A mathematical model is developed to describe the conditions for buckling of steel strip between transport rolls due to strip camber, together with conditions necessary for the subsequent “ironing-in” of the buckle as it passes over the downstream roll. For a permanent crease to form, the buckle must be sufficiently stable so that it is prohibited by friction to spread laterally, and the stresses from the buckle defect must be large enough for plastic deformation to occur as it travels over the downstream roll. Once the conditions to produce a permanent crease are known they can be avoided in plant operations

Dixon, A.

Yuen, W. Y. D.

University of Queensland eSpace

5th Australasian Congress on Applied Mechanics, ACAM 2007  
10-12 December 2007, Brisbane, Australia 
Crease Formation in the Processing of Thin Web Material 
 
A. Dixon and W.Y.D. Yuen  
 
BlueScope Steel Research, P.O. Box 202, Port Kembla, 2505, Australia  
Abstract: A mathematical model is developed to describe the conditions for buckling of steel strip 
between transport rolls due to strip camber, together with conditions necessary for the subsequent 
ironing-in of the buckle as it passes  over the downstream roll. For a permanent crease to form, the 
buckle must be sufficiently stable so that it is prohibited by friction to spread laterally, and the stresses 
from the buckle defect must be large enough for plastic deformation to occur as it travels over the 
downstream roll. Once the conditions to produce a permanent crease are known they can be avoided 
in plant operations. 
Keywords: buckling, creasing, elastica, thin strip, wide strip, web mechanics. 
1 Introduction 
Rolls are commonly used to transport and guide steel strip through processing lines. Strip defects 
known as tension crease occasionally occur in these production units, being a narrow deformation 
typically 30 mm wide but extending many metres along the length of the strip as shown in Figure 1. 
The origin of these defects is not fully understood at present, but current research [1,2] indicates that 
the tension crease is formed from a strip buckle upstream of a transport roll. The buckle is then 
permanently ironed in as the buckle passes over the roll. For a tension crease to occur, then: 
· There must be sufficient transverse compressive stress upstream of the roll to form buckles. 
This may occur from roll misalignment [1,2] or, as is discussed here, from strip camber. 
· There must be sufficient frictional stress between the strip and roll to prevent the buckle from 
spreading transversely, and be ironed in as it passes over the roll. 
 
Good et al [1] and Hashimoto [2] investigated the creation of a tension crease for the cases of 
polyester film and coated paper-web respectively. Tension crease was experimentally produced by 
sufficient misalignment between the processing direction and a downstream roll. Ripples were 
initially observed upstream [2] which increased in magnitude with roll misalignment. These ripples 
were explained by an induced compressive stress in the transverse direction of the strip due to the roll 
misalignment. A model for the prediction of the onset of these buckles for misaligned rolls was 
developed [1,2]. 
 
An alternative mechanism for buckling is investigated here for the case of steel strip where the 
transverse compressive stresses is caused by camber, a term for strip curvature arising from one strip 
edge being longer than the other. The stability of the buckle ahead of the transport roll is then  
 
 
 
 
Figure 1. Tension crease defect in steel strip. 
  
