Slab Analysis of Ring Rolling Assuming

Constant Shear Friction

A. Parvizi, K. Abrinia, and M. Salimi
(Submitted October 3, 2009; in revised form October 30, 2010)
In this article, an analytic solution for ring rolling process based on the slab method theory is presented, in
which the non-uniformity of the normal and shear stresses across the section of the deforming material
throughout the plastic region is considered. The friction factor multiplied by the shear yield strength
(s = mk) is used to present friction between the main roll and the ring. The inuence of the process
parameters such as friction factor, feed speed, main roll rotational speed, and radii of the main roll and
mandrel on process outputs is investigated. Complete expressions for the ring rolling pressure, force, and
torque are obtained, and the position of neutral point is predicted. Comparison of the analytic results of this
model with the experimental results of other investigators and FEM analysis show that they are in good
agreement.
Keywords analytic solution, pressure distribution, ring rolling,
slab method
1. Introduction
Ring rolling process is widely used to produce annular
seamless rings from different materials in a wide range of
dimensions. It is most often used to form steel rings, but has
also been used for aluminum alloys, titanium alloys, composite
metal rings, and polymers. These rings are manufactured from a
bulk pierced billet through a complex sequence of radial or
axial-radial rolling operations carried out in hot and cold
working conditions (Ref 1). Large rings like those used in
power generation plants, aircraft engines, or large cylindrical
vessels are usually manufactured by hot ring rolling and small
ones such as inner and outer bearing races by cold ring rolling.
The advantages of ring rolling include short production
time, efcient use of material, uniform quality, smooth surfaces,
and close geometrical tolerances which lead to considerable
cost savings in terms of energy and labor. The main challenges
in ring rolling technology are categorized into three branches:
achievement of dimensional accuracy, surface quality, and a
awless product; achievement of different proles from simple
pre-forms, any material, right rst time die design; and
achievement of low total cost, low scrap, fast cycle times, low
labor and energy input; little post-processing (Ref 2).
Over the years, some analytic, physical, and numerical
modeling methods including experimental formulae (Ref 3),
volume mapping (Ref 4), slip line elds method (Ref 5, 6),
upper bound method (Ref 7, 8), and FE analysis (Ref 9, 10) are
used by many researchers to predict the nal results of ring
rolling processes. The experimental formulae are generally
limited to each class of ring rolling operations with specic
A. Parvizi and K. Abrinia, School of Mechanical Engineering,
College of Engineering, University of Tehran, P. O. Box 11365-4563,
Tehran, Iran; and M. Salimi, Department of Mechanical Engineering,
Isfahan University of Technology, P. O. Box 84154, Isfahan, Iran.
Contact e-mails: aliparvizi@ut.ac.ir and ali.parvizi@yahoo.com.
List of symbols
h Variable ring thickness at the roll gap, m
h
1
, h
2
Upper and lower element heights with respect to
horizontal axis, respectively, m
h
i
, h
o
Ring thicknesses at entry and exit of the roll gap,
respectively, m
Dh Decrease in ring thickness
k Mean yield shear stress of the ring, pa
m Friction factor of the main roll
n Rotational speed of main roll, rad/s
r Reduction in thickness, %
v Feed speed (linear speed of mandrel), m/s
w Width of the ring, m
x Horizontal distance from the exit point in the roll
gap, m
x
n
Neutral point of main roll, m
y Vertical distance from the centerline of the ring, m
F
1
, F
2
Main roll and mandrel rolling forces per unit width,
respectively, N/m
L The contact length, m
P
1
, P
2
Main roll and mandrel rolling pressures,
respectively, pa
P
1,II
, P
1,II
Main roll pressures at the zones I and II,
respectively, pa
R
1
, R
2
Radii of main roll and mandrel, respectively, m
R
3
Inner radius of ring, m
R
eq
Equivalent work roll radius
R
m
Mean radius of ring, m
T Rolling torque of the ring rolling process, N m
r
1
, r
2
Normal stresses on vertical side of element at the top
and bottom of the slab, respectively, pa
s Shear stress on vertical side of element, pa
JMEPEG (2011) 20:15051511 ASM International
DOI: 10.1007/s11665-010-9824-9 1059-9495/$19.00
Journal of Materials Engineering and Performance Volume 20(9) December 20111505
materials and processing parameters; therefore, results cannot
be generalized for other ring rolling operations. The volume
mapping is capable of predicting the diametric growth in closed
pass ring rolling operations but cannot predict roll forces or
torques. The upper bound involves nding a kinematically
admissible velocity eld, which leads to a minimum forming
energy. However, it is difcult to express an admissible velocity
eld for problems involving complex shapes and general
material ow especially in transient conditions (Ref 11).
As the ring rolling process is non-steady state throughout
and a large number of ring rotations are required to nish the
product, it is different from the conventional rolling process.
Therefore, the number of increments requires in numerical
simulation of the process normally is several times that required
for a regular metal-forming application. Although the FEM
may be used to simulate this process accurately, it, however,
involves excessive computation time and convergence prob-
lems due to the highly nonlinear nature of the process and the
existence of complicated contact conditions (Ref 12).
The slab method can predict rapidly the rolling force and
torque, and a large amount of CPU time can be saved.
Therefore, this method is suitable for the ring rolling industry.
Although numerous studies have used the slab method to
investigate metal-forming processes like symmetric and asym-
metric rolling characteristics (Ref 12-17), but this method has
never been applied to ring rolling process. The major reason
may have been the non-steady state characteristic throughout
this process. In the previous studies on at rolling (Ref 12), the
slab method has been mainly applied with uniform distribution
in shear stress. In the authors previous study (Ref 13), this
approach has been modied to non-uniform and linear
distribution of stress while the constant shear friction was
assumed. This approach has been adopted for the present study,
applying the slab method to ring rolling process for the rst
time.
In this article, a solution based on the slab method of
analysis is presented for ring rolling process. At rst, a
mathematical model for the stress eld acting on a vertical slab
in the roll gap is assumed in which the non-uniformity of the
normal and shear stresses across the section of the deforming
material is considered. Next, imposing the force balance on the
slab with a differential thickness, one-dimensional differential
equation for pressure at the roll gap is obtained. Finally,
imposing relevant boundary conditions on the differential
equation provides an analytic solution for the rolling pressure.
In the analytic solution, a variety of factors for ring rolling
process such as friction factor, feed speed (mandrel speed),
main roll rotational speed, and radii of the main roll and
mandrel have been used to investigate the behavior of the
process. Based on this analysis, complete expressions for the
ring rolling pressure, force, and torque are obtained, and
the position of neutral point is determined.
2. Mathematical Models
The ring cross section at roll gap is rectangular with uniform
unit thickness and heights, h
i
and h
o
, at entry and exit,
respectively. The horizontal distance from the exit point in the
roll gap is taken as the x-axis, and the origin is its intersection
with the normal plane passing by the main roll and mandrel
centers (Fig. 1).
The deforming region is dened as shown in Fig. 2 bounded
by two at surfaces, at the entry and exit and the surfaces of the
main roll and the mandrel.
In the development of the mathematical model, it was
assumed that the rolls are rigid and the material being rolled is
rigid perfectly plastic. The plastic deformation occurs under
plane strain conditions, so that material spread in transverse
direction is ignored. The friction factor multiplied by the shear
Fig. 1 Schematic illustration of ring rolling process
Fig. 2 Ring rolling geometry
1506Volume 20(9) December 2011 Journal of Materials Engineering and Performance
yield strength (s = mk) is used to represent the interface friction
between the main roll and the ring. The friction factor between
the main roll and the ring is assumed to be constant along the
contact length. As mandrel is free to rotate, it cannot afford any
frictional moments. For simplicity, the friction at the interface
between the mandrel and ring is neglected. The contact length
is small as compared with the roll circumference and can be
obtained from the following relationship (Ref 18):
L R
2
1

