ANALYSIS OF STRIP RESIDUAL CURVATURES IN ANTI
CROSSBOW CASSETTE IN TENSION LEVELING PROCESS
M. Jamshidian
1
, A. Beheshti
1
, A. Sadeghi Dolatabadi
2
, M. Olfat Nia
1
, M. Salimi
1
1
Department of Mechanical Engineering, Isfahan University of Technology, P.O. Box 84154,
Isfahan, Iran
2
Mobarakeh Steel Company, Isfahan 84815161, Iran
ABSTRACT
Tension leveling is a process used in steel
industry in order to remove any shape
defects present in temper rolled strip.
Although strip bad shapes come into sight
as edge waves, centre buckles and quarter
buckles can be corrected giving the strip a
suitable amount of elongation in work rolls
cassette, such defects of socalled crossbow
and coil set known as residual curvatures
are inevitable because in the process the
strip undergoes repeated bending. These
undesired curvatures can be corrected in
anticrossbow and anticoil set cassettes,
giving the strip an optimum amount of
curvature. In this paper a theoretical
analysis based on incremental theory of
plasticity and considering mixed mode
hardening for material behavior has been
employed in order to study the effect of
primary process parameters namely work
curvature and tension on strip residual
curvatures coming out of the work rolls
cassette and then to analyze variation of
these curvatures with respect to second
bending curvature in anticrossbow
cassette. Review of the results presents a
method for determining the optimal work
curvature in anticrossbow cassette to
remove residual curvatures especially
transverse residual curvature or crossbow.
1. INTRODUCTION
Tension levelling is an indispensable
process to improve the flatness of metallic
strip products in order to satisfy the
stringent requirements of the costumers.
Tension levelling is a metal forming
technique to produce perfectly flat metallic
strips [1]. The tension levelling process is
performed to elongate the strip plastically in
a combination of tensile and bending strain
so that all longitudinal fibres in the strip are
of approximately the same length. The
tension levelling process is designed
primarily to remove shape defects resulting
from unequal elongation over the width of
rolled strips such as wavy edges, side
buckles and centre buckles. This
improvement occurs in work rolls cassettes
in a typical tension leveller as shown in
Figure 1. The repeated bending and
stretching in this step however, tends to
induce subsequent curl or coil set and gutter
or crossbow which are kinds of global
shape defects. Crossbow and coil set are
usually corrected in the second stage of the
process when the strip passes through anti
crossbow and anticoil set cassettes
respectively.
Figure 1. A typical tension leveller machine
Because of its crucial role in delivering
perfect flat strips to customers, tension
levelling has been studied by many
researchers. Misaka and Masui studied
analytically the mechanism of the process
for curling and guttering after repeated
bending and developed a method for
calculating the curl and gutter [2]. The
amount of strip elongation, power loss and
tension that occurs in the process were
studied by Patula [3]. Kawaguchi studied
numerically the actual strip curvature on a
roll and the strip deflection between rolls in
the process [4]. Hattori et al. developed a
mathematical model to predict the curl and
crossbow induced in a strip during the
process[5]. Kajihara et al. developed an
analytical method of the work radius,
elongation, curling and guttering of strips
[6]. Hira et al. introduced theoretical
analysis of deformation behaviour under
repeated bending in strip processing lines
[7]. Hibino devised a practical equation for
estimating the levelling strain from
experimental results and revealed that the
extreme point of the strain did not lie at the
contact point between the sheet and the roll
[8]. Yoshida and Urabe carried out a
computer aided process design for tension
levelling with finite element analysis
involved with a sophisticated constitutive
model of cyclic plasticity [9]. Morris et al.
performed a series of low cycle fatigue tests
under constant strain amplitude control
[10]. The results from their experiments
suggest that a mixed mode hardening
process is present in tension levelling, with
the application of a Bauschinger effect. Huh
et al. performed a simulationbased process
design for the tension levelling of metallic
strips based on the elasticplastic finite
element analysis [11]. As in tension
levelling process the material undergoes
repeated bending unbending deformation, it
experiences cyclic tension compression
loadings under which the effect of
Bauschinger effect dominates. Meanwhile
the effect of isotropic hardening in cyclic
loading affects the material behaviour as
well. In this study a theoretical analysis
based on incremental theory of plasticity
and considering mixed mode hardening for
material behaviour has been employed in
order to study the effect of primary process
parameters on strip residual curvatures and
then to analyze variation of these curvatures
with respect to second bending curvature in
anticrossbow cassette. Finite element
simulations have also been employed to
verify the proposed solution. The Results
are used to determine the optimal work
curvature in anticrossbow cassette that can
remove residual curvatures especially
transverse residual curvature or crossbow.
