Journal of Materials Processing Technology 121 (2002) 273±284

A study on seamless tube in the planetary rolling process

Chih-Kang Shih, Ray-Quen Hsu, Chinghua Hung*
Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu, Taiwan, ROC
Received 4 February 2001

Abstract

In this paper, the ®nite element method and the dual-stream functions upper bound method are selected to analyze the tube rolling process
with a planetary rolling mill. Three-dimensional elastic±plastic ®nite element simulations with special emphasis on the determination of the
roller pro®le and realistic contact conditions between the rollers, workpiece and mandrel are used to reveal the deformation characteristics of
seamless tube during the rolling process. In addition, the kinematically admissible velocity ®eld of the deforming tube is described in terms of
two stream functions so that the velocity components of the deforming tube can be obtained through the upper bound analysis. In both analysis
methods, the equation of meshing is used to determine the contact condition between the rollers and the workpiece. The results from both
methods with different variables are observed and compared. # 2002 Elsevier Science B.V. All rights reserved.

Keywords: Planetary rolling process; Finite element; Upper bound; Dual-stream functions

1. Introduction positioned by the ring, which encloses the rollers and rotates
in the direction opposite to that of rollers. Besides ®xing the
The planetary rolling mill (PlanetenshraÈgwalzwerk, rollers, the main purpose of the ring is to counteract the
PSW), as shown in Fig. 1 [1], has been operated successfully twisting deformation of the workpiece during the rolling
in the roughing rolling process. The advantages of PSW process and to keep the tubes round after rolling.
include ¯exibility of workpiece size, low rolling load, low Most papers about PSW have been focused on the rod
lateral spread of material [2,3] and low temperature drop rolling process, with only a few on the seamless tube rolling
between the leading and tail ends of the rolled workpiece [4]. process. For the rod rolling process, Aoyagi and Ohta [7]
Because of its high reduction and continuity, the PSW is also observed the ¯ow of the material, the load and the torque on
able to manufacture seamless tubes from pierced billets. A the rollers, and the pressure distribution on the rollers during
set of PSWs used as a elongation mill combined with a the rolling process with different cross-section reduction
sizing or stretch-reducing mill in tube production forms one ratios. In addition, they studied the in¯uence of the offset
continuously operating unit (Fig. 2) [5] and permits the angle on the exit velocity of the rods. Nishio et al. [8]
omission of the cooling and reheating operations that are discussed the quality characteristics of rolled products with
usually required after elongation for the conventional elon- different offset angles and roller pro®les. In analytical
gation mill. Thus cost can be saved because reheating research, Hwang et al. [9] used dual-stream functions for
furnaces are not necessary and the noise level is reduced investigating the plastic deformation behavior of the rod
because there are no cold tubes clashing against the mandrel. during the planetary rolling process. In numerical analysis,
Further, the PSW provides close wall thickness tolerances Li [10] used the elastoplastic ®nite element method to
through transverse rolling at the elongation stage [6] and analyze the elementary deformation behavior of the plane-
increased yield. tary rolling process. Shih and Hung [11] employed three-
The PSW mainly contains three conical rollers, an exter- dimensional ®nite element analysis and an optimum method
nal ring and a mandrel. As shown in Fig. 3, the rollers are to obtained the rolling deformation and the resulting stress
inclined and located equally around the axis of workpiece and strains. For the tube rolling process, Siebke and Gartner
with their axes intersecting that of the workpiece by an off- [12] used elementary mechanics to analyze the force acting
set angle. This arrangement provides a feed movement on the rollers and the mandrel. However, few works have
as the rollers rotate around the workpiece. The rollers are been done on studying the tube rolling process with three-
dimensional ®nite element deformation analysis or with an
*
Corresponding author. Fax: ‡886-35-720-634. upper bound solution through dual-stream functions. The
E-mail address: chhung@cc.nctu.edu.tw (C. Hung). purpose of this study is thus to use the ®nite element method

0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 1 2 6 5 - 1
274 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284

to further analyzes the planetary rolling process with a

systematic study of the effects of rolling parameters and
also compare the results to those derived from dual-stream
functions. Because the simulation of the planetary rolling
process is three-dimensional with characteristics of non-
linearity and large deformation, the numerical calculation
demands ef®cient and powerful software. The vectorized
explicit ®nite element code LS-DYNA3D [13] has thus been
selected as the simulation tool.