R
1
R
2
h
o
h
i

2
R
2
1
R
2
2
2 R
1
R
2
h
o
h
i

_ _
2
_
_
_
_
1
2
Eq 1
where R
1
and R
2
are the radii of main roll and mandrel,
respectively. Identifying the ring rolling loading effects is the
rst step in the development of equation of equilibrium for a
differential vertical element in the roll gap. In passing through
the roll gap (Fig. 2), the ring thickness is steadily reduced,
and the ring radius and velocity progressively increased from
entry to exit. The main roll surface speed is equal to an inter-
mediate value between the ring entry and exit speeds. The
point at which the ring and the main roll surface periphe-
ral speeds are equal is called a neutral point. The ring is
advanced by the frictional force acting toward the neutral
point at entry and opposed by the frictional force acting at
the exit region. Thus, the plastic deformation zone can be
divided into two distinct regions: zone I from the exit to the
neutral point; and zone II from the neutral point to the entry.
Subscripts 1 and 2 are designated to refer to quantities at the
main roll and mandrel, respectively. From simple geometry,
h
1
h
0
=2 x
2
_
2R
1
h
2
h
0
=2 x
2
_
2R
2
Eq 2
where h
1
and h
2
are the upper and lower element heights with
respect to horizontal axis, respectively. Hence in general
h h
1
h
2
h
0
x
2
_
R
eq
; R
eq
2R
1
R
2
= R
1
R
2