2. THEORETICAL ANALYSIS
To simplify and simulate the complicated
behavior of strip in tension leveling
process, the following assumptions are
considered in the present analysis:
1. The strip elongates under plane strain
conditions. Transverse strain is neglected
because the strip width is sufficiently large
compared to the strip thickness.
2. The stress in thickness direction is
ignored with respect to large in plane stress
values, that is plane stress condition in
thickness direction dominates.
3. The shear stresses and shear strains in
longitudinal (rolling) and transverse
directions are neglected.
4. The applied tension on strip is assumed
to be constant as strip passes through roll
gaps i.e. longitudinal tension on strip is
constant during bending over free rotating
rolls.
5. The tensile stress is applied uniformly
across the strip width.
6. The material behaviour is considered to
be in mixed or combined hardening mode
i.e. both isotropic and kinematic hardening
modes are deemed to be present as material
deforms plastically.
From the assumptions 2 and 3 and Mises
yield criterion, the equivalent stress ߪത is
given by:
ߪത
ൌ
ඨ
3
2
ቂ
ሺ
ܵ
௫
െߙ
௫
ሻ
ଶ
൫ܵ
௬
െߙ
௬
൯
ଶ
ሺ
ܵ
௭
െߙ
௭
ሻ
ଶ
ቃ
(1)
Where: ݔ,ݕ,ݖ – subscripts indicating
longitudinal or rolling, transverse and
thickness direction of the strip, respectively,
ܵ
௫
,ܵ
௬
,ܵ
௭
– Deviatoric stress components in
ݔ,ݕ,ݖ directions, respectively, ߙ
௫
,ߙ
௬
,ߙ
௭
–
Back stress components in ݔ,ݕ,ݖ directions,
respectively. The equivalent plastic strain
increment ݀ߝҧ
is expressed by the
following expression:
݀ߝ
ҧ
ൌ
ඨ
2
3
ቀ݀߳
௫
ଶ
݀߳
௬
ଶ
݀߳
௭
ଶ
ቁ
(2)
Where: ݀߳
௫
, ݀߳
௫
, ݀߳
௫
– Plastic strain
increments in ݔ,ݕ,ݖ directions,
respectively. It is assumed that the
relationship between stress and strain is
governed by the Hooke’s law for elastic
deformation. Then the following expression
relates stress increments to strain
increments while the deformation occurs
elastically:
ቐ
݀߳
௫
݀߳
௬
݀߳
௭
ቑ
ൌ
ሾ
ܦ
ሿ
ቐ
݀ߪ
௫
dߪ
௬
݀ߪ
௭
ቑ
(3)
Where: ݀߳
௫
,݀߳
௬
,݀߳
௭
– Total strain
increments in ݔ,ݕ,ݖ directions,
respectively, ݀ߪ
௫
,݀ߪ
௬
,݀ߪ
௭
– Stress
increments in ݔ,ݕ,ݖ directions,
respectively. The matrix
ሾ
ܦ
ሿ
is defined as:
ሾ
ܦ
ሿ
ൌ
1
ܧ
1 െߴ െߴ
െߴ 1 െߴ
െߴ െߴ 1
൩
(4)
Where: ߴ – Poisson’s ratio, ܧ – Elastic
modulus. In case of plastic deformation, the
associated flow rule together with Mises
yield criterion as the yield function is used
to relate stress and strain increments. The
yield function is defined by the following
equation:
݂
ൌ
3
2
ሺሼ
ܵ
ሽ
െ
ሼ
ߙ
ሽሻ
்
ሺሼ
ܵ
ሽ
െ
ሼ
ߙ
ሽሻ
െߪ
ଶ
ൌ 0
(5)
Where: ߪ
– Current yield stress or flow
stress.
