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A simplified model of maximum cross-section flattening in continuous rotary straightening process of thin-walled circular steel tubes
Journal of Materials Processing Technology 238 (2016) 305–314
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
A simplified model of maximum cross-section flattening in continuous rotary straightening process of thin-walled circular steel tubes Z.Q. Zhang ∗ , Y.H. Yan, H.L. Yang School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110004, China
a r t i c l e
i n f o
Article history: Received 21 September 2015 Received in revised form 11 July 2016 Accepted 18 July 2016 Available online 19 July 2016 Keywords: Straightening Thin-walled circular steel tube Cross-sectional ovalization Maximal flattening Plastic bending
a b s t r a c t Cross-sectional ovalization of thin-walled circular steel tube because of large plastic bending, also known as Brazier effect, usually occurs during the initial stage of tube’s continuous rotary straightening process. The maximal cross-section flattening (ovalization) is an important technical parameter in tube’s straightening process to control tube’s bending deformation and prevent buckling. However, for the lack of special analytical model, it is determined in accordance with the specified charts developed by experienced operators on the basis of experimental data, thus it is inevitable that the local buckling will occur during some actual straightening operation. In the present paper, simplified normal strain component formulas are derived based on the thin shell theory. Then strain energy of thin-walled tube (per unit length) is obtained subsequently using the elastic-plastic theory. Finally a rational model for predicting the maximal section flattening of the thin-walled circular steel tube under its straightening process is presented by the principle of minimum potential energy. The simplified model is validated by experiments and numerical simulations. The results show that this model agrees well with the experiments and the numerical simulations with less than 10% error. This new model is expected to find its potential application in thin-walled steel tube straightening machine design. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The high-precision thin-walled steel cylinders, referred to as thin-walled tubes or pipes, have a wide variety of applications in pipe networks, aerospace structures, military and petrochemical industry. But they are usually curved in rolling and transporting process due to the external or internal force or temperature variation, so straightening is an important process in order to eliminate the tube’s undesirable geometric imperfections. Currently, the multiple cross roll straightening machine is usually used to straighten thin-walled tubes. It employs a set of roll system, consisting normally four or five pairs of work rolls placed along a longitudinal axis, as illustrated in Fig. 1a. Each paired rolls have the same circular profile of a single curvature radius rather than the hyperbolic profile, one with a convex surface and the other a concave surface, and are mounted at an oblique angle. This kind of roll system has been validated to be suitable for straightening thinwalled tubes by Zhang et al. (2013). During straightening operation,
∗ Corresponding author at: P.O. Box 319, School of Mechanical Engineering and Automation, Northeastern University, No. 3-11, Wenhua Road, Heping District, Shenyang 110004, China. E-mail address: [email protected] (Z.Q. Zhang). http://dx.doi.org/10.1016/j.jmatprotec.2016.07.034 0924-0136/© 2016 Elsevier B.V. All rights reserved.
the thin-walled tube is moved forward whilst being rotated in a spiral direction. It is straightened by alternating bending applied by the roll system. When the tube is in a paired working rolls gap, it is bent by the roll’s profile, with which the radius of curvature of the tube’s axis equals to the roll profile’s, as shown in Fig. 1b. At the initial stage of straightening, the thin-walled tube is usually subjected to a large bending in order to unify its non-uniform initial curvature along its longitudinal axis. This large bending creates large nonlinear plastic deformation, causing the tube cross-sectional ovalization, known as Brazier effect. This section ovalization or flattening, as a major technical parameter for the straightening process, shall be controlled so that no buckling occurs. Therefore, it is very important for the straightening process of thin-walled circular steel tube to analyze Brazier effect response and to predict its maximum crosssection flattening accurately. Since Brazier (1927) firstly described and modeled this highly nonlinear cross-section flattening phenomenon observed by tube bending experiments with simplified method, numerous studies investigating the responses of circular steel tubes under pure bendings have been published. Reissner (1959) generated a nonlinear equation by applying a small-strain bending theory. His equation, even though could not be solved analytically at that time, but was solved by Fabian (1977) using numerical method, with a conclusion that the numerical solution of Reissner’s equation converged
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Fig. 1. The continuous rotary straightening process of thin-walled tube. (a) The working roll system employed by multiple cross roll straightening machine (b) The stress-strain state of thin-walled tube subjected to large bending in working rolls gap (c) The tube’s cross-sectional shape before and under bending
well with Brazier’ s simplified solution except at the limit point. Then Levyakov (2014) proposed another numerical method to solve Reissner’ s nonlinear equations in terms of two unknown functions and two unknown parameters.Using a similar method, Rotter et al. (2014) investigated in detail the effect of cylinder length on the nonlinear elastic buckling behaviour of clamped cylindrical tubes under global bending, covering a very wide range of lengths. However, the studies mentioned above were all assuming that the material were in elastic range. So the more studies have been done in plastic range in order to describe the mechanic behaviour of the circular tube under large plastic bending. Ades (1957) firstly proposed an iterative numerical method, and analyzed the tube ovalization in the plastic range by applying the energy principle. Munz and Mattheck (1982) considered the ver-
tical component of the axial bending stresses as primary cause of cross-sectional ovalization. Following this Prinja and Chitkara (1984) derived a theoretical solution for tube cross-section flattening under plastic bending, by modeling the initially circular section subjected to static compression using four stationary hinges. Then this model has been widely used. It was employed by Poonaya et al. (2009) to analyze thin-walled circular tube’s plastic collapse due to bending. It was also used by Mentella and Strano (2012) to predict the quality of the cross-section of small hydraulic pipings in the rotary draw bending process. Especially, according to this model Arabzadeh and Zeinoddini(2013) analyzed cross-sectional ovalization by considering the tube resting on a flexible bed. Beside this, the Mamalis kinematics model was employed by Elchalakani et al. (2002) to develop a closed-form solution for moment-rotation
Z.Q. Zhang et al. / Journal of Materials Processing Technology 238 (2016) 305–314
response of the tube under pure bending using two local plastic mechanisms (star and diamond shapes). But all models mentioned above are used to predict the post-buckling response, and the tube’s cross-sectional shape observed during the buckling experiments does not resemble the real shape during the continuous straightening process. Moreover, if the post-buckling occurred during bending, the tube could hardly be straightened subsequently. Furthermore, Yu and Zhang (1996) developed a general method to deal with tube’s ovalization of arbitrary cylindrical cross-section. To apply this method into the straightening process, however, lots of technical issues are still unsolved. Wierzbicki and Sinmao (1996) proposed a simplified theoretical model to predict bending moment and cross-section deformation. Using a similar approach, Daxin et al. (2013) deduced a series of fomulations to calculate the stresses and the cross-section flattening ratio of tubes subjected to combined bending and additional tension with linear hardening law. But these models are valid only for very large sectional distortion, and are not suitable for the straightening process. Tatting et al. (1997) established governing equations for nonlinear bending response of finite length composite tubes, which exhibited cross-section deformation associated with Brazier effect by using semi-membrane constitutive theory. Following this Shima et al. (2014) proposed the equations to describe circular crosssection ovalization under pure bending especially for multilayered cylinders using thin-shell theory. But these equations are too complicated to be used in a straightening machine routine design. To conclude, there is no established method available for estimation of the maximum cross-section flattening for straightening process of thin-walled circular steel tube. The maximum section flattening obtained from the experimental charts sometimes causes that the design radius of the work roll is small enough to induce tube’s local buckling during straightening operation. In this paper, a simplified model of maximum cross-section flattening in rotary straightening process of thin-walled circular steel tube is proposed and its validation is investigated.
307
(1) The material is assumed that to be isotropic and homogeneous. The Bauschinger effect and the hysteresis loop in reloading and unloading, are ignored. (2) For thin-walled circular carbon steel tube, based on the conclusion by Zhang et al. (2013), the offset of the neutral surface to the tube’s centroid surface is so small that it is neglected. Therefore, it is assumed that the cross section neutral axis (the axis in the cross section with zero stress and zero strain) coincides with its centroid axis. (3) Tube thickness is small and is assumed to be constant during the straightening process. Therefore, the tube’s radial strain is zero. The radial stress normal to the middle surface is small compared with the other stress components, and is also neglected in the stress and strain relationships (Zhang et al., 2013). (4) The cross section which is plane and normal to the longitudinal axis of the tube before deformation remains plane and normal after deformation. There is no twist of the adjacent cross sections, so shear stress and shear strain components are considered to be small and are neglected. Thus the stress-strain state of the deformation region can be simplified as shown in Fig. 1b. (5) The volume of material remains unchanged during plastic deformation. (6) Based on the generalized Hooke’s law, in the elastic range, the relationship between the effective stress and the effective strainε is = Eε. And in the plastic range, the material follows bilinear strain-hardening diagram. The stress-strain relationship is defined as = s + E1 (ε − s /E), where E is the modulus of elasticity, E1 is the plastic strain-hardening coefficient, s is the yield stress.