Eq 3
where h is variable ring thickness at the roll gap, and R
eq
is
equivalent work roll radius. If the slip between the main roll
and the ring workpiece is neglected, then the circumference of
the outer surface of the ring workpiece in one rotation is equal
to the perimeter of the driven roll. Considering this fact, the
decrease in ring thickness in any rotation is given by (Ref 19)
Dh h
i
h
o

v
n

R
3
h
o

R
1
Eq 4
where v and n are the feed and rotational speeds of main roll,
respectively, and R
3
is inner radius of ring. The equivalent
stress eld chosen for this purpose is shown in Fig. 3. This
gure includes the non-uniformities of the stresses in the
through thickness direction. The shear stresses on the vertical
sides of the differential elements increase from zero at the
bottom to the maximum values at the top which are equal to
the ring-main roll interface friction. The normal and shear
stresses on the vertical sides of the slab are assumed to be
linearly distributed.
2.1 Ring Rolling Pressure
From the horizontal and vertical force equilibria based on
the stress eld of Fig. 3 and noting Fig. 1 and 2, the following
relationships are obtained:
r
1
r
2

dh
2
dr
1
dr
2

h
2
P
1
x
R
1
_ _
P
2
x
R
2
_ _
mk
_ _
dx 0 (Eq 5)
P
1
P
2
mk
1
R
1

1
R
eq
_ _
x 0 Eq 6
where r
1
and r
2
are normal stresses on the vertical side of
element at the top and bottom of the slab respectively, P
1
and
P
2
are the main roll and mandrel rolling pressures, respec-
tively, m is the friction factor of the main roll, and k is the
mean yield shear stress of the ring. The plus and minus signs
in Eq 5 and 6 are designated to the zones I and II, respec-
tively. The plane strain condition at any point in the plastic
region gives
s
xz
s
yz
0; e
z
0 Eq 7
where s
xz
and s
yz
are the shear stresses on the vertical side of
the element, and e
z
is the normal strain in the z direction.
Imposing the plane strain condition on the ow rule
e
z
1=E r
z
1=2 r
x
r
y
_ _ _ _ _ _
, results in
r
z
r
x
r
y
_ __
2 Eq 8
Using Eq 7 and 8 gives a two-dimensional Von-Mises yield
criterion as follows [13]:
r
x
r
y
_ __
2

k
2
s
2
xy
_
Eq 9
Considering the method of the authors previous study [16]
and assuming small contact angle of the ring and the rolls,
there are three different loading conditions for elements in the
vertical slabs of Fig. 2. There are two elements on the top of
any vertical slab and an element at the bottom. These ele-
ments are shown in Fig. 4.
Using the loading conditions of Fig. 3 for Eq 9, the yield
criteria for the top and bottom elements on the vertical slabs
take the following form:
r
1
P
1
2k

1 m
2
p
r
2
P
2
2k Eq 10
Substituting Eq 10 and 6 into 5 and solving the subsequent
one-dimensional differential equation, the main roll pressure
distribution in zones I and II can be expressed as
Fig. 3 Stress eld acting on a vertical element at (a) entry zone II,
and (b) exit zone I
Journal of Materials Engineering and Performance Volume 20(9) December 20111507
P
1
x A
1
x A
2
Ln
x
2
R
eq
h
o
_ _
A
3
arctan
x

h
o
R
eq
_
_ _
C
1
Eq 11
where C
1
is a constant which will be derived later and
A
1
mk
1
R
1

1
R
eq
_ _
3
2

R
eq
R
2
_ _
A
2
k 1

1 m
2
p
_ _
A
3
mk
1
R
1

1
R
eq
_ _
1
R
eq
R
2
_ _

1
h
o
_ _

h
o
R
eq
_
Eq 12
where plus and minus signs in A
1
and A
3
are designated to
zones I and II, respectively. It is assumed that no axial forces
are applied at the exit side of the roll gap, and so the bound-
ary conditions on the entry and exit are

F
x
r
1
r
2

dh
2
0 at x 0 and x L Eq 13
Substituting Eq 6 and 11 into 10 and using the subsequent
results with Eq 13, the constants C
1
for zones I and II are
obtained as follows:
Zone I
C
1;I
k 1