ሼ
ܵ
ሽ
and
ሼ
ߙ
ሽ
are defined as:
ሼ
ܵ
ሽ
ൌ
ቐ
ܵ
௫
ܵ
௬
ܵ
௭
ቑ
(6)
ሼ
ߙ
ሽ
ൌ ൝
ߙ
௫
ߙ
௬
ߙ
௭
ൡ
(7)
The associated flow rule using the above
yield function is:
ቐ
݀߳
௫
୮
݀߳
௬
݀߳
௭
ቑ
ൌ ݀ߣ ൜
߲݂
߲
ሼ
ߪ
ሽ
ൠ
(8)
Where: ݀ߣ – Constant coefficient of
associated flow rule. ቄ
డ
డ
ሼ
ఙ
ሽ
ቅ is defined as:
൜
߲݂
߲
ሼ
ߪ
ሽ
ൠ ൌ
ቐ
߲݂ ߲ߪ
௫
⁄
߲݂ ߲ߪ
௬
⁄
߲݂ ߲ߪ
௭
⁄
ቑ
(9)
In order to relate stress and strain
increments during plastic deformation, an
equation to evaluate the yield locus
movement in terms of plastic deformation is
essential. The simplest equation is Prager’s
linear kinematic hardening rule that for our
case is defined as [12]:
ቐ
݀ߙ
௫
݀ߙ
௬
݀ߙ
௭
ቑ
ൌ ܥ
ቐ
݀߳
௫
݀߳
௬
݀߳
௭
ቑ
(10)
Where: ݀ߙ
௫
,݀ߙ
௬
,݀ߙ
௭
– Back stress
increments in ݔ,ݕ,ݖ directions,
respectively, ܥ – kinematic hardening
modulus. Coefficient ܥ is a scalar that
indicates the kinematic hardening rate or
the rate at which yield locus moves during
the development of plastic deformation. If
the relation between ߪത and ݀ߝҧ
is known as:
σ
ഥ
ൌ
݂
൬
න݀ߝ
ҧ
൰
(11)
Then H, the tangent modulus or isotropic
hardening modulus will be defined as:
ܪ ൌ
݀ߪ
ത
݀ߝ
ҧ
(12)
In the present study the isotropic hardening
is approximated by the exponential law as
the following [13]:
σ
ൌ ߪ
ܳ
ஶ
൫1 െ݁
ିఌ
ത
൯
(13)
Where: ߪ
– Yield stress at zero plastic
strain, ܳ
ஶ
– Material parameter indicating
the maximum change in the size of yield
surface, ܾ – Material parameter that defines
the rate at which size of the yield surface
changes as plastic straining develops, ߝҧ
–
Accumulated equivalent plastic strain.
Differentiating equation (13) gives:
ܪ ൌ ܳ
ஶ
ܾ݁
ିఌ
ത
(14)
Finally doing some algebra, the following
equation will be obtained as the relationship
between stress increments and total strain
increments during plastic deformation:
ሾ
ܫ
ሿ
െ
3ܥ
2ܪߪ
ത
ሺሼ
ܵ
ሽ
െ
ሼ
ߙ
ሽሻ
൜
߲ߪ
ത
߲
ሼ
ߙ
ሽ
ൠ
்
ቐ
݀߳
௫
݀߳
௬
݀߳
௭
ቑ
ൌ
ሾ
ܦ
ሿ
3
2ܪߪത
ሺሼ
ܵ
ሽ
െ
ሼ
ߙ
ሽሻ
ቆ൜
߲ߪത
߲
ሼ
ߪ
ሽ
ൠ
்
െܥ ൜
߲ߪ
ത
߲
ሼ
ߙ
ሽ
ൠ
்
ሾ
ܦ
ሿ
ቇ
൩
ቐ
݀ߪ
௫
݀σ
௬
݀ߪ
௭
ቑ
(15)
Where
ሾ
I
ሿ
is the three dimensional identity
matrix and ቄ
డఙ
ഥ
డ
ሼ
ఈ
ሽ
ቅ and ቄ
డఙ
ഥ
డ
ሼ
ఙ
ሽ
ቅ are defined as:
൜
߲ߪ
ത
߲
ሼ
ߙ
ሽ
ൠ ൌ
ቐ
߲ߪ
ത
߲ߙ
௫
⁄
߲ߪ
ത
߲ߙ
௬
⁄
߲ߪ
ത
߲ߙ
௭
⁄
ቑ
(16)
൜
߲ߪ
ത
߲
ሼ
ߪ
ሽ
ൠ ൌ
ቐ
߲ߪ
ത
߲ߪ
௫
⁄
߲ߪ
ത
߲ߪ
௬
⁄
߲ߪ
ത
߲ߪ
௭
⁄
ቑ
(17)
In Eq. (15) the transverse strain increment
dԖ
୷
and the stress increment in thickness
direction dσ
are set to zero according to
assumptions 1 and 2 respectively. Therefore
if strain increment ݀߳
௫
is known at a given
time, the incremental stresses ݀ߪ
௫
and ݀ߪ
௬
and strain increment ݀߳
௭
can be calculated
thorough some computational efforts. The
method used for identification of elastic and
plastic boundary and method of calculating
reverse direction stress in this study is
similar to what has been proposed by Hira
et al. [7]. Generally the procedure involves
strain increment decomposition into elastic
and plastic parts. When through thickness
distributions of stresses ߪ
௫
and ߪ
௬
are
calculated as above, the bending moments
in longitudinal and transverse directions,
ܯ
௫
and ܯ
௬
are calculated by integration.