3. Model development 3.1. Ovalised tube section strain-displacement relationship
2. Brazier effect and assumptions For a thin-walled circular steel tube under bending, there are three distinct deformation stages, as observed in the experiments by Elchalakani et al. (2002), namely, elastic, ovalization plateau and structural collapse. A tube at its initial stage of the straightening process is normally subjected to a large nonlinear bending without collapse. This deformation belongs to the second deformation stage. So the cross section under this stage deforms from a circular shape to an oval shape shown in Fig. 1c based on the ovalization assumption. Here taking the quarter cross section AB into account, as depicted in Fig. 2a, it is a circular arc of radius r before bending, with the assumption that the initial cross section’s imperfection is small and can be neglected. During bending process the distance between the points O and A (semi minor axis length) is shortened, and the distance between the points O and B (semi major axis length) is elongated. The initial circular shape of the cross section deforms into an oval. The deformed elliptic arc is defined as A1 B1 . The length change of the circular diameter is defined as the crosssectional flattening. From Fig. 2a, it is obviously that the maximum flattening of all the diameters in any direction appears on the major or minor axis. According to the results obtained by Yu and Zhang (1996), the length increase of the major axis is assumed equal to the length decrease of the minor axis. So here the maximum flattening is considered as the length change of the minor axis. Meanwhile it is also assumed that there are no circumferential displacements at the points A and B. To analyze the cross-section flattening, the following assumptions are made in this paper:
When a thin-walled circular steel tube undergoes a large bending, based on Brazier’s effect, the circular section is flattened to an oval shape. To be able to analyze the strain-displacement relationship of a point on an arbitrary cylindrical surface, which is a radial distance z away from its middle surface, as depicted in Fig. 2b. It is necessary to analyze the strain-displacement relationship of a point on the tube’s middle surface at first. Here the middle surface is defined as halfway between the inner and outer edges of the tube, shown in Fig. 2b. The strain components of a point on the middle surface can be described as functions of the point displacements v, u and w, in which v is the axial displacement, and its positive direction is when the bending angle of the tube is increasing; u is the circumferential displacement and its positive direction is when is increasing; w is the radial displacement, and its positive direction is when its radius is increasing, as depicted in Fig. 1b. For the circumferential strain of a point on the middle surface, due to the assumption (4), it is assumed that circumferential strain is not related to the axial displacement v. Therefore, the circumferential strain is a function of circumferential displacement u and radial displacement w only. Consider an initial circular arc segment PQ on the middle surface with a radius r and an infinitesimal angle d (Fig. 2a). It is defined by the central angle between OP and OB. After deformation, PQ is displaced to an elliptic arc segment P1 Q1 . The length increase of PQ is a combination of the length increases owing to the circumferential displacement and the radial displacement. In its circumferential direction, the point P is first moved to the point P due to a circumferential displacement u, and at the same time, point Q is moved to the point Q due to a circumferential displacement u + ∂u d. ∂
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Fig. 2. The deformation and shape of thin-walled tube cross section under bending.
The circumferential length increase of the arc segment PQ can be expressed as: 1 = u +
∂u ∂u d − u = d ∂ ∂
In its radial direction, with a radial displacement w, the point P is moved to the position P independently, as well as the point Q is moved to the position Q . The length of circular arc segment PQ is changed from rd to (r − w)d. The circumferential length increase of the arc segment PQ due to radial displacement can be expressed as: 2 = P Q − PQ = (r − w)d − rd = −wd The total circumferential length increase of the arc segment PQ is the sum of 1 and 2 . The circumferential strain is then: ε =
1 1 + 2 = r rd
∂u −w ∂
Rwp w − Rw w Rw w
shape before bending. If the cross section is ovalized, the point is moved from the position P to the position P1 due to the circumferential and radial displacements. Considering the geometrical relationship shown in Fig. 2a, Rwp is changed to: Rwp = PT + P T + Rw = Rw + (r − w) sin + u cos
ε =
(r − w) sin + u cos Rw
where, Rwp is the axial radius of curvature of the arbitrary point P on the middle surface. As shown in Fig. 2a, it is originally described as Rwp = Rw + PT1 = Rw + r sin when the cross section is a circular
(4)
As described by He and Shen (1993), there is a one-to-one correspondence between the strain components on an arbitrary revolution surface at a radial distance z from the middle surface of the shell and the strain components on the middle surface. It can be expressed as: εi (z) =
(2)
(3)
Substituting Eq. (3) into Eq. (2), the axial strain component acting on the middle surface becomes:
(1)
Then, the axial strain of an arbitrary point on the middle surface can be developed as follows. During the straightening process, the tube is bended by the gap of working rolls. Let Rw denote the radius of curvature of the deformed tube’s centerline axis, and w the central angle of rotation of the axis. From the assumption (2), it is assumed that the neutral axis coincides with the centroid axis. Therefore there is no length change in the tube’s centroid axis. The length of the initial tube is Rw w . Sequentially the axial strain component of a point on the middle surface may be expressed as: ε =
(a) The displacement of segment PQ (b) The element of point P
1 εi [u(z) , w(z) ], (i = , ) 1 + Rz i
where, z is the distance of the surface to the middle surface in Fig. 2b. u(z) ,w(z) are the circumferential and radial displacements acting on the arbitrary cylinder surface at a radial distance z from the middle surface of the tube. Ri is defined as the radius of principal curvature. From Fig. 2a in this paper, the circumferential radius of principal curvature R equals to r, and the radial one R should be Rw + r sin . Then by the use of Eqs. (1) and (4), the strain components on an arbitrary cylinder surface at a radial distance z from the middle surface of the tube can be expressed as:
⎧ r 1 ∂u(z) (z) ⎪ (z) ⎪ ε = w − ⎨ r r+z ∂
(z) (z) ⎪ ⎪ ⎩ ε(z) = (r − w ) sin + u cos Rw + r sin
Rw
Rw + r sin + z
(5)
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In which u(z) ,w(z) can be expressed as follows (He and Shen 1993):
⎧ ⎨ u(z) = u + z u − 1 ∂w ⎩
r
A ∂
(6)
w(z) = w
where, Ais the circumferential Lame parameter, and here also equals to r base on the geometric description of Lame parameter. Thus substituting Eq. (6) into Eq. (5), the following relationships are obtained:
from the neutral surface. When bending of tubes induces crosssectional ovalization, deformation may be so large as to cause the material to become plastic. Comparing with the plastic deformation region, the elastic deformation region is very small and could be neglected. Hence J2 deformation theory can be employed to describe the stress-strain relationship of the whole cross section deformation as follows:
⎧ 1 1 (z) (z) (z) ⎪ ⎨ ε = E − 2
⎪ ⎩ ε(z) = 1 (z) − 1 (z)
⎧ ⎫ ⎧ E ∂w z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ u + r (u − ∂ ) ⎬ 2 ⎪ ⎪ 1 1 1 1 ∂u z ∂u z∂ w ⎪ (z) ⎪ ε = = ( w − w) − + − ⎪ z ⎪ z ⎪ r ∂ r ∂2 ∂ ⎪ ⎪ 1+ r ⎪ 1 + r ∂ ⎪ ⎨ ⎩ ⎭ r r (7) ⎪ 1 u ∂ w ⎪ ⎪ (r − w) sin + u + z( − ) cos ⎪ ⎪ r r ∂ ⎪ 1 (z) ⎪ ε = × ⎪ ⎪ z Rw ⎩ 1+
Rw + r sin
In view of assumption (3), the tube is so thin that the term z/r (z) in Eq. (7) can be assumed to be 0, thus the formula of ε in Eq. (7) can be simplified as (z)
ε ≈
1 ∂u ( − w) = ε r ∂
(8)
In addition, because the value of z is small comparing with Rw , so it can be assumed that 1 + R +rz sin ≈ 1. On the other hand, conw
sidering the following expressionz
u r
−
1 ∂w r ∂
cos <<
(z) Rw , ε
in
(z)
ε ≈
(r − w) sin + u cos = ε Rw
(9)
From Eqs. (8) and (9), it is concluded that the strains on the arbitrary cylindrical surface at a radial distance z from the middle surface can be replaced approximately with the strains on the middle surface. Furthermore the circumferential and radial displacements in Eqs. (8) and (9) are expanded in terms of trigonometric series given by:
⎧ n ⎪ ⎪ ⎪ u = r i sin(2i) ⎪ ⎨ i=1 n
(10)
⎪ ⎪ ⎪ ⎪ ⎩ w = r i cos(2i)
⎧ 2 (z) (z) (z) ⎪ ⎨ = 3 E 2ε + ε
⎪ ⎩ (z) = 2 E 2ε(z) + ε(z)
3
(14)
In view of assumption (6), the effective stress-strain relationship can be expressed as:
s E
(15)
Hence, the modulus of plasticity E can be obtained by the use of Eq. (15): E =
s − = ε ε
E1 s E
+ E1
(16)
In accordance of the assumption (5), the volume of material elements remains constant during plastic deformation, thus: (z)
(z)
ε + ε = 0
(17)
1 (z) 2 (z) (z) ε = √ |ε − ε | = √ |ε | 3 3
(18)
E =
where, i , i are the dimensionless coefficients. Here Eq. (11) is truncated after only the first term in order to simplify u and w as: u = r sin 2
(11)
w = r cos 2 Substituting Eq. (11) into Eqs. (8) and (9) results in
⎧ 1 ⎪ ≈ ε = (2r cos 2 − r cos 2) = (2 − ) cos 2 ⎨ ε(z) r ⎪ ⎩ ε(z) ≈ ε = (r − r cos 2) sin + r sin 2 cos Rw
Substituting Eq. (18) into Eq. (16) results in:
i=1
2
(13)
where, E = /ε is the modulus of plasticity, which is a function of the material properties and the loading history. From Eq. (13) the stress components can be obtained:
= s + E1 ε −
Eq. (7) also can be simplified to:
309
(12)
Rw
E1 s E (z) 2 √ |ε | 3
s −
+ E1
(19)
Substituting Eqs. (19) and (17) into Eq. (14) results in:
⎧ ⎛ ⎞ E1 s ⎪ ⎪ − ⎪ 2⎜ s ⎪ (z) E + E ⎟ ε(z) ⎪ = ⎝ ⎪ 1⎠ ⎪ 2 (z) 3 ⎪ ⎨ √ |ε | ⎛ 3 ⎞ E1 s ⎪ ⎪ ⎪ ⎪ (z) 2 ⎜ s − E ⎟ (z) ⎪ = ⎝ + E1 ⎠ ε ⎪ ⎪ 2 (z) 3 ⎪ ⎩ √ |ε | 3
(20)
3.2. Ovalised tube section stress-strain relationship
3.3. Thin-walled tube maximum section flattening model
During the straightening process, the cross section deformation of the thin-walled circular steel tube can be divided into two parts by the yielding surface. One is the elastic region adjacent to the neutral surface, and the other is the plastic region at the far end away
As previously mentioned, the maximum flattening of the thinwalled tube cross section equals to the length change of the minor axis, which is just two times of the radial displacement at point A in Fig. 2a. So if the coefficients i , i in Eq. (11) can be obtained,
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here some parameters below are defined as U m1 =
2
ε2 d, Um3 =
0
2
ε d, Um4 =
0
2
0
2
ε2 d, Um2 =
0
ε d. Substituting the axial
strain component from Eq. (12), Um1 becomes:
2
Um1 =
ε2 d =
0
0
2 r2 2 Rw
−4( − )2 cos6 +
+ 4( − )2 + 4 (1 + ) − 4(1 + ) cos4 +
4(1 + ) − 4 (1 + ) − (1 + )2 cos2 + (1 + )2 d
where, Fig. 3. YC10GJ—70 twelve cross rolls thin-walled tube straightening machine for actual experiments.