1 m
2
p
_ _
1 Ln h
o
0 x x
n

Zone II
C
1;II
A
1
L A
2
1 Ln
L
2
R
eq
h
o
_ _ _ _
A
3
arctan
L

h
o
R
eq
_
_ _

1
2
mkL
1
R
1

1
R
eq
_ _
x
n
x L (Eq 14)
where A
1
and A
3
(with plus sign), and A
2
are given by (12).
2.2 Ring Rolling Force
If the contact angle of the rolls and that of the ring is
assumed to be small, then the rolling force is given by
F
1

_
x
n
0
P
1;I
mk
x
R
1
_ _
dx
_
L
xn
P
1;II
mk
x
R
1
_ _
dx
Eq 15
where P
1,I
and P
1,II
are the main roll pressures at the zone I
and II, respectively, and x
n
is the neutral point where the
pressures at zones I and II become equal. Substituting Eq 11
into 15 and integrating the terms, the rolling force per unit
width of the ring is obtained as follows:
F
1
A
1

mk
R
1
_ _
x
2
n

L
2
2
_ _
A
2
_
L Ln
L
2
R
eq
h
o
_ _
2
_ _
2

h
o
R
eq
_
arctan
L

h
o
R
eq
_
_ __
A
3
_
1
2

h
o
R
eq
_
:Ln 1
L
2
h
o
R
eq
_ _
L arctan
L

h
o
R
eq
_
_ _
2x
n
arctan
x
n

h
o
R
eq
_
_ _

h
o
R
eq
_
Ln 1
x
2
n
h
o
R
eq
_ _
_
C
1;I
x
n
C
1;II
L x
n
(Eq 16)
where A
1
and A
3
(with plus sign), and A
2
are given by (12)
and C
1,I
and C
1,II
given by (14), respectively.
2.3 Ring rolling Torque
The rolling torque is an important parameter which deter-
mines the energy consumption of the ring rolling mill. As
mandrel is idler, the total torque is exerted on the ring
individually by the main roll. The ring rolling torque per unit
width of the ring can be calculated by integrating the moment
of the frictional shear force along the arc length of contact
around the roll axis as follows:
T
1
R
1

_
x
n
0
mk dx
_
L
xn
mk dx
_
_
_
_
R
1
mk
_
x
n
0
dx
_
L
xn
dx
_
_
_
_
R
1
mk L 2x
n
(Eq 17)
3. Results and Discussion
From the mathematical model presented above, the effects
of factors on the ring rolling process are investigated. The effect
of friction factor on rolling pressure distributions for the main
roll and the mandrel are considered in Fig. 5. It is seen that with
increasing friction factor, the specic rolling pressure of the
main roll and the mandrel decrease with a positive rate and the
neutral point approaches the entry. Furthermore, a jump in
mandrel pressure distribution at the neutral point is noticeable.
An important physical reason for this is the direction of friction
force at the main roll and ring interface which is suddenly
changed at this point. Therefore, the magnitude of the mandrel
pressure is consequently changed abruptly to hold the equilib-
rium condition in both the vertical and horizontal directions.
The normal stresses on the vertical side of element at the top
and bottom of the slab for various friction factors are shown in
Fig. 6. Unlike the results shown in Fig. 3, the normal stresses at
the top and bottom of the slab have negative and positive signs,
respectively. The magnitudes of normal stresses at both the top
and bottom of the slab increase with increasing friction factor.
Fig. 4 Elements on the (a) top of the vertical slab at entry zone II,
(b) top of the vertical slab at exit zone I, and (c) bottom of the verti-
cal slab
1508Volume 20(9) December 2011 Journal of Materials Engineering and Performance
Furthermore, from entry and exit to the neutral point, the
magnitudes of normal stresses at the top increase and at the
bottom decrease. An uneven residual stress distribution at the
resulting rings is one of the main problems which have negative
economical implications. From this point of view, the lower the
friction factor, the better the outcome. According to Eq 9, since
the normal stress at the bottom of the slab depends directly on
the mandrel pressure, there is also a jump in its distribution
along the contact length.
Fig. 6 Normal stresses on vertical side of element (a) at the top of
the slab and (b) at the bottom of the slab for various friction factors
R
1
100 mm; R
2
50 mm; R
3
70 mm; n 2 rev/s; h
0
10 mm;
v 1 mm/s
Fig. 7 The specic ring rolling pressure of (a) the main roll,
(b) the mandrel for various feed speeds R
1
100 mm; R
2