2. Mathematical model

2.1. Simplification of the PSW model

Fig. 1. Cross-section of PSW (source: H. Marten, New approaches in plant The operation of the PSW includes the revolution of three
technology to increase quality and productivity, MPT Metall. Plant rollers, the rotation of the rollers and the rotation of the
Technol. 9 (5) (1986) 39±55). external ring. The major boundary conditions include the
contacts between three rollers and the workpiece, the con-
tacts between the rollers and the ring and the contacts
between the workpiece and the mandrel. Thus, a complete
and realistic model will make the ®nite element analysis
very complicated. In addition, the control over the rotation
of each component and the analyses of contacts between
each component will incur huge computational time. There-
fore, it is reasonable to simplify the simulation model. If the
axes of the rollers are ®xed and the orbital revolution of the
rollers is ignored, the external ring can be omitted from the
PSW model. The orbital motion provided by the external
ring can then be taken care of by the relative motion of the
workpiece with free rotational degree of freedom. Thus the
key points on the analysis of the simpli®ed model contain
Fig. 2. Layout of a tube production plant with PSW (source: E. only the rotation of the rollers and the contact conditions
Bretschneider, New process for the production of seamless tubes-PSW between the rollers, the workpiece and the mandrel.
now also in the tube sector, MPT Metall. Plant Technol. 6 (6) (1983)
44±49). 2.2. Finite element model

A third-order polynomial was used to de®ne the pro®les

of the geometry of the rollers. For a point RR on a roller, the
radius rR was de®ned as

rR ˆ rR x† ˆ ax3 ‡ bx2 ‡ cx ‡ d (1)

Thus the coordinates of RR on the roller's coordinate system
SR can be represented as
R R ˆ ‰ xR yR zR 1 ŠT (2)
where

xR ˆ ff u; yR ˆ rR cos y;
zR ˆ rR sin y; ff ˆ const: (3)
where 0  y < 2p; 0  u < U; U ˆ const:.
Fig. 4 shows the projective pro®le of the roller on the local
roller's coordinate system. The revolution of this curve with
Fig. 3. Rollers of PSW (courtesy of Walsin Cartech Specialty Steel respect to the axis XR makes the complete pro®le of a roller.
Corporation). The global coordinate system of the simulation model is
 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284 275

Fig. 4. Roller profile on the coordinate system SR.

de®ned as the coordinate system of the workpiece. The then the corresponding mesh system was generated, as
geometric relation between the local coordinate system shown in Fig. 6.
SR XR ; YR ; ZR † and the global coordinate system S0 X0 ; Low-alloy-contained stainless steel in the tube rolling
Y0 ; Z0 † is shown in Fig. 5. Thus, a point RR in SR can process was analyzed. The effective stress vs. effective strain
be represented as R0 in S0 by using the coordinate transfor- curve of 304L stainless steel at 1000 8C as shown in Fig. 7
mation [14] was adopted as the material property in the ®nite
element analysis. The effective stress is assumed to be
R0 ˆ M 0R RR (4) constant as the effective strain is greater than 0.7. At high
The transformation matrix is temperature, the material is sticky and the coef®cient of
2 3 friction in the contact surface was taken as m ˆ 0:5 [15].
cos b cos a sin b cos a sin a 0
6 sin b cos b 0 dr 7 2.3. Contact analysis between the roller and the workpiece
M 0R ˆ 6
4 cos b sin a sin b sin a
7 (5)
cos a 05
0 1 0 1 By observing the geometrical relationship of the roller
and the workpiece it is found that because the surface of the
where a represents the offset angle and b represents the
workpiece is formed by a curve rotating around the X0-axis,
inclined angle. Through the transformation matrix M 0R ,
the normal vector of any point on the workpiece surface will
point R0 becomes
pass through the center axis. Similar condition prevails in the
R0 ˆ ‰ x y z 1 ŠT (6) rollers also. Thus on the contact point, the normal vector
must simultaneously pass through the axis of the workpiece
x ˆ cos b cos a xR ‡ cos a sin b yR sin a zR ; and the axis of the roller.
yˆ sin b xR ‡ cos b yR ‡ dr; The point RR is represented as R0 in the coordinate system
z ˆ cos b sin a xR ‡ sin b sin a yR ‡ cos a zR (7) S0 (Eqs. (6) and (7)); therefore the normal vector N of R0 is
represented as
With the above procedure, the authors built all three rollers
with 1208 separation together with a hollow workpiece and a @R0 @R0
N ˆ N Nx ; Ny ; Nz † ˆ  (8)
cylindrical mandrel. A complete model was established and @u @y

