Experimental and Computational Investigation of the Roll Forming Process

DOCTORA L T H E S I S

Division of Material Mechanics

Experimental and Computational Investigation of the Roll Forming Process

Michael Lindgren

ISSN: 1402-1544 ISBN 978-91-7439-031-5

Lule University of Technology 2009

Michael Lindgren E

xperimental and C

omputational Investigation of the R

oll Forming Process

ISSN: 1402-1544 ISBN 978-91-86233-XX-X Se i listan och fyll i siffror dr kryssen r

Experimental and Computational Investigation

of the Roll Forming Process

Michael Lindgren

Lule University of Technology Division of Material Mechanics

Printed by Universitetstryckeriet, Lule 2009

ISSN: 1402-1544 ISBN 978-91-7439-031-5

Lule 2009

www.ltu.se

Preface

This work has been carried out at Dalarna University. The nancial supportwas provided by the Swedish Foundation for Knowledge and Competence De-velopment (KK-stiftelsen), ORTIC AB, Jernkontoret and Dalarna University.

I would like to thank the following people:

My supervisor, professor Lars-Erik Lindgren for his experienced guidance andcontinual support.

Dr Lars Ingvarsson, for sharing his great experience in roll forming with me.

All colleagues at Dalarna University and ORTIC AB, for their friendship andfor making the workplace a great place to be at.

Finally, I would like to thank my family, Jenny, Fanny and Sandra for alwaysbeing there.

Borlange, November 2009

Michael Lindgren

i

Abstract

One of the rst questions to consider when designing a new roll forming lineis the number of forming steps required to produce a prole. The numberdepends on material properties, the cross-section geometry and tolerance re-quirements, but the tool designer also wants to minimize the number of form-ing steps in order to reduce the investment costs for the customer. There areseveral computer aided engineering systems on the market that can assist thetool designing process. These include more or less simple formulas to predictdeformation during forming as well as the number of forming steps. In recentyears it has also become possible to use nite element analysis for the designof roll forming processes.

The objective of the work presented in this thesis was to answer the fol-lowing question:

How should the roll forming process be designed for complex geometries and/orhigh strength steels?

The work approach included both literature studies as well as experimentaland modelling work. The experimental part gave direct insight into the processand was also used to develop and validate models of the process. Starting withsimple geometries and standard steels the work progressed to more complexproles of variable depth and width, made of high strength steels. The resultsobtained are published in seven papers appended to this thesis.

In the rst study (see paper 1) a nite element model for investigating theroll forming of a U-prole was built. It was used to investigate the eect on lon-gitudinal peak membrane strain and deformation length when yield strengthincreases, see paper 2 and 3. The simulations showed that the peak straindecreases whereas the deformation length increases when the yield strength in-creases. The studies described in paper 4 and 5 measured roll load, roll torque,springback and strain history during the U-prole forming process. The mea-surement results were used to validate the nite element model in paper 1. Theresults presented in paper 6 shows that the formability of stainless steel (e.g.AISI 301), that in the cold rolled condition has a large martensite fraction,

iii

can be substantially increased by heating the bending zone. The heated areawill then become austenitic and ductile before the roll forming. Thanks tothe phenomenon of strain induced martensite formation, the steel will regainthe martensite content and its strength during the subsequent plastic strain-ing. Finally, a new tooling concept for proles with variable cross-sections ispresented in paper 7.

The overall conclusions of the present work are that today, it is possible tosuccessfully develop proles of complex geometries (3D roll forming) in highstrength steels and that nite element simulation can be a useful tool in thedesign of the roll forming process.

iv

The doctoral project hasresulted in following

The ThesisThis thesis consists of a survey and the following appended papers:

1. M. Lindgren, Finite Element Model of Roll Forming of a U-Channel Pro-le, Presented at the international conferences on technology of plastic-ity, Verona, Italy October 2005.

2. M. Lindgren, Cold Roll Forming of a U-channel Made of High StrengthSteel, Journal of Materials Processing Technology 186 (2007) 77-81.

3. M. Lindgren, An Improved Model for the Longitudinal Peak Strain inthe Flange of a Roll Formed U-Channel developed by FE-Analyses, SteelResearch Int. 78 (2007) No. 1.

4. M. Lindgren, Experimental Investigation of Roll Load and Roll Torquewhen High Strength Steel is Roll Formed, Journal of Materials Process-ing Technology, 191 (2007) 44-47.

5. M. Lindgren, Validation of Finite Element Model of Roll Forming, Pre-sented at international deep drawing research group, Olofstrom, Sweden,June 2008.

6. M. Lindgren, U. Bexell, L.Wikstrom, Roll Forming of Partially HeatedCold Rolled TRIP Steel, Journal of Materials Processing Technology,209 (2009) 3117-3124.

7. M. Lindgren, L-O. Ingmarsson, 3D Roll-forming of Hat-section WithVariable Depth and Width, Presented at the 1st international congresson roll forming, Bilbao, Spain, October 2009.

v

Author contribution to the publications

1. Single author

2. Single author

3. Single author

4. Single author

5. Single author

6. Writing and done all work in close co-operation with the co-authors

7. Everything except measuring the shape of the produced sections

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Notation

Symbols

a Flange lengthA Integration constantb Web widthB Integration constantBB The width of the at strip used to formed the nished prolec Forming lengthC Relative sliding velocityDS Diameter of the shaftD Tolerance distance or horizontal distance between the forming stepsE Youngs moduluse Longitudinal engineering strain in the edge of the angeep Longitudinal peak strain in the edge of the angeet Transverse longitudinal strain in the angeej Extra pass.f Yield criterionfs Tolerance factor.Fn Normal forceFt Tangential forceh The distance from the neutral layer to the inner side of the bendhl The vertical distance from where the prole will hit the lower

tool to the roll gap between the upper and lower toolHC , d Horizontal distance between two forming stepsH, g Height of the nished proleI Second moment of inertiakl Constants used in transverse bendingL Deformation lengthLS Length of a shaftn Number of forming stepsp Perpendicular moment armR1 Female tool radius

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r Distance from bendr0 Radius to the neutral layers Parallel moment armsij Deviatoric Cauchy stresssf Shape factor.t The thickness of the materialUs Ultimate tensile strength.VC The vertical distance between centre of upper and lower toolvr Sliding velocityWb Plastic work due to transverse bendingWs Plastic work due to longitudinal bending per unit volumeWsl Plastic work due to longitudinal bending per unit lengthWt Total plastic work for one bending

Y, y Yield strengthz Coordinate in longitudinal directionx Coordinate in transverse directionX Distance between centre of the roll station and

there the prole will hit the lower toolz Pre-punched hole Bend angle for the active bend

1,2 Bend angle for the female tool Bend angle increment

The derivate of the bend angle

d Total degree of formed angle. Angle between the web and the innite small element Bend angle for the inactive bend