After successive bending and unbending the
strip is released or is unloaded. Unloading
means that the integrated longitudinal stress
through thickness ߪ
்
and the longitudinal
and transverse bending moments, ܯ
௫
and
ܯ
௬
, are reduced to zero. ߪ
௫
and ߪ
௬
at
ߪ
்
ൌ 0 are first calculated by varying the
݀߳
௫
in the middle of the strip thickness so
as to set ߪ
்
to zero. Subsequently,
calculations are made to reduce the bending
moments to zero. When the curvature of the
strip is zero under loading of bending
moments ܯ
௫
and ܯ
௬
, the residual
longitudinal and transverse curvatures after
unloading, ߢ
௫
and ߢ
௬
are given by the
following equations, respectively:
ߢ
௫
ൌ
ܯ
௫
െߥܯ
௬
ܦ
ሺ
1 െߥ
ଶ
ሻ
(18)
ߢ
௬
ൌ
ܯ
௬
െߥܯ
௫
ܦ
ሺ
1 െߥ
ଶ
ሻ
(19)
Where: t – strip thickness. ܦ is a constant
being defined by the following equation:
ܦ ൌ
ܧݐ
ଷ
12
ሺ
1 െߥ
ଶ
ሻ
(20)
Since the present constitute equations for
strip behaviour are in terms of longitudinal
strain increment ݀߳
௫
, here an equation is
needed to calculate through thickness
b
endin
g
ݐ is
b
longitu
d
longitu
d
middle
by:
߳
௫
ൌ ߢ
ߟ
Using
increm
e
b
een d
e
b
ehavi
o
the pur
p
present
e
previo
u
model
unbend
i
simulat
e
ABAQ
U
b
e elas
hardeni
n
b
revity
present
e
present
Figure
2
distrib
u
exampl
e
distrib
u
like Fi
g
agreem
e
and acc
u
Figure
2
strain p
r
solutio
n
Distancefrommiddlefibre(m)
g
strain. W
h
b
ent to a
d
inal strai
n
d
inal strai
n
fibre in th
i
ߟ
߳
above e
q
e
ntal theor
y
e
veloped i
n
o
ur in tensi
o
p
ose of co
n
e
d here;
d
u
s works [2
,
presented
i
ng proces
s
e
d using t
h
U
S [13]. T
h
ticplastic
n
g behavi
the sim
u
ed
here;
d
authors’
2
shows th
e
u
tion afte
r
s
e
of the res
u
u
tion result
s
g
ure 2 w
e
e
nt, which
u
racy of b
o
2
. Compari
s
r
ofile from
n
s
Equiv
a
h
en a strip
w
curvature
n
in middl
e
n
߳
௫
at a d
i
i
ckness dir
e
q
uation t
o
y
of plastici
n
order to
o
n levellin
g
n
ciseness
d
d
etails can
,
7]. In orde
here, the
s
of steel
h
e finite el
e
h
e steel w
a
with isotr
o
our. For
u
lation de
t
d
etails can
previous
e
equivalen
t
s
traightenin
u
lts. All St
r
s
from bo
t
e
re seen t
o
demonstrat
e
o
th approac
h
s
on of Equi
v
F.E.M. an
d
a
lent plastic
s
w
ith thick
n
ߢ with
e
fibre ߳
,
i
stance ߟ fr
o
e
ction is gi
v
(
2
o
gether
w
ty, a code
h
simulate s
t
g
process.