Fig. 5. Relation between roller coordinate system and workpiece coordinate system.
276 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284

Thus the vector from M to R0 can be written as

xm x 0 y 0 z
ˆ ˆ (10)
Nx Ny Nz
so that
f u; y† ˆ Nz y Ny z ˆ 0 (11)
The equation above is called the equation of meshing [16].
From Eq. (11), we have
Ny
yˆz (12)
Nz
If there exists xm ˆ x u†, the corresponding y, y and z can be
obtained. Then the distance from the contact point to the axis
of workpiece is
s
 2
p Nz
rm ˆ 2 2
y ‡ z † ˆ jzj 1‡ (13)
Ny
By connecting each contact points, a contact line can be
obtained. Rotating this contact line around the axis of
workpiece, generates a special pro®le for the workpiece,
which can perfectly ®t to the roller through contact. A
Fig. 6. Mesh system of PSW.
similar procedure can be used to derive a special pro®le
of roller if the deformation of the workpiece is pre-given.
Assume that R0 is the contact point between the roller and
the workpiece and that the corresponding normal vector 2.4. Dual-stream functions
extends through point M on the axis of the workpiece. The
coordinate of M is The upper bound method in which velocity components
are derived from dual-stream functions is used in the fol-
M ˆ ‰ xm 0 0 1 ŠT (9) lowing analysis. The concept that an admissible velocity
®eld for an incompressible body in a three-dimensional
where xm is a constant. space can be represented in terms of two stream functions

Fig. 7. Effective stress±effective strain curve of 304L stainless steel (source: H.E. Boyer, Atlas of Stress±Strain Curves, American Society of Metals, Metals
Park, OH, 1987, p. 301).
 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284 277

Fig. 8. The tube, the roller and the mandrel.

was ®rst proposed by Nagpal [17]. The velocity components distributed [9] and can be written as
in cylindrical coordinates can be written as the cross-product
of the gradients of two stream functions: z y
cˆ (17)
l 2p
V ˆ V Vr ; Vy ; Vz † ˆ rc  rf (14) where l is the moving distance along the z-axis as the hollow
where c and f are the stream functions and Vr ; Vy ; Vz † are tube makes a complete rotation.
the velocity components. Then the velocity components in cylindrical coordinates
The velocity ®eld can be determined by choosing a ¯ow can be derived from Eqs. (14), (15) and (17) as
function that makes the tool boundary a stream function  
[18]. In this paper, the rollers are considered rigid and the v0 r 2 R2i † R2w R2i †R z† @R z†
Vr ˆ ;
material properties of hollow tubes as rigid±plastic and 2 R z†2 R2i †2 rp @z
isotropic. Thus the ¯ow pattern of the deforming material R2w R2i †r v0 R2w R2i
in the r±z-plane can be written as [19] (Fig. 8) Vy ˆ v0 2
; Vz ˆ (18)
R z† 2
Ri †l 2p R z†2 R2i
R2w R2i r 2 R2i
f ˆ v0 (15)
2 R z†2 R2i
where R z† represents the workpiece pro®le in the roll gap
and can be de®ned (Fig. 9) as
R z† ˆ R1 z† ‡ R2 z† (16)
The workpiece in the roll gap is divided equally into three
parts corresponding to the three rollers, Fig. 9 showing the
con®guration of one of such parts. In the ®gure, R1 repre-
sents the deforming pro®le, which is deformed by the roller
and can be obtained from the equation of meshing, whilst R2
denotes the free surface of the deforming workpiece.
The contact (deforming) area can be determined from the
intersection of two rigid bodies. By assuming that both the
roller and the workpiece are rigid, the interfered geometry
can be derived once the workpiece is positioned with respect
to the roller. The range of interference can be represented in
terms of two curves, i.e., CU and CL in Fig. 9. Thus, the
corresponding yCL and yCU can be obtained and can then be
represented as functions of z.
In the y±r-plane, the ¯ow pattern is assumed to be that the
circumferential velocity along the radius direction is linearly Fig. 9. A workpiece in the roll gap.
278 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284