Ls, x Longitudinal stretching of the angeLb Longitudinal bending of the anget Strain due to transverse bending in the bending zoney Transverse strain of the angepij Plastic strain Plastic parameter Friction coecient Eective stress

, xy Shear strain in the ange Forming angle

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List of Figures

1.1 Roll formed products have many applications in the automotivesector, buildings, domestic appliances etc. . . . . . . . . . . . . 1

2.1 The strip is formed in several steps, beginning with an unde-formed strip and ending with the nished prole, Lund et al.[41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The material in the ange will travel a longer distance than thematerial in the bending zone. The dierence will cause strain(e) in the ange. In this gure (H) represents the ange length,(L) the deformation length and (e) the strain in the ange, Lundet al. [41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Dierent defects that can occur in the nished prole if the rollforming process is not well designed. From left: twist, are andoil canning, Ingvarsson [27]. . . . . . . . . . . . . . . . . . . . . 4

2.4 Lead gate is used to guide the prole into the tools and checkedlead gate is used to control the strip laterally. . . . . . . . . . . 5

2.5 3D roll formed proles in paper 7, Lindgren and Ingmarsson[39]. From the top: a straight section, a hat-section with awaist on one side and nally a hat-section conical in depth andwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.6 U-proles roll formed in the experimental 3D roll forming ma-chine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.7 a) 3D roll forming simulation of the tooling concept in Paper7, Lindgren and Ingmarsson [39]. b) The real tools which aremodelled in a). . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 This gure shows the strain histories of the longitudinal webstrain and the longitudinal membrane ange strain for two form-ing stations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 The denition of the parameters for the bend angle and thebend angle increment. . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Deformation length for one forming step. . . . . . . . . . . . . . 123.4 The prole will be reversely bent since the prole will be curved

down and then lifted up (hl) by the next roll station. . . . . . . 133.5 The forming angle method. . . . . . . . . . . . . . . . . . . . . 143.6 The ower method: the engineer starts with a nished prole

and progressively unfolds it to a at strip. . . . . . . . . . . . . 153.7 Front view and side view of the roll forming mill used in the

Algorithm, equation 3.6 to 3.11, [41]. . . . . . . . . . . . . . . . 15

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3.8 u is the distance from the edge of a at strip to the edge of thenished prole. In this case u is used to calculate the numberof forming steps, equation 3.11. . . . . . . . . . . . . . . . . . . 17

3.9 An element strip between two roll forming passes. . . . . . . . . 193.10 View in z-direction. . . . . . . . . . . . . . . . . . . . . . . . . . 193.11 The bend angle is divided into three stages: stage one when the

angle does not change, stage two when the angle changes butthe strip is not in contact with the tool and stage three whenthe strip is contact with the tool. . . . . . . . . . . . . . . . . . 22

3.12 The V-prole made of high strength steel and mild steel wereroll formed in four forming steps: 15o, 30o, 45o and 60o. TheV-prole made of high strength steel was straight after formingbut the prole made of mild steel was not. . . . . . . . . . . . . 24

3.13 Delta Flare is the dierence between the springback in the cen-tre of the prole and the end of the prole. This is a plot for 4dierent ultra high strength steels with various yield strength. . 25

3.14 To the left a C-prole is roll formed in 8 passes and to the right atop-hat prole is roll formed in 5 passes. The C-prole is formedmainly by bending whereas the top-hat prole is formed bothby bending and drawing [13]. . . . . . . . . . . . . . . . . . . . 26

3.15 This gure shows three dierent ways of forming a 90o prole.One way is to let the tools have the same radius in every form-ing step and dierent arc length. Another way is to let the arclength be the same in every forming step but decrease the toolradius. These methods give dierent spring back and longitu-dinal residual stresses in the bending zone. The constant arclength is the method most commonly used. . . . . . . . . . . . 27

3.16 The deformed sheet between roll stand (i) and roll stand (i+1).The neutral layer of the sheet is described by the shape function,Equation 3.32. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.17 Shape functions for dierent parameters (q). . . . . . . . . . . . 283.18 To the left: the unfolded C-prole, the blank of sheet metal.

To the right: two dierent cross-sections of the C-prole. . . . . 333.19 A 3D roll-formed U-prole. Tension stress acts on the transition

zone where the U-prole is small and compression stress actson the transition zone where the prole is wider. . . . . . . . . 33

4.1 A universal, spindle type roll-forming mill where the lower toolsare driven with universal joint driven shafts. . . . . . . . . . . . 36

4.2 Geometry of the tools used in the experiment. The tools havea bend angle of 20o, 40o, 60o and 80o. . . . . . . . . . . . . . . 37

x

4.3 This gure shows only half of the U-channel because the sym-metry and the rosette strain gage is bonded on the top surfaceclose to the edge of one ange. The longitudinal direction isthe same as x-direction and the transverse direction is same asy-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4 The roll load in z-direction was measured with two donut cells.The torque sensor was used to measure the roll torque and itwas mounted between the lower tool and power transmission.The equipment measured the forces of each forming step, oneby one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.5 The spring back, width and depth between the dierent formingsteps were measured with a sliding caliper. . . . . . . . . . . . . 39

4.6 The picture shows the prototype resistance heating machine andthe steel strip which was fed through the machine by wheelsmade of copper. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.7 A sketch of the electrical circuit of the prototype resistanceheating machine. Wikstrom [56]. . . . . . . . . . . . . . . . . . 41

4.8 The tools and the ower pattern. The V-section was roll formedin 6 forming steps, from 15o to 120o. . . . . . . . . . . . . . . . 42

4.9 The experimental equipment used for research and prototypingof 3D roll formed proles. . . . . . . . . . . . . . . . . . . . . . 43

4.10 The experimental equipment has 6 forming stands. Each standhas 4 units (tool holders) that can rotate and translate. . . . . 44

4.11 View from the top: The prole is roll formed in six passes andto do that the prole must go through the machine two times.Forming stands number 1, 3 and 5 form the left side and formingstands number 2, 4, and 6 form the right side. . . . . . . . . . . 44

4.12 View from the back. The geometry of the tools is the same forall forming stands. The only dierence is that the tools for theange are moving up and closer to the tools that hold the web,for example bend angles 30o and 60o. Tools with a constantradius have been used, Chaing [9]. . . . . . . . . . . . . . . . . 45

4.13 The xture used. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1 The nite element model described in paper 5, Lindgren [37]. . 475.2 The geometry of the tools and the corresponding ower pattern

described in paper 5. . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Four forming stands were used. The two rst stands were used

as a belt feeder, the other two forming stands have the samebend angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.4 The tensile test data for the materials used in paper 5. . . . . . 51

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5.5 The used tolerance distance D1 and D2 in paper 5 is 5 % of thethickness. The bias factor is default 0 in MARC which meansthat D1 = D2. In paper 5 is a bias factor of 0.25 used whichmeans D1 < D2. . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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Contents