analysed as it passes over the roll. The buckle is modelled by the Elastica Equation, which describes 
the out-of-plane bending of a strip for moderate to large curvatures. Under low frictional stresses 
between the strip and roll, the buckle will spread out by transverse slipping as it passes over the rolls 
and no crease will form. Under high frictional stresses, either from large friction coefficients or high 
strip tensions (so that the strip tightly grips the roll), the buckle will not spread. This buckle will 
eventually turn into a crease if plastic deformation occurs as the strip passes over the transport roll. 
Predictions from this theory compare favourably with empirical guidelines used by operators. In 
particular tension crease increases with: 
1. Lower strip thickness. 
2. Increased strip width. 
3. Lower strip yield stress. 
4. Increased strip tensions. 
5. Increased roll-strip friction. 
2 Determining Camber Tolerances to Avoid Upstream Strip Buckling 
A strip typically has a small degree of camber that may give rise to in-plane shear stress between 
rolls. For sufficient camber buckling may occur across the strip width. Consider a strip section with 
camber, as shown in Figure 2, with the strip exiting the upstream roll at an angle normal to the roll 
axis. The strip edges are normally described by a broad circular arc in an unrestrained state, but as it 
is fed into the downstream roll then, providing there are sufficient strip tension and no slippage 
between the strip and roll, the strip is bent such that it must enter the second roll at an angle 
perpendicular to the roll axis [1]. 
The resulting strip shearing stress can be approximately estimated using beam theory. Let us assume 
that the variation in the edge position )(xy along the strip length for the unstressed cambered strip is 
approximately given by (see Figure 2) 
,22
1 xy k»     with the angle of the edge to the processing direction x
dx
dy
kf =»                  (1) 
where k is the curvature of the cambered strip. In addition, a straight strip that is bent due to a 
transverse force F  at the second roll has an edge profile )(xy , and angle of the edge to the 
processing direction q , given by [3] 
( ) ( )xl
EI
Fx
dx
dyxl
EI
Fxy --==--= 2
2
,3
6
2
q              (2) 
where I is the moment of inertia of the strip being bent in the transverse direction. A strip bending 
solution for cambered strip such that the strip enters perpendicularly to the second roll can be 
obtained by superposition with the angles from camber and strip bending balancing as shown in 
Figure 2b: 
,0=+ fq   at  lx = ,  or  lEIF /2 k= .                (3) 
The corresponding shear stress in the strip of thickness h and widthW can be approximated by;  
l
WE
lWh
IE
hW
F y kkt
2
6
2
==»  since  
12
3 hWI = .             (4) 
If it is assumed that the normal stress in the direction of the strip width ys  is zero [1,2] then the 
expressions for the principal normal stresses become 
( ) ( ) ( ) ( )
x
xxxxx s
t
tsssstsss
2
22
2
1
2
1
2
22
2
1
2
1
1 , -»+-=»++=       (5) 
for small shear stresses compared to the applied longitudinal stress. It should be noticed that 2s  is 
compressive and, for large enough magnitudes, is capable of causing strip buckling. The resulting 
  
buckle would be parallel to the direction of 1s , being approximately at the angle of xsta » [4] for 
small shears compared to the longitudinal tension, see Figure 2b.  
Conditions of buckling can be found using standard theory [1,5] where the strip deformation in the out-
of-plane direction w  is described by 
222
2
1
4
Y
wh
X
whwD
¶
¶
+
¶
¶
=Ñ ss   with  ( )2
3
112 n-
=
Eh
D       (6) 
where the buckling is being described in the directions of the principal stresses ( )YX , as shown in 
Figure 2b, and D  is the flexural rigidity with n  the Poissons ratio. 
Periodic buckles in the transverse Y direction are precursors to tension crease [2]. As we are looking 
for buckling solutions that are periodic across the strip width, but long in the processing direction, let 
us assume 
( ) ÷
ø
ö
ç
è
æ
÷
ø
ö
ç
è
æ=
W
Yn
l
XaYXw n
pp sinsin, ,                  (7) 
with n  being an integer. Upon substitution into Eqn.(6) one obtains 
1
222
2 1 sss +÷
÷
ø
ö
ç
ç
è
æ
÷
ø
ö
ç
è
æ+=÷
ø
ö
ç
è
æ-
W
nl
W
nl
e   with       2
2
l
D
e
p
s = .                (8) 
 
 
 
 
 
 
 
 
 
 
(a) Strip path between guiding rolls 
 
 
 
 
 
 
 
 
 
(b) Formation of buckle on strip 
Figure 2. Cambered strip between guiding rolls together with oblique buckle. 
l  
x  
y  
X
 
Y
 
Buckle 
a
Free cambered strip 
Upstream 
Roll 
Downstream 
Roll l
 
x  
y
Strip 
width 
W 
Cambered strip 
fed into downstream roll  f
 
  
0
0.0002
0.0004
0.0006
0 10 20 30
Stress in Processing Direction (MPa)
0
10
20
30
Bu
ck
le 
An
gle
 (d
eg
)
Curvature
Buckle Angle
No Buckling
 Buckled
Ca
mb
er
 C
ur
va
tu
re
 (/m
)
  