F
d
etails are
n
be found
r to verify
bending
a
sheets is
a
e
ment pack
a
a
s assume
d
o
pickinem
a
the sake
t
ails are
n
be found
works [
1
t
plastic st
r
g as a typ
i
r
ess and st
r
t
h simulati
o
o
be in cl
o
e
s the vali
d
h
es.
v
alent plas
t
d
analytical
s
train ߝ
ҧ
n
ess
the
the
o
m
v
en
2
1)
w
ith
h
as
t
rip
F
or
n
ot
in
the
a
nd
a
lso
a
ge
d
to
a
tic
of
n
ot
in
1
4].
r
ain
i
cal
r
ain
o
ns
o
se
d
ity
t
ic
3.
In
us
c
u
le
v
ar
e
p
a
th
i
T
a
si
m
E
2
3.
1
p
a
Pr
in
pr
i
w
o
w
o
C
a
i.
e
Fi
g
by
u
n
ot
h
tr
a
fo
r
u
n
o
f
M
Fi
g
re
s
o
f
Residualcurvatures
1/m
RESUL
T
this sectio
n
ed to si
m
u
rvatures el
i
v
eller as s
h
e
made fo
r
a
rameters
a
i
ckness of
1
a
ble 1.
m
m
ulations (
S
E
ߥ
2
00e6 0.3
1
. Effect
o
a
rameters
o
ior to anal
y
tension le
i
mary proc
e
o
rk curvat
u
o
rk rolls
a
lculation
r
e
. crossbo
w
g
ure 3 for
a
y
a curvatu
r
n
der consta
n
h
er han
d
i
a
nsverse re
s
r
a stri
p
n
bending p
r
f
25 (1/m)
u
P
a to 200
M
g
ure 3. Ef
f
s
idual cur
v
f
100 MPa
Residual
curvatures
,
1/m
T
S AND DI
n
the simul
a
m
ulate pr
o
i
mination i
h
own in Fi
g
r
a steel
s
a
s shown
1
mm.
m
aterial p
a
S
I Units)
ߪ
ܳ
200e6 5
0
o
f primar
y
o
n residua
l
y
se the ant
i
velling pr
o
e
ss parame
t
u
re on res
i
cassette
h
r
esults for
w
and coil
s
a
strip that
r
e varying
n
t tension
o
i
n Figure
4
s
idual curv
p
that u
n
r
ocess by a
u
nder a ten
s
M
Pa.
f
ect of cur
v
v
atures und
e
Curvatur
e
SCUSSIO
N
a
tion code
h
o
cess of
n a typica
l
g
ure 1. Cal
c
s
heet with
in table
a
rameters
u
ܳ
ஶ
b
0
e6 500
process
l
curvature
i
crossbow
o
cess, the
e
t
ers i.e. ten
s
i
dual curv
a
h
as been
residual c
u
s
et are pre
s
is bent an
d
from 0 to
o
f 100 MP
a
4
longitud
i
atures are
n
dergoes
b
constant
c
s
ion varyin
g
v
ature vari
a
e
r constan
t
e
ߢ, 1/m
N
h
as been
residual
l
tension
c
ulations
material
1 and
u
sed in
C
10e9
s
cassette
e
ffect of
s
ion and
a
tures in
studied.
u
rvatures
ented in
d
unbent
60 1/m
a
. On the
i
nal and
depicted
b
ending
c
urvature
g
from 0
a
tion on
t
tension
Figure
residua
l
unbend
i
These
residua
l
the ap
p
with re
s
κ
୷
. I
n
curvatu
r
large c
h
value t
h
with i
n
curvatu
r
b
endin
g
sudden
increas
e
3.2.
E
bendin
g
The eli
m
the stri
p
rolls ca
This u
n
applica
t
crossbo
longitu
d
a direc
directio
that cro
factor t
h
shows
t
curvatu
r
strip is
b
of κ
ଵ
straight
e
Residual curvatures (1/m)
4. Effect
l
curvature
s
i
ng curvatu
r
figures
s
l
curvature
p
lied
b
end
i
s
pect to tra
n
n
other
w
r
e even s
m
h
ange in
κ
h
敲攠楳漠
n
捲敡c楮朠κ.
r
e κ
୷
in
c
g
curvatur
e
change in
e
in κ
୶
.