1
. 4
cos6 d =
r2
Um =
1 2
(x εx + y εy )dV =
1 2
(V )
=
r 2 2 8Rw
(z) (z)
(21)
3
, 16
cos4 d =
2
0
cos2 d =
1 1
+ (1 + )2
4 2
[(1 + )2 + (1 + )2 ]
In the same way, Um3 can be expressed as:
( ε + ε )rddz (S)
2
0
[4(1 + ) − 4 (1 + ) − (1 + )2 ] ·
2
Um3 = (z) (z)
5 3
+ [4( − )2 + 4 (1 + ) − 4(1 + )]
+ 32 16
−4( − )2
2 Rw
5
, 32
Integrate Um1 with respect to , and it is reduced to
Um1 =
the maximum flattening will be calculated by the use of Eq. (11). In order to proceed, the appropriate form of the coefficientsi and i should be deduced at first. Here by using the strain energy principles, the expression for the internal plastic strain energy of per unit length tube is defined as follows:
2
0
r Rw
2
ε d =
0
0
2 (r − r cos 2) sin + r sin 2 cos d = Rw
( 2 cos2 − − 1) − 2cos2 d cos =
0
r 1 2 ( + 1 + ) Rw 3 3
Substituting the circumferential strain component from Eq. (12), and integrate Um2 with respect to , then it is reduced to As the cross section is symmetric with respect to the major axis as well as the minor axis (double symmetric cross section), the internal plastic work can be calculated by taking into account of the quarter of the circular cross section in extrados-wall region, as shown in Fig. 2a. In this region, although the cross section is bent to an oval shape, there is no “kink” or “local buckle”. So the state of strain is still determined by the axial bending, which induces that the extrados-wall of the tube is in tension. So the axial strain (z) component is positive ε > 0, and from equation (17) the circum(z)
2
Um2 =
3
E
Similarly, Um4 is given by:
Um4 =
4rt 3
2
2
0
ε d =
0
2
(2 − ) cos 2d
0
2
(2cos2 − 1)d = (2 − )(
− )=0 2 2
Substituting all of the Umi into Eq. (23) results in (22) Um =
1 E1 s 2 2 (z) (z) −√ s − + E1 ε ε t E 3 3 0 − 2 1 E1 s 2 (z) (z) ) + E1 ε ε + √ (s − ddz = E 3 3
Um = 2r
0
Substituting Eq. (22) into Eq. (21), and combining with Eq. (12), the internal plastic strain energy of per unit length tube can be expressed as:
t
2
= (2 − )
3
(2 − )2 (4cos4 − 4cos2 + 1)d =
0
3 1 1 (2 − )2 ( − + ) = (2 − )2 4 2 4
(z)
2 ⎧ 1 E1 s (z) (z) ⎪ + E1 ε ⎨ = − √3 s − E 3
⎪ ⎩ (z) = √1 s − E1 s + 2 E1 ε(z)
2
=
0
ferential strain component is negative ε < 0ε < 0. Thus Eq. (20) can be simplified as:
ε2 d
√ 3s E1 (1 − )(ε − ε ) d E1 (ε 2 + ε 2 ) + 2 E
4rt E1 3
r 2 2 8Rw
(2 + 2 + 2 + 2 + 2 )+
2rt 1 (4 2 − 4 + 2 ) + √ 4 3
s −
E1 s E
×
r Rw
1 3
+1+
2 3
(24)
Then, it is postulated that for a given Rw , the dimensionless coefficients , adjust themselves so as to minimize the strain energy. So the following relationship will be derived:
∂Um ∂Um = =0 ∂ ∂
(25)
Substituting Eq. (24) into Eq. (25) results in: (23)
⎧ ∂Um ⎪ ⎨ ∂ ⎪ ⎩ ∂Um ∂
=
4 E1 3
=
4 E1 3
r 2 2 4Rw
r 2 2 4Rw
2 (1 + ) + (2 − ) + √ 3
(1 + ) +
s −
1 2 (−2 + ) + √ 2 3
E1 s E
s −
r 2 · =0 Rw 3
E1 s E
r 1 · =0 Rw 3
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(26) 2, here some parameters S 1 = E1 r 2 /3Rw S2 = 2E1 /3, √ S3 = 2s r(E − E1 )/3 3ERw are defined for simplifying. Then Eq. (26) can be expressed as:
!
(S1 + 4S2 ) − 2S2 = −2S3 − S1 −2S2 + (S1 + S2 ) = −S3 − S1
(27)
Finally, solving the above equation set, the coefficients , can be obtained from the following formulas if the tube’s material properties, geometric parameters and the bending radius of curvature are given.
⎧ (S3 + S1 )2S2 + (2S3 + S1 )(S1 + S2 ) ⎪ ⎨ = 4S22 − (S1 + S2 )(S1 + 4S2 ) (2S 3 + S1 )2S2 + (S3 + S1 )(S1 + 4S2 ) ⎪ = ⎩
(28)
4S22 − (S1 + S2 )(S1 + 4S2 )
with obtained coefficients , , and by applying = /2 into Eq. (11), the relative displacement at point A from its original position along the minor axis can be obtained. This is the half of the maximum section flattening. So the maximum cross-section flattening ıa can be then obtained as: ıa = 2ı = | − 2r |
Table 1 The experiment measurements of the maximal cross-sectional flattening during actual experiments. Specimen no.
Section flattening/mm
Specimen no.
Section flattening/mm
1 2 3 4 5
2.26 2.03 2.33 2.36 1.88
6 7 8 9 10
2.52 2.28 2.31 2.42 2.15
Repeated the steps for the rest of the specimens and the results were summarized in Table 1. Removed the maximum and minimum values, the arithmetic mean value of the testing measurements was equal to 2.27 mm as the actual value of the maximum flattening. Meanwhile calculated the theoretical value with Eq. (29), the theoretical value of the flattening was equal to 2.59 mm. The comparison of the experimental solution with the result of theoretical calculation implies that the analytical solution is larger than the experimental data. It is because that the second pair of rolls could not be opened during measuring procedures. It is hard to determine the accurate direction of the maximum flattening and also to obtain the maximum flattening measurement. But from the comparison here, it shows that the theoretical model still provides a practice value with only 10% away from the experimental results.
(29)
where, ı is half of the maximum flattening, equals to subtracting OA1 from OA. Obviously the maximum section flattening is a function of tube longitudinal bending curvature radius, material properties and geometric parameters. To prevent the tube from buckling, the tube maximum section flattening shall be less than the allowed limit section flattening ılim , which can be obtained through stability analysis or actual experiments. So the following expression must be satisfied: ıa = | − 2r | ≤ ılim
311
(30)
For a given thin-walled steel tube, making the substitution of the parameter into Eq. (30) will give us a design equation which can be used to determine the reasonable radius of curvature of working rolls for the safe design of straightening machine. Under this bending radius the tube maximum cross-section flattening will be below the allowable limit, in order to prevent the local buckling. 4. Model validation 4.1. Actual experiments In order to verify the proposed model, thin-walled circular steel tube straightening experiments are conducted by the straightening machine YC10GJ-70, which has six pairs of cross work rolls, as shown in Fig. 3. The second and fifth paired roll profiles are circular curves, and the other paired roll profiles are hyperbolic curves. Thereinto the second pair of rolls applies the required large plastic bending to the specimen, unifying its initial existing non-uniform longitudinal curvature. So the maximum flattening measurements can be obtained at this stage. The specimens are 10 thin-walled tubes made of 1Cr18Ni9Ti steel with similar initial curvatures. The tube outside diameter d = 21 mm, tube wall thickness tn = 1 mm. The tube’s material properties are assumed s = 205 MPa, E = 206GPa, E1 = 2 GPa. The test set-ups are as follows: (1) Pushed a specimen into the straightening machine; (2) When the specimen passed the third pair rolls, shut down the machine; (3) Chose the cross section at the middle point of the second pair of rolls, whose diameter of curvature was equal to 4.26 m, and measured the flattening of the cross section at vertical direction.
4.2. Numerical simulations One set of material model parameters discussed for experiments did not provide a perfect validation for the formulas. There must be more experiments for several materials of different parameters. However, when the material parameters changed, a corresponding change in structure of the straightening machine must be done, such as the roll profile, the intermesh and the diameter of roll. Meanwhile it was difficult to measure the maximum flattening accurately by experimental method. Therefore the numerical simulations instead of experiments are employed to further validate the theoretical model. Here an accurate FEM model have been developed using ANSYS/LS-DYNA program, as shown in Fig. 4. It was a typical two-roll straightening system, consisting of a thin-walled tube, a paired working rolls with the circular profile of 4.26 m diameter and a slit guide system which had two guide sleeves and a paired guide plates to play a role of feeding of the tube into the rolls. The tube was modeled by the 3D solid element with an elastoplastic material. The rolls and the guide system were modeled by the rigid body elements. During the simulation, each roll rotated about its own centroid axis with an angular velocity of 280 rmp, the tube subjected to an initial feeding velocity of 0.28 m/s was pushed into the rolls. Then it was rotated at the same time moved along the z direction (as illustrated in Fig. 5a). Every pitch the tube moved forward, twice bending it was subjected to (bending in y direction and reverse bending respectively).Because the roll profile was four times the length of the scroll pitch, so throughout the whole simulation the tube would undergo the plastic bending in y direction eight times rather than once. Considered the moment when the tube was completely in the rolls gap, as shown in Fig. 5a, it contacted with the working rolls completely. So the diameter of curvature of the tube axis was equal to the rolls’ diameter. Here the tube was bent by the large curvature. From Mises stress distribution as illustrated in Fig. 5b, the stress level was much higher. Then chose a cross section m–m in the center of the profile as shown in Fig. 5a, the shape of the cross section was described in Fig. 5c, it was shown that the cross section was deformed to an oval shape under bending. In order to measure the maximum flattening of the cross section accurately, chose two nodes C and D on the section m–m (also
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Fig. 4. The FEA simulation model.