The strain rates derived from Eq. (18) are 3. Discussions of simulation results
 
@Vr v0 R2w R2i †R z† r 2 ‡ R2i † @R z†
e_ rr ˆ ˆ ; 3.1. Contact analysis
@r 2 p R z†2 R2i †2 r 2 @z
 
1 @Vy Vr v0 R2w R2i † r 2 R2i †R z† @R z† The pro®le of a typical PSW roller is described by two
e_ yy ˆ ‡ ˆ ;
r @y r 2 R z†2 R2i †2 r 2 p @z straight lines as shown in Fig. 10(a). Fig. 10(b) shows the
  contact pro®le of the workpiece corresponding to this
@Vz R2w R2i †R z† @R z†
e_ zz ˆ ˆ v0 (19) roller. The inclination and offset of the roller lead to a
@z R z†2 R2i †2 p @z
V-shape pro®le for the workpiece. In reality, the right-side
and, from Eq. (19): part of the V-shape will never occur, instead, a horizontal
e_ rr ‡ e_ yy ‡ e_ zz ˆ 0 (20) straight line will replace it because that represents the part
of workpiece that had undergone deformation. If the cross-
showing that volume constancy is satis®ed and the con- section of the roller is described as in Fig. 10(c), the
structed velocity ®eld is kinematically admissible. generated pro®le of the workpiece will be a straight line
as in Fig. 10(d).
2.5. Upper bound solution The front part of the roller usually will not contact with
the deformed workpiece during rolling. The remaining part
Among all kinematically admissible velocity ®elds, the of the roller, which deforms the workpiece through contact,
actual one minimizes the expression [20] is called the deforming zone [4]. The roller can be arranged
Z  1=2 Z Z in such a way that makes the deformed workpiece remain
 2 1
J ˆ p s0 e_ ij e_ ij dV ‡ tjDvj dS Ti vi dS constrained by the front part of the roller and the workpiece
3 V 2 SV St
will then move forward smoothly before departing from the
(21) roller. This special front part of roller is then called the
The ®rst term on the right-hand side is the internal strain smoothing zone.
energy rate in the deformation zone, whilst the second term Based on this characteristic, the front part pro®le of the
accounts for the shear loss within the material over the roller can be modi®ed by using the equation of meshing, to
surface of velocity discontinuities Gs and the friction loss make the smoothing zone perfectly ®t to the deformed
dissipated on the contact surfaces Gf. It can be represented as workpiece to assure the accurate radius of the workpiece
Z Z Z through the rolling process.
1 m
tjDvG j dS ˆ p s0 jDvGS j dS ‡ p s0 jDvGf j dS
G 3 GS 3 Gf
3.2. Results of finite element analysis
(22)
where DvGS and DvGf are the velocity discontinuities across The vertical cross-section of a deforming tube and its
the boundary surfaces and m denotes the friction factor on con®guration are shown in Figs. 11 and 12, respectively. The
the contact surface. The velocity discontinuity DvGS along workpiece was twisted by the interaction of the rollers and
G1 and G2 can be represented as Dv1 and Dv2 : the mandrel, and thus has spiral marks on its surface. Due to
  the rotation and feed movement brought about by the rollers,
v0 r 2 R2i † R2w R2i †R0 @R L†
Dv1 ˆ Vr jzˆL ˆ the material ¯ow was not uniform between the internal and
2rp R20 R2i †2 @z external part of the tube. The ¯ow velocity on the external
(23) part is faster than that near to the mandrel.
2
  Figs. 13 and 14 are the contours of the effective stress and
v0 r R2i †Rw @R 0†
Dv2 ˆ Vr jzˆ0 ˆ (24) effective strain of the workpiece during the rolling process. It
2rp R2w R2i @z can be seen that the maximum von Mises stress occurs at the
and DvGf along G3 and G4 are written as Dv3 and Dv4 : parts contacting with the rollers. The largest deformation
s
 2 occurs in the part of the tube where the rolling began, as
q v R 2
R 2
@R z†
Dv3 ˆ Vr2 ‡ Vz2 ˆ
0 w i
1‡ indicated by the effective strain contours.
rˆR z† 2p R z†2 R2i @z For this analysis, the wall thickness of the tube was
(25) reduced from 100 mm to ®ve different sizes with the rota-
tional speed of the roller kept at 150 rpm. The inclined
Dv4 ˆ Vz v0 (26) angles were set to 508, which is the typical value for
The last term in Eq. (21) is the additional power to overcome industrial practice. The offset angle of the roller has impor-
the back-pull of the tube, which is written as tant effects upon the tube rolling process, especially on the
Z exit velocity of the deformed workpiece.
Ti vi dS ˆ p R2w R2i †sxb (27) Fig. 15 shows that the exit velocity increased with the
St
magnitude of the offset angle and decreased with respect to
It is assumed that sxb ˆ 0:3s0 [9]. the wall thickness of the deformed tube. In addition, the
 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284 279