E
liminatio
n
g
in antic
r
m
ination o
f
p
due to be
n
ssette is in
v
n
desired cu
r
t
ion of s
e
w cassett
e
d
inal curva
t
tion oppo
s
n in work
r
ss
b
ow is re
m
h
at causes
t
he calcul
a
r
es i.e. cro
s
b
ent and st
r
ൌ 50 1/m
e
ned in th
e
T
e
of tension
s
under co
n
r
e of 25 1/
m
s
how that
κ
୶
is mor
e
i
ngunben
d
n
sverse resi
d
w
ords app
l
m
all, leads
t
κ
୶
and afte
r
perceptible
But trans
v
c
reases s
m
e
κ, with
a point af
t
n
of crossb
o
r
ossbow ca
s
f
crossbow
n
dingunbe
n
v
estigate
d
i
r
vature is r
e
e
cond ben
d
e
. Second
t
ure applie
d
s
ite to the
r
olls casset
t
m
oved by
u
its appear
a
a
tion result
s
s
sbow and
c
r
aightened
b
and the
n
e
reverse
d
e
nsion ߪ
்
(
1
/
variation
n
stant bend
i
m
longitudi
e
sensitive
d
ing curvat
u
d
ual curvat
u
l
ication o
f
t
o a relati
v
r
a maxim
u
change i
n
v
erse resi
d
m
oothly
w
a relati
v
t
er the sud
d
o
w b
y
sec
o
s
sette
introduce
d
n
ding in w
o
n this secti
e
moved by
d
ing in a
n
bending i
s
d
to the stri
p
last bend
i
t
e. The tric
k
u
sing the s
a
a
nce. Figur
e
s
for resi
d
c
oil set whe
b
y a curvat
u
n
bent
a
d
irection b
y
/
m
)
on
i
ng
nal
to
u
re
u
re
f
a
v
ely
u
m
n
κ
୶
d
ual
w
ith
v
ely
d
en
o
nd
d
to
o
rk
on.
the
n
ti
s
a
p
in
i
ng
k
is
a
me
e
5
d
ual
n a
u
re
a
nd
y
a
se
c
0
t
o
b
(1
)
sa
m
re
s
w
o
sa
m
di
r
c
o
a
s
w
h
m
o
th
a
in
t
d
u
ca
be
o
p
w
h
cr
o
cr
o
w
h
c
o
it
s
m
el
i
w
o
re
m
se
t
cr
o
le
v
in
a
dj
Fi
g
w
i
Residualcurvatures(1/m)
c
ond be
n
t
o 50 1/m
b
tained Fro
m
)
When th
e
m
e that is
κ
s
楤畡氠捵i
v
o
牤猠牥獩r
u
m
攠慢獯e
u
r
散瑩潮⸠⠲
)
o
楬整牥
o
s
灥捩慬p
o
h
楬攠捯楬
e
o
牥敮獩瑩
v
a
渠捲潳獢
o
t
牯摵捴楯渠
o
u
攠瑯⁴桥
獳整瑥⸠⠴⤠
B
e
敭潶敤o
p
瑩t畭
u
h
楣栠楳i
d
o
獳扯眠潲
o
獳扯眠 κ
୷
h
ile coil se
t
o
il set is mo
r
reaches ze
r
m
aller than
i
mination.
W
o
uld be a
n
m
ove
d
b
y
a
t
cassette
o
ssbow c
a
v
eller. The
n
both casse
t
dj
usting cor
r
g
ure 5. Va
r
i
th respect t
Residual
curvatures
(1/m)
Seco
n
ding cu
r
. The
fo
m
Figure 5:
e
curvature
κ
ଵ
ൌ κ
ଶ
ൌ
5
v
atures are
u
al curvat
u
u
te value
)
The sign
s
o
t necessar
i
o
unt of κ
ଶ
,
c
e
t is negat
i
v
e to secon
d
o
w, as w
h
o
f residual
first
b
end
i
B
oth crossb
by revers
e
u
rvature o
f
d
ifferent f
o
coil set. I
n
is zero
w
t
disappears
r
e sensitive
r
o in an a
m
what is n
e
W
hen cros
s
n
egative coi
a
similar pr
o
that is
m
a
ssette in
n
ecessary
a
t
tes can be
r
esponding
r
iation of cr
o second b
e
nd bending
c
r
vatu
r
e o
f
fo
llowing
s κ
ଵ
and κ
ଶ
5
0 1/m, th
e
reversed.