1 Introduction 11.1 Design of the roll forming process . . . . . . . . . . . . . . . . . 11.2 Research Question and Approach . . . . . . . . . . . . . . . . . 2

2 The roll forming process 32.1 Traditional roll forming . . . . . . . . . . . . . . . . . . . . . . 32.2 3D roll forming . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Literature Survey 93.1 Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Strain histories in roll forming . . . . . . . . . . . . . . 93.1.2 Longitudinal membrane strain . . . . . . . . . . . . . . 93.1.3 Deformation length . . . . . . . . . . . . . . . . . . . . . 113.1.4 Roll load . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Theoretical work . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.1 Number of forming steps . . . . . . . . . . . . . . . . . . 133.2.2 Deformation types . . . . . . . . . . . . . . . . . . . . . 173.2.3 Deformation length . . . . . . . . . . . . . . . . . . . . . 193.2.4 Longitudinal membrane strain . . . . . . . . . . . . . . 213.2.5 Geometrical restriction from the female tool . . . . . . . 223.2.6 Roll forming of high strength steel . . . . . . . . . . . . 233.2.7 Bending method . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Computer simulation of roll forming . . . . . . . . . . . . . . . 263.4 3D roll forming, proles with variable cross-section . . . . . . . 32

4 Experimental setup 354.1 The roll forming experiment . . . . . . . . . . . . . . . . . . . . 35

4.1.1 The roll forming machine and data acquisition equipment 354.1.2 The tools . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.3 Strain measurement . . . . . . . . . . . . . . . . . . . . 364.1.4 Roll load and roll torque . . . . . . . . . . . . . . . . . . 384.1.5 Spring back . . . . . . . . . . . . . . . . . . . . . . . . . 39

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xiv Contents

4.2 Partial heating experiment . . . . . . . . . . . . . . . . . . . . . 404.2.1 Furnace experiment . . . . . . . . . . . . . . . . . . . . 404.2.2 Prototype resistance heating machine . . . . . . . . . . 414.2.3 Roll forming machine and tools . . . . . . . . . . . . . . 414.2.4 Sample preparation . . . . . . . . . . . . . . . . . . . . 42

4.3 3D Roll forming experiment . . . . . . . . . . . . . . . . . . . . 424.3.1 The 3D roll forming machine . . . . . . . . . . . . . . . 424.3.2 The tooling concept . . . . . . . . . . . . . . . . . . . . 434.3.3 Measuring equipment . . . . . . . . . . . . . . . . . . . 45

5 Computational model 475.1 Nonlinear solution procedure and convergence criteria . . . . . 485.2 The geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Shell element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.4 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5 Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Summary of appended papers 536.1 Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Paper 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.3 Paper 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.4 Paper 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.5 Paper 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.6 Paper 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.7 Paper 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7 Discussions and conclusions 57

Bibliography 61

Chapter 1

Introduction

1.1 Design of the roll forming process

Roll forming is a metal forming process which is spread throughout the world.Today, roll forming products have numerous applications, for example in build-ings, airplanes and the automotive sector as well as in furniture and domesticappliances, see Figure 1.1. Roll forming is a highly productive process and itsuse increases every year, Halmos [19]. Compared to other metal forming pro-cesses the benet of this process is that auxiliary operations, such as punching,welding, clenching etc. can be included, which makes it possible to produceproles that are ready to use directly.

Figure 1.1. Roll formed products have many applications in the automotive sector,buildings, domestic appliances etc.

1

2 Introduction

When a new roll forming machine is designed the tool designer must decidehow many forming steps are required to form the prole. The number of stepsdepends on the cross-section, tolerances, nish of the surface and materialproperties. Today there are several computer aided engineering (CAE) sys-tems, for example ORTIC System [27], COPRA RF [15], PROFIL [55], thatcan support the tool designer in creating tools. CAE systems use more orless simple formulas and rules of thumb for predicting the number of formingsteps and suggesting geometry of the tools. Some CAE programs also includecomputerized simulation techniques, for example based on the nite dierencemethod, Duggal[12], for investigating tool design proposed by the program.The advantage of this type of analysis is that the simulation time can be veryshort. A few years ago nite element analysis was not used for designing rollforming processes or roll formed proles. Today, however, there are examplesof new proles successfully developed with the help of nite element simu-lations. This research was devoted to nite element simulations as well asexperiments, and great eort was put into building experimental equipment.The experiments were carried out in parallel with the building of nite elementmodels and simpler models.

1.2 Research Question and Approach

The research question of the work can be stated as:

How should the roll forming process be designed for complex geometries and/orhigh strength steels?

The approach used was a combination of literature studies, and experimentaland modelling work. The experimental part gave direct insight into the processand was also used to develop and validate models of the process. The workstarted with simple geometries and standard steel and progressed with morecomplex proles of variable depth and width, made of high strength steel.

Chapter 2

The roll forming process

2.1 Traditional roll forming

In roll forming bending is done in several steps, beginning with an undeformedstrip and ending with the nished prole, see Figure 2.1. Forming is a geomet-rically complex process since the forming takes place not only in the formingtools, but also between each forming stand. As a result of the latter, the mate-rial in the ange will travel a longer distance than the material in the bendingzone, see Figure 2.2. This causes a longitudinal strain in the ange, whichshould not be plastic in order to avoid large, longitudinal residual stresses inthe nished prole.

Figure 2.1. The strip is formed in several steps, beginning with an undeformed stripand ending with the nished prole, Lund et al. [41].

3

4 The roll forming process

All materials that can be bent, such as aluminium, steel, stainless steel, copper,can also be roll formed and the material can be pre-painted or pre-coated. Thespeed at which a prole can be produced varies between 15 m/min and 185m/min, Tool and Manufacturing Engineers Handbook [3], depending on thetolerance of the cross-section, the material and how fast the machine can befed with raw material or how fast the nished product can be removed fromthe run out table. The thickness of material that can be roll formed rangesfrom 0.15 mm to 19 mm, Kolev [31]. The roll forming process is very robustprovided it is set up correctly, and the geometry of the proles produced showsonly small variations. However, the prole can have defects as bows, twist,are, spring back and oil-canning, see Figure 2.3, if the roll forming process isnot well designed. Many defects depend on the number of forming steps beingtoo small, which gives residual stresses in the prole that cause unwanteddeformations.

Figure 2.2. The material in the ange will travel a longer distance than the materialin the bending zone. The dierence will cause strain (e) in the ange. In this gure(H) represents the ange length, (L) the deformation length and (e) the strain in theange, Lund et al. [41].

Figure 2.3. Dierent defects that can occur in the nished prole if the roll formingprocess is not well designed. From left: twist, are and oil canning, Ingvarsson [27].

2.2 3D roll forming 5

Today simulations are used more and more in the industry and especially soin the automotive sector for industrial engineering, crash tests, sheet metalforming and so on. The need of simulation also includes roll formed products.Important advantages of the nite element analysis are that not only can itprovide the same information as computerized simulations, it also includes theeect of lead gates, see Figure 2.4, and end eect of pre-cut material amongother things.