Fig. 5. The numerical simulation for straightening the thin-walled tube causing ovalization to the cross section (a) The bending state of the tube in working rolls gap (b) The tube Mises stress distribution under bending (c)The tube cross-sectional shape under bending.
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313
Fig. 6. Variation in the coordinate values of nodes C and D in y direction with time.
Table 2 The flattening of the cross section m–m at different times. Time/s
Angular displacement/rad
Section flattening/mm
Time/s
Angular displacement/rad
Section flattening/mm
0.1 0.2 0.3 0.4
2 3 4
2.13 2.32 2.37 2.41
0.5 0.6 0.7 0.8
5 6 7 8
2.42 2.43 2.45 2.45
shown in Fig. 5c), and traced their y-displacements, then the variation in the mean vertical diameter of cross section m–m (measured by the distance y between node C and node D) with the deformation time t was described in Fig. 6. The maximum flattening measured by the minimum y-distance between nodes C and D for t = 0.1, 0.2, 0.3. . .0.8 s (accordingly the angular displacement of the working roll is , 2, 3. . .8 rad) were summarized in Table 2. It is shown that the cross section flattening increases as the number of bending circle increases, so the maximum flattening throughout the whole deformation process is around 2.45 mm. This result agrees well with both the theoretical calculation and the experimental data. It is concluded that the numerical simulations can be employed instead of experiments for further validations. Using above numerical method, three sets of similar simulations have been conducted. The first set tested the same tubes (1Cr18Ni9Ti, d = 21 mm, t = 1 mm) at the different bending diameter of curvature, ranging from 3m ∼ 12m. The numerical results were summarized in Fig. 7. Then they were fitted into a curve by the interpolation polynomial shown in Fig. 7. At the same time the theoretical results for each set of parameters were obtained from Eq. (29) also shown in Fig. 7. The comparison of the numerical solutions with the results of theoretical calculation implies that the analytical solutions agree well with the numerical datum with only 10% error of the numerical datum. It is also seen that the error is big as the bending radius increase to a certain extent. But for these radii, the values of maximum flattening are so small that can be neglected in actual straightening progress. In the second set of simulation, the similar tubes (1Cr18Ni9Ti, d = 21 mm) of different thicknesses, ranging from 0.5mm–2 mm, were bent by the same bending diameter 4m. The numerical results and the fitted curve were plotted in Fig. 8. At the same time the theoretical results were calculated according to Eq. (29) also shown in Fig. 8. The comparison of the numerical solutions with the results of theoretical calculation shows that the analytical solutions agree well with the numerical datum with 10% error of the numerical datum except one data point for the thickness tn = 2 mm. Taking
Fig. 7. Comparison of theoretical calculations with numerical results for the same tubes at different bending radii.
this data point into account, the ratio of the diameter to the thickness is about 10. Thus the tube does not belong to the thin-walled tube, and some assumptions for the modeling are strongly violated. So the result of this set of data should be neglected. Final set of simulation tested the tubes with the same geometric parameters, but differing in material properties. Both the numerical and theoretical results were summarized in Table 3. From Table 3 it is shown that the analytical solutions agree well with the numerical datum with 10% error except the fifth set of data. Taking this set of data into account, the material is aluminium alloy 6061 which usually follows the power law, such as Kyriakides and Ju, (1992) employed the stress-strain curve fitted with Ramberg Osgood fit in his experiments. So the bilinear stress-strain relationship used in this paper may cause the theoretical calculation is not sufficiently accurate.
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Table 3 Comparison of theoretical calculations with numerical results for the tubes with different materials. Material
Yield stress s /MPa
Modulus of elasticity E/GPa
Diameter d/m
Thickness tn/m
Bending radius Rw /m
Theoretical result/mm
Numerical result/mm
Error/%
1Cr18Ni9Ti Q235 65Mn 40Cr 6061
205 235 430 785 41.4
206 206 206 206 60.33
0.034
0.0005
2
5.03 4.79 4.02 2.93 6.22
4.7 4.4 3.8 3.2 7.9
7 8.86 5.79 8.44 21.3
Acknowledgements The authors would like to acknowledge the support of National Natural Science Foundation of China (Grant No. 51374063) and the fund of basic scientific research for Chinese center university (Grant No. N140303009). References
Fig. 8. Comparison of theoretical calculations with numerical results for the tubes of different thicknesses under the same bending diameter.
Above all, the theoretical model is validated to be correct and suitable for the large plastic bending stage in thin-walled steel tubes straightening process. The maximum flattening can be predicted reasonably. Although it is seen that the theoretical results are larger than the numerical results generally from Figs. 7 and 8, and Table 3, especially at the large bending radii of curvature, these predictions are better safe for straightening machine design. 5. Conclusions The following conclusions are drawn in this paper: (1) In thin-walled tube’s continuous straightening process, the tube is usually subjected to a large bending causing the crosssectional ovalization. In this paper a simplified theoretical model is proposed to predict the value of the maximum flattening in terms of curvature, material properties and geometric parameters of a thin-walled tube. It is validated to be approximately correct by both actual experiments and numerical simulations. (2) The comparison of the numerical solutions with the results of theoretical calculations implies that although the theoretical results are usually larger than the numerical ones, they still agree well with the numerical datum within only 10% error of the numerical datum. (3) The proposed model is able to determine the important technological parameter for the safe design of straightening machine, but it still needs to be further verified and possibly modified by more experiments and industrial practices.
Ades, C.S., 1957. Bending strength of tubing in the plastic range. J. Aeronau. Sci. 24 (8), 605–610. Arabzadeh, H., Zeinoddini, M., 2013. A closed-form solution for lateral indentation of pressurized pipes resting on a flexible bed. Int. J. Mech. Sci. 75, 189–199. Brazier, L.G., 1927. On the flexure of thin cylindrical shells and other thin sections. J. Proc. R. Soc. Lond. Ser. A 116 (773), 104–114, Containing Papers of a Mathematical and Physical Character. Daxin, E., Jisheng, Chen, Cai, Yang, 2013. Plane strain solution and cross-section flattening analysis in tube bending with linear hardening law. J. Strain Anal. Eng. Des. 48 (3), 198–211. Elchalakani, M., Zhao, X.L., Grzebieta, R.H., 2002. Plastic mechanism analysis of circular tubes under pure bending. Int. J. Mech. Sci. 44, 1117–1143. Fabian, O., 1977. Collapse of cylindrical elastic tubes under combined bending, pressure and axial loads. Int. J. Solids Struct. 13, 1257–1270. He, Fubao, Shen, Yapeng, 1993. In: Xipan, M. (Ed.), Plate Theory. , 1st ed. Jiaotong University Press, Xipan (In Chinese). Kyriakides, S., Ju, G.T., 1992. Bifurcation and localization instabilities in cylindrical shells under bending—I. Experiments. J. Int. J. Solids Struct. 29, 1117–1142. Levyakov, S.V., 2014. Evaluation of Reissner’s equations of finite pure bending of curved elastic tubes. J. Appl. Mech. Trans. ASME 81 (4), 1–9. Mentella, A., Strano, M., 2012. Rotary draw bending of small diameter copper tubes: predicting the quality of the cross section. J. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf. 226 (B2), 267–278. Munz, D., Mattheck, C., 1982. Cross-sectional flattening of pipes subjected to bending. Int. J. Press. Vessels Pip. 10, 421–429. Poonaya, S., Teeboonma, U., Thinvongpituk, C., 2009. Plastic collapse analysis of thin-walled circular tubes subjected to bending. J. Thin-Walled Struct. 47 (6-7), 637–645. Prinja, N.K., Chitkara, N.R., 1984. Post-collapse cross-sectional flattening of thick pipes in plastic bending. J. Nucl. Eng. Des. 83, 113–121. Reissner, E., 1959. On finite bending of pressurized tubes. J. Appl. Mech. 26, 386–392. Rotter, Micheal J., Michael, Sadowski, Adam, J., Lei, Chen, 2014. Nonlinear stability of thin elastic cylinders of different length under global bending. Int. J. Solids Struct. 51 (15–16), 2826–2839. Shima, Hiroyuki, Sato, Motohiro, Park, Sung-Jin, 2014. Suppression of brazier effect in multilayered cylinders. Adv. Condens. Matter Phys., 1–11. Tatting, B.F., Gürdal, Z., Vasiliev, V.V., 1997. The brazier effect for finite length composite cylinders under bending. Int. J. Solids Struct. 34 (12), 1419–1440. Wierzbicki, Tomasz, Sinmao, V. Monique, 1996. A simplified model of Brazier effect in plastic bending of cylindrical tubes. Int. J. Pressure Vessels Pip. 71 (1), 19–28. Yu, Tongxi, Zhang, Liangchi, 1996. In: World, M. (Ed.), Plastic Bending: Theory and Application. , 1st ed. Scientific Publishing Co Pte Ltd, Singapore. Zhang, Ziqian, Yang, Yuilin, Tian, Yongli, 2013. Offset modeling and analysis of strain neutral surfaces for straightening a thin-walled tube. J. China Mech. Eng. 24 (10), 1390–1395 (In Chinese).