50 mm; R
3
70 mm; n 2rev/s; h
o
10 mm; m 0:6
Fig. 8 The specic ring rolling (a) force and (b) torque for various
feed speeds with respect to the current mean radius of ring R
1

275 mm; R
2
45 mm; R
3
60 mm; R
m
83:75 mm; n 0:78 rev/s;
m 0:4; h
o
47:5 mm
Fig. 5 The specic ring rolling pressure of (a) the main roll and
(b) the mandrel for various friction factors R
1
100 mm; R
2

50 mm; R
3
70 mm; n 2 rev/s; h
o
10 mm; v 1 mm/s
Journal of Materials Engineering and Performance Volume 20(9) December 20111509
In Fig. 7, the effect of the feed speed on the rolling pressure
distributions for the main roll and the mandrel is observed. It is
seen that the specic rolling pressure for the main roll and the
mandrel increase with increasing feed speed, and the neutral
point also approaches the entry side. In addition, a jump in
mandrel pressure distribution at the neutral point is noticeable.
The force and torque with respect to the current mean radius
of ring are shown in Fig. 8 for different values of feed speeds.
Some other analytic results at the end of process are illustrated
in Table 1. For any value of the feed speed, the force and torque
increase slowly with increase in the ring mean radius. The slope
of the curve at higher feed speed is larger than that at lower
ones. It could also be seen that the force and torque increase
with increasing feed speed, which indicates that a much higher
force and torque are necessary to maintain the ring rolling
process in the cases of relatively high pressing velocity and
large ring mean radius.
In Fig. 9, the results of the slab method analysis and those of
experimental studies by Ryoo and Yang (Ref 18) are compared.
The effects of feed speed and radius of the mandrel on the ring
rolling torque are evaluated in parts (a) and (b), respectively. It
could be seen that the values for torque obtained from the slab
method are in good agreement with experimental ones. For
different feed speeds, the differences between two values
increase with increasing ring mean radius, while the maximum
deviations at 0.9 and 0.7 mm/s feed speeds are 12.5 and 9.8%,
respectively. According to the other case, the slab method has
more accurate results for the mandrel with the bigger radius. It
means that the maximum differences between theoretical and
experimental results for 45 and 55 mm mandrel radii are 19 and
6%, respectively.
Figure 10 shows results for ring rolling force and torque
with respect to thickness reduction from the present modied
slab method analysis and the FEM simulation (Ref 20). Again,
it could be seen that the values for force and torque obtained
from the present method are in good agreement with FEM
results.
4. Conclusions
Using different feed speeds and friction factors, ring rolling
process was investigated by applying the slab method. It was
found that with increasing friction factor or decreasing feed
speed, specic rolling pressure of main roll and mandrel
decrease, and the neutral points approach the entry and exit,
respectively, but uneven residual stresses increase. Further-
Table 1 Analytic results of ring rolling process for different feed speeds
Feed speed,
mm/s
Number
of rotations
Contact
length, mm
Neutral point,
mm
Reduction in
thickness, %
Mean ring
radius, mm
0.5 90 4.5 1.7 35 120.2
0.7 80 6.1 2.3 49.5 123.8
0.9 70 7.8 3.0 61.3 216.25
Fig. 9 Comparison of results for slab analysis and experimental
studies. Effect of (a) feed speed and (b) radius of the mandrel on the
ring rolling torque. R
1
275 mm; R
2
45 mm; R
3
60 mm; v
0:7 mm/s; n 0:78 rev/s; h
0
47:5 mm; m 0:4; w 52 mm
Fig. 10 Comparison of the specic ring rolling (a) force, (b) torque
with respect to thickness reduction for modied slab analysis
and FEM simulation R
1
114:3 mm; R
2
34:9 mm; R
3
38:1 mm;
n 0:52 rev/s; m 0:3; v 0:297 mm/s; h
o
25:4 mm Material:
aluminum - alloy HE 30 (0:8% Mg, 1% Si, 0:7% Mn)
1510Volume 20(9) December 2011 Journal of Materials Engineering and Performance
more, a jump in mandrel pressure distribution as a result of
change in the direction of frictional force at the neutral point is
noticeable. For any value of the feed speed, the force and
torque required for ring rolling process increase slowly with the
ring mean radius. The accuracy of this analysis was veried
using reference (Ref 18) experimental data and the authors
FEM analysis (Ref 20).
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