0
0.0002
0.0004
0.0006
0 10 20 30
Stress in Processing Direction (MPa)
Ca
mb
er
 C
ur
va
tu
re
 (/m
)
Strip Thickness = 0.50 mm
 0.35 mm
No Buckling
 Buckled
 
                         (a)                                          (b) 
Figure 3. (a) Predicted critical camber curvature to prevent buckling, together with the corresponding 
buckle angle in the processing direction. (b) Predicted camber curvature for the onset of 
buckling for different strip thicknesses: Strip width: 1m, distance between rolls 5.0 m. 
 
Note that an increase in the longitudinal tensile stress 1ss »x  will require a larger transverse 
compressive force for buckling. The value of n  is chosen to minimize the necessary transverse 
compressive stress for buckling and leads to higher modes being chosen for higher strip tensions, 
giving buckles with shorter wavelengths in the width direction [1]. 
For a given strip tension and camber then 1s  may be calculated, together with the value for 2s  and 
the corresponding value of n  for the minimum compressive force to obtain transverse buckles, from 
Eqn.(5) and Eqn.(8) [1]. The shear stress for buckling may then be found from Eqn.(5) and the 
necessary camber from Eqn.(4). These results are shown in Figure 3a where, for various applied 
tensions the necessary strip camber for the onset of transverse buckles is predicted, together with the 
angle of the buckle to the processing direction. Higher strip tensions will prevent buckling from 
occurring. Figure 3b shows that for thinner strip less camber is required for  buckling. 
3 Stability of Buckle over a Roll 
Upstream buckling alone will not produce a permanent crease in the strip. For this to occur the buckle 
must be ironed-in when it passes over a roll, namely, from plastic deformation. Consider a strip, 
containing a localised buckle, being in contact with the roll away from the buckle. The Elastica 
Equation [6] describing the bending of strip from the applied moments can be approximated by: 
R
sT
H
ds
dD x--» qq2
2
                        (9)  
for small buckle heights, where s  is the distance along the strip width from the buckle centre and q  is 
the angle of the strip to the roll surface along its axis, as is shown in Figure 4. The term H is the 
horizontal component of the strip compressive force along the roll length while the second term on the 
right-hand side represents the downward moment into the roll from the strip curvature in the 
processing direction, with R being the roll radius and xT  the hoop force per unit strip width across the 
buckle.  Away from the buckle the strip conforms to the roll so at the buckle edges: 
2/bs ±=  then 0== dsdqq ,                                             (10)  
where b is the buckle width. For a symmetric buckle about 0=s  this gives the solution  
( )
( ) D
Hb
F
F
bsF
b
s
RH
bTx
4
,
sin
/sin2
2
2
=÷÷
ø
ö
çç
è
æ
-=q ,         (11) 
  