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
A simplified model of maximum cross-section flattening in continuous rotary straightening process of thin-walled circular steel tubes Z.Q. Zhang ∗ , Y.H. Yan, H.L. Yang School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110004, China
a r t i c l e
i n f o
Article history: Received 21 September 2015 Received in revised form 11 July 2016 Accepted 18 July 2016 Available online 19 July 2016 Keywords: Straightening Thin-walled circular steel tube Cross-sectional ovalization Maximal flattening Plastic bending
a b s t r a c t Cross-sectional ovalization of thin-walled circular steel tube because of large plastic bending, also known as Brazier effect, usually occurs during the initial stage of tube’s continuous rotary straightening process. The maximal cross-section flattening (ovalization) is an important technical parameter in tube’s straightening process to control tube’s bending deformation and prevent buckling. However, for the lack of special analytical model, it is determined in accordance with the specified charts developed by experienced operators on the basis of experimental data, thus it is inevitable that the local buckling will occur during some actual straightening operation. In the present paper, simplified normal strain component formulas are derived based on the thin shell theory. Then strain energy of thin-walled tube (per unit length) is obtained subsequently using the elastic-plastic theory. Finally a rational model for predicting the maximal section flattening of the thin-walled circular steel tube under its straightening process is presented by the principle of minimum potential energy. The simplified model is validated by experiments and numerical simulations. The results show that this model agrees well with the experiments and the numerical simulations with less than 10% error. This new model is expected to find its potential application in thin-walled steel tube straightening machine design. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The high-precision thin-walled steel cylinders, referred to as thin-walled tubes or pipes, have a wide variety of applications in pipe networks, aerospace structures, military and petrochemical industry. But they are usually curved in rolling and transporting process due to the external or internal force or temperature variation, so straightening is an important process in order to eliminate the tube’s undesirable geometric imperfections. Currently, the multiple cross roll straightening machine is usually used to straighten thin-walled tubes. It employs a set of roll system, consisting normally four or five pairs of work rolls placed along a longitudinal axis, as illustrated in Fig. 1a. Each paired rolls have the same circular profile of a single curvature radius rather than the hyperbolic profile, one with a convex surface and the other a concave surface, and are mounted at an oblique angle. This kind of roll system has been validated to be suitable for straightening thinwalled tubes by Zhang et al. (2013). During straightening operation,
∗ Corresponding author at: P.O. Box 319, School of Mechanical Engineering and Automation, Northeastern University, No. 3-11, Wenhua Road, Heping District, Shenyang 110004, China. E-mail address: [email protected] (Z.Q. Zhang). http://dx.doi.org/10.1016/j.jmatprotec.2016.07.034 0924-0136/© 2016 Elsevier B.V. All rights reserved.
the thin-walled tube is moved forward whilst being rotated in a spiral direction. It is straightened by alternating bending applied by the roll system. When the tube is in a paired working rolls gap, it is bent by the roll’s profile, with which the radius of curvature of the tube’s axis equals to the roll profile’s, as shown in Fig. 1b. At the initial stage of straightening, the thin-walled tube is usually subjected to a large bending in order to unify its non-uniform initial curvature along its longitudinal axis. This large bending creates large nonlinear plastic deformation, causing the tube cross-sectional ovalization, known as Brazier effect. This section ovalization or flattening, as a major technical parameter for the straightening process, shall be controlled so that no buckling occurs. Therefore, it is very important for the straightening process of thin-walled circular steel tube to analyze Brazier effect response and to predict its maximum crosssection flattening accurately. Since Brazier (1927) firstly described and modeled this highly nonlinear cross-section flattening phenomenon observed by tube bending experiments with simplified method, numerous studies investigating the responses of circular steel tubes under pure bendings have been published. Reissner (1959) generated a nonlinear equation by applying a small-strain bending theory. His equation, even though could not be solved analytically at that time, but was solved by Fabian (1977) using numerical method, with a conclusion that the numerical solution of Reissner’s equation converged
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Fig. 1. The continuous rotary straightening process of thin-walled tube. (a) The working roll system employed by multiple cross roll straightening machine (b) The stress-strain state of thin-walled tube subjected to large bending in working rolls gap (c) The tube’s cross-sectional shape before and under bending
well with Brazier’ s simplified solution except at the limit point. Then Levyakov (2014) proposed another numerical method to solve Reissner’ s nonlinear equations in terms of two unknown functions and two unknown parameters.Using a similar method, Rotter et al. (2014) investigated in detail the effect of cylinder length on the nonlinear elastic buckling behaviour of clamped cylindrical tubes under global bending, covering a very wide range of lengths. However, the studies mentioned above were all assuming that the material were in elastic range. So the more studies have been done in plastic range in order to describe the mechanic behaviour of the circular tube under large plastic bending. Ades (1957) firstly proposed an iterative numerical method, and analyzed the tube ovalization in the plastic range by applying the energy principle. Munz and Mattheck (1982) considered the ver-
tical component of the axial bending stresses as primary cause of cross-sectional ovalization. Following this Prinja and Chitkara (1984) derived a theoretical solution for tube cross-section flattening under plastic bending, by modeling the initially circular section subjected to static compression using four stationary hinges. Then this model has been widely used. It was employed by Poonaya et al. (2009) to analyze thin-walled circular tube’s plastic collapse due to bending. It was also used by Mentella and Strano (2012) to predict the quality of the cross-section of small hydraulic pipings in the rotary draw bending process. Especially, according to this model Arabzadeh and Zeinoddini(2013) analyzed cross-sectional ovalization by considering the tube resting on a flexible bed. Beside this, the Mamalis kinematics model was employed by Elchalakani et al. (2002) to develop a closed-form solution for moment-rotation
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response of the tube under pure bending using two local plastic mechanisms (star and diamond shapes). But all models mentioned above are used to predict the post-buckling response, and the tube’s cross-sectional shape observed during the buckling experiments does not resemble the real shape during the continuous straightening process. Moreover, if the post-buckling occurred during bending, the tube could hardly be straightened subsequently. Furthermore, Yu and Zhang (1996) developed a general method to deal with tube’s ovalization of arbitrary cylindrical cross-section. To apply this method into the straightening process, however, lots of technical issues are still unsolved. Wierzbicki and Sinmao (1996) proposed a simplified theoretical model to predict bending moment and cross-section deformation. Using a similar approach, Daxin et al. (2013) deduced a series of fomulations to calculate the stresses and the cross-section flattening ratio of tubes subjected to combined bending and additional tension with linear hardening law. But these models are valid only for very large sectional distortion, and are not suitable for the straightening process. Tatting et al. (1997) established governing equations for nonlinear bending response of finite length composite tubes, which exhibited cross-section deformation associated with Brazier effect by using semi-membrane constitutive theory. Following this Shima et al. (2014) proposed the equations to describe circular crosssection ovalization under pure bending especially for multilayered cylinders using thin-shell theory. But these equations are too complicated to be used in a straightening machine routine design. To conclude, there is no established method available for estimation of the maximum cross-section flattening for straightening process of thin-walled circular steel tube. The maximum section flattening obtained from the experimental charts sometimes causes that the design radius of the work roll is small enough to induce tube’s local buckling during straightening operation. In this paper, a simplified model of maximum cross-section flattening in rotary straightening process of thin-walled circular steel tube is proposed and its validation is investigated.
307
(1) The material is assumed that to be isotropic and homogeneous. The Bauschinger effect and the hysteresis loop in reloading and unloading, are ignored. (2) For thin-walled circular carbon steel tube, based on the conclusion by Zhang et al. (2013), the offset of the neutral surface to the tube’s centroid surface is so small that it is neglected. Therefore, it is assumed that the cross section neutral axis (the axis in the cross section with zero stress and zero strain) coincides with its centroid axis. (3) Tube thickness is small and is assumed to be constant during the straightening process. Therefore, the tube’s radial strain is zero. The radial stress normal to the middle surface is small compared with the other stress components, and is also neglected in the stress and strain relationships (Zhang et al., 2013). (4) The cross section which is plane and normal to the longitudinal axis of the tube before deformation remains plane and normal after deformation. There is no twist of the adjacent cross sections, so shear stress and shear strain components are considered to be small and are neglected. Thus the stress-strain state of the deformation region can be simplified as shown in Fig. 1b. (5) The volume of material remains unchanged during plastic deformation. (6) Based on the generalized Hooke’s law, in the elastic range, the relationship between the effective stress and the effective strainε is = Eε. And in the plastic range, the material follows bilinear strain-hardening diagram. The stress-strain relationship is defined as = s + E1 (ε − s /E), where E is the modulus of elasticity, E1 is the plastic strain-hardening coefficient, s is the yield stress.
