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 84815-161, 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 so-called 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

anti-crossbow and anti-coil 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 anti-crossbow

cassette. Review of the results presents a

method for determining the optimal work

curvature in anti-crossbow 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 anti-coil 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 simulation-based process

design for the tension levelling of metallic

strips based on the elastic-plastic 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

anti-crossbow cassette. Finite element

simulations have also been employed to

verify the proposed solution. The Results

are used to determine the optimal work

curvature in anti-crossbow 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

tic-plastic

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

pic-kinem

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 anti-c

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

ng-unben

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

ding-unbe

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

anti-coil

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 anti-crossbow 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 anti-coil set cassette subsequent

to anti-crossbow 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 anti-coil 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. 475-485.

3. Patula E.J., “Tension-roller-levelling

process; elongation and power loss”,

Trans ASME, Journal of Engineering

Industry, vol. 101, 1979, pp. 269-277.

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. 807-814.

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. 34-41.

6. Kajihara T., Furumoto H., Takemasa

T., Taniguchi N., “Study in levelling

process”, Technical report in

Mitsubishi Heavy Industry, vol. 25,

1988, pp. 315-322.

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.

54-62.

8. Hibino F., “Fundamental nature of

tension levelling”, Journal of the Japan

Society for Technology of Plasticity,

vol. 37, 1996, pp. 431-439.

9. Yoshida F., Urabe M., “Computer

aided process for the tension levelling

of metallic strip”, Journal of Materials

Processing Technology, 89, 1999, pp.

218-223.

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. 93-100.

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. 714-719.

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. 187-196.

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