PREDICTION OF WEB WRINKLING INDUCED BY
ROLLER DEFLECTION
By
PEDRO J. ARIAS
Bachelor of Engineering
Universidad Central de Venezuela
Caracas, Venezuela
1981
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
July, 1998
OKLAHOMA STATE UNIVERSITY
PREDICTION OF WEB WRINKLING INDUCED BY
ROLLER DEFLECTION
Thesis Approved:
~AQ~~
~Th;sisad'A~
~ t: ~/l' \ _
~
~]; 76lfk...;.&..Ji__
Dean of the Graduate College
ii
ACKNOWLEDGMENTS
I would like to thank my adviser, Dr. J. K. Good for his guidance, support
and patience during my graduate studies and in the course of this research work.
I would also like to thank to Dr. C. E. Price and Dr. H. B. Lu for serving in my
committee.
I would like to extend many thanks the school of Mechanical and
Aerospace Engineering and the Web Handling Research Center for the research
assistantship given to me during this research and those fellows who provided
suggestions and help: Ronald Markum, Pat Straughn and Carrie Reynolds.
I specially thank and love my wife Licelot for her support and taking care
of our children since I have returned to school, and my parents and family for
their constant encouragement and moral support during my graduate studies.
iii
TABLE OF CONTENTS
Page
CHAPTER 1: INTRODUCTION 13
CHAPTER 2: THEORETICAL ANALySiS 17
2.1 TRACTION BOUNDARY VALUED PROBLEM FOR A WEB SPAN 18
2.2 BEAM THEORY TO PREDICT ROLLER DEFLECTION DUE TO A WEB TENSION 25
2.3 THEORY OF LATERAL COMPRESSION CAUSED BY ROLLER DEFLECTION 31
2.4 MfNlMUM PARABOLIC TRACTION THAT CAUSES WEB WRfNKLING DUE TO ROLLER CURVATURE 35
CHAPTER 3: EXPERIMENTAL SETUP AND PROCEDURE 40
3.1 TESTING MACHINE 40
3.2 WEB PROPERTIES AND DIMENSIONS 43
3.3 ROLLER PROPERTIES AND DIMENSIONS 44
3.4 EXPERIMENTAL PROCEDURE 45
3.5 WEB AND ROLLER SURFACE ROUGHNESS 46
3.6 COEFFICIENT OF FRICTION BETWEEN WEB AND ROLLER 47
CHAPTER 4: EXPERIMENTAL RESULTS 48
4.1 EXPERIMENTAL WRINKLING CONDITIONS 48
4.2 EXPERIMENTAL WEB AND ROLLER ROUGHNESS 55
4.3 COEFFICIENT OF FRICTION BETWEEN WEB AND ROLLER 56
CHAPTER 5: ANALYSIS OF EXPERIMENTAL DATA 57
5.1 INITIAL THEORETICAL COMPUTATIONS 57
5.2 COMPARISON BETWEEN EXPERIMENTAL RESULTS AND INITIAL THEORETICAL COMPUTATJONS .. 60
5.3 MODIFIED STRESS WRINKLING CONDITION 76
CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS 88
REFERENCES 90
iv
Figure
LIST OF FIGURES
Page
1. WEB SPAN MODELED AS A THIN PLATE 18
2. A ROLLER SUBJECTED TO A PARABOLIC WEB TENSION f MODELED AS A SIMPLY SUPPORTED BEAM 26
3. FORCES ACTING OVER A ROLLER DUETO WEB TENSION 27
4. TEST SECTION 28
5. SCHEMATIC OF AN ELEMENTAL WEB PIECE AT THE CONTACT LINE WEBROLLER. , 32
6. SCHEMATIC OF AN ELEMENTAL PARABOLIC WEB TRACTION d" APPLIED ON AN ELEMENTAL WEB
WIDTH 35
7. IMAGINARY POINTS ALONG THE CONTACT LINE WEBROLLER 36
8. TESTING MACHINE IN THE WHRC AT OSu. AO
9. TESTING MACHINE IN THE WHRC ATOSU. 41
10. TESTING MACHINE IN THE WHRC ATOSU 41
11. GENERAL LAYOUT OF THE WEB LINE ON THE TESTING MACHINE 42
12. 31.75MM (11/2 IN.) ROLLER ASSEMBLED WITH THE TWO MICROMETERS ON THE TESTING MACHINE.A5
13. TROUGHS OR WRINKLES AT THE WEB SPAN BEFORE WRINKLING FORMATION IN THE WEB WRAPPING
THE ROLLER. 49
14. INITIAL WRINKLING FORMATION AT THE ROLLER CENTER 50
15. WEB WRINKLING OVER THE ROLLER 50
16. INITIAL WRINKLING FORMATION ON THE 38.10 MM (1 1/2 IN.) ROLLER 51
17. WEB WRINKLING OVER THE 38.10 MM (1 1/2 IN.) ROLLER 51
18. WEB WRINKLING OVER THE 50.8 MM (2 IN.) ROLLER. 52
19. THEORETICAL STRESS WRINKLING CONDITION 58
20. THEORETICALCOMPATlBILlTY WRINKLING CONDITION . 59
21. THEORETICAL FRICTIONAL WRINKLING 59
22. MINIMUM THEORETICAL LOAD RJheor. ACCORDING TO INITIAL CALCULATIONS VS.
EXPERIMENTAL LOADS Rw exp. FoR31.75MM (1 %IN.) ROLLER, POLYESTER GAUGE 48, SPAN =
381 MM AT14, 29,42 M/MIN 61
23. MINIMUM THEORETICAL LOAD Rwtheor. ACCORDING TO INITIAL CALCULATIONS VS.
EXPERIMENTAL LOADS Rw exp. FOR 31.75MM (1% IN.) ROLLER, POLYESTER GAUGE 48, SPAN =
508 MM AT 13,29 AND 41 M/MIN 62
24. MINIMUM THEORETICAL LOAD Rwtheor. ACCORDING TO INITIAL CALCULAnONS VS.
EXPERIMENTAL LOADS Rw exp. FOR 38.1 0 MM (1 112 IN.) ROLLER, POLYESTER GAUGE 92, SPAN =
381 MM AT14, 27 AND 41 M/MIN 62
v
25. MINIMUM THEORETICAL LOAD R",theor. ACCORDING TO INITIAL CALCULATIONS VS.
EXPERIMENTAL LOADS R... expo FOR38.10 MM (1 V2IN.) ROLLER, POLYESTER GAUGE 92, SPAN =
508 MM AT 13, 28 AND41 M/MIN 63
26. MINIMUM THEORETICAL LOAD R",fheor. ACCORDING TO INITIAL CALCULATIONS VS.
EXPERIMENTAL LOADS R... expo FOR 38.10 MM (1 V2IN.) ROLLER, POLYESTER GAUGE 142, SPAN =
381 MM AT 13, 30 AND 40 M/MIN 63
27. MINIMUM THEORETICAL LOAD R,Jheor. ACCORDING TO INITIAL CALCULATIONS VS.
EXPERIMENTAL LOADS R", exp. FOR 38.10 MM (11/2 IN.) ROLLER, POLYESTER GAUGE 142, SPAN =
508 MM AT 12,30 AND 39 M/MIN 64
28. THEORETICAL ROLLER CENTER DEFLECTION UTrheor. ACCORDING TO INITIAL COMPUTATIONS VS.
EXPERIMENTAL RELATIVE DEFLECTIONS U rexp. FOR 31.75 MM (1 % IN.) ROLLER, POLYESTER
GAUGE 48, SPAN =381 MM AT 14,29 AND 42 M/MIN 65
29. THEORETICAL ROLLER CENTER DEFLECTION U rtheoT. ACCORDING TO INITIAL COMPUTATIONS YS.
EXPERIMENTAL RELATIVE DEFLECTIONS U r  exp FOR 31.75 MM (1 1/4IN.) ROLLER, POLYESTER
GAUGE 48, SPAN = 508 MM AT 13,29 AND 40 M/MIN 65
30. THEORETICAL ROLLER CENTER DEFLECTION U rtheoT ACCORDING TO INITIAL COMPUTATIONS YS.
EXPERIMENTAL RELATIVE DEFLECTIONS Ur  exp FOR 38.10 MM (11/2 IN.) ROLLER, POLYESTER
GAUGE 92, SPAN = 381 MM AT14, 27 AND 41 M/MIN 66
31. THEORETlCAL ROLLER CENTER DEFLECTION U rtheur ACCORDING TO INITIAL COMPUTAT10NS YS.
EXPERIMENTAL RELATIVE DEFLECTION Ur  exp FOR 38.10 MM (1 1/2 IN.) ROLLER, POLYESTER
GAUGE 92, SPAN = 508 MM AT 13,28 AND 41 M/MIN 66
32. THEORETICAL ROLLER CENTER DEFLECTION Urtheor ACCORDING TO TNlTIAL COMPUTATIONS VS.
EXPERIMENTAL RELATIVE DEFLECTIONS Ur  exp . FOR 38.10 MM (11/2IN.) ROLLER, POLYESTER
GAUGE 142, SPAN = 381 MM AT 13,30 AND 40 M/MIN 67
33..fJlEORETICA!. ROLLER CENTER DEFLECTION U rthellT ACCORDING TO INITIAL COMPUTATIONS VS.