I
u
res occur
but in
s
of cross
b
i
ly the sam
e
c
rossbow is
i
ve. (3) C
o
d
bending
c
h
at was
t
curvatures
i
ng in w
o
b
ow and coi
e
bending
f
second
o
r elimin
a
n
this spe
c
w
hen κ
ଶ
؆
at κ
ଶ
؆ 5
1
to reverse
b
m
ount of
κ
e
敤敤潲e
c
s
扯眠楳⁺
e
氠獥琠瑨lt
h
o
捥摵牥渠
m
潵湴敤o
t
愠瑹灩捡氠
a
浯畮琠潦m
c
慰灬楥搠瑯a
牯汬湴r牭
e
潳獢潷o
d
e
湤楮朠捵n
v
c
urvature ߢ
ଶ
f
κ
ଶ
ൌ
can be
ଶ
牥 瑨攠
e
楧渠潦
I
渠潴桥爠
批⁴桥b
潰灯獩瑥o
b
潷湤o
e
.攮潲e
灯p楴楶e
o
楬整i
c
畲癡瑵牥u
t
牵攠景爠
楮瑲楰
o
牫潬汳
氠獥琠捡l
睩瑨w
扥湤楮朠
a
瑩潮t
c
楡氠捡獥i
13 1/m
1
/m. As
b
ending,
κ
ଶ
that is
c
rossbow
e
ro there
h
ould be
anticoil
t
e
r
anti
tension
c
urvature
strip by
e
sh.
d
coil set
v
ature
(1/m)
3.3. Ef
f
on resi
d
After
q
elimina
t
import
a
and stri
of κ
ଶ
h
a
same p
r
Figures
variatio
elimina
t
how th
e
Figure
coil se
t
Figure
variati
o
4. CO
N
In this
p
b
y its
n
simulat
i
simulat
e
process
Coil set (1/m)
Coil set
(
1/m
)
f
ect of ten
s
d
ual curva
t
q
uantitativ
e
t
ion, here
a
nt process
p thicknes
s
a
s been in
v
r
ocedure u
s
6 and 7 s
h
ns on
t
ion and F
i
e
optimum
s
6. Effect
o
t
eliminati
o
8. effe
c
o
ns on coil
N
CLUSIO
N
p
aper a the
o
n
umerical i
m
i
on progr
a
e
strip beh
a
. Finite el
Second b
e
Second b
e
s
ion and s
t
t
ures elimi
e
analysis
the effect
paramete
r
s
on the op
t
v
estigate
d
,
s
ed for plo
t
h
ow the ef
f
residual
i
gures 8 a
n
s
econd ben
d
o
f tension
v
on
c
t of str
i
set elimina
N
S
o
retical an
a
m
plementa
t
a
m was
d
a
viour in te
n
ement si
m
e
nding curva
t
e
nding curva
t
t
rip thickn
nation
of crossb
of two ot
h
r
s i.e. tens
i
t
imum amo
u
following
t
ting Figur
e
f
ect of tens
i
curvat
u
n
d 9 illust
r
d
ing curvat
u
v
ariations
o
i
p thickne
tion
a
lysis follo
w
t
ion throug
h
d
eveloped
n
sion levell
i
m
ulations
w
t
ure ߢ
ଶ
(1/m
)
t
ure ߢ
ଶ
(1/m
)
ess
ow
h
er
i
on
u
nt
the
e
5.
i
on
u
res
r
ate
u
re
o
n
ss
w
ed
h
a
to
i
ng
w
ere
v
a
th
e
a
n
a
m
c
u
re
s
re
m
be
c
u
fa
c
be
te
n
F
c
r
F
v
e
m
a
n
pr
o
o
f
T
h
re
s
)
)
a
ries with s
t
e
se figures
n
d strip thi
c
m
ount of
u
rvatures
d
s
idual cur
v
m
oved by
e
nding cur
v
u
rvature an
d
c
tor in resi
e
nding cur
v
n
sion is sli
g
F
igure 7. e
f
r
ossbow el
i
F
igure 9.
v
ariations o
n
m
ployed t
o
n
alytical s
o
o
gram was
f
crossbow
h
e calcul
a
s
idual c
u
Crossbow
(1/m)
Sec
o
Crossbow (1/m)
Sec
o
t
rip thickn
e
it is clear
t
c
kness inc
r
κ
ଶ
to
e
d
ecreases.
v
atures in
greater a
m
v
ature. Furt
h
d
tension s
h
dual curva
t
v
ature bec
a
g
ht.
