ¼ tgðy=2Þ = sin a. By applying small angle approximation and from Eq. (8), g is given by   ðy=2Þ 1 H 1 Hy 1=2 ¼ a)g¼ g¼ . (9) a 2a 2 a By substituting (8) and (9) into (6), (7), and (4) we have 0 2paH E_ int ¼ E_ 1 þ E_ 2 þ 4E_ 3 ¼ M 0 a_ @pR þ 1:63 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s   p2 R2 H 2 A. þ2 ð10Þ þH 1:63R 4

Substituting (3) and (10) into (2) and using small angle assumption for angle a, the instantaneous impact force P can be calculated as 0 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  ffi 2 R2 pR p p 1 A. þ þ þ H2 P ¼ M0@ 2Ha 1:63 1:63Ra 4

Fig. 6. Instantaneous crushing force P.

(11)

According to the research of Zheng and Wierzbicki [11] H ¼ 1:84Rðt=2RÞ1=2 ,

(12)

where t is the tube’s thickness. Eq. (12) is an analytical solution of H for a bent circular tube, which was obtained by minimizing the absorbed energy with respect to H. This equation was developed based on plastic materials and for the relatively ‘‘thick’’ tube (R/to30) so that it deformed in the plastic range [11]. By substituting (12) and (8) into (11),

445

Fig. 7. Moment–angle curves for thin-walled circular tube.

Author's personal copy ARTICLE IN PRESS 446

Y. Liu, M.L. Day / Thin-Walled Structures 46 (2008) 442–450

Similarly, the M(y) calculated here is also an instantaneous value and is a function of the rotation angle y for a given thin-walled tube model. The relationship between the moment M(y) and y is plotted in Fig. 7. From that figure, it can be found that the M(y)y relationship derived based on small angles are used for angles of rotation beyond 2 rad. Later, it will be verified that Eq. (14) can correctly predict the bending behavior of the circular tubes even it is derived by taking small angle approximations. 3. Validation In this section, the circular tube’s moment–angle relationship derived here is validated by comparing it to a published moment–angle response for tubular structures [11]. Zheng and Wierzbicki obtained an analytical solution Eq. (15) for the bending of tubular structures based on the plastic hinge theory, which was verified by experimental data:  11=12  ! 4 R 1 2 pffiffiffi MðyÞ ¼ 4t M 0  þ 2:6117 . (15) 3 t y In (14) and (15), M0 is the fully plastic moment per unit length of the section wall, which equals (s0t2)/4. For a practical curved thin-walled circular tube model, whose R is 50 mm and t is 1.5 mm, its moment–angle curves obtained from (14) and (15) are plotted in Fig. 7. From Fig. 7, it can be seen that the derived M(y)y curve shows good agreement with the published tubular structure’s moment–angular response. Therefore, it is verified that the global energy equilibrium method is reliable for properly deriving the bending resistance for most of the thin-walled structures. To apply this method, an appropriate collapse mode of the objective structure has to be referenced and the results can be directly derived following the method illustrated here and in [1]. 4. Simplified modeling The derived bending resistance Eq. (14) then can be applied for developing simplified thin-walled circular tube model, and the developed simplified model is then used for crash analyses and verified by comparing the crash results

to those of the detailed tube model. In the crash analysis, a thin-walled circular tube model impacted a rigid wall with an initial velocity and collapsed. From [14], when a straight thin-walled beam impacted the rigid wall, it would buckle axially and no bending collapse appeared (similar conclusions can be found in [2,5,7–10]. However, when a curved thin-walled beam impacted the wall, it would bend and the bending collapse was concentrated only on its ‘‘plastic hinges’’ ([1,3,4]). Therefore, the curved thin-walled circular tube model is used for crash analysis to show the bending collapse. Both detailed and simplified curved circular tube models subjected to bending moment during the crash analyses are created and the derived bending resistance Eq. (14) is applied. Similarly, the tube models’ material is steel and the material properties are listed in Table 1. One thing that needs to be mentioned is that even though Eq. (14) was derived for straight tubes, it is still applicable to the curved beams because the straight and curved beams show similar bending collapse mechanisms (from [1,12–14]). Fig. 8 displays the original detailed model composed of shell elements and Fig. 9 shows the developed simplified model, which was created following the modeling method of Liu and Day [14]. In the simplified model, the straight beam segments are modeled using Hughes–Liu beam elements, and the developed nonlinear rotational spring elements are used for modeling the local plastic hinges. To define the nonlinear rotational spring element in LSDYNA, a created rotational angle–moment table is input, which contains a series of moment values in terms of different rotational angles y that were calculated based on Eq. (14) derived in Microsoft Excel. Both detailed and simplified circular tube models presented here are used for crashworthiness analyses in order to validate the developed simplified model. In deciding the simplified model’s boundary conditions, the boundary conditions of the detailed model are referenced. In the simplified model, the impact-end node of the curved beam is fully constrained by fixing all six degrees of freedom, and releasing only the longitudinal translational degree of freedom for the rear-end node, where the initial velocity is imposed. Additionally, since the curved beam is defined along a 2D plane, additional constraints have to be applied in order to prevent the out-of-plane motion.

Fig. 8. Detailed model for thin-walled curved circular tube.

Author's personal copy ARTICLE IN PRESS Y. Liu, M.L. Day / Thin-Walled Structures 46 (2008) 442–450

447<...