EXPERIMENTAL RELATIVE DEFLECTIONS U r  exp FOR 38.10 MM (11/2IN.) ROLLER, POLYESTER
GAUGE 142, SPAN = 508 MM AT 12,30 AND 39 M/MIN.................. . 67
34. THEORETICAL INTERNAL STRESS IN eMD CF y YS. CRITICAL BUCKLING STRESS CFcr AT
EXPERIMENTAL NET FORCES R.. exp.FOR31.75 MM (1 % IN.) ROLLER, POLYESTER GAUGE 48
AND SPAN = 381 MM. . 69
35. THEORETICAL INTERNAL STRESS IN CMD CF y VS. CRIT1CAL BUCKLING STRESS CFer AT
EXPERIMENTAL NET FORCES R", expo FOR 31.75 MM (1 V4IN.) ROLLER, POLYESTERGAUGE48
AND SPAN= 508 MM 70
36. THEORETICAL INTERNAL STRESS IN CMD CF y VS. CRITICAL BUCKLING STRESS CFcr AT
EXPERIMENTAL NET FORCES R", expo FOR 38.10 MM (1 112 IN.) ROLLER, POLYESTER GAUGE 92
AND SPAN = 381 MM 70
37. THEORETICAL INTERNAL STRESS IN CMD CF y VS. CRlTICAL BUCKLING STRESS CF cr AT
EXPERIME TAL NET FORCES R", exp. FOR 38.10 MM (11/2 IN.) ROLLER, POLYESTER GAUGE 92
AND SPAN = 508 MM 71
vi
38. THEoRETIcAL INTERNAL STRESS IN CMD a y VS. CRITICAL BUCKLING STRESS U cr AT
EXPERIMENTAL NET FORCES R", exp.FOR38.10 MM (11/2 IN.) ROLLER, POLYESTER GAUGE 142
AND SPAN = 381 MM 71
39. THEORETICAL INTERNAL STRESS IN CMD a y VS. CRITICAL BUCKLING STRESS a cr AT
EXPERIMENTAL NET FORCES R"" exp. FOR 38.10 MM (1 1f2 IN.) ROLLER, POLYESTER GAUGE 142
AND SPAN =508 MM 72
40. THEORETICAL ( Ur.theor. ) VS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS AT
EXPERIMENTAL LOADS Rw expo FOR 31.75 MM (IV, IN.) ROLLER, POLYESTER GAUGE 48 AND
SPAN = 381 MM AT 14,29 AND 42 M/MIN 73
41. THEORETICAL (ur.theor.) VS. EXPERIMENTAL (Ur exp.) ROLLER CENTER DEFLECTIONS AT
EXPERIMENTAL LOADS Rw exp. FOR 31.75 MM (1 1/, IN.) ROLLER, POLYESTER GAUGE 48 AND
SPAN =508 MM AT 13,29 AND 41 M/M[N 73
42. THEORETICAL ( Ur.theor. ) VS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS AT
EXPERIMENTAL LOADS Rw exp. FOR 38.10 MM (1 112IN.) ROLLER, POLYESTER GAUGE 92 AND
SPAN = 381 MM AT 14,27 AND 41M/MIN , ' , 74
43. THEORETICAL (U r. theor. )VS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS AT
EXPERIMENTAL LOADS Rw exp. FOR 38.10 MM (1 112IN.) ROLLER, POLYESTERGAUGE92 AND
SPAN = 508 MM AT 13, 28 AND41M/MIN ,74
44. THEORETICAL ( Ur. theor. ) VS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS AT
EXPERIMENTAL LOADS Rw exp. FOR 38.10MM (1 1/2 IN.) ROLLER, POLYESTER GAUGE 142 AND
SPAN = 381 MM AT 13, 30 AND40 M/MIN 75
45. THEORETICAL(Ur.theor.) VS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS AT
EXPERIMENTAL LOADS Rw exp. FOR 38.10 MM (1 V2 IN.) ROLLER, POLYESTER GAUGE 142 AND
SPAN = 508 MM AT 12,30 AND 39 M/MIN 75
46. MINIMUM THEORETICAL LOAD Rwlheor. CONSIDERING TOTAL STRESS IN CMD a ty VS.
EXPERIMENTAL LOADS Rw exp. FOR 31.75 MM (11/4 IN.) ROLLER, POLYESTER GAUGE 48, SPAN =
381 MM AT 14,29 AND42 M/MIN , 78
47. MI IMUM THEORETICAL LOAD Rwtheor. CONSIDERINGTOTALSTRESSINCMD a ty VS.
EXPERIMENTAL LOADS Rw exp. FOR 31.75 MM (11/4 IN.) ROLLER, POLYESTER GAUGE 48, SPAN =
508 MM AT 13,29 AND41 M/MIN "." " " , , " 79
48. MINIMUM THEORETICAL LOAD R,)heor. CONSIDERINGTOTALSTRESSINCMD a ty VS.
EXPERIMENTAL LOADS R., exp. FOR 38.10 MM (1 112IN.) ROLLER, POLYESTER GAUGE 92, SPAN =
381MM AT 14,27 AND 41 M/MIN , " , 79
49. MINIMUM THEORETICAL LOAD Rwlheor. CONSIDERING TOTAL STRESS IN CMD a ty VS.
EXPERIMENTAL LOADS Rw expo FOR38.10MM(1 1/2JN.)ROLLER,POLYESTERGAUGE92,SPAN=
508 MM AT 13,28 AND 41 M/MIN " 80
vii
50. MINIMUM THEORETICAL LOAD R.jheor. CONSIDERING TOTAL STRESS IN CMD (jty VS.
EXPERIMENTAL LOADS R", exp. FOR 38.10 MM (1 1I2IN.) ROLLER, POLYESTER GAUGE 142, SPAN =
381 MM AT 13,30 AND40 M/MIN 80
51. MINIMUM THEORETICAL LOAD R",theor. CONSIDERING TOTAL STRESS IN CMD (jry VS.
EXPERIMENTAL LOADS R... expo FOR 38.10 MM (1 1I2IN.) ROLLER, POLYESTER GAUGE 142, SPAN =
508 MMAT 12, 30 AND39 M/MIN 81
52. THEORETICAL ( Urtheor .) YS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS
CONSIDERING TOTAL STRESS IN CMD (jty FOR 31.75 MM (11/( IN.) ROLLER, POLYFSTER GAUGE 48
AND SPAN = 381 MM AT 14,29 AND 42 M/MIN 81
53. THEORETICAL ( U rtheor.) VS. EXPERIMENTAL ( U r exp.) ROLLER CENTER DEFLECTIONS
CONSIDERING TOTAL STRESS IN CMD (jty FOR 31.75 MM (11/41N.) ROLLER, POLYESTER GAUGE 48
AND SPAN = 508 MM AT 13,29 AND 41 M/MIN 82
54. THEORETICAL (Urtheor. ) YS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS
CONSIDERING TOTAL STRESS IN CMD (j,y FOR 38.10 MM (11/2IN.) ROLLER, POLYESTER GAUGE 92
ANDSPAN= 381 MM AT 14, 27 AND 41 M/MIN 82
55. THEORETICAL (U rtheor.) VS. EXPERIMENTAL (U r exp.) ROLLER CENTER DEFLECTIONS
CONSIDERING TOTAL STRESS IN CMD (jIY FOR 38.10 MM (11/2IN.) ROLLER, POLYESTER GAUGE 92
ANDSPAN = 508 MM AT 13, 28 AND 41 M/MIN 83
56. THEORETICAL (u rtheor.) YS. EXPERIMENTAL ( Ur exp.) ROLLER CENTER DEFLECTIONS
CONSIDERING TOTAL STRESS IN CMD (jly FOR 38.10 MM (1 1/2IN.) ROLLER, POLYESTER GAUGE 142
AND SPAN =381 MM AT 13,30 AND 40 M/MIN 83
57. THEORETICAL (U rtheor .) VS. EXPERIMENTAL ( Ur exp.) ROLLER CENTER DEFLECTlONS
CONSIDERING TOTAL STRESS IN CMD (jly FOR 38.10 MM (1 1/2 IN.) ROLLER, POLYESTER GAUGE 142
AND SPAN = 508 MM AT 12,30 AND 39 M/MIN 84
58. LATERAL SURFACE FORCE.r VS. INTERNAL FORCE ./: CONSIDERIN 0ty FOR 31.75 MM (1 1/4 fN.)
"
ROLLER AND POLYESTER GAUGE 48 AT THE TESTED VELOCITIES 86
59. LATERAL SURFACE FORCE f,. VS. INTERNAL FORCE ./: CONSIDERING 0ty FOR 38.10 MM (1 1/2 tN.)
ROLLER AND POLYESTER GAUGE 92 AT THE TESTED VELOCITIES 87
60. LATERAL SURFACE FORCE fl. VS.INTERNALFORCE.r. CONSIDERING 0ty FOR 38.10 MM (11/2IN.)
ROLLER AND POLYESTER GAUGE 142 AT THE TESTED YELOCITIES 87
viii
Table
LIST OF TABLES
Page
1. Experimental Data. Minimum load required to get wrinkling for 31.75mm
(1% in.) roller, web: polyester gauge 48 and span: 381 rom 53
2. Experimental Data. Minimum load required to get wrinkling for 38.10 mrn
(11/2 in.) roller, web: polyester gauge 48 and span: 381 mm 54
3. Web and roller surface roughness 55
4. Coefficient of friction web  roller 56
5. Theoretical wrinkling loads vs. mean values for experimental loads 85
ix
a
f
.M
MD
P
PVC
q
NOMENCLATURE
half of web span
half of web width
cross machine direction
modulus of elasticity of the roller material
modulus of elasticity of the web material
distributed tension on the roller due to the web traction
feet per minute
lateral surface force per unit length
lateral internal force per unit length
air film layer between web and roller
area moment of inertia for thin roller
bending moment
machine direction
pressure onto the roller due to the web traction
polyvinyl chloride
parabolic web traction
x
R
Rq
Rq,roller
Rq,web
Sl&S2
T
Ur expo
urtheor.
v
x
y
outer radius of the roUer
equivalent root mean square roughness
roller surface roughness
web surface rouglmess
net force due to the web traction
experimental net force due to the web traction
theoretical net force due to the web traction
numerical coefficients of the parabolic web traction
web tension
web thickness
roller deflection in x direction
experimental roller deflection in x direction
theoretical roller deflection in x direction
shear force
distance along the machine direction
distance along the cross machine direction
aI, a2 & a3 constants of the airy stress function
web strain in the machine direction
¢(x,y) stress function
xi
qJw wrapping angle
J.1 coefficient of friction between web and roller
J.151 static coefficient of friction between web and roller
v web velocity
J[ potential energy
B slope of the deflected roller
[) dynamic viscosity of the air
(Jx normal web stress in the machine direction
(Jxy web shear stress
(J y normal web stress in the cross machine direction
v web's poison ratio
xii
CHAPTER1
INTRODUCTION
A web is any material that is produced as a continuous, flexible, thin sheet
able to withstand a high tensile stress, but unable to support a low compressive
stress (1). Examples of webs include: material foil, polyester film, paper, and
textiles.
Web manufacturing includes equipment functions requiring the web to be
unwound, transported by rollers through different production steps, (such as
printing, coating, laminating, etc.), and wound on a roller to be stored (2). During
processing there are many opportunities for the web to obtain permanent defects
that can affect the quality of the final product, thus producing high costs and
material loss. Among those defects are wrinkles.
Wrinkles can be formed in the machine direction and/or in a direction not
parallel to the machine direction. Wrinkles not aligned with the machine
direction are called "shear wrinkles"(3). Good, Gehlbach and Kedl (3) have
demonstrated that shear wrinkles are induced by lateral deformations, and hence
shear on the web. Their research indicates the main reasons for deformations are
misaligned rolls, controlled guide rolls, interaction rollerweb, and web twist.
In the case of machine direction web wrinkling, Shelton (4) indicated that
a lateral compressive strain/ stress is responsible for wrinkle formation. This
13
compressive strain/ stress may be produced by a roller deflection, a decrease in
the tension across a driven roller, an increase in temperature and moisture, and
the bending of a wound roll.
Shelton developed a theory of lateral compression for web wrinkles (4).
According to Shelton's theory (4), in the case of web wrinkling due to roller
deflection, there are two types of compressive stresses in the cross machine
direction. The first type of the compressive stress causes wrinkles in a tensioned
free web span. This stress is determined by the theory of elastic stability
researched by Timoshenko and Gere (5) for a rectangular plate with uniform
distributed load in the machine direction. The second stress is the lateral
compressive stress that produces wrinkles in the web wrapping a roller. This
stress is based on the theory of buckling of pressurized cylindrical shells (5).
Shelton found that the critical compressive stress that buckles a web wrapping a
roller is greater than the compressive stress that wrinkles a web span. For that
reason, he concluded that one requirement of getting wrinkles in the web over a
roller is to reach the buckling stress for a cylindrical shell without internal
pressure.
Shelton's theory (4) goes on to indicate that the other condition required
to maintain web wrinkles along the roller surface is the lateral surface force. This
force is due to frictional contact with the web on the roller being greater than the
compressive internal force due to buckling stress.