f
fect of ten
i
mination
effect o
f
n
crossbow
o
verify
o
lution. T
h
used to e
x
in a typic
a
tion resu
l
u
rvatures
o
nd bending
o
nd bending
e
ss variatio
n
t
hat as bot
h
r
ease the n
e
liminate
This me
a
thinner s
m
ounts o
f
h
ermore c
o
h
ows that
t
t
ures elimi
n
a
use the
e
sion variat
i
f
strip th
eliminatio
n
accuracy
h
en the si
m
x
amine eli
m
al tension
l
ts show
e
are in
t
curvature ߢ
ଶ
curvature ߢ
ଶ
n
s. From
h
tension
ecessary
residual
a
ns that
trip are
f
secon
d
o
mparing
t
he main
n
ation is
e
ffect of
i
ons on
ickness
n
of the
m
ulation
m
ination
leveller.
e
d how
t
roduced
ଶ
(1/m)
ଶ
ㄯ洩
simultaneously to a strip due to longitudinal
bending and straightening and how they
vary with bending curvature variations in
work rolls cassette. Furthermore,
elimination of residual curvatures by
second bending in anticrossbow cassette
was investigated. It was revealed that an
optimal amount of second bending exists at
which crossbow is removed while coil set
occurs in opposite direction; so the
additional anticoil set cassette subsequent
to anticrossbow cassette is required to
eliminate this undesired curvature. Finally
effect of tension and strip thickness on
crossbow elimination was studied and it
was found that greater the tension and
thicker the strip, the smaller second bending
curvature is required to eliminate residual
curvatures. The same procedure can be
followed to determine the optimum
operating conditions in anticoil set
cassette. In general this simulation program
can also be used to determine optimum
process parameters in process online
control.
5. ACKNOWLEDGMENTS
The support of Mobarakeh Steel Company
is gratefully acknowledged. The authors
also wish to thank Mr. Rasouli for his
contributions to this work.
6. REFERENCES
1. Roberts W.L., “Cold rolling of steel”,
Marcel Dekker, New York, 1978.
2. Misaka Y., Masui T., “Shape correction
of steel strip by tension leveller”, Trans
ISIJ, vol. 18, 1978, pp. 475485.
3. Patula E.J., “Tensionrollerlevelling
process; elongation and power loss”,
Trans ASME, Journal of Engineering
Industry, vol. 101, 1979, pp. 269277.
4. Kawaguchi, K., “A numerical study on
the wrapping angle of strip in the
tension leveller”, Journal of the Japan
Society for Technology of Plasticity,
vol. 21, 1980, pp. 807814.
5. Hattori S., Maeda K., Matsushita T.,
Murakami S., Hata J., “Strip curvature
in tension levelling”, Journal of the
Japan Society for Technology of
Plasticity, vol. 28, 1987, pp. 3441.
6. Kajihara T., Furumoto H., Takemasa
T., Taniguchi N., “Study in levelling
process”, Technical report in
Mitsubishi Heavy Industry, vol. 25,
1988, pp. 315322.
7. Hira T., Abe S., Azuma, S., “Analysis
of sheet metal bending deformation
behaviour in processing lines and its
effectiveness”, Kawasaki Steel
Engineering Report, vol. 19, 1988, pp.
5462.
8. Hibino F., “Fundamental nature of
tension levelling”, Journal of the Japan
Society for Technology of Plasticity,
vol. 37, 1996, pp. 431439.
9. Yoshida F., Urabe M., “Computer
aided process for the tension levelling
of metallic strip”, Journal of Materials
Processing Technology, 89, 1999, pp.
218223.
10. Morris J.W., Hardy S.J., Lees A.W.,
Thomas J.T., ”Cyclic behaviour
concerning the response of material
subjected to tension levelling”,
International Journal of Fatigue, vol.
22, 1999, pp. 93100.
11. Huh H., Lee H., Park S., Kim G., Nam
S., “The parametric process design of
the tension levelling with an elasto
plastic finite element method”, Journal
of Materials Processing Technology,
vol. 113, 2001, pp. 714719.
12. Khan A.S., Huang S., “continuum
theory of plasticity”, John Wiley &
Sons Inc., New York, 1995.
13. Hibbit, Karlsson, Sorensen,
“ABAQUS Standard User’s Manual”,
ABAQUS Inc, 2003.
14. Salimi M., Jamshidian M., Beheshti
A., Sadeghi Dolatabadi A., “Bending
unbending analysis of anisotropic sheet
under plane strain condition”,
Esteghlal Journal of Engineering, vol.
26(2), 2008, pp. 187196.
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