3. Model development 3.1. Ovalised tube section strain-displacement relationship
2. Brazier effect and assumptions For a thin-walled circular steel tube under bending, there are three distinct deformation stages, as observed in the experiments by Elchalakani et al. (2002), namely, elastic, ovalization plateau and structural collapse. A tube at its initial stage of the straightening process is normally subjected to a large nonlinear bending without collapse. This deformation belongs to the second deformation stage. So the cross section under this stage deforms from a circular shape to an oval shape shown in Fig. 1c based on the ovalization assumption. Here taking the quarter cross section AB into account, as depicted in Fig. 2a, it is a circular arc of radius r before bending, with the assumption that the initial cross section’s imperfection is small and can be neglected. During bending process the distance between the points O and A (semi minor axis length) is shortened, and the distance between the points O and B (semi major axis length) is elongated. The initial circular shape of the cross section deforms into an oval. The deformed elliptic arc is defined as A1 B1 . The length change of the circular diameter is defined as the crosssectional flattening. From Fig. 2a, it is obviously that the maximum flattening of all the diameters in any direction appears on the major or minor axis. According to the results obtained by Yu and Zhang (1996), the length increase of the major axis is assumed equal to the length decrease of the minor axis. So here the maximum flattening is considered as the length change of the minor axis. Meanwhile it is also assumed that there are no circumferential displacements at the points A and B. To analyze the cross-section flattening, the following assumptions are made in this paper:
When a thin-walled circular steel tube undergoes a large bending, based on Brazier’s effect, the circular section is flattened to an oval shape. To be able to analyze the strain-displacement relationship of a point on an arbitrary cylindrical surface, which is a radial distance z away from its middle surface, as depicted in Fig. 2b. It is necessary to analyze the strain-displacement relationship of a point on the tube’s middle surface at first. Here the middle surface is defined as halfway between the inner and outer edges of the tube, shown in Fig. 2b. The strain components of a point on the middle surface can be described as functions of the point displacements v, u and w, in which v is the axial displacement, and its positive direction is when the bending angle of the tube is increasing; u is the circumferential displacement and its positive direction is when is increasing; w is the radial displacement, and its positive direction is when its radius is increasing, as depicted in Fig. 1b. For the circumferential strain of a point on the middle surface, due to the assumption (4), it is assumed that circumferential strain is not related to the axial displacement v. Therefore, the circumferential strain is a function of circumferential displacement u and radial displacement w only. Consider an initial circular arc segment PQ on the middle surface with a radius r and an infinitesimal angle d (Fig. 2a). It is defined by the central angle between OP and OB. After deformation, PQ is displaced to an elliptic arc segment P1 Q1 . The length increase of PQ is a combination of the length increases owing to the circumferential displacement and the radial displacement. In its circumferential direction, the point P is first moved to the point P due to a circumferential displacement u, and at the same time, point Q is moved to the point Q due to a circumferential displacement u + ∂u d. ∂
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Fig. 2. The deformation and shape of thin-walled tube cross section under bending.
The circumferential length increase of the arc segment PQ can be expressed as: 1 = u +
∂u ∂u d − u = d ∂ ∂
In its radial direction, with a radial displacement w, the point P is moved to the position P independently, as well as the point Q is moved to the position Q . The length of circular arc segment PQ is changed from rd to (r − w)d. The circumferential length increase of the arc segment PQ due to radial displacement can be expressed as: 2 = P Q − PQ = (r − w)d − rd = −wd The total circumferential length increase of the arc segment PQ is the sum of 1 and 2 . The circumferential strain is then: ε =
1 1 + 2 = r rd
∂u −w ∂
Rwp w − Rw w Rw w
shape before bending. If the cross section is ovalized, the point is moved from the position P to the position P1 due to the circumferential and radial displacements. Considering the geometrical relationship shown in Fig. 2a, Rwp is changed to: Rwp = PT + P T + Rw = Rw + (r − w) sin + u cos
ε =
(r − w) sin + u cos Rw
where, Rwp is the axial radius of curvature of the arbitrary point P on the middle surface. As shown in Fig. 2a, it is originally described as Rwp = Rw + PT1 = Rw + r sin when the cross section is a circular
(4)
As described by He and Shen (1993), there is a one-to-one correspondence between the strain components on an arbitrary revolution surface at a radial distance z from the middle surface of the shell and the strain components on the middle surface. It can be expressed as: εi (z) =
(2)
(3)
Substituting Eq. (3) into Eq. (2), the axial strain component acting on the middle surface becomes:
(1)
Then, the axial strain of an arbitrary point on the middle surface can be developed as follows. During the straightening process, the tube is bended by the gap of working rolls. Let Rw denote the radius of curvature of the deformed tube’s centerline axis, and w the central angle of rotation of the axis. From the assumption (2), it is assumed that the neutral axis coincides with the centroid axis. Therefore there is no length change in the tube’s centroid axis. The length of the initial tube is Rw w . Sequentially the axial strain component of a point on the middle surface may be expressed as: ε =
(a) The displacement of segment PQ (b) The element of point P
1 εi [u(z) , w(z) ], (i = , ) 1 + Rz i
where, z is the distance of the surface to the middle surface in Fig. 2b. u(z) ,w(z) are the circumferential and radial displacements acting on the arbitrary cylinder surface at a radial distance z from the middle surface of the tube. Ri is defined as the radius of principal curvature. From Fig. 2a in this paper, the circumferential radius of principal curvature R equals to r, and the radial one R should be Rw + r sin . Then by the use of Eqs. (1) and (4), the strain components on an arbitrary cylinder surface at a radial distance z from the middle surface of the tube can be expressed as:
⎧ r 1 ∂u(z) (z) ⎪ (z) ⎪ ε = w − ⎨ r r+z ∂
(z) (z) ⎪ ⎪ ⎩ ε(z) = (r − w ) sin + u cos Rw + r sin
Rw
Rw + r sin + z
(5)
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In which u(z) ,w(z) can be expressed as follows (He and Shen 1993):
⎧ ⎨ u(z) = u + z u − 1 ∂w ⎩
r
A ∂
(6)
w(z) = w
where, Ais the circumferential Lame parameter, and here also equals to r base on the geometric description of Lame parameter. Thus substituting Eq. (6) into Eq. (5), the following relationships are obtained:
from the neutral surface. When bending of tubes induces crosssectional ovalization, deformation may be so large as to cause the material to become plastic. Comparing with the plastic deformation region, the elastic deformation region is very small and could be neglected. Hence J2 deformation theory can be employed to describe the stress-strain relationship of the whole cross section deformation as follows:
⎧ 1 1 (z) (z) (z) ⎪ ⎨ ε = E − 2
⎪ ⎩ ε(z) = 1 (z) − 1 (z)
⎧ ⎫ ⎧ E ∂w z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ u + r (u − ∂ ) ⎬ 2 ⎪ ⎪ 1 1 1 1 ∂u z ∂u z∂ w ⎪ (z) ⎪ ε = = ( w − w) − + − ⎪ z ⎪ z ⎪ r ∂ r ∂2 ∂ ⎪ ⎪ 1+ r ⎪ 1 + r ∂ ⎪ ⎨ ⎩ ⎭ r r (7) ⎪ 1 u ∂ w ⎪ ⎪ (r − w) sin + u + z( − ) cos ⎪ ⎪ r r ∂ ⎪ 1 (z) ⎪ ε = × ⎪ ⎪ z Rw ⎩ 1+
Rw + r sin
In view of assumption (3), the tube is so thin that the term z/r (z) in Eq. (7) can be assumed to be 0, thus the formula of ε in Eq. (7) can be simplified as (z)
ε ≈
1 ∂u ( − w) = ε r ∂
(8)
In addition, because the value of z is small comparing with Rw , so it can be assumed that 1 + R +rz sin ≈ 1. On the other hand, conw
sidering the following expressionz
u r
−
1 ∂w r ∂
cos <<
(z) Rw , ε
in
(z)
ε ≈
(r − w) sin + u cos = ε Rw
(9)
From Eqs. (8) and (9), it is concluded that the strains on the arbitrary cylindrical surface at a radial distance z from the middle surface can be replaced approximately with the strains on the middle surface. Furthermore the circumferential and radial displacements in Eqs. (8) and (9) are expanded in terms of trigonometric series given by:
⎧ n ⎪ ⎪ ⎪ u = r i sin(2i) ⎪ ⎨ i=1 n
(10)
⎪ ⎪ ⎪ ⎪ ⎩ w = r i cos(2i)
⎧ 2 (z) (z) (z) ⎪ ⎨ = 3 E 2ε + ε
⎪ ⎩ (z) = 2 E 2ε(z) + ε(z)
3
(14)
In view of assumption (6), the effective stress-strain relationship can be expressed as:
s E
(15)
Hence, the modulus of plasticity E can be obtained by the use of Eq. (15): E =
s − = ε ε
E1 s E
+ E1
(16)
In accordance of the assumption (5), the volume of material elements remains constant during plastic deformation, thus: (z)
(z)
ε + ε = 0
(17)
1 (z) 2 (z) (z) ε = √ |ε − ε | = √ |ε | 3 3
(18)
E =
where, i , i are the dimensionless coefficients. Here Eq. (11) is truncated after only the first term in order to simplify u and w as: u = r sin 2
(11)
w = r cos 2 Substituting Eq. (11) into Eqs. (8) and (9) results in
⎧ 1 ⎪ ≈ ε = (2r cos 2 − r cos 2) = (2 − ) cos 2 ⎨ ε(z) r ⎪ ⎩ ε(z) ≈ ε = (r − r cos 2) sin + r sin 2 cos Rw
Substituting Eq. (18) into Eq. (16) results in:
i=1
2
(13)
where, E = /ε is the modulus of plasticity, which is a function of the material properties and the loading history. From Eq. (13) the stress components can be obtained:
= s + E1 ε −
Eq. (7) also can be simplified to:
309
(12)
Rw
E1 s E (z) 2 √ |ε | 3
s −
+ E1
(19)
Substituting Eqs. (19) and (17) into Eq. (14) results in:
⎧ ⎛ ⎞ E1 s ⎪ ⎪ − ⎪ 2⎜ s ⎪ (z) E + E ⎟ ε(z) ⎪ = ⎝ ⎪ 1⎠ ⎪ 2 (z) 3 ⎪ ⎨ √ |ε | ⎛ 3 ⎞ E1 s ⎪ ⎪ ⎪ ⎪ (z) 2 ⎜ s − E ⎟ (z) ⎪ = ⎝ + E1 ⎠ ε ⎪ ⎪ 2 (z) 3 ⎪ ⎩ √ |ε | 3
(20)
3.2. Ovalised tube section stress-strain relationship
3.3. Thin-walled tube maximum section flattening model
During the straightening process, the cross section deformation of the thin-walled circular steel tube can be divided into two parts by the yielding surface. One is the elastic region adjacent to the neutral surface, and the other is the plastic region at the far end away
As previously mentioned, the maximum flattening of the thinwalled tube cross section equals to the length change of the minor axis, which is just two times of the radial displacement at point A in Fig. 2a. So if the coefficients i , i in Eq. (11) can be obtained,
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here some parameters below are defined as U m1 =
2
ε2 d, Um3 =
0
2
ε d, Um4 =
0
2
0
2
ε2 d, Um2 =
0
ε d. Substituting the axial
strain component from Eq. (12), Um1 becomes:
2
Um1 =
ε2 d =
0
0
2 r2 2 Rw
−4( − )2 cos6 +
+ 4( − )2 + 4 (1 + ) − 4(1 + ) cos4 +
4(1 + ) − 4 (1 + ) − (1 + )2 cos2 + (1 + )2 d
where, Fig. 3. YC10GJ—70 twelve cross rolls thin-walled tube straightening machine for actual experiments.