14
Duvall (6) performed several tests on a model based on Shelton's theory of
buckling due to lateral forces caused by a deflecting roller. He found that the
strains induced due to roller curvature were extremely high compared to the
critical buckling strain. Duvall concluded that the maximum compressive strain
due to bending on the top surface of the roller, matched with the experimental
results.
Predicting web wrinkling is valuable in the web handling industry due to
frustration, excessive costs, and the time involved with these defects. The
purpose of this study is to develop a model to predict wrinkles in the machine
direction in a web wrapping over a roller induced by roller deflection. Focusing
on practical applications, the motivation is to develop a procedure based on
simple equations. This application can be used in industry without requiring
highpowered computers and sophisticated technology. To achieve that goal,
and based on previous research, a web span is considered subjected to an
assumed traction that causes a parabolic roller deflection. Expressions for web
stress, web deformation in the machine direction, and roller deflection, among
others, will be determined using energy methods and classical solid mechanics.
Finally, theoretical wrinkling conditions will be established by enforcing
compatibility between webroller deformations and between the lateral
compressive stress and buckling stress.
The experimental part of this study consists of building several flexible
rollers for testing and a frame which was sufficient to support the required
15
instruments on the testing machine. Each roller will be deflected by a web
tension obtaining webwrinkling conditions over the roller. The correlation given
among roller dimensions, roller properties, web dimensions, web properties,
web span, web velocity and roller deflections will allow us to compare the
experimental results to the theoretical computations and to determine a grade of
accuracy of the proposed prediction model.
16
CHAPTER 2
THEORETICAL ANALYSIS
In web handling, roller deflections can be caused by web traction.
Considering the roller as a beam, for reasons of simple analysis and derivations,
it has been verified that the parabolic shape is a reasonable approximation, even
though the deflection curve is not exactly in the form of parabola (4). This
geometry transfers a parabolic traction to the web when it conforms to the roller.
The objective of this chapter is to find a mathematical expression for the
parabolic traction that produces simultaneously the same deformation at the
roller and at the web under wrinkling conditions. To achieve that goal, we
assume a polynomial expression for the web traction, this expression is
composed of the numerical coefficients 51 and 52 which are functions of
material properties and dimensional characteristics. Web span and roller
deflection problems will be analyzed using the theory of elasticity. Based on
Shelton's theory of lateral compression caused by roller deflection and other
studies, the assumed parabolic traction will be verified for the inducement of
wrinkle existence in the web wrapping over the roller.
17
2.1 Traction Boundary Valued Problem for a Web Span
Web spans can be analyzed using the criteria of twodimensional elasticity
in plane stress problems.
The geometry of a web is basically a thin plate or membrane with one
dimension (thickness) much smaller than the others. The distributed load is
applied over the thickness in the machine direction. In this case we consider no
body forces in the plate problem.
The exact solution of twodimensional problems in elasticity requires
satisfying the following conditions: boundary conditions (kinematically
admissible state), equilibrium conditions and compatibility equations(7).
..
......
..
.. bJ.._+;" ..... ~ x ..
i
... .. .. .. ..
...... ..
q=Sl+S2y 2
Figure 1. Web span modeled as a thin plate.
18
Figure 1 shows a web span modeled as a thin plate subjected to a
parabolic distributed traction q. This web traction is assumed to have the
following mathematical expression:
q=Sl+S2y 2
where Sl and S2 are numerical coefficients.
Rivello (8) obtained an approximate solution for a plane stress plate
[1]
problem with a pure parabolic traction in the form of a polynomial term (the
parabolic traction) plus an infinite series. Rivello's stress function was modified
in order to introduce our more general parabolic web traction as follows:
where al a2 and a3 are undefined constants.
The two first terms satisfy the essential (traction) boundary condition, and
the third term assures no more stresses on the boundary. The series was
truncated after the first three terms al a2 and a3.
From classic solid mechanics (7), we know that a stress function ¢ (x, y) is
related to stress as follows:
19
The normal stress in the machine direction (MD) is determined by:
[3]
The normal stress in the cross machine direction (CMD) is determined by:
[4]
and the shear stress can be found by:
[5]
Now, the problem is oriented to find the values for the unknown
coefficients in the approximate solution ¢ (x, y) for the plane stress problem.
2.1.1 Principle of complementary strain energy. RayleighRitz method.
We shall now consider how the energy principles can be used to obtain
the values of the coefficients al,a2&a3 in the stress function ¢ (x, y) [2]. The
procedure, known as the RayleighRitz method will be used with the principle of
20
complementary strain energy. The web will be considered as an isotropic
homogeneous linearly elastic material.
The expression for the Complementary Energy II " II in the case of the
plane stress problem without body forces is given by Rivello (8). Considering the
web span as a region limited by a ~ x ~ a and b ~ Y ~ b, this expression
becomes:
[6]
where Ew is the modulus of elasticity of the web span and u is the web's
Poisson ratio.
By using equations [2], [3], [4] and [5], we derive the expressions for
CY \' (JI. & (J" xy which are then substituted into equation [6]. After integration, an
expression is obtained for the complementary energy:
,,=1
2Ew
32768 32b9 9 32768 65 13 22 131072 a a + a a + 67 a7a 12+131072 b7
3675 75075 11025 121275
aIIa22 + 131072 a32blla7+ 32768 a 32bl3a5+ 65536 b9a7a2a+1
121275 75075 11025
1024a3b7S2a 5 + 65536 a 3b7 Q IIa 2+4 b5S22 a+ 32768 a 2269a9
1575 121275 5 3675
4bS1 2 32768 65 9 12 32768 6+ a+ a a + 9a 5a12+ 65536 bll a5a 3a 1+
1575 1575 17325
1024 65 a 5a 2a1+1024+ 65536 bll a 7a 2a 3+ 65536 b7 a9a 3a 1+
225 1575 121275 11025
~b3SIS2a+ 65536 65a"a2a1
3 17325
21
[ 7}
The RayleighRitz method employs the theory of minimum total
complementary potential (8). The total complementary strain energy is
minimized with respect to the unknown coefficients in the stress function.
Minimizing expression [7] with respect to al,a2 &a3, we get a set of three
algebraic equations as follows:
EQ = Btr = 0
1 Bal
Btr
EQ2 ==0
Ba2
[8]
[9]
[10]
Solving the system of equation above yields the expressions for the
coefficients aI, a2 & a3 as follows:
al =_ 77 (1430a s +9477a6b2 +74219a 4b4 +9477a 2b6 +1430bS !S'2
64 (25025a 12 +129740b 2a 'o +911998b 4a8 + 726044b 6a6 +J
911998b sa 4 +129740b 1o a 2 + 25025b l2
a2 =_1001 (715a 6 + 1235b 2a 4 +170b4a 2 + 22b 6 }S'2
64 (25025a 12 +129740b 2a10 +911998b 4a8 + 726044b 6a6 +J
911998b 8a 4 +129740b ,oa2 +25025b J2
a3 =_1001 (22a 6 +170b 2a4 +1235b4a2 +715b 6 }s'2
64 (25025a 12 +129740b 2
a
1o +911998b4a8 + 726044b 6a6 +J
911998b 8a4 +129740b'Oa 2 +25025b 12
22
[11]
[12]
[13]
By substituting expressions [11], [12] and [13] into the equation [2] we will
have determined an approximate solution for a web span subjected to a parabolic
traction.
2.1.2 Expressions for normal stresses, strain and deformation in the machine
direction for a web span.
Expressions for internal normal stresses in this web span problem can be
found by substituting the above determined approximate solution ¢ (XI y) 12] into
equations [3] and [4] as follows:
The internal stress in the machine direction (MD) is:
ax =51 +S2/ +~X2 _a2)2 /(al+a2x2+a3l )+1~x2 _a2Y(i b2)+a3/
+4{x2 _a2y(i _b2 Xal+a2x2 +a3/)+2(x2_a2y&2 _b2 ya3
The internal stress in the cross machine direction (CMD) is:
CY y =8X2
(y2 b2 Y(al+a2x 2a3y 2)+16(x 2 a2 Xy 2 b2Ya2x 2+
4(x 2 a2 Xy 2 b 2 Y(al+a2x 2a3y 2)+2(x 2 _a 2Y(y2 b2Ya2
114]
[15]
To find the expression for the strain in the machine direction, we use the
stressstrain relationship for a plane stress problem:
23
[16]
Substituting equations [14] and [15] into [16] yields:
1
Ii =J
E",
Sl + S2y 2+ 8(x 2 a2yy2 (al +a2x2+a3y 2)
+16(x 2_ a 2y(y2 _ b2)+a3y 2+ 4(x 2_a 2y(y2 b 2)
(al +a2x2 +a3y 2 )+2(x 2 _a 2 Y(y2 _b 2 Ya3
8x 2 (y2 b2Y(al+a2x 2a3y 2)+16(x 1 _a 2)
v (y2 b2Ya2x 2+4(x 2a2Xy 2b 2Y(al+a2x 2a3y 2)
+ 2(x 2 _a 2y(y2 _b2ra2
[17]
Finally, the straindisplacement relationship in the machine direction is
determined by the following equation:
ou [; =w
x ox [18]
From equation [18], we find that the web deformation is expressed by:
[19]
where f(y)is a function only of y. Solving equation [19], taking into
account that f(y)=O due to the symmetry of the web span about the y axis (at
x=O, U", =0), leads to the following expression for web deformation in the
machine direction:
24
u
I
E
[20]
2.2 Beam theory to predict roller deflection due to a web tension
To find the expression for the roller deflection caused by a web tension,
we consider the equilibrium of a flexible beam subjected to a distributed load. A
roller can be satisfactorily modeled as a beam simply supported at its ends.
When no load is applied, the neutral axis of the roller lies along the y axis. Due to
25
the lateral web behavior, it is convenient to fix the origin of coordinates at the
roller center (Figure 2).
x •
j 1j f
y 1~ ~ ~
~. ,, .........................
h b
Figure 2. A roller subjected to a parabolic web tension f modeled as a simply
supported beam.
Figure 3 shows a free body diagram for a roller subjected to a web
tensionT. The experimental apparatus used in this study set the wrapping angle
¢w at 180 degrees (figures 4 and 8 to 11).
26
.....
<) !P.
_r~",.
f
T T
Figure 3. Forces acting over a roller due to web tension.
The assumed parabolic web traction q is varying with respect to the y
axis. Hence, the expression for the distributed web tension f due to the web
traction" q " becomes:
where f w is the web thickness.
27
[21]
Idler roller
y
Figure 4. Test section.
28
Distributed load over the roller " l "
Deflecting roller
rp w =1800
Web spans
In order to find the roller deflection, we will use the equilibrium equations
for a distributed load on a beam, shear, bending moment and slope. For a roller
of constant cross section, the beam equations are expressed as follows(7):
E,I, dd> =f
El d
3
u,=_V
, , dy 3
E 1
d2U,
, '2 =M
dy
du, =f:)
dy
[22]
[23]
[24]
[25]
where u, is the roller deflection, E, is the modulus of elasticity of the
roller material, 1, is the moment of inertia of the roller, V is the shear force, Mis
the bending moment and f:) is the slope.