Figure 2.4. Lead gate is used to guide the prole into the tools and checked leadgate is used to control the strip laterally.

In the last 10 years a number of research papers have been published wheremore or less simplied simulation models have been compared with experi-ments, for instance Hellborg [24], Bui and Ponthot [8], Sukmoo et al. [53] andpaper 5 in this work, Lindgren [37]. These studies show that the nite elementmethod is a very valuable tool in designing the roll forming process. Some ofthe simplied models also show the possibility of shortening the simulationtime, for example by ignoring the friction between sheet and tooling. Todaythere are commercial programs that can reduce the time spent on modellingthe process, COPRA RF [15]. In combination with increased computer capac-ity this makes the nite element modelling a standard tool in designing theroll forming process.

2.2 3D roll forming

The use of the roll forming process grows every year thanks to the possibil-ity of forming complicated products in dierent metals in combination withhigh productivity. Until eight years ago a disadvantage of roll forming wasthat only sections with a constant cross-section could be produced. Todaythe problem is solved, ORTIC [1], and 3D roll forming technology is used inthe building industry to produce panels with variable cross-sections and vari-able longitudinal curvature. The method is very exible which means thatpanels can be produced with the same set-up of roll forming tools. For ex-


ample, the Budapest Arena is covered with about 4700 dierent, individuallyshaped panels. The automotive industry has become interested in 3D rollforming as a result of the methods exibility making it particularly suitablefor components made of high strength steels. However, the automotive in-dustry demands design rules, simulations and prototypes to be convinced ofthe benets. To meet their demands, probably one of the worlds rst 3Droll-forming machines, with two translational and two rotational degrees offreedom per axis, was built within this research work (see Chapter 4). Expe-riences and results obtained are presented in depth in paper 7, Lindgren andIngmarsson [39], of this thesis. Examples of manufactured proles are shownin Figure 2.5 and Figure 2.6. For the same purpose more or less advancedexperimental equipment have been built by other research institutes and uni-versities also, including the Labein Tecnalia [46], the University of Technologyin Darmstadt [16], Takushoku University in Tokyo [44].

Figure 2.5. 3D roll formed proles in paper 7, Lindgren and Ingmarsson [39]. Fromthe top: a straight section, a hat-section with a waist on one side and nally a hat-section conical in depth and width.

Today, only a few papers describe nite element simulations of 3D roll forming,Gulceken et al. [18]. The main dierence between traditional roll forming and3D roll forming is that modelling the 3D process requires much more worksince the motion of every tool must be exactly described. In the nal paper ofthis thesis, Lindgren and Ingmarsson [39], a new tooling concept is describedwhere hat-proles with variable depth and width are roll formed. In parallelwith the tooling concept, a rst nite element model was developed and usedto simulate the process, Sagstrom [48], see Figure 2.7. These results are notincluded in the thesis.

2.2 3D roll forming 7

Figure 2.6. U-proles roll formed in the experimental 3D roll forming machine.

Figure 2.7. a) 3D roll forming simulation of the tooling concept in Paper 7, Lindgrenand Ingmarsson [39]. b) The real tools which are modelled in a).

Chapter 3

Literature Survey

A literature survey of research concerned with the roll forming process ispresented below. The chapter is divided into four parts: experimental works,theoretical work, computer simulations and 3D roll forming.

3.1 Experimental work

3.1.1 Strain histories in roll forming

The strain history, when roll forming a U-prole, has been measured in severalexperiments, [9], [4], [51]. These experiments show, see Figure 3.1, that thelongitudinal ange strain starts between the forming stations and increasesrapidly to a maximum value just before the centre of the forming station. Theweb strain exhibits the same behaviour but the strain is more compressive.These strains decrease rapidly when approaching the roll centre. Some strainscause residual stresses that give rise to defects on the nished prole. Chiang[9] also measured the longitudinal membrane strain across the strip whenforming a U-prole and a V-prole. The measurement results showed that thestrain is largest at the ange edge.

3.1.2 Longitudinal membrane strain

Bhattacharyya and Smith [4] investigated the longitudinal strain by use ofa single roll station as well as multiple roll stations. They also investigatedthe strain variation with dierent bend angles and bend angle increments, seeFigure 3.2.

The results obtained were as follows:

In the case with a single roll station (0oo1 ) the strain increased almostlinearly with the bend angle, o1 .

9

10 Literature Survey

Figure 3.1. This gure shows the strain histories of the longitudinal web strain andthe longitudinal membrane ange strain for two forming stations.

When multiple roll stations were used (0oo10o), where the rst andlast forming steps were at rolls, the longitudinal strain was reduced by10 15 % more than in the case with a single station.

With multiple roll stations (0o o1 o2 ) the strain level was on thesame level as in the case (0o o1 0o).

The level of peak strain depends on the bend angle increment =(o2 o1 ) and not the roll angle used at the roll station.

Figure 3.2. The denition of the parameters for the bend angle and the bend angleincrement.

Chiang [9] investigated how ange length (a), web width (b) and bend angle() eect the web strain and the longitudinal membrane strain in the ange of

3.1 Experimental work 11

a U-prole. A single roll station was used to study the bend angle, and the re-sult obtained was similar to those obtained by Bhattacharyya and Smith. Thelongitudinal strain increased almost linearly with the bend angle but the varia-tion for the web strain was small. In the experiment with varying ange lengththe longitudinal strain decreased when the ange length increased. When theweb width increased the longitudinal strain slightly decreased, see Table 3.1.

Table 3.1. The peak strain in the ange edge for various ange lengths and webwidths.

Web(b) Flange(a) Average peak strain (103)20 10 3.7520 15 3.1020 20 2.7030 10 3.7030 15 2.8530 20 2.60

In an experiment Zhu [57] investigated the inuence of ange length (a), ma-terial thickness (t), the bend angle () and bend angle increment () on thelongitudinal strain distribution. The results obtained were as follows:

The longitudinal strain increases in the beginning with a ange lengthshorter than 15 mm, but with a ange length over 15 mm the strainstarts to decrease.

When the material thickness increases the longitudinal strain increasesas well.

An increasing bend angle increment increases the longitudinal strain. An increasing bend angle at a constant bend angle increment decreases

the longitudinal strain. For example, the longitudinal strain is higher for0o 20o than for 20o 40o. This was not obtained by Bhattacharyyaand Smith [4].

3.1.3 Deformation length

In a study Bhattacharyya et al. [5] formed a U-prole and measured the defor-mation length, see Figure 3.3, for mild steel and aluminium. The results werecompared with a model, equation 3.1, which they had developed. The deriva-tion of the deformation length will be reviewed in the section on theoreticalwork.

L =

8a3

3t(3.1)


Figure 3.3. Deformation length for one forming step.

Their conclusions were as follows:

There is a good agreement between the experiment results and Equation3.1. In most cases the discrepancy was around 6 %.

Based on the good agreement they concluded that the deformation length(L) depends on three variables: ange length (a), bend angle increment() and material thickness (t), whereas it is independent of materialproperties.

3.1.4 Roll load

Bhattacharyya et al. [6] also investigated the roll load when U-proles of mildsteel with thicknesses of 0.6 mm, 1.0 mm and 1.2 mm, as well as a U-proleof aluminium with a thickness of 0.6 mm, were roll formed. The investigatedbend angles () ranged from 20o to 40o at intervals of 5o, and to simulatemultiple roll pass situations a roll schedule of 0o o 0o where o waschosen as above. The width of 40 mm to 60 mm was formed depending onwhich ange length (a) was used. The roll load was measured with a loaddynamometer. The roll clearance was set to give an initial bite of 200 N andthis roll load was subtracted from the measured roll load. The result fromthe experiment was compared with a model, equation 3.2, which they haddeveloped. The derivation of equation 3.2 is extensive and can be found inBhattacharyya et al. [6]. The following is a brief explanation: the rst termin equation 3.2 is load caused by bending and stretching of the prole. Thesecond term is from the reverse bending casing by the prole will curve downafter the rst roll station and then be lifted up by the next roll station, seeFigure 3.4. The second term comes from small deection theory.

3.2 Theoretical work 13

P = Y

2t33a3 sin2

+3hlEI

(D X)3 (3.2)

Their conclusions were as follows:

The experiments and equation 3.2 agreed quite well, the discrepancybeing 0 - 20 %.

The theory showed that the roll load depends on yield strength (Y ) ofthe material, thickness (t), bend angle () and ange length (a).

The theory also showed that when multiple roll stations are used theroll load depends on an extra load, the second term in equation 3.2 ,This term includes the Youngs modulus (E) of the material, the secondmoment of inertia (I) of the prole and machine data (D,hl,X).

Figure 3.4. The prole will be reversely bent since the prole will be curved downand then lifted up (hl) by the next roll station.

3.2 Theoretical work

3.2.1 Number of forming steps

The number of forming steps that a prole requires depends on material prop-erties and the complexity of the prole. Other factors such as the part width,horizontal centre distance between the passes and the part tolerances must alsobe taken into account. However, a standard method to calculate the number


of forming steps does not exist, Halmos [19], so below a number of dierentapproaches are described. A common method used to calculate the number offorming steps is the forming angle method [3], see Figure 3.5. In the followingexample the calculation of a simple section is described.

To determine the number of passes you take the height (g) of the nishedprole and the distance between the forming stands (d) and use the derivedequation 3.5 below. The method is based on the amount of forming per length.That a conservative forming angle (C) is 1.5o can be kept in mind as rule ofthumb.

Figure 3.5. The forming angle method.

The following notation and equations 3.3 - 3.5 are used to calculate the num-ber of forming steps:

n = Number of forming stepsc = The forming lengthg = Height of the nished prole = Forming angled = Horizontal distance between the forming steps

c = g cot. (3.3)


c = (n 1)d. (3.4)3.3 and 3.4

g cot = (n 1)d n = gdcot + 1. (3.5)

If the prole has multiple bends the forming angle method must be appliedto every bend independently and then combine the bends where it is possibleand approximate how many passes that are needed to nish the prole.

Figure 3.6. The ower method: the engineer starts with a nished prole andprogressively unfolds it to a at strip.

In the ower method, [9] the tool engineer starts with a nished section andgradually unfolds the prole to a at strip, see Figure 3.6. The way the prolewill be unfolded and the number of forming steps is up to the tool designerto decide. However, a successful use of this method requires a great deal ofexperience, and the method can be very time consuming when applied to morecomplex shapes.

Figure 3.7. Front view and side view of the roll forming mill used in the Algorithm,equation 3.6 to 3.11, [41].

A method to roughly estimate the machine design and number of forming stepfor a prole is proposed in [41]. Figures 3.7 - 3.8 and the section below present


the notations used to calculate the machine design.

HC = Horizontal distance between the forming standsVC = Vertical distance between the centre of upper and lower toolsLS = Length of the shaftDS = Diameter of the shaftH = Height of the nished proleBB = Width of the at strip used to form the prolen = Number of forming stepsu = Distance from the edge of the at strip to the edge

of the nished prole, see Figure 3.8.

The logic Algorithm, Equation 3.6 - 3.11 in [41] consists of the rules belowthat should be fullled by a design. It is used to roughly estimate the machinedesign.

LS 1.2BB. (3.6)

DS LS7 min 30mm. (3.7)

VC 1.5DS + 2H. (3.8)

HC 2VC (pre-cut material). (3.9)

HC 2.5VC (post-cut material). (3.10)

n >75uHC

. (3.11)

The drawback to this method is that thickness and yield strength of the ma-terial used are not taken into account.

To predict the number of forming steps Halmos [19] proposes an empiricalequation to be used as a guide. The equation 3.12 was not fully tested at thepublishing date of [19].

n = [0.237H0.8 +0.834t0.87

+d90

][Y 2.1

0.003Us]0.15s(1+0.5z)+ ej + fs +5zsf . (3.12)


Figure 3.8. u is the distance from the edge of a at strip to the edge of the nishedprole. In this case u is used to calculate the number of forming steps, equation 3.11.

wheren = Number of forming steps.Y = Yield strength.H = Height of the prole.t = Thickness of the material.Us = Ultimate tensile strength.d = Total degree of formed angle.sf = Shape factor.fs = Tolerance factor.z = Pre-punched holeej = Extra pass.

3.2.2 Deformation types

In a bending operation the major part of the deformation occurs in a trans-verse direction. In roll forming, which also is a bending operation althoughthe bending is made gradually, other deformation types besides deformation inthe transverse direction can occur. Paton et al. [45] propose four fundamentaldeformation types: longitudinal stretching, longitudinal bending, transversebending and shear.

The following assumptions were made to derive the deformation types, seeFigure 3.9 and Figure 3.10:

The thickness of the strip is small compared to other geometrical dimen-sions.

Bending only takes place in the fold line of the active bend. Deformation at the inboard side is neglected.


The outboard region remains constant in the cross-section and rotatesaround the active bend.

Transverse sections of the strip remain plane and the prole is bent asa beam. The latter means that cross-sections remain orthogonal to thecentreline along the prole.

With help of the innitesimal element of length dz, see Figure 3.9 and 3.10,the following strain models were derived:

Longitudinal stretching

Ls =12r2(

d

dz)2. (3.13)

where (r) is the distance from the bend.

Longitudinal bending

Lb = h(s(d2

dz2 p(d

dz)2)). (3.14)

where (h) is the distance from the neutral layer to the inner side of the bend,(s) is parallel moment arm and (p) is the perpendicular moment arm.

Shear

= p(d

dz). (3.15)

Transverse bending

t =h

r0. (3.16)

where (r0) is the radius to the neutral layer.

The conclusion drawn by Paton et al. [45] was that the shear strain is largerthan the longitudinal strain in sections with inactive bends and must thereforebe considered as a factor in the roll forming process. The shear strain is smallfor a simple section such as a U-prole without inactive bends.


Figure 3.9. An element strip between two roll forming passes.

Figure 3.10. View in z-direction.

3.2.3 Deformation length

Bhattacharyya et al. [5] derived an expression for the deformation length (L),see Figure 3.3, by minimising the total plastic work (Wt) for the bending intransverse and longitudinal direction. They made the following assumptions:

The material is rigid perfectly plastic.

The bend only takes place along the fold line.

Out of plane bending of the ange and the longitudinal bending of theweb are neglected.

The ange adopts a shape that minimises the plastic work.

Plastic work due to transverse bending


Wb =14Y t2. (3.17)

where (Y) is the yield strength.

Plastic work due to longitudinal bending per unit volume

Ws =12Y x2(

d

dz)2. (3.18)

where (x) is coordinate in the transverse direction.

Plastic work per unit length

Wsl = a0

12Y x2(

d

dz)2(tdx) =

16Y a3t(

d

dz)2. (3.19)

The total work done for one bend is

Wt = L0

[14Y t2 +

16Y a3t(

d

dz)2]dz. (3.20)

The function (z) that minimises the Equation 3.20 satises the Euler Equa-tion 3.21

dF

d d

dz(F

) d

2

dz2 3t

4a3= 0. (3.21)

where F = F (z, , ) = t4 +a3

6 (ddz )

2

the general solution is

(z) =3t8a3

z2 + Az + B. (3.22)

The end conditions give the integration constants (A) and (B), (0) = 0,(L) = , (0) = 0 and (L) = 0.

The end conditions and Equation 3.22 give the deformation length

L =

8a3

3t. (3.23)


where () is the bend angle increment, (t) is the thickness of the materialand (a) is the ange length.

3.2.4 Longitudinal membrane strain

Chiang [9] developed two models, (A) and (B) for the longitudinal membranestrain. Model (A) was based on a publication by Bhattacharyya et al. [5],equation 3.24, and a model (B) that Chiang derived by geometrical consider-ations, equation 3.27.

equation 3.13 and equation 3.22 give model (A)

e =932

(t2

a6)r2z2|0ra0zL. (3.24)

The peak strain and transverse strain be obtained from Equation 3.24 in model(A). The peak strain for the ange edge, (r = a, z = L), is written:

ep =34(t

a). (3.25)

The transverse longitudinal strain at any position z is written:

et = k1r2. (3.26)

where k1 = 932(t2

a6)z2

Chiang compared model (A) with an experiment and concluded that the modeloverestimated the strain three times when approaching the roll station.

Model (B) based on geometry for the engineering strain in the ange edge iswritten:

e =

1 +

3t4a

(1 cos) 1. (3.27)

The model predicts that the strain is uniform in the deformation zone but givesno information about the transverse strain. Compared with an experiment,model (B) gave a closer approximation of the value of the peak strain thanmodel (A).


3.2.5 Geometrical restriction from the female tool

Zhu [57] studied how the geometry of the tool inuences the bend angle in theange of a U-prole. The forming was divided into three stages, see Figure3.11, the rst stage being when the bend angle does not change, stage twowhen the bend angle changes but the strip is not in contact with the tool andthe third stage being when the strip is contact with the tool.

Figure 3.11. The bend angle is divided into three stages: stage one when the angledoes not change, stage two when the angle changes but the strip is not in contactwith the tool and stage three when the strip is contact with the tool.

Zhu studied both horizontal and vertical rolls, but it is the study of horizontaltool that is discussed here. Zhu developed models applicable to three cases:

a) the roll fully overlaps the outer edge of the ange.

b) the outer edge of the ange overlaps the roll at any position.

c) the outer edge of the ange overlaps the roll initially.

Only case a) is presented below since in practise case b) and c) are not utilized.The model obtained for case a) is

z = L

a2 cos2 cos2 2

+ 2aR1sin(2 )

cos 2 a2. (3.28)

Zhu assumed that the derivative for equation 3.28 could be used to predictforming severities at the point where the strip get in contact with the femaletool for the rst time. The derivative is


d

dz=

L za2 sin 22 cos2 2

+ aR1cos(2)

cos 2

. (3.29)

The theoretical work was compared with the experiment and the conclusionswere as follows:

A concept, the bend angle curve, was proposed and there was a goodagreement between the predicted bend angle distribution and the exper-imental results.

The longitudinal membrane strain reaches a maximum when the stripgets in contact with the female tool for the rst time.

An increased ange length will decrease the longitudinal membrane strain. Increased tool radius will decrease the longitudinal membrane strain. The longitudinal peak membrane strain increases with increasing bend

angle increment.

Constant bend angle increment and an increasing bend angle decreasesthe longitudinal peak membrane strain. However, when the bend angleis close to 90o the opposite eect will occur.

3.2.6 Roll forming of high strength steel

One project [54] involved the roll forming of both a V-prole made of highstrength steel, yield strength 1000 MPa, and a V-prole made of mild steel,yield strength 250 MPa. In both cases the V-channel was roll formed in foursteps: 15o, 30o, 45o and 60o. The V-prole made of high strength steel wasstraight after the forming whereas the prole made of mild steel was curved,see Figure 3.12.

Ingvarsson [28] derived a simple formula, equation 3.30, based on geometryfor the longitudinal engineering strain in the ange of the V-channel used inthe project [54].The simple formula was:

e =

a2 sin2 V 2c Vc

Vc. (3.30)

where (Vc) is the vertical distance between the centre of upper and lower tool.In this case the deformation length (L) was simplied with (Vc).


Figure 3.12. The V-prole made of high strength steel and mild steel were rollformed in four forming steps: 15o, 30o, 45o and 60o. The V-prole made of highstrength steel was straight after forming but the prole made of mild steel was not.

Ingvarsson calculated the longitudinal strain, equation 3.30 for the case withthe V-prole and compared it with the maximum elastic strain, equation 3.31,for both mild and high strength steel used in the experiment [54]. The resultshowed that in the case with mild steel the strain in the ange was larger thanthe maximum elastic strain, and therefore both elastic and plastic strain werepresent during the forming. In the case with high strength steel only elasticstrain was present. Plastic strain will give residual strain and in this casethe channel made of mild steel was curved. The conclusion drawn was thatit is advantageous to use high strength steel in roll forming since it will havesmaller plastic strain in the anges and as a result smaller residual strain aswell.

eelastic =Y

E. (3.31)

where E is Youngs modulus and Y is the yield strength for the material.

Sagstrom et al. [49] studied the inuence of roll forming on the tolerances ofpre-punched holes in a hat-prole made of dierent ultra high strength steels.


Furthermore, the amount of springback and are for these steels was observed.They concluded by measurements that the holes maintained their shape andthat they could be pre-punched, with high tolerances requirements, close to theange edges before the roll forming started. Their investigation also showedthat the are in the end of a prole will be smaller with increasing yieldstrength, see 3.13, and a parameter for this phenomenon called Delta Flarewas introduced. The denition of Delta Flare is the dierence between thespringback at the end of a prole and the springback in the centre of the sameprole. A suggested explanation to decreasing are was that increasing yieldstrength gives less residual stresses that could cause are in the proles.

Figure 3.13. Delta Flare is the dierence between the springback in the centre ofthe prole and the end of the prole. This is a plot for 4 dierent ultra high strengthsteels with various yield strength.

Galdos et al. [14] studied the inuence of material properties on the rollforming process when a U-prole was formed. The main objectives were toinvestigate the longitudinal strain, forming force, springback and longitudinalbow for steels ranging from mild to ultra high strength steels. Both niteelement simulations and experiments were used in the study. Their conclu-sion was that the maximum longitudinal peak strain decreases when the yieldstrength increases. The experiment also showed that increasing yield strengthgives a slight, longitudinal bow, which possibly is the result of a smaller peakstrain in the prole.

3.2.7 Bending method

Two dierent proles are shown in Figure 3.14: one C-prole and one top-hatprole. The C-prole can be roll formed mainly by bending, but forming thetop-hat prole requires both bending and drawing in the bending zone [13].When bending is used to form a prole the spring back will be large and thelongitudinal residual stresses in the bending zone will be small. In the case


when both bending and drawing are used to form the prole the spring backwill be smaller whereas the longitudinal residual stresses in the bending zonewill be larger, Ingvarsson [26], Lindgren [32]. In roll forming the eect ofdrawing can be reduced depending on which bending method that is used,see Figure 3.15. For example, if the most common method called constantarc length is used, the thinning of the material will be reduced. This givesless longitudinal residual stresses in the bending zone, than when the constantradius-method is used. The latter method, on the other hand, gives morestretch forming and thus a smaller spring back. Another method called thevariable radius and variable arc length- method, gives a large spring back butsmall longitudinal residual stresses.

Figure 3.14. To the left a C-prole is roll formed in 8 passes and to the right atop-hat prole is roll formed in 5 passes. The C-prole is formed mainly by bendingwhereas the top-hat prole is formed both by bending and drawing [13].

3.3 Computer simulation of roll forming

Kiuchi et al. [30] introduced a normalized shape function, equation 3.32, todescribe a spatial locus of an element of the strip between two forming stands(roll stand = i to roll stand = i + 1), see Figure 3.16.

S(U) = sin(

2(

U

HC)q). (3.32)

HC = (U2 U1). (3.33)where HC is the horizontal distance between two forming stands and U is thecoordinate in longitudinal direction, Figure 3.16

3.3 Computer simulation of roll forming 27

Figure 3.15. This gure shows three dierent ways of forming a 90o prole. Oneway is to let the tools have the same radius in every forming step and dierent arclength. Another way is to let the arc length be the same in every forming step butdecrease the tool radius. These methods give dierent spring back and longitudinalresidual stresses in the bending zone. The constant arc length is the method mostcommonly used.

The shape function has a built in parameter (q, q > 1), i.e. a low value ofq gives a surface with a curvature that gradually increases when the strip isapproaching the forming tools whereas a larger value describes a curvaturethat increases slowly in the beginning and faster just before the tools, see Fig-ure 3.17. To determine the value of (q) a minimization of the total power ofdeformation in the strip between the forming stands is done through a math-ematical procedure.

To describe the deformed curved surface, the following equation is used:

U = U(u, v). (3.34)

W = W1(v) + [W2(v)W1(v)] S(U). (3.35)

V = V1(v) + [V2(v) V1(v)] S(U). (3.36)

U = U1 U2. (3.37)


Figure 3.16. The deformed sheet between roll stand (i) and roll stand (i + 1). Theneutral layer of the sheet is described by the shape function, Equation 3.32.

Figure 3.17. Shape functions for dierent parameters (q).


where

W1, V1 = W and V coordinates of the cross section of the prole at the rollstand (i) at U = U1W2, V2 = W and V coordinates of the cross section of the prole at the rollstand (i + 1) at U = U2

Boundary conditions and Equation 3.35 and Equation 3.36 give:

When U = U1, S(U) = 0, V = V1(v) and W = W1(v)When U = U2, S(U) = 1, V = V2(v) and W = W2(v)

The equations 3.34 - 3.37 are purely geometric descriptions of the 3D shape ofthe deformed strip, which do not take into account the stresses. To calculatestress and strain a steady-state deformation is assumed between roll stand (i)and roll stand (i+1). The strip is divided into a suitable number of elementsand a incremental theory of plasticity is used to calculated the stress andstrain in the strip. The details are given in Hallmos [19]. Kiuchi et al. [30]developed a computed aided design system that includes the above mentionmethod. The system has successfully been used to design tools for circulartubes.

Duggal et al. [12], [11] used the computer aided simulation program RF-PASS in a study. The program is based on the mathematical model developedby Kiuchi et al. [30] and a nite dierence method. The program uses anelastic-plastic formulation and can be used for analysing multiple roll sta-tions. In the study a simulation of a U-prole was compared with an exper-iment done by Bhattacharrya and Smith [4]. Duggal et al. concluded thatthe peak longitudinal strain deviated less then 2 % between the simulationand the experiment. The longitudinal residual strains obtained from the RF-PASS dropped more then measured strains, which can be explained by theassumption of a sinusoidal shape for the movement of the sheet. RFPASS canbe used, at least for simple sections, to assist the tool designer. Furthermore,other surface models than the sinusoidal shape function will be investigatedin future studies.

Han et al. [22] developed a B3-spline nite strip method to simulate the rollforming process. The displacement of the strip was modelled with two typesof shape functions. In the longitudinal direction the strip was modelled withB3-spline functions and in the transverse direction it was modelled with Her-mitian cubic polynomials. Han et al. [22] introduced the updated-Lagrangianmethod of the nite element theory to the nite strip method. The modelused accounted for large deformation strain and displacement. Prandtl-Reussplasticity ow theory and the von Mises yield criteria were also applied. Han


et al. have written several articles, for example [20] and [21], where they de-scribe how the method was used to analyse simple sections and parametersthat eect the design. The conclusion drawn was that the results from thesimulations agree with observations of roll forming in practise.

Rebelo et al. [47] compared the relative eectiveness of implicit and explicitnite element analysis in metal forming. The simulated U-prole was mod-elled with 4-node shells - 20 through the width and 40 through the length. Thestrip was pulled through three roll stations and the rigid rolls rotated freely.The material was modelled as an elastic-plastic material with a yield strengthof 229 MPa and with Youngs moduls of 206.7 GPa. The implicit nite ele-ment analysis was almost three times faster (47 CPU hours) than the explicitanalysis (125 CPU hours). The conclusion was the implicit formulation has arelative advantage, since the problem is rather one dimensional and thereforehas a small wave front.

Brunet et al. [7] developed a master 2D cross-section model with a slave3D analysis. The 2D analysis was a generalised plane-strain analysis and forthe 3D analysis a thick shell element was used. The tools were modelled asrigid surfaces and instead of rotating tools they were modelled as rigid surfacescontinuously moving from one forming station to the next station. The frictionbetween the sheet and the tools was modelled as Coulomb friction. Both the3D and the 2D analyses included Hills anisotropic model of initial anisotropywith isotropic hardening. The computed longitudinal deection was comparedwith the measured deection in order to validate the model. Brunet et al.concluded that there was a reasonable dierence between the model and theexperiment.

Heislitz et al. [23] used the explicit code PAM-STAMP to simulate rollforming. The strip was pulled through the rolls with constant speed. Therolls were not rotating and the friction between the strip and the rolls wasignored. Two dierent elements were tried: 8-node brick elements and fournode shell elements. In the nal simulations, however, they used 8-node brickelement. The tools were modelled with rigid 4-node shell elements. The massdensity was increased with a factor 100 without inertia eects aecting theresult. The material model used was Swifts isotropic strain hardening andHookes law. The simulations were compared with an experiment and themaximum deviation was about 10%. The conclusions from the simulationswere as follows:

The adaptive mesh renement can help to speed up the simulation.

At the current status of development, the simulation of roll forming byusing PAM-STAMP is not very ecient due to the required CPU time.The simulation time for a U-channel was 250 CPU hours.


FEM code PAM-STAMP can be used to accurately simulate roll form-ing. It is possible to produce both the strain distribution and the nalgeometry after spring back.

Sukmoo et al. [53] used the nite element program (COPRA FEA-RF) tosimulate the roll forming of a U-prole and compared the deformation lengthwith an experiment described in literature. The FE-program COPRA FEA-RF is a rigid-plastic nite element analysis that uses a combined 2D and a3D algorithm. One conclusion from the study was that the work hardeningexponent has the most signicant eect on the forming length. Increasingwork hardening exponent gives increasing forming length. Sukmoo et al. alsoconcluded that it was a good agreement between the simulation results andthe experiment.

Alsamhan et al. [2] utilised the FE-code EPFEP3, which is a 3D implicitelastic-plastic FE program, to developed a remeshing technique for simulatingroll forming. A dual mesh was used: one for storing the deformation historyand another for FE computational. The conclusion drawn from the simulationwas that the computational time was much less compared to a conventionalFE-model. The result from the simulation was also acceptable compared tocomputations without remeshing.

Sheu [52] used LS-DYNA explicit nite element code to simulate a rollformed U-channel. The study comprised the inuence of friction, corner ra-dius, rolling speed and changing rate of bend angle on the ange angle de-viation and the ange length deviation of a 90o U-prole. In the simulationthe front end of the strip was xed and the tools were moved towards thestrip with a given velocity curve. To minimise the number of trails, in thiscase 9 trail runs, a reduced factorial design L9(34) was used. The conclusiondrawn from this study was that the bend angle changing rate and rolling speedare important factors for the bend angle deviation. As concerns the lengthdeviations all factors are equally important.

Salmani et al. [50] used ABAQUS implicit FE-package to investigate local-ized edge buckling in a symmetric channel section when dierent bend angleswere used. Here only the model itself will be described. The tools were mod-elled as rigid surfaces and they rotated to drive the strip forward. A frictioncoecient of 0.2 was assumed. The strip was modelled with a four nodeshell element and ve integration points through the thickness were used. Anelastic-plastic material model was implemented as a table in the software. Acomparison was made between the model and an experiment done by Bhat-tacharyya and Smith [4]. The conclusion was that the deformation history inthe ange edge of the prole was similar to the deformation history in exper-iment and the error between the membrane peak strain ranged from 0 to 12% depending on the bend angle.


Bui and Ponthot [8] used the in-house nite element code Metafor to sim-ulate a roll formed U-prole and compared the result with experiments inliterature. The strip was modelled with a 8-noded brick element and withrenement of the mesh in the bending zone. The tools were modelled as rigidsurfaces. The mechanical behaviour of the material was described by Swiftisotropic strain hardening law. The speed of the rolls and the friction betweenthem and the sheet were studied. Coulomb friction = 0 and = 0.2 weretested. The result showed that the friction did not inuence the springbackand the longitudinal strain to any great extent, and the speed inuenced theresult only moderately. Bui and Ponthot concluded that the nite elementmodels can be simplied especially for complex proles, for example by ig-noring the friction and thereby saving computer simulation time. They alsoconcluded that the study conrmed the potential of the nite element simu-lation in designing the roll forming process.

3.4 3D roll forming, proles with variable cross-section

In 2001 ORTIC AB [1] developed a new roll forming technology, 3D roll form-ing, which made it possible to roll form proles with variable cross-sections.Since then the technology has been used to produce panels that cover buildingsworld wide, Ingvarsson [29].

In [33] Lindgren describes how to produce a C-prole with 3D roll formingtechnology. Figure 3.18 shows two cross-sections for a C-prole with variabledepth and width and the corresponding blank of sheet metal. The automaticcontrol system controls the tools so they are always perpendicular to the toolpaths. The tool path is a mathematical description of the bending line whichis generated from the cross-section in the longitudinal direction of the channel.One way to roll form the C-prole according to Figure 3.18 is to start formingalong the outer bending line and then continue along the inner bending line.

Groche et al. [16] integrated a single exible roll forming stand with atraditional roll forming line. The objective was to produce U-proles withvariable cross-sections. The stand had tools that could rotate and translatewith the help of servomotors. The control of the motors depended on the feedrate of the prole which was measured by an encoder. The tools followed abending line created in a CAD-System which the control program could read.

In [44] Ona describes a single exible roll forming stand and a slitter rollstand. The units were controlled with a CPU. The tools and the slitter couldrotate and translate in and out. The equipment was used for producing U-proles with variable cross-sections in the longitudinal direction. The exper-iments showed that the material in the ange were compressed or stretched,see Figure 3.19, if the ange was too high.

3.4 3D roll forming, proles with variable cross-section 33

Figure 3.18. To the left: the unfolded C-prole, the blank of sheet metal. To theright: two dierent cross-sections of the C-prole.

Figure 3.19. A 3D roll-formed U-prole. Tension stress acts on the transition zonewhere the U-prole is small and compression stress acts on the transition zone wherethe prole is wider.

Groche et al. [17] also developed a semi-empirical model for designing 3D rollformed U-channels. The model was based on mechanics of buckling of platesand nite element analyses. The model focused on the compressed area in theange of a U-prole with variable cross-section, see Figure 3.19, and it can beused for feasibility checks without simulation or experimental tests.


Guleceken et al. [18