1
. 4
cos6 d =
r2
Um =
1 2
(x εx + y εy )dV =
1 2
(V )
=
r 2 2 8Rw
(z) (z)
(21)
3
, 16
cos4 d =
2
0
cos2 d =
1 1
+ (1 + )2
4 2
[(1 + )2 + (1 + )2 ]
In the same way, Um3 can be expressed as:
( ε + ε )rddz (S)
2
0
[4(1 + ) − 4 (1 + ) − (1 + )2 ] ·
2
Um3 = (z) (z)
5 3
+ [4( − )2 + 4 (1 + ) − 4(1 + )]
+ 32 16
−4( − )2
2 Rw
5
, 32
Integrate Um1 with respect to , and it is reduced to
Um1 =
the maximum flattening will be calculated by the use of Eq. (11). In order to proceed, the appropriate form of the coefficientsi and i should be deduced at first. Here by using the strain energy principles, the expression for the internal plastic strain energy of per unit length tube is defined as follows:
2
0
r Rw
2
ε d =
0
0
2 (r − r cos 2) sin + r sin 2 cos d = Rw
( 2 cos2 − − 1) − 2cos2 d cos =
0
r 1 2 ( + 1 + ) Rw 3 3
Substituting the circumferential strain component from Eq. (12), and integrate Um2 with respect to , then it is reduced to As the cross section is symmetric with respect to the major axis as well as the minor axis (double symmetric cross section), the internal plastic work can be calculated by taking into account of the quarter of the circular cross section in extrados-wall region, as shown in Fig. 2a. In this region, although the cross section is bent to an oval shape, there is no “kink” or “local buckle”. So the state of strain is still determined by the axial bending, which induces that the extrados-wall of the tube is in tension. So the axial strain (z) component is positive ε > 0, and from equation (17) the circum(z)
2
Um2 =
3
E
Similarly, Um4 is given by:
Um4 =
4rt 3
2
2
0
ε d =
0
2
(2 − ) cos 2d
0
2
(2cos2 − 1)d = (2 − )(
− )=0 2 2
Substituting all of the Umi into Eq. (23) results in (22) Um =
1 E1 s 2 2 (z) (z) −√ s − + E1 ε ε t E 3 3 0 − 2 1 E1 s 2 (z) (z) ) + E1 ε ε + √ (s − ddz = E 3 3
Um = 2r
0
Substituting Eq. (22) into Eq. (21), and combining with Eq. (12), the internal plastic strain energy of per unit length tube can be expressed as:
t
2
= (2 − )
3
(2 − )2 (4cos4 − 4cos2 + 1)d =
0
3 1 1 (2 − )2 ( − + ) = (2 − )2 4 2 4
(z)
2 ⎧ 1 E1 s (z) (z) ⎪ + E1 ε ⎨ = − √3 s − E 3
⎪ ⎩ (z) = √1 s − E1 s + 2 E1 ε(z)
2
=
0
ferential strain component is negative ε < 0ε < 0. Thus Eq. (20) can be simplified as:
ε2 d
√ 3s E1 (1 − )(ε − ε ) d E1 (ε 2 + ε 2 ) + 2 E
4rt E1 3
r 2 2 8Rw
(2 + 2 + 2 + 2 + 2 )+
2rt 1 (4 2 − 4 + 2 ) + √ 4 3
s −
E1 s E
×
r Rw
1 3
+1+
2 3
(24)
Then, it is postulated that for a given Rw , the dimensionless coefficients , adjust themselves so as to minimize the strain energy. So the following relationship will be derived:
∂Um ∂Um = =0 ∂ ∂
(25)
Substituting Eq. (24) into Eq. (25) results in: (23)
⎧ ∂Um ⎪ ⎨ ∂ ⎪ ⎩ ∂Um ∂
=
4 E1 3
=
4 E1 3
r 2 2 4Rw
r 2 2 4Rw
2 (1 + ) + (2 − ) + √ 3
(1 + ) +
s −
1 2 (−2 + ) + √ 2 3
E1 s E
s −
r 2 · =0 Rw 3
E1 s E
r 1 · =0 Rw 3
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(26) 2, here some parameters S 1 = E1 r 2 /3Rw S2 = 2E1 /3, √ S3 = 2s r(E − E1 )/3 3ERw are defined for simplifying. Then Eq. (26) can be expressed as:
!
(S1 + 4S2 ) − 2S2 = −2S3 − S1 −2S2 + (S1 + S2 ) = −S3 − S1
(27)
Finally, solving the above equation set, the coefficients , can be obtained from the following formulas if the tube’s material properties, geometric parameters and the bending radius of curvature are given.
⎧ (S3 + S1 )2S2 + (2S3 + S1 )(S1 + S2 ) ⎪ ⎨ = 4S22 − (S1 + S2 )(S1 + 4S2 ) (2S 3 + S1 )2S2 + (S3 + S1 )(S1 + 4S2 ) ⎪ = ⎩
(28)
4S22 − (S1 + S2 )(S1 + 4S2 )
with obtained coefficients , , and by applying = /2 into Eq. (11), the relative displacement at point A from its original position along the minor axis can be obtained. This is the half of the maximum section flattening. So the maximum cross-section flattening ıa can be then obtained as: ıa = 2ı = | − 2r |
Table 1 The experiment measurements of the maximal cross-sectional flattening during actual experiments. Specimen no.
Section flattening/mm
Specimen no.
Section flattening/mm
1 2 3 4 5
2.26 2.03 2.33 2.36 1.88
6 7 8 9 10
2.52 2.28 2.31 2.42 2.15
Repeated the steps for the rest of the specimens and the results were summarized in Table 1. Removed the maximum and minimum values, the arithmetic mean value of the testing measurements was equal to 2.27 mm as the actual value of the maximum flattening. Meanwhile calculated the theoretical value with Eq. (29), the theoretical value of the flattening was equal to 2.59 mm. The comparison of the experimental solution with the result of theoretical calculation implies that the analytical solution is larger than the experimental data. It is because that the second pair of rolls could not be opened during measuring procedures. It is hard to determine the accurate direction of the maximum flattening and also to obtain the maximum flattening measurement. But from the comparison here, it shows that the theoretical model still provides a practice value with only 10% away from the experimental results.
(29)
where, ı is half of the maximum flattening, equals to subtracting OA1 from OA. Obviously the maximum section flattening is a function of tube longitudinal bending curvature radius, material properties and geometric parameters. To prevent the tube from buckling, the tube maximum section flattening shall be less than the allowed limit section flattening ılim , which can be obtained through stability analysis or actual experiments. So the following expression must be satisfied: ıa = | − 2r | ≤ ılim
311
(30)
For a given thin-walled steel tube, making the substitution of the parameter into Eq. (30) will give us a design equation which can be used to determine the reasonable radius of curvature of working rolls for the safe design of straightening machine. Under this bending radius the tube maximum cross-section flattening will be below the allowable limit, in order to prevent the local buckling. 4. Model validation 4.1. Actual experiments In order to verify the proposed model, thin-walled circular steel tube straightening experiments are conducted by the straightening machine YC10GJ-70, which has six pairs of cross work rolls, as shown in Fig. 3. The second and fifth paired roll profiles are circular curves, and the other paired roll profiles are hyperbolic curves. Thereinto the second pair of rolls applies the required large plastic bending to the specimen, unifying its initial existing non-uniform longitudinal curvature. So the maximum flattening measurements can be obtained at this stage. The specimens are 10 thin-walled tubes made of 1Cr18Ni9Ti steel with similar initial curvatures. The tube outside diameter d = 21 mm, tube wall thickness tn = 1 mm. The tube’s material properties are assumed s = 205 MPa, E = 206GPa, E1 = 2 GPa. The test set-ups are as follows: (1) Pushed a specimen into the straightening machine; (2) When the specimen passed the third pair rolls, shut down the machine; (3) Chose the cross section at the middle point of the second pair of rolls, whose diameter of curvature was equal to 4.26 m, and measured the flattening of the cross section at vertical direction.
4.2. Numerical simulations One set of material model parameters discussed for experiments did not provide a perfect validation for the formulas. There must be more experiments for several materials of different parameters. However, when the material parameters changed, a corresponding change in structure of the straightening machine must be done, such as the roll profile, the intermesh and the diameter of roll. Meanwhile it was difficult to measure the maximum flattening accurately by experimental method. Therefore the numerical simulations instead of experiments are employed to further validate the theoretical model. Here an accurate FEM model have been developed using ANSYS/LS-DYNA program, as shown in Fig. 4. It was a typical two-roll straightening system, consisting of a thin-walled tube, a paired working rolls with the circular profile of 4.26 m diameter and a slit guide system which had two guide sleeves and a paired guide plates to play a role of feeding of the tube into the rolls. The tube was modeled by the 3D solid element with an elastoplastic material. The rolls and the guide system were modeled by the rigid body elements. During the simulation, each roll rotated about its own centroid axis with an angular velocity of 280 rmp, the tube subjected to an initial feeding velocity of 0.28 m/s was pushed into the rolls. Then it was rotated at the same time moved along the z direction (as illustrated in Fig. 5a). Every pitch the tube moved forward, twice bending it was subjected to (bending in y direction and reverse bending respectively).Because the roll profile was four times the length of the scroll pitch, so throughout the whole simulation the tube would undergo the plastic bending in y direction eight times rather than once. Considered the moment when the tube was completely in the rolls gap, as shown in Fig. 5a, it contacted with the working rolls completely. So the diameter of curvature of the tube axis was equal to the rolls’ diameter. Here the tube was bent by the large curvature. From Mises stress distribution as illustrated in Fig. 5b, the stress level was much higher. Then chose a cross section m–m in the center of the profile as shown in Fig. 5a, the shape of the cross section was described in Fig. 5c, it was shown that the cross section was deformed to an oval shape under bending. In order to measure the maximum flattening of the cross section accurately, chose two nodes C and D on the section m–m (also
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Fig. 4. The FEA simulation model.