The net force F over the roHer due the uniform distributed tension l is
determined by:
b ( \ 1 (S2b
2 F =  J2tw SI + S2y2 flY = 4twb SI+J
b 3
and the reactions at the simple supports becomes:
F (S2b2 F ==2t b 81+J
x 2 w 3
[26]
[27]
Substituting equation [21] into [22], integrating and then applying the
boundary condition at y = b, V = expression [27], yields:
29
[28]
The expression for the bending moment M or curvature can be found
integrating equation [28] and using the boundary condition at y =b,M =0 as
follows:
dIu, = _1_[_21 [Sly2 + S2y
4 : + t ",b
I
(6S] + s2b 2 )1 [29]
dy 2 EJ, W 2 12 6 J
The slope of the deflection curve is found by integrating equation [29] and
enforcing the boundary condition of no slope at y = 0 I as follows:
du, =21 .[Sly3 +S2y
5 J+ t",b
2
(6S1+S2b 2 )y
dy "6 60 6
[30]
Finally, the equation for the roller deflection is determined by integrating
equation [30] and enforcing the boundary condition u, =0 at y =b, as follows:
30
2.3 Theory of lateral compression caused by roller deflection
Based on Timoshenko and Gere's theory of elastic stability (5), Shelton (4)
reasoned that the lateral compression is caused by roller curvature; and that the
lateral compressive stress responsible for wrinkles in a web, due to roller
deflection, is the compressive stress that buckles the web wrapping a cylinder.
Timoshenko and Gere (5) studied the axial compression of a curved sheet
panel. They determined, for the case of thin cylindrical shape, that the critical
stress for symmetrical buckling with respect to the axis of the cylinder has the
following expression:
[32}
where R is the nominal radius of the curved sheet and, also, of the outer
surface of the roller.
The use of the above equation has been justified by Shelton (4) in the case
of a web wrapping a roller, or the outer wraps of a wound roll.
Shelton, also, considered the effect of friction between a web and roller (4).
For a web wrinkling over a roller, a reaction force is required to sustain the
wrinkled web upon the roller surface (9). This force is denominated as the lateral
surface force f p and must be greater than the internal force J, due to the lateral
31
compressive buckling stress acting on the web, for a wrinkle to be sustained in
the web as it crosses a roller.
When the pressure P between the web and roller is constant, the
maximum lateral surface force IIJ is equal to the pressure P multiplied by the
contact area and by the coefficient of friction J.1 between web and roller(4).
dllJ
P
Figure 5. Schematic of an elemental web piece at the contact area webroller.
Figure 5 shows an elemental piece of web of the contact area between the
web and roller; using the given definition for the lateral surface force, we get:
di'l =flPdy [33]
where P is given by the equilibrium equations for a thin wall pressure
vessel, as follows:
32
p =_G'_xf_w =,(S_1_+_S_2_y_2,,}l,,Y
R R
[34]
Therefore, substituting equation [34] into [33] and considering symmetry
about the x axis, we can get the expression for the lateral surface force .I~ per
unit wrap length as follows:
[35]
The expression for the maximum lateral compressive internal force If due
to buckling is given by:
[36]
where G'cr is the critical buckling stress obtained by equation [32].
2.3.1 Effect of the air film layer ho on the coefficient of friction between web
and roller.
Knox and Sweeney (10) demonstrated the existences of an air film layer ho
between web and roller due to hydrodynamic lubrication. They developed the
following expression:
[37J
33
where.9 is the dynamic viscosity ofthe air (3.077*107 Nmin/m2 @
27°C), v is the web velocity (m/min) and Tw. is the web tension.
In the case of parabolic web traction" q ", the equation [37] becomes:
2
[
129v ]3 ho = 0.6SR ( , }
81 +S2y w
[38]
Good, Kedl and Shelton (9) analyzed the effect of the air film layer ho on
the behavior of the coefficient of friction Ji . They developed an algorithm which
relates ho' the equivalent root mean square roughness R'I
(Rq =JR'I.roller 2 + R'I.weh 2 ) and the static coefficient of friction Jist to Ji. The
algorithm is expressed as follows:
Ji =Ji.w
Ji =0
[39]
Therefore, f)l per equation [351 may be affected depending on the value of
the air film ho along the web width.
Duvall (6) found that his experimental results did not obey the theory
developed by Shelton. The experimental results matched better with the
assumption that the web assumes the shape of the deflected roller and hence
induced bending stresses in the shell of web upon the roller. Therefore, an
34
additional condition of compatibility between web and roller deformations
should be considered for wrinkle formation.
2.4 Minimum parabolic traction that causes web wrinkling due to roller
curvature.
dq
Figure 6. Schematic of an elemental parabolic web traction dq applied on an
elemental web width.
Recalling the assumed expression for the parabolic web traction (equation
[1]) and based on figure 6, the expression for the net force applied, Rw ' due to
the parabolic distributed traction can be determined as follows:
b[ 82b
Z
R ) w = !/" (S 1 + 82 y 2)iy = 2 t"b 8 1 + 3
[40]
Equation [40] represents the relationship between the necessary
load to transport the web Rw and the coefficients 81 and 82 which define
the parabolic distribution of the traction along the web width.
35
2.4.1 Methodology to predict the minimum parabolic web traction that causes
wrinkling due to roller curvature.
The objective of this part of the research is to describe the proposed model
to predict the critical parabolic web traction for wrinkling due to roller
deflection. The iterative method was used during the theoretical computation in
this work. To perform it, a spreadsheet in the software EXCEL was utilized and
the iterative process was accomplished using the command SOLVER.
Step 1: Imaginary points along the contact line webroller.
We will consider divided the contact line between the web span and the
deflected roller into equidistant npoints. This will assure full contact between
the roller length and the web width.
Deflecting roller
........ ...... ..'
Contaclline webroller
Web span
Parabolic web traction "q"
Figure 7. Imaginary points along the contact line webroller.
36
Step 2: Estimation of R", and S2
Now, an iterative process will be performed. It begins with the
assumption of a load R... and a coefficient S2. The coefficient Sl can be
determined using equation [40].
Step 3: Maximum compressive stress in CMD (J'y:
With 81 & 82 from step 2 and using equation [15], the internal web stress
in the cross machine direction for each point PI ,P2 ,...Pn can be computed. It is
expected that the maximum (J' y being located at the center point on the contact
line webroller.
Step 4: Critical buckling stress eTa'
The critical compressive stress for buckling will be determined by
equation [32].
Step 5: Web deformation at the contact line webroller u...
With 81 & 82 from step 2 and using equation [20], the web deformation
for each point PI' P2 ,...p" can be found.
Step 6: Roller deflection ur •
Substituting 81 & 82 into equation [31] will allow us to compute the roller
deflection for each point PI' P2 ,...p" along the contact line webroller.
Step 7: Compatibility between roller deflection ur and web deformation
37
Next, we determine the error or difference between the web deformation
and the roller deflection for each point PI' P2 ,...p" . The summation of errors will
be referred to as compatibility error or sumerror.
Step 8: Lateral surface force per unit length Ill.
Substituting 81 & 82 into equation [38], the air film thickness ho for each
point PI' P2 ,···Pn will be found. According to the algorithm [39], the lateral
surface force per unit length can be determined using equation [35].
Step 9: Lateral compressive force per unit length f, .
The lateral compressive force per unit length can be calculated by
equation [36].
Step 10: Stress wrinkling condition.
Is the maximum CY y obtained in step 3 equal to CYer given by step 4?
If YES go to step 11.
If NO go to step 2.
Step 11: Compatibility wrinkling condition.
Is sum'error or compatibility error in step 7 negligible (less than 12%)?
If YES go to step 12.
If NO go to step 2 and assume a new value for S2.
Step 12: Frictional wrinkling condition.
Is .ill obtained in step 8 greater than I, given in step 9?
38

If YES, Rw is the minimum net wrinkling load to be applied in web
handling equipment. 81 & 82 are the coefficients that when placed into the
equation [1], describe the minimum parabolic web traction responsible to
wrinkle the web that wraps a roller due to roller deflection.
If NO go to step 2 and increment R",
39
CHAPTER 3
EXPERIMENTAL SETUP AND PROCEDURE
The objective of this part of the study is to describe the steps taken during
the experimental process of this research. It covers a brief description of the
testing machine, material selection, roller design based on initial estimations, and
a description of the experimental procedure.
3.1 Testing machine
Web wrinkling experiments were performed in the Web Handling
Research Center at OSU. The testing machine is shown below in Figures 8 to 10.
Figure 8. Testing Machine in the WHRC at OSU.
40
Figure 9. Testing Machine in the WHRC at OSU.
Figure 10. Testing Machine in the WHRC at OSU.
41
The general layout of the web on the testing machine is shown in Figure
11.
Deflecting roller
Unwinding roller
(braked)
Adjustable web
span
R1
Idler rollers
Powered winding
roller
Web tension load
cell
o
Tachometer
Figure 11. General layout of the web line on the testing machine.
The experimental testing machine is able to produce a maximum tensile
load of 89 N (20 lb±). This load is created by an adjustable magnetic brake system
at the unwinding roller. The desired value of the load is the result of the tension
difference between the powered winding roller and the required brake during
the unwinding operation. The applied tensile load over the web can be
monitored in a digital tension readout indicator with a resolution no less than
1/10 lbf. This indicator receives the corresponding signal from a tension
transducer located at the idler roller Rl (Figure 11).
42
The web velocity is set in a digital panel that controls the speed of the
powered winding roller. During the experiments, the minimum testing velocity
was of the order of 13 m/min (40 fpm) and the maximum of the order of 40
m/min (130 fpm). The web span between the deflecting roller and the idler
rollers can be adjusted manually. This testing machine has a minimum web span
of 254 mrn (10 in.) and a maximum value of 610 nun (24 in.); however, those web
span limits may vary depending on the roller sizes. The web spans chosen in this
study were 381 nun (15 in.) and 508 mm (20 in.). As mentioned previously, all
experiments were done using an angle of wrap of 180 degrees.
3.2 Web properties and dimensions.
Web wrinkling tests were performed on films with the following
characteristics:
Material: Polyester.
Modulus of elasticity ( Ew.): 4140 MPa (600468 psi )
Poisson's ratio: 0.3
Web width: 152.4 nun (6 in.)
Web thickness: a  Gauge 48 (12 Microns or 4.72 *104 in.)
b Gauge 92 (23 Microns or 9'" 104 in.)
c Gauge 142 (36 Microns or 1.4 '" 103 in.)
43
3.3 Roller properties and dimensions.
Roller design started with the material selection. Several factors had an
influence in making this decision. Among them were: a desired considerable
material flexibility, testing machine limitations (e.g. max. tensile load =89 N (20
lbf.)), costs, availability, and weight. PVC pipe having a Young's modulus of
approximately 1800 MPa (261000 psi) was selected.
In order to find the proper outer diameter and wall thickness of the roller,
initial estimations were made taking into consideration the following conditions:
maximum testing load = 89 N (20 lbf), roller length equals to the web width, a
desired low roller stiffness, web spans of 381 mm (15 in.) and 508 rnm (20 in.),
and the material's ability to cause wrinkles.
The result was the following three rollers: a) Nominal diameter of 31.75
mm (1 % in.), schedule 20, b) Nominal diameter of 38.10 mm (1 V2 in.) schedule
20, and c) Nominal diameter of 50.80 rnm (2 in.), schedule 20.