where F must satisfy the relation FF =tan  for the conditions of Eqn.(10) to be satisfied. The 
term F  is therefore not arbitrary but has discrete values corresponding to the buckling modes, with 
the lowest mode having 16.201 »F . This term relates the transverse compressive force for buckling 
to the strip flexural rigidity and buckle width. The strip curvature in the transverse direction Sk  and the 
buckle height variation w may be found from  
ds
d
S
q
k =  and ( ) ò
-
»
s
b
dssw
2/
q ,                  (12) 
where the profiles across the buckle width for the first mode being shown in Figure 4 for a unit buckle 
height and normalised width. For the buckle to be stable, and not spread across the roll face, then the 
compressive horizontal force H  must be sufficient to prevent slippage between the roll and strip. 
From the transverse force equilibrium equation this compressive force is maintained by friction:    
p
dy
dH
m£     or  
R
hlplH xsmm =£    as   R
hp xs=  .                 (13) 
Here p  is the pressure between the strip and roll, m  is the static friction coefficient, xs  is the applied 
tension stress in the processing direction and l  is the distance of the buckle from the strip edge. The 
greater the applied tension in the processing direction the greater the pressure between the strip and 
the roll. For the buckle to be stable as it travels over the roll then the frictional stress must be greater 
than the compressive stress necessary for buckling: 
2
1 /4 bDFHR
hlx =³ms                                   (14)  
for the first mode using Eqn.(11). Large friction and high applied tensions in the processing direction 
are therefore needed to maintain the buckled strip. The necessary conditions for the ironing-in of the 
buckles then become: 
( )
135.0
131
1
22
»
-
<÷
ø
ö
ç
è
æ
Fb
h
l
RE
x
n
sm
, for steel as  ( )2
3
112 n-
=
EhD         (15) 
implying that thin wide strip under high tensions and high frictions are more prone to the buckles being 
ironed-in as creases when the strip passes over the roll. This is consistent with the experimental 
finding outlined previously. For a typical case with E = 210,000 MPa, m = 0.2, sx = 20 MPa, 
R = 200 mm and h = 0.35 mm, Eqn.(16) predicts stable buckles for widths of the order of 100 mm. 
Buckles with widths of 30 mm, as seen in defects, should only occur only for abnormally high tensions 
or friction conditions.  
 
Extra strip tension occurs in the buckle region as it is wrapped around the roll, as the radius of the 
buckle region around the roll centre will be the roll radius plus the buckle height, causing the strip 
buckle region to stretch for the same roll-strip contact angle. The extra tension stress in the buckle 
region is then: 
R
wEx =s  
where w is the buckle height. For a roll radius of 200 mm, only a buckle height of 0.5 mm is necessary 
to produce yielding of strip with a yield stress of 500 MPa, being consistent with the buckle heights 
measured after ironing-in. Lower buckle heights will be sufficient for the ironing-in of the buckle for 
softer strip. 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 4. Co-ordinate system ( )q,s  for Elastica Equation under compressive horizontal force H  
and vertical force RsTx / .  
 
4  Conclusions 
A mathematical model to predict line conditions for the occurrence of a strip defect known as tension 
crease has been developed. Results of this model are consistent with observations where it is 
predicted that tension crease may be caused by strip camber and is more likely under conditions of 
thin wide strip under high friction conditions, according to the simple formula given by Eqn.(15). It is 
also predicted that if the tension is sufficiently high such that upstream buckles do not occur then the 
crease may be avoided, as is shown in Figure 3. 
 
5  Acknowledgements 
The authors wish to thank the management of BlueScope Steel Research for permission to publish 
the material contained in this paper, and participants of MISG2007 for stimulating discussions.   
References 
[1] J.K. Good, D.M. Kedl and J.J. Shelton, Shear Wrinkling in Isolated Spans, Proc. 4th Int. Conf. Web 
Handling, Ed. J.K. Good, 1997, pp. 462-479. 
[2] H. Hashimoto, Prediction Model for Paper-Web Wrinkling and some Numerical Calculation Examples with 
Experimental Verification, Proc. 8th Int. Conf. Web Handling , June 6-8, 2005. 
[3] W. Young, R.J. Roarks Formulas for Stress and Strain , 6 th Ed. McGraw-Hill, 1989, pp. 100. 
[4] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, 3 rd Ed. McGraw-Hill, pp. 22. 
[5] S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, 2nd Ed. McGraw-Hill, 1961, Ch.9. 
[6] C.Y. Wang, A Critical Review of the Heavy Elastica, Int. J. Mech. Sci., Vol. 28, No. 8, pp.549- 559.  
0
0.25
0.5
0.75
1
-0.5 -0.25 0 0.25 0.5
Distance from Buckle Centre / Buckle Width
B
uc
kl
e 
Pr
of
ile
 (u
ni
t h
ei
gh
t)
s
 
q  
2/bs =  2/bs -=
H
 
RsTx /  


Crease Formation in the Processing of Thin Web Material

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Crease Formation in the Processing of Thin Web Material

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