Fig. 5. The numerical simulation for straightening the thin-walled tube causing ovalization to the cross section (a) The bending state of the tube in working rolls gap (b) The tube Mises stress distribution under bending (c)The tube cross-sectional shape under bending.
Z.Q. Zhang et al. / Journal of Materials Processing Technology 238 (2016) 305–314
313
Fig. 6. Variation in the coordinate values of nodes C and D in y direction with time.
Table 2 The flattening of the cross section m–m at different times. Time/s
Angular displacement/rad
Section flattening/mm
Time/s
Angular displacement/rad
Section flattening/mm
0.1 0.2 0.3 0.4
2 3 4
2.13 2.32 2.37 2.41
0.5 0.6 0.7 0.8
5 6 7 8
2.42 2.43 2.45 2.45
shown in Fig. 5c), and traced their y-displacements, then the variation in the mean vertical diameter of cross section m–m (measured by the distance y between node C and node D) with the deformation time t was described in Fig. 6. The maximum flattening measured by the minimum y-distance between nodes C and D for t = 0.1, 0.2, 0.3. . .0.8 s (accordingly the angular displacement of the working roll is , 2, 3. . .8 rad) were summarized in Table 2. It is shown that the cross section flattening increases as the number of bending circle increases, so the maximum flattening throughout the whole deformation process is around 2.45 mm. This result agrees well with both the theoretical calculation and the experimental data. It is concluded that the numerical simulations can be employed instead of experiments for further validations. Using above numerical method, three sets of similar simulations have been conducted. The first set tested the same tubes (1Cr18Ni9Ti, d = 21 mm, t = 1 mm) at the different bending diameter of curvature, ranging from 3m ∼ 12m. The numerical results were summarized in Fig. 7. Then they were fitted into a curve by the interpolation polynomial shown in Fig. 7. At the same time the theoretical results for each set of parameters were obtained from Eq. (29) also shown in Fig. 7. The comparison of the numerical solutions with the results of theoretical calculation implies that the analytical solutions agree well with the numerical datum with only 10% error of the numerical datum. It is also seen that the error is big as the bending radius increase to a certain extent. But for these radii, the values of maximum flattening are so small that can be neglected in actual straightening progress. In the second set of simulation, the similar tubes (1Cr18Ni9Ti, d = 21 mm) of different thicknesses, ranging from 0.5mm–2 mm, were bent by the same bending diameter 4m. The numerical results and the fitted curve were plotted in Fig. 8. At the same time the theoretical results were calculated according to Eq. (29) also shown in Fig. 8. The comparison of the numerical solutions with the results of theoretical calculation shows that the analytical solutions agree well with the numerical datum with 10% error of the numerical datum except one data point for the thickness tn = 2 mm. Taking
Fig. 7. Comparison of theoretical calculations with numerical results for the same tubes at different bending radii.
this data point into account, the ratio of the diameter to the thickness is about 10. Thus the tube does not belong to the thin-walled tube, and some assumptions for the modeling are strongly violated. So the result of this set of data should be neglected. Final set of simulation tested the tubes with the same geometric parameters, but differing in material properties. Both the numerical and theoretical results were summarized in Table 3. From Table 3 it is shown that the analytical solutions agree well with the numerical datum with 10% error except the fifth set of data. Taking this set of data into account, the material is aluminium alloy 6061 which usually follows the power law, such as Kyriakides and Ju, (1992) employed the stress-strain curve fitted with Ramberg Osgood fit in his experiments. So the bilinear stress-strain relationship used in this paper may cause the theoretical calculation is not sufficiently accurate.
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Table 3 Comparison of theoretical calculations with numerical results for the tubes with different materials. Material
Yield stress s /MPa
Modulus of elasticity E/GPa
Diameter d/m
Thickness tn/m
Bending radius Rw /m
Theoretical result/mm
Numerical result/mm
Error/%
1Cr18Ni9Ti Q235 65Mn 40Cr 6061
205 235 430 785 41.4
206 206 206 206 60.33
0.034
0.0005
2
5.03 4.79 4.02 2.93 6.22
4.7 4.4 3.8 3.2 7.9
7 8.86 5.79 8.44 21.3
Acknowledgements The authors would like to acknowledge the support of National Natural Science Foundation of China (Grant No. 51374063) and the fund of basic scientific research for Chinese center university (Grant No. N140303009). References
Fig. 8. Comparison of theoretical calculations with numerical results for the tubes of different thicknesses under the same bending diameter.
Above all, the theoretical model is validated to be correct and suitable for the large plastic bending stage in thin-walled steel tubes straightening process. The maximum flattening can be predicted reasonably. Although it is seen that the theoretical results are larger than the numerical results generally from Figs. 7 and 8, and Table 3, especially at the large bending radii of curvature, these predictions are better safe for straightening machine design. 5. Conclusions The following conclusions are drawn in this paper: (1) In thin-walled tube’s continuous straightening process, the tube is usually subjected to a large bending causing the crosssectional ovalization. In this paper a simplified theoretical model is proposed to predict the value of the maximum flattening in terms of curvature, material properties and geometric parameters of a thin-walled tube. It is validated to be approximately correct by both actual experiments and numerical simulations. (2) The comparison of the numerical solutions with the results of theoretical calculations implies that although the theoretical results are usually larger than the numerical ones, they still agree well with the numerical datum within only 10% error of the numerical datum. (3) The proposed model is able to determine the important technological parameter for the safe design of straightening machine, but it still needs to be further verified and possibly modified by more experiments and industrial practices.
Ades, C.S., 1957. Bending strength of tubing in the plastic range. J. Aeronau. Sci. 24 (8), 605–610. Arabzadeh, H., Zeinoddini, M., 2013. A closed-form solution for lateral indentation of pressurized pipes resting on a flexible bed. Int. J. Mech. Sci. 75, 189–199. Brazier, L.G., 1927. On the flexure of thin cylindrical shells and other thin sections. J. Proc. R. Soc. Lond. Ser. A 116 (773), 104–114, Containing Papers of a Mathematical and Physical Character. Daxin, E., Jisheng, Chen, Cai, Yang, 2013. Plane strain solution and cross-section flattening analysis in tube bending with linear hardening law. J. Strain Anal. Eng. Des. 48 (3), 198–211. Elchalakani, M., Zhao, X.L., Grzebieta, R.H., 2002. Plastic mechanism analysis of circular tubes under pure bending. Int. J. Mech. Sci. 44, 1117–1143. Fabian, O., 1977. Collapse of cylindrical elastic tubes under combined bending, pressure and axial loads. Int. J. Solids Struct. 13, 1257–1270. He, Fubao, Shen, Yapeng, 1993. In: Xipan, M. (Ed.), Plate Theory. , 1st ed. Jiaotong University Press, Xipan (In Chinese). Kyriakides, S., Ju, G.T., 1992. Bifurcation and localization instabilities in cylindrical shells under bending—I. Experiments. J. Int. J. Solids Struct. 29, 1117–1142. Levyakov, S.V., 2014. Evaluation of Reissner’s equations of finite pure bending of curved elastic tubes. J. Appl. Mech. Trans. ASME 81 (4), 1–9. Mentella, A., Strano, M., 2012. Rotary draw bending of small diameter copper tubes: predicting the quality of the cross section. J. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf. 226 (B2), 267–278. Munz, D., Mattheck, C., 1982. Cross-sectional flattening of pipes subjected to bending. Int. J. Press. Vessels Pip. 10, 421–429. Poonaya, S., Teeboonma, U., Thinvongpituk, C., 2009. Plastic collapse analysis of thin-walled circular tubes subjected to bending. J. Thin-Walled Struct. 47 (6-7), 637–645. Prinja, N.K., Chitkara, N.R., 1984. Post-collapse cross-sectional flattening of thick pipes in plastic bending. J. Nucl. Eng. Des. 83, 113–121. Reissner, E., 1959. On finite bending of pressurized tubes. J. Appl. Mech. 26, 386–392. Rotter, Micheal J., Michael, Sadowski, Adam, J., Lei, Chen, 2014. Nonlinear stability of thin elastic cylinders of different length under global bending. Int. J. Solids Struct. 51 (15–16), 2826–2839. Shima, Hiroyuki, Sato, Motohiro, Park, Sung-Jin, 2014. Suppression of brazier effect in multilayered cylinders. Adv. Condens. Matter Phys., 1–11. Tatting, B.F., Gürdal, Z., Vasiliev, V.V., 1997. The brazier effect for finite length composite cylinders under bending. Int. J. Solids Struct. 34 (12), 1419–1440. Wierzbicki, Tomasz, Sinmao, V. Monique, 1996. A simplified model of Brazier effect in plastic bending of cylindrical tubes. Int. J. Pressure Vessels Pip. 71 (1), 19–28. Yu, Tongxi, Zhang, Liangchi, 1996. In: World, M. (Ed.), Plastic Bending: Theory and Application. , 1st ed. Scientific Publishing Co Pte Ltd, Singapore. Zhang, Ziqian, Yang, Yuilin, Tian, Yongli, 2013. Offset modeling and analysis of strain neutral surfaces for straightening a thin-walled tube. J. China Mech. Eng. 24 (10), 1390–1395 (In Chinese).