The study of the roller deflections was necessary in this research. The
experimental roller deflections were measured with two micrometers with a
resolution of 1/100 rnm or 5/10000 inch. One micrometer was located so that it
could measure the deflection at roller center. The other micrometer was able to
detect the deflection at the roller edge, note Figure 12.
44
Figure 12. 31.75 mm (1 112 in.) roller assembled with the two micrometers on the
testing machine. Web is Polyester gauge 92.
3.4 Experimental Procedure.
Having fully defined web and roller characteristics, each roller was tested
on the testing machine using different thickness polyester films. The
experimental procedure follows:
For a given roller size, polyester film and a fixed web span, each test
began by setting the web tensile load and web velocity. Then, progressively, the
web tension was increased in intervals of approximately 2lbf until a wrinkle
formed in the web wrapping the deflecting roller. The wrinkling load was
45
recorded from readings in the digital tension readout and the roller deflections
from the micrometers.
After that, the applied tension was reduced to a minimum value. The
velocity was set to a new value and the web load was increased again until the
new minimum load to form wrinkles was found.
The same procedure was repeated for each roller size, polyester caliper
and web span. The data obtained were processed in order to compare with the
theoretical computations obtained under the same conditions.
3.5 Web and roller surface roughness
The surface roughness of the tested polyester films is an important
property used to determine the lateral surface force j/.1 by algorithm [39}. For
that reason each type of polyester film (gauge 48,92 and 142) was tested in the
Web Handling Research Center using a" Surftest Analyzer and Surftest 402" a
surface profilometer manufactured by Mitutoyo Corporation. Pieces of web were
located on a very smooth surface (glass) and several surface roughness readings
were obtained.
Similar to the surface roughness tests done on the polyester films, the
surface roughness of the rollers were obtained using the Surfatest Analyzer. Data
46
along lines parallel to the cylindrical axis at the external surface of the rollers
were recorded.
3.6 Coefficient of friction between web and roller.
Several tests were conducted in order to measure the experimental
coefficient of friction between the web and roller surfaces. These tests consisted
of registering the maximum load just when the slippage impediment between
the web and roller was broken. Using the brandbrake equation, we could find
the value of the respective coefficient of friction.
1 PI
II =In [41]
,S( P
CfJw 2
where PI is the maximum force to break the slippage impediment, P2 is the
weight of a dummy body, CfJ",is the angle of wrap of the web over the roller and
iiI( is the static coefficient of friction.
47
CHAPTER 4
EXPERIMENTAL RESULTS
4.1 Experimental Wrinkling Conditions
After performing several experiments, web wrinkles were formed on each
of the tested rollers and the polyester films at different spans and velocities. The
only exception was the 50.80 mm (2 in.) roller. This roller only allowed wrinkle
existence for polyester gauge 48, web span of 381 mm (15 in.) and speed of 13
mjmin (42 fpm).
It was noted that prior to wrinkle formation at the web over the roller,
many troughs appeared at the web span (Figure 13). It confirmed what Shelton
(4) established. The compressive stress that buckles the web wrapping a roller is
greater than the compressive stress that wrinkles a web span.
48
Figure 13. Troughs at the web span before wrinkling formation in the web
wrapping the roller. Note that even though the span has troughs, the web
wrapping the roller is uncorrugated. Roller: 31.75 mm (1 % in.), web: Polyester
gauge 48, span: 508 mm (20 in.) and velocity: 14 m/min (47 fpm).
During the tests, wrinkles over the deflecting roller initially appeared
approximately at the central region of the roller. Then, they moved from the
roller center to the left roller edge or right roller edge. At the edges, the wrinkles
disappeared to repeat the cycle with a new wrinkle. Only wrinkles in the
machine direction were formed during these experiments. Figures 14 to 18 show
several examples of web wrinkles that were formed during the experimental part
of this research.
49
Figure 14. Initial wrinkling formation. Wrinkle initially appeared at the roller
center. Roller: 31.75 mm (1 % in.), Polyester gauge 48, span: 508 mm (20 in.) and
velocity: 14 m/min (47 fpm).
Figure 15. Web wrinkling over the roller. Note how the wrinkle initially at the
roller center is moving to the roller edge. Roller: 31.75 mm (1 % in')1 Polyester
gauge 481 span: 508 mm (20 in.) and velocity: 14 m/min (47 fpm).
50
Figure 16. Initial wrinkling formation on the 38.10 nun (1 1/2 in.) roller, Polyester
gauge 92, span: 508 mm (20 in.) and velocity: 29 m/min (95 fprn). Wrinkles were
initially formed at the center of the web.
Figure 17. Web wrinkling over the 38.10 mm (11/2 in.) roller, Polyester gauge 48,
span: 508 mm (20 in.) and velocity: 29 m/min (95 fpm).
51
Figure 18. Web wrinkling over the 50.80 nun (2 in.) roller, Polyester gauge 48,
span: 381 nun (15 in.) and velocity 13 m/min (42 fpm).
The experimental data obtained from the performed tests were recorded
in tables in order to be analyzed and compared with the theoretical calculations.
The recorded data were the minimum experimental tensile loads requir d
to wrinkle the web films over the roller and the deflections at the roller edge and
at the center.
Each of the tables shows the experimental data for a particular web span
and web caliper tested to a set of three different web speeds. The relative center
deflections (difference between roller deflections at the center and at the edge)
are also indicated. As an example, Tables 1 and 2 are displayed in the following
pages. They indicate the experimental data for 31.75 mm (1 % in.) and 38.10 mm
(11/2 in.) rollers at web span of 381 mm (15 in.).
52
Table 1
Minimum Load required to get Web Wrinkling
Experimental Data
Roller: 31.75 mm(1 V4")
Web: Polyester Gauge
48
Web Span: 381 mm {IS")
a) Web velocity= 14.32 mlmin
Test Experimental Center deflection Edge deflection Relative center
load deflection
(N) (mm) (mm) (mm)
1 15.12 0.03 0.01 0.02
2 13.80 0.03 0.01 0.02
3 12.45 0.03 0.01 0.02
4 16.45 0.03 0.01 0.02
5 16.01 0.03 0.01 0.02
6 19.26 0.04 0.01 0.03
7 17.35 0.04 0.01 0.03
b) Web velocity= 29 mlmin
Test Experimental Center defl.ection Edge deflection Relative center
load deflection
(N) (mm) (mm) (mm)
1 21.00 0.04 0.01 0.03
2 22.24 0.04 0.01 0.03
3 22.00 0.04 0.01 0.03
4 17.40 0.04 0.01 0.03
5 22.70 0.04 0.01 0.03
6 18.00 0.04 0.01 0.03
7 18.70 0.04 0.01 0.03
c) Web velocity= 42.4 mlmin
Test Experimental Center defl.ection Edge defl.ection Relative center
load deflection
(N) (mm) (mm) (mm)
1 23.13 0.04 0.01 0.03
2 24.02 0.04 0.01 0.03
3 21.00 0.04 0.01 0.03
4 22.70 0.04 0.01 0.03
5 22.24 0.04 0.01 0.03
6 21.35 0.04 0.01 0.03
7 23.60 0.04 0.01 0.03
53
Table 2
Minimum Load required to get Web Wrinkling
Experimental Data
Roller: 38.10mm
(11/2")
Web: Polyester Gauge 48
Web Span: 381 mm (15")
a) Web velocity= 13 mlmin
Test Experimental Center deflection Edge Relative center
load deflection deflection
(N) (mm) (mm) (mm)
1 18.23 0.03 0.02 0.01
2 19.60 0.04 0.02 0.02
3 21.00 0.04 0.02 0,02
4 24.02 0,05 0.02 0.03
5 20.50 0,04 0.02 0.02
6 16.46 0.03 0.02 om
b) Web velocity= 27 mlmin
Test Experimental Center deflection Edge Relative center
load deflection deflection
(N) (mm.) (mm) (mOl)
1 22.70 0.04 0.02 0.02
2 25.35 0.04 0.02 0.02
3 25.80 0.05 0.02 0.03
4 25.35 0.04 0.02 0.02
5 23.57 0.04 0.02 0.02
6 26.24 0.05 0.02 0,03
c) Web velocity= 41 mlmin
Test Experimental Center deflection Edge Relative center
load deflection deflection
(N) (mm) (mm) (mm)
1 29.36 0.05 0.02 0.03
2 26.24 0.04 0.02 0.02
3 25.35 0.04 0.02 0.02
4 27.57 0.04 0.02 0.02
5 28.47 0.05 0.02 0.03
6 22.70 0.05 0.02 0.03
54
4.2 Experimental Web and Roller Roughness
Table 3 registers the experimental data obtained for the surface roughness
of the different web calipers and tested rollers:
Table 3
Web surface roughness It Rq, web It
(104 mm)
Test gauge 48 gauge 92 gauge 142
1 6.6 6.6 3.3
2 3.0 2.5 3.6
3 2.3 9.4 5.3
4 6.1 10.2 7.6
5 4.8 1.8 4.1
6 3.0 2.0
7 3.3 5.8
Average= 4.2 5.s 4.8
Roller surface roughness It Rq, roller It
(104 mm)
Test 31.75mm diam. 38.10 mm diam. 50.80 mm diam.
1 15.2 10.9 8.9
2 15.0 19.8 14.5
3 17.5 12.2 18.3
Average= 15.9 14.3 13.9
55
4.3 Coefficient of Friction between Web and Roller
The following experimental table was obtained from the test performed
on each roller and web caliper and using the brandbrake equation:
Table 4
Coefficient of friction webroller
Test
1 1/4"
1 1/2 "
2 "
56
Static Coeff.
0.24
0.27
0.30
CHAPTERS
ANALYSIS OF EXPERIMENTAL DATA
5.1 Initial theoretical computations.
Initial theoretical computations were done based on the method described
in 2.4.1 of Chapter 2. The goal of these calculations was to find simultaneous
fulfillment of the theoretical wrinkling conditions (stress, compatibility and
frictional) .
The mathematical expressions to perform the initial computations for each
roller diameter, web caliper, span and web speed are indicated in Chapter 2. The
iterative process explained in the proposed methodology was accomplished by
using a spreadsheet in the software EXCEL.
As an example of the initial computations, results and graphs for the case
of 31.75 mm (1% in.) roller, polyester gauge 48, span of 381 mm (15 in.) and
velocity of 14.32 m/min (47 fpm) are given below:
57
Results of Theoretical Computations
Roller: PVC Diam.: 31.75 mm, Web: Polyester, gauge 48, 5pan: 381 mm
Web veloc.= 13 m/min
Min. Load (Rw theor.)=
Traction Coeff. 51=
911
20.69
Lbf or
N/mm"2
0Al0
40.51
and 52=
Newtons
0.0008 N/mm"4
·'00
1 60
web w idlh/roller length (mm)
100
Ocr
Figure 19. Theoretical stress wrinkling condition. Lateral compressive stress due
to web traction (J" y vs. Critical buckling stress (J"tr
58
'00
,_uw'(mm) =ut (mm) I
00'00
סס. 0 OO I
..0 ·20 20 '0
·0.0100 .