Fig. 10. The roller profile and the generated front end profile of the workpiece (unit: mm): (a) typical roller profile; (b) generated workpiece profile; (c) a
roller profile; (d) the workpiece profile generated from (c), where y ˆ 1:57874  10 6 x3 0:00326815x2 ‡ 1:71973x ‡ 90:9101.

Fig. 11. Cross-section of the deforming tube.

280 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284

Fig. 12. Configuration of the deforming tube.

rolling load increased with the offset angle and also rates calculated from the dual-stream functions are dis-
increased with the tube thickness, as shown in Fig. 16. played in Fig. 18 and are compared with the results from
the ®nite element analysis. The tendencies from both ana-
3.3. Results of upper bound approach lyses are very similar, as shown in the ®gure. The energy
dissipation rates all increase with the reduction of cross-
Velocity components obtained from dual-stream func- sectional area.
tions are shown in Fig. 17(a) while Fig. 17(b) shows the The differences between the two methods of analysis
velocity components on the tube surface from the ®nite come mainly from the oversimpli®ed assumptions of the
element analysis. The velocity distribution from the dual- dual-stream functions. In fact, the material ¯ow on the roll
stream functions clearly indicates the rotational move- gap is not uniform, as can be seen in Figs. 11 and 12. The
ment of the workpiece: similar results are found in the circumferential velocity along the radius direction should
®nite element calculation. However, the amount of advan- not be treated as a linear distribution, especially in the
cing movement of the deforming tube is not as obvious as deforming zone. In addition, the lack of consideration of
that obtained from ®nite element analysis. The energy the velocity discontinuities along the interface between the

Fig. 13. Distribution of von Mises stress in the deforming tube (unit: Pa).
C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284 281

Fig. 14. Distribution of effective strain in the deforming tube.

Fig. 15. Relation between the offset angle and the exit velocity of the workpiece.
282 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284

Fig. 16. Relation between the offset angle and the rolling load.

Fig. 17. Velocity field of the roll gap: (a) upper bound approach; (b) finite element analysis.
 C.-K. Shih et al. / Journal of Materials Processing Technology 121 (2002) 273±284 283

Fig. 18. Energy rate.

deforming surface and the free surface in the roll gap also increase the efficiency of the rolling process, but can
leads to deviations from the ®nite element simulation. increase the rolling load and energy dissipation at the
same time.
4. Combining the dual-stream functions and upper bound
4. Conclusions approach provides a feasible way to analyze three-
dimensional metal-forming problems analytically. The
Applying the PSW in the tube rolling process could help results show similar tendencies with those from the
in increasing the working ef®ciency and obtaining econom- finite element method in respect of the energy rate, but
ical bene®ts. This research tried to provide the solutions of with some discrepancies in magnitude.
the problem in the planetary rolling process for tube produc- 5. The model of the dual-stream function from which the
tion by using both the ®nite element method and dual-stream velocity field is derived can be modified further to
functions. The conclusions are as follows: approach the behavior of the real material during
deformation.
1. The geometric model of PSW has been built by using
the coordinate transformation method. In addition, the
finite element simulation model has been reasonably References
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