·0.0200 I
·0.0300 I
0 oeoo J
web width/roUer length(mm)
Figure 20. Theoretical compatibility wrinkling condition. Web deformation uw
vs. roller deflection ur .
0.25
02
EE~
.s: 015
Ii
" .!
'c
" 0.1
i..
"~ o.os
o .
0 10 20 JO 50 1IO 70
web width/roller length (mm)
Figure 21. Theoretical frictional wrinkling condition. Lateral surface force f~ VS.
lateral internal force fi .
59
Similar theoretical computations and graphs were done for each roller size
and web caliper at fixed spans and velocities. For all calculations the following
were noted:
1) The frictional wrinkling condition is overly satisfied when the
maximum compressive internal stress O"y (at the roller center) is equal
to the critical buckling stress 0" cr ' and web deformation at the contact
line is very close to roller deflection. Therefore, meeting the stress and
compatibility wrinkling conditions, the frictional condition was
automatically satisfied.
2) For each roller size, web span, and web caliper, the results for
wrinkling conditions did not vary for the velocities used in the
experiments.
5.2 Comparison between experimental results and initial theoretical
computations.
A comparative study between the experimental data and the initial
calculations was performed in order to analyze the results and accuracy of the
proposed model.
60
5.2.1 Minimum load Rw required to get web wrinkling.
Figures 22 to 27 compare the experimental minimum load required to get
wrinkles Rw exp. and the corresponding theoretical value. These are based on the
initial computations Rjheor. for 31.75 mm (11/( in.) roUer, polyester gauge 48
and 38.10 rom (1 1/2 in.) roller, polyester gauges 92 and 142. Although the figures
below represent only six cases out of thirteen computed and tested in the lab,
they describe the pattern observed for all cases.
It was noted that the minimum theoretical wrinkling loads were, always,
more than double the experimental values.
45
40 xxxxxxx
~ JS
~ JO ..'9" 25
<II
~ 20
c:::
~ 15
Q:i Z 10
o
o 6
Tests
. RWI'''r29m!mln
Rwl'.r....2m/min
xRwthtor.
Figure 22. Minimum theoretical load R.•Jheor. according to initial calculations vs.
experirnentalloads R.... exp. for 31.75 mm (1 % in.) roller, polyester gauge 48,
span = 381 rom at 14, 29 and 42 rn/ min.
61
70
60
10
o·
o
)( )( )( )( )( )( X
 ~13m'rrin I
 ~29m1ninl I
~= = =l RMxp41 IlV'mn  3 6 8
Tests
Figure 23. Minimum theoretical load R...theor. according to initial calculations Ys.
experimental loads Rw expo for 31.75 mm (1% in.) roller, polyester gauge 48,
span = 508 rnrn at 13, 29 and 41 m/min.
120
\00
)( )( )( )( )( )(
20
o
o
.. .
Tesls
I
:+ AWIILp..104 mlmln
I R.....p.27 mtmln
__ RweI~'mlmln'
~RWIheo,
Figure 24. Minimum theoretical load Rwtheor. according to initial calculations vs.
experimental loads Rw exp. for 38.10 mm (11/2 in.) roller, polyester gauge 92, span
=381 mm at 14, 27 and 41 m/min.
62

'60
140
120
"0 '00
CO
..Q
~ 80
'iii
c:
~ 60
Q)
Z
40
20
o
o
)( )( )(
Tests
)( )( )(
1 Rwhp13m1mln I
I _ Rwhp28 mlmln I
I  RwOxp41 mlmln II
)1
Figure 25. Minimum theoretical load R•.,theor. according to initial calculations vs.
experimentalloadsRw expo for 38.10 mrn (11/2 in. ) roller, polyester gauge 92,
span =508 mm at 13,28 and 41 m/min.
'10
160 I
140
)( )( )( )( )( )(
~ 120
"0
~ 100
~
.~ 80
~
QJ eo z
40
20
o 'o
8
Tests
M Rwtneor
Figure 26. Minimum theoretical load Rwtheor. according to initial calculations vs.
experimentalloadsR", expo for 38.10 mrn (11/2 in.) roller, polyester gauge 142,
span =381 mrn at 13, 30 and 40 m/min.
63
300
250
z
:;200
'"o~
1501
'ii.j
<=
CD
 1001 4i
Z
50.
o
o
)( )( )( )( )( )(
' _JlP12m1mln II
+ R'wexl>)Q mtmln
I ~ •~ • I~ _Rwlh"'"
6
Tests
Figure 27. Minimum theoretical load Rwtheor. according to initial calculations vs.
experimental loads Rw exp. for 38.10 mm (11/2 in. )roller, polyester gauge 142,
span = 508 rom at 12, 30 and 39 m/min.
5.2.2 Roller deflection at the center U r('('lIIer under wrinkling conditions.
Figures 28 to 32 exhibit the comparison between theoretical roller center
deflections according to initial computations and the experimental relative
deflections obtained in WHRC for the same cases mentioned in 5.2.1.
For all cases displayed, the theoretical roller center deflections were very
high when compared with the experimental relative roller deflections at the
center. The deviations between theoretical and experimental values were in the
range from 2 to 4.
64
0.06
xxxxxxx 0.05
E 00. .s
c
o 003
~
"" OJ
Cl 0.02 : : : : ~ •
+ ur expH m/m,,," ,
t ~ .. r cllp29 m/min
'. ur4"~p42m/m'n
I
X urLtw.'Or.
o .o
• Tests
6
Figure 28. Theoretical roller center deflection U ,Ihm, according to initial
computations vs. experimental relative deflections u,_exp for 31.75 mm (1 %. in.)
roller, polyester gauge 48, span = 381 mm at 14, 29 and 42 m/min.
0.08
)( )( )( )( )( )( )(
007
0.06
E 005  Uf up13 rrvmln .s
c ur .xp.28 mlmln i 0 0.0<  U
QJ
0;: ur IUtp.4 t mfmln I QJ 0.03
0
002  \Jr Iheo,
001
0
0 6
Tests
Figure 29. Theoretical roller center deflection U'fhe", according to initial
computations vs. experimental relative deflections u,exp for 31.75 mm (1 %. in.)
roller, polyester gauge 48, span =508 mm at 13, 29 and 40 fpm.
65
0.1
0.011
0.08
0.07
)( )( )( )( )( )(
E.s 0.06
c:
.2 0.05 I 004
0.03
0.02
001
o
o
Tests
_ur~'~mIlnIn I
I
_ ur .~27 mtmIn
1. urup41 rnImk"l
I_IKlIlOor.
Figure 30. Theoretical roller center deflection U,,hear according to initial
computations vs. experimental relative deflections ur  exp for 38.10 mm (1 V2 in.)
rollerJ polyester gauge 92J span =381 mm at 14J 27 and 41 rn/min.
0.1.
0.12
01
x )( )( )( )( )(
E.s 0.0&
c: o~
006
c;:::
ell o
O~
002
o
o
Tests
+ Uf ftllp13 mlmln
ur expo.' mlmln
Hur theor
Figure 31. Theoretical roller center deflection UrIlwor according to initial
computations vs. experimental relative deflection ur  exp for 38.10 mm (1 1/2 in.)
rollerJ polyester gauge 92J span = 508 mm at 13J 28 and 41 m/min.
66
0.18
0.14
012
)( )( )( )( )( )(
I 01
c:
o ooe
~
~ 0.08 1
O.a.
0.02
o
o
Tests
+ urelCP 13m1min
. Uf e~O I'T'I/mln
I ..ur e.xp40 mtmin
_urll\eor.
Figure 32. Theoretical roller center deflection UHiI,'or according to initial
computations vs. experimental relative deflections ur  exp for 38.10 mm (11/2 in.)
roller, polyester gauge 142, span =381 mrn at 13, 30 and 40 rn/min.
0.25
02 I
)( )( )( )( )( )(
E
§. 0'5
c: a~
01
Ql o
0.05
o
o
Tests
~ urexp12m1mln
~ ur elCp30 mlmln
ur up3G mlmln
~lKlheOf
Figure 33. Theoretical roller center deflection Urthcur according to initial
computations vs. experimental relative deflections ur  cxp for 38.10 mm (11/2 in.)
roller, polyester gauge 142, span = 508 mm at 12,30 and 39 m/min.
67
5.2.3 Discussion
The comparison between the theoretical loads and deflections, based on
the initial computations, and the experimental values obtained in the lab did not
match. These differences make the predictions unacceptable.
Three (3) important factors may be observed:
a A large difference between the minimum wrinkling theoretical loads
R.,)heor. and the experimental values Rw exp .. The theoretical loads
were very high compared with the experimental ones, in some cases
the predicted load would be impossible to reach at the testing machine,
but / during the corresponding experiments, we acquired wrinkles
under less loading conditions.
b Very high theoretical roller deflections at the center Urt""or compared
with the experimental relative deflections Urexp . This notable
theoreticalexperimental deviation for the roller deflection can be
interpreted as follows: the roller should be subjected to a load greater
than the minimum required to wrinkle the web in order to meet the
theoretical stress and compatibility wrinkling conditions.
c The large high values for the lateral surface force 111 at the roller
center, with respect to the internal forces ~ / seem to support the
inference about an excessive computed load. A high RJheor.impIies
68
high Sl and S2 that increase the value of fIJ according to equation
[35].
In order to analyze in detail our hypothesis, we computed the internal
web stress in CMD (Y y (equation [15]) and the roller center deformation UrCClller
(equation [31]) using the experimental loads obtained during the tests.
Figures 34 to 39 show the comparison between the internal web stress
with the roller under the experimental loads, and the respective critical buckling
stress for roller sizes 31.75 rnrn (1 % in.) and 38.10 nun (11/2 in.) and web
thickness of gauge 48, 92 and 142 as follows:
..8 ,. )( )( )( )( )( )( )(
'.4
___...__.,. lh.....' '2 rn/mlil
04
02
o·
o
Tests
Figure 34. Theoretical internal stress in CMD (Y y vs. critical buckling stress (Y cr at
experimental net forces R... exp. for 31.75 mm (1 % in.) roller, polyester gauge 48
and span = 381 nun.
69
16 )( )( )( )( )( )( )(
1.2 i
~'
~ 08,
.I:l
U) 06"t o. 1
0.2 !
o
o
~:=:l
Tests
I
II
~.)'lMa,,·')m/ll\in I
_'Y"·,,,,·29O'/"'" I
asy lht..or.&1 mJmin I _ocr I!
II
Figure 35. Theoretical internal stress in CMD a y vs. critical buckling stress a r
at experimental net forces Rw expo for 31.75 nun (11/.~ in.) roller, polyester gauge
48 and span= 508 mm.
)( )( )( )( )( )(
2.'
'0"
::::E
~ " <£fl , • • •~
O.
0
0 '"
Tests
Figure 36. Theoretical internal stress in CMD a y vs. critical buckling stress a"r at
experimental net forces R", exp. for 38.10 mm (11/2 in.) roller, polyester gauge 92
and span =381 mm.
70
_w,

300
,.'"
_ ~.OO
III
tl.
::!; "i "0
~
Vi 100
0"" .
000 •o
)( )( )(
Tests
)( )( )(
  I
~.y~..U"'I""!1
___._.ytMo.... lI_/mlJl II
"'1) u. .... "/"'''''
i
II
Figure 37. Theoretical internal stress in CMD a y YS. critical buckling stress a c,
at experimental net forces Rw expo for 38.10 mrn (11/2 in.) roller, Polyester gauge
92 and span= 508 mm.
• 50
4.00
3.50 .
)( )( )( )( )( )(
~ 3.00
III
tl.
~ 250
~'" 200
(/) 1.50
1.00 I
050 I
0.00 .o
Tests
~.ylh'Ot.'3mtmln
____ .~ thlar.·30 mtmln
Iy Iheor.40 mlmtn ,I
'_Ia II
I
Figure 38. Theoretical internal stress in CMD a y YS. critical buckling stress a,., at
experimental net forces Rw expJor 38.10 mm (11/2 in.) roller, Polyester gauge 142
and span= 381 mrn..
71
'00
"" 1
)( )( )( )( )( )(
oso J
000 •
o
Tests
_·, ...... ·"·'·.. 11
=~::::::ol
_N' !
I J
I
Figure 37. Theoretical internal stress in CMD a y vs. critical buckling stress a a
at experimental net forces Rw exp. for 38.10 nun (11/2 in.) roller, Polyester gauge
92 and span= 508 nun.
<.50
350
)( )( )( )( )( )(
_300
a'".
~ 250
'"~ 200
U5 150
1,00 ~
050
0.00 .
o
Tests
+&y IMOf 13 mlmln
_.ythOOf.·30mlmln 0II
_IaIy the« "'0 rntmln
Figure 38. Theoretical internal stress in CMD a y vs. critical buckling stress a L ,. at
experimental net forces R..o exp.for 38.10 nun (1 1/2 in.) roller, Polyester gauge 142
and span= 381 nun.
71

4.00
3.50
)( )( )( )( )( )(
3.00 rn a.
~ 2.50.
<II
~200~
ii5
1.!.O ;
100
0.50
0.00 +o
Tests
__..yyIheo<.·12~ 11_.·30 mlmln _.yU.·3V mlmln
loc<
Figure 39. Theoretical internal stress in CMD a y vs. critical buckling stress a cr at
experimental net forces Rw exp .for 38.10 mm (1 V2 in.) roller, polyester gauge 142
and span = 508 mm.
As can be seen from figures 34 to 39, the internal web stress a y is not
large enough, itself, to produce simultaneously stress, compatibility, and
frictional wrinkling conditions. For that reason, the iterative predictive
computation on our spreadsheet needed to assume a greater load (than the
minimum required for wrinkling) to achieve the simultaneous presence of the
three mentioned wrinkling conditions.
The theoretical roller center deflections with the roller under the
experimental loads, and the experimental relative deflections are compared in
Figures 40 to 45.
72

0.035
~url"ItOf'.I04 mI"*,
~ur1h40r.• ~ mJrnIn
. ur 1xp28 rntmin
_ uro"",14""_ I!
I
_____ ur exp..t2 ""min
+ur lheor 412 mtmin
~ 0025
EE~
002
co
'~ 0.015
0:
ell
Q 0,01
0.03
0.005
o
o
Tests
Figure 40. Theoretical (ur.theor.) vs. experimental (ur exp.) roller center
deflections at experimental loads Rw exp. for 31.75 rnm (1 % in.) roller,
polyester gauge 48 and span = 381 mm at 14,29 and 42 m/min.
0,0~5
003
10025
~ O.ol
c:
o
'lj
lj 00'5
0:: 8 00,
0.005
o
o
. ur"JlpIl m/ml/\ '
,I
'Ut IhPor41 m/mln I
I
Tests
Figure 41. Theoretical (ur.theor.) vs. experimental (ur exp.) roller center
deflections at experimental loads Rw exp. for 31.75 nun (1% in.) roller, polyester
gauge 48 and span =508 nun at 13, 29 and 41 m/min.
73

0.06
0.05
_urup.UmImin,
_urur.1._'
_ur_.·27_
 .... exp4,
_",_.41 mlmIn
001
o
a
Tests
Figure 42. Theoretical (U r. theor. ) vs. experimental (u r exp.) roller center
deflections at experimental loads Rw exp. for 38.10 nun (11/2 in.) roller,
polyester gauge 92 and span = 381 mm at 14,27 and 41 m/min.
006
005
E 0.04
.§.
"o 003
'D
~
<;;: Cl 002
0.01
o
o
+ ur bp13 mfmin I
____ urlhfoo,.·13 "l/mm
~urIheor.28 m/nlln
_____ u, t'llp41 m/rmn
'u.r lheo,.tl nl/min
Figure 43. Theoretical (U r.thear. ) vs. experimental (u r exp.) roller center
deflections at experimental loads Rw exp. for 38.10 nun (11/2 in.) roller,
polyester gauge 92 and span =508 nun at 13, 28 and 41 rn/ min.
74
oOIl
0.08 •
0.07 .
0.02
0.01 ~
I
o !
o
Tests
,I
... url'xp13m/rnm
41.11 ' ...."'Or.·U mjmm
. ur t'Jl:pJO rn/nUn
_"nn.or.JOm/m;n I
I _.r..pI(lm/m<n :
I_·,''''",·~m/mln II
I
Figure 44. Theoretical (ur.theor.) vs. experimental (u r exp.) roller center
deflections at experimental loads Rw expo for 38.10 mm (11/2 in.) roller, polyester
gauge 142 and span = 15 inches at 13, 30 and 40 m/min.
0.09
0.06
0.07
E 006 e
7 005
o
.~ 0.000·
~ 0.ll3
Cl
0.02
001
o
o
Tests
1_ .rap12m/min II
 ur theor.·12 mlmln
urt"X~JO rn/m'n
W ur thoor .·30 n1/ nvn
I _ur""p:l9m/lTln
I "urlheor.•3'9m/rnln I
Figure 45. Theoretical (ur.theor.) vs. experimental (u r exp.) roller center
deflections at experimental loads Rw expo for 38.10 mm (11/2 in.) roller, polyester
gauge 142 and span =508 mm at 12, 30 and 39 m/min.
75
From Figures 40 to 45, we can observe that under the experimental loads,
the proposed method gives us an acceptable correlation between theoretical and
experimental roller deflections at its center. Now, the problem is focused on
correcting the necessary stress that satisfies the stress wrinkling condition and
correlates with the experimental data.
5.3 Modified Stress Wrinkling Condition.
Duvall (6) found, in his experimental research, that the theoretical roller
bending strain matched better than Shelton's proposed strain (strain caused by
steering of the deflected roller) for his particular model.
In our case, we can not simply substitute the internal web stress in CMD,
0"\ ' by the roller bending stress because of the likeness between theoretical and
experimental roller center deflections shown in figures 40 to 45.
The theoretical roller deflection u, (equation [31]) is a function of roller
properties and dimensions, web thickness and the numerical coefficients 81
and 52 of the parabolic web traction q. This web traction is responsible, at the
same time, for the roller deflection and the web deformation. Therefore our
assumption is based on the web is subjected simultaneously to the internal stress
76
a y that is stretching it, plus the roller bending stress (J' h that is imparted to the
web in contact with the roller, so that the web can assume the same roller shape.
The maximum compressive stress due to roller bending is located on the
top surface of the roller and contributes, with the internal web stress, to surpass
the critical stress for buckling.
Now, the stress wrinkling condition becomes:
[42]
where a ty is the total stress in the cross machine direction and is given by
the following expression:
[43]
where a,. is the internal web stress in CMD (equation [15]) and a h is the
maximum bending roller stress.
The expression for the maximum compressive stress on the roBer surface
due to bending is:
)
d2ur
a h =EwCMI) (R +t w 2
dy
[44]
where Ew_cMDis the modulus of elasticity in the cross machine direction.
77
In the development of this study, we have considered the web as an
isotropic material; accepting that this assumption is not rigorously true, we can
assume:
E w  CMD == EwMD {45]
New computations based on the modified stress wrinkling conditions
(equation [42]) in the proposed method were done to find the minimum
theoretical loads to achieve wrinkles and the roller center deflections. These
theoretical values are compared with the experimental ones in figures 46 to 57 as
follows:
JO
z,
"0
Cll 20 g
~
.~ 15
$
ii z 10
o
o
Tests
! e RWllxp4 14 mImi" 1_ Rw.'r29m/ m'n I
HW('J(r42 m/min
)(RwlN.'or
Figure 46. Minimum theoretical load R..Jheor. considering total stress in CMD
(Y,y vs. experimental loads R", expo for 31.75 mm (1 % in.) roller, polyester gauge
48, span =381 rnrn at 14, 29 and 42 mjmin.
78
25
20·
.. ~"~
II
1~~2IIfTVnW1,
__, ....... 1 
o
o
Tests
Figure 47. Minimum theoretical load R,)heor. considering total stress in CMD
eYe> vs. experimental loads Rw expo for 31.75 rnm (1% in.) roller, polyester gauge
48, span =508 mrn at 13, 29 and 41 rn/ min.
70
eo!
"'.
____ R~21~
Q)
z
20 _R_
10
o
o
Tests
Figure 48. Minimum theoretical load Rwtheor. considering total stress in CMD
eY')' vs. experirnentalloads R... exp. for 38.10 nun (1 V2 in.) roller, polyester gauge
92, span = 381 rnm at 14,27 and 41 rn/min.
79
70
'"
 10,
o
o
Tests
Figure 49. Minimum theoretical load R•.,theor. considering total stress in CMD
(Jty vs. experimental loads Rw exp. for 38.10 mm (11/2 in.) roller, polyester gauge
92, span = 508 mm at 13,28 and 41 m/min.
..
'"
70
~ eo,
"C '
~ so i
~
.~ 40 I
~
OJ 'OJ
Z
'0
o
o
Tests
"
I
I I
Figure 50. Minimum theoretical load R,.,theor. considering total stress in CMD
(J't)' vs. experimental loads Rw expo for 38.10 mm (1 1/2 in.) roller, polyester gauge
142, span =381 mm at 13,30 and 40 m/min.
80
90
00
'0
~oo
".0. .2 SO
.!!
~ 40
a:; 30
Z
,0  10
o
o
Tests
Figure 51. Minimum theoretical load R.,.,theor. considering total stress in CMD
0"1)' vs. experimental loads Rw expo for 38.10 nun (11/2 in.)roHer, polyester gauge
142, span =508 nun at 12, 30 and 39 m/min.
0035
003
0025
E
§. 002
co
'g 0.015
~
o 001
0005
o
o
;;;;;z:
Tests
)C l.Ir 1n.·Hr
Figure 52. Theoretical (Ur (heor.) vs. experimental (u r exp.) roller center
deflections considering total stress in CMD 0",)' for 31.75 mm (1% in.) roller,
polyester gauge 48 and span = 381 mm at 14,29 and 42 m/min.
81
00.
0.02~ )( )( )( )( )( )( )(
E 002
.§.
c:
0 O.O'~
~
0::
Ql a 0.01
0.005
0
0
Tests
Figure 53. Theoretical (U r theor.) vs. experimental (u r exp.) roller center
deflections considering total stress in CMD (Y,y for 31.75 mm (1 % in.) roller,
polyester gauge 48 and span =508 mm at 13, 29 and 41 rn/ min.
006
0.05
E 00•
.§.
c: o 003
"t~
0'"::
~ 002
00'
o
o
Tests
____ur ex.p21 mlmin
ur e_p.4' m/mlfl
~urlhtl'or
Figure 54. Theoretical (U r theor.) vs. experimental (ur exp.) roller center
deflections considering total stress in CMD a ty for 38.10 mm (11/2 in.) roller,
polyester gauge 92 and span =381 mm at 14, 27 and 41 m/min.
82
0.06
0.05 •
E 0.04 .s
c
.2 003
~
~
~ 002
001
o .o
Tests
~ ur exp13 mlmin
~ ur exp:ze mlmln
. Uf'~1 mtmln
~urtheor.
1
Figure 55. Theoretical (U r theor.) vs. experimental (u r exp.) roller center
deflections considering total stress in CMD (jIY for 38.10 rom (1 1/2 in.) roller,
polyester gauge 92 and span =508 rom at 13, 28 and 41 m/min.
009
0.08
0.01 .
E 0.06·
E~
0.05
o~
0.04
~r3 0.03
002
001
o
o
Tests
+ IoH' ellp13 mJm",
____ ur u~30 mlmln
ur upo.CO mlmln
H ur IMor
Figure 56. Theoretical (U r theor.) vs. experimental (u r exp.) roller center
deflections considering total stress in CMD (jly for 38.10 mm (11/2 in.) roller,
polyester gauge 142 and span =381 mm at 13, 30 and 40 m/min.
83
0.09
0.0Il
0.07
E 0.0iI
E
;; 0.05
o .'" ~ 0.04
0::
~ 0.0.
0.02
0.0'
a
a
T.lta
_ ,",_12m/min
____ ur .~)O mtrrMn
~urlheor
Figure 57. Theoretical (urtheor.) vs. experimental (ur exp.) roller center
deflections considering total stress in CMD CJ(y for 38.10 nun (1 1/2 in.) roller,
polyester gauge 142 and span = 508 mm at 12,30 and 39m/min.
Table 5 shows the comparison between the theoretical wrinkling loads
and the mean values for the experimental wrinkling loads obtained from the
testing machine. The standard deviations for the experimental values are also
indicated.
84
Table 5
Theoretical wrinkling loads Rw theor. vs. mean values for the experimental loads Rw expo
(for all tested cases)
Mean values for Rwexp Stand. dey. of Rwexp
Case Description Rwtheor. Rwexp. SD Rwexp.
(Newtons) (Newtons) (Newtons)
1 Roller: 31.75 mm, Web: 17.17 19.51 0.77
gauge 48, span:381 mm.
2 Roller: 31.75 mm, Web: 19.70 19.00 0.58
gauge 48, span:508 mm.
3 Roller: 31.75 mm, Web: 34.38 37.26 1.69
gauge92, span:381 mm.
4 Roller: 31.75 mm, Web: 39.05 39.57 1.03
gauge 92, span:S08 mm.
5 Roller: 31.75 mm, Web: 55.29 56.61 1.84
gauge 142, span:381 mm.
6 Roller: 31.75 mm, Web: 62.76 71.04 2.56
gauge 142, span:508 mm.
7 Roller: 38.10 mm, Web: 21.13 23.80 0.80
gauge 48, span:381 mm.
8 Roller. 38.10 mm, Web: 23.93 23.62 0.73
gauge 48, span:508 mm.
9 Roller. 38.10 mm, Web: 42.08 46.53 1.51
gauge 92, span:381 mm.
10 Roller. 38.10 mm, Web: 46.62 48.31 1.43
gauge 92, span:508 mm.
11 Roller. 38.10 mm, Web: 66.05 63.38 2.23
gauge 142, span:381 mm.
12 Roller. 38.10 mm, Web: 74.82 67.88 1.67
gauge 142, span:508 mm.
13 Roller: 58.10 mm, Web: 90.29 94.74 0.98
gauge 48, span:381 mm.
As can be seen in the figures 46 to 57 and Table 5, the assumption that the
web is bearing simultaneously the internal stress in CMD and the bending stress,
agrees more closely with the experimental results. This gives a better relationship
between the theoretical wrinkling loads and roller center deflections and the
experimental ones.
85
d
Figures 58 to 60 show how the frictional wrinkling condition is satisfied
for the cases studied. No relevant change in the lateral surface force f was
IJ
observed at the different tested velocities for each roller.
01200 .
,. 01000 • j 00800 .
.~ 0.D600
10.0<'00 i 0.0200 +.j+t"Ik1
tfu14 mlmLn.•pan011 mQl
{u·19 m/m.""plln~11mm
ru·42 m/mln.,pan"'llmm
I
.. (u·13m /mlA.•'P an "Qlmm I
_ru·19"'m,".•pan~<llmm I
___fll'~ I m/mlnipin 08mm '
0.0000 .
o 10 20 3D ~o 50 60 70 80
web wkhhfro lief length (mm)
+r.
Figure 58. Lateral surface force f" vs. internal force .I: considering O'ty for 31.75
rom (1 % in.) roller and polyester gauge 48 at the tested velocities.
86
03000
10.2500 ~
........
Z
~02000 .
.J:;
~
c:
~ 01500
'r:
:l
Qj 01000
c.. j 0.0500
סס. 0 OO •o
w w ~ ~ ~ ~ ro ~
web widthjroller length (rrun)
~(\l·1" m/m'" 'p"n,l8'lmrn
............ fu·'D m/min .....nJI·llII\m
(un m/mln lfpanJ81mm
.. 'ul:lm/nul'l......n~mm
41,,28 tn/min 'rn508mm
e.fu....l m/mln.p.n..nmm
+1,
Figure 59. Lateral surface force f
p
vs. internal force f considering ()ty for 38.10
mm (11/2 in.) roller and polyester gauge 92 at the tested velocities.
04500
"
E 0.4000
E ,~
.......... 0.3500
Z
~ 03000
.J:;
~ 0.2500
~
.~ 02000
~ 015OO+++'l1c++
~
~ 01000
~
..E 0.0500
סס 0 oo
o 10 20 30 40 ~ ~ 70 ~
web widthjro~erlength (rrun)
4ru.1' m/mln8rnVUmm
__.__'u.1I1 m/mtnllran.131mm I
fu ...llm!rnlnlip.ln18lmm I
f,,12 m/nlln ,.,.,..YJI"nm
..,,,1(1 m/,nln IJran!l)lmm
+'u.3"1m/mfn!lpan.~mmI
+10
Figure 60. Lateral surface force f
p
vs. internal force f, considering (Jty for 38.10
nun (1 1/2 in.) roller and polyester gauge 142 at the tested velocities.
87
•
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
The intent of this study was to develop a practical model to predict web
wrinkling due to roller deflection based on the theory of elasticity, by using
Shelton's theory and previous works in this area. The result was a theoretical
procedure that allows an acceptable accuracy within the experimental data. Some
of the important conclusions that were drawn from the theoretical and
experimental work described above are given below.
1. A parabolic web traction in the machine direction is responsible for the
roller deflection and web deformation in the same direction. When this
traction reaches a critical value, wrinkles start appearing within the
web wrapping the deflected roller.
2. The minimum parabolic web traction that produces web wrinkling
over the roller is unique and its distribution along the web width is
governed by two (2) numerical coefficients developed in this study
(51&52).
3. As a consequence of that parabolic web traction, three (3) simultaneous
conditions are required to be present for web wrinkles over a deflected
roller, they are:
88
..
• The maximum total web stress in the cross machine direction should be
greater or equal than the critical buckling stress for a cylindrical web shell
without internal pressure. This maximum total stress is the result of the
maximum internal web stress due to the parabolic web traction, and the
maximum roller stress due to bending.
• The web must assume the shape of the deflected roller. For this reason, at the
contact area, the web deformation in the machine direction should match
with the roller deflection.
• The frictional force between web and roller should be greater than the
internal force due to web buckling. This frictional force is affected by the
coefficient of friction webroller and by the thickness of the air film between
web and roller.
The model given above describes the general situation of the wrinkling of
webs due to roller curvature. Further studies should consider additional web and
roller sizes and materials, spans, and velocities that can have a major effect on
the web lateral slippage over the roller.
Since most webs are not isotropic materials, the effect of web anisotropy
on the web wrinkles due to roller curvature should also be investigated in future
studies.
89
REFERENCES
1. Good, J. K. and Delahoussaye, R. D., "Analysis of Web Spreading Induced by
The Concave Roller", Proceedings of the Second International Conference on
Web Handling, June 69, 1993, Oklahoma State University, Stillwater, OK.
2. Hakiel, 2., "From predictive Models to Profitability in The WebHandling
Industry", Proceedings of the Third International Conference on Web
Handling, June 1821, 1995, Oklahoma State University, Stillwater, OK.
3. Good, J. K., Gehlbach, L. S. and Kedl, D. M., "Predicting Shear Wrinkles in
Web Spans", TAPPI Journal, August, 1989.
4. Shelton, J. J., "Machine Direction Troughs in Web Spans and Corrugations in
Wound Rolls", Web Handling Research Center Oklahoma State University,
August, 1991.
5. Timoshenko, S. P. and Gere, J. M., "Theory of Elastic Stability" McGrawHill
Book Company, New York, Second Edition, 1961.
6. Duvall, M. G. " A Study of Web Wrinkling Due to Roller Curvature", Thesis,
Department of Mechanical and Aerospace Engineering, Oklahoma State
University,1997.
7. Ugural, A. C. and Fenster, S. K., "Advanced Strength and Applied Elasticity"
Elsevier North Holand Publishing Co., New York, Third Edition, 1975.
8. Rivello, R. M., "Theory and Analysis of Flight Structures", McGrawHill Book
Company, New York, First Edition, 1969.
9. Good, J. K., Kedl, D. M. and Shelton,}. J., "Shear Wrinkling in Isolated
Spans", Proceedings of the Fourth International Conference on Web
Handling, June 14, 1997, Oklahoma State University, Stillwater, OK.
10. Knox, K. L. and Sweeney, T. L. , "Fluid Effects Associated with Web
Handling", Industrial Engineering Chemical Process Design development,
Vol. 10, No 2, pp. 201205.
90
VITA
Pedro Arias
Candidate for the degree of
Master Science
Thesis: PREDICTION OF WEB WRINKLING INDUCED BY ROLLER
DEFLECTION
Major Field: Mechanical Engineering
Biographical:
Personal Data: Born in Barcelona, Venezuela, On June 4, 1959, the son of
Pedro Arias and Ana Diaz de Arias.
Education: Graduated from T. A. Calatrava High School, Puerto La Cruz,
Venezuela in June 1976; received Bachelor of Science degree in
Mechanical Engineering from Universidad Central de Venezuela,
Caracas, Venezuela in June of 1981. Completed the requirements for
the Master of Science degree in Mechanical engineering at
Oklahoma State University in July, 1998.
Experience: Mechanical Engineer assigned to Gas Dpt. in Meneven SA,
Venezuela from 1981 to 1984, Project Engineer in Polar, Venezuela
from 1984 to 1987, Mechanical Engineer in Petroleos de Venezuela,
S.A. (PDVSA) from 1987 to 1996, Graduate Research Assistant in the
Web Handling Research Center at Oklahoma State University from
summer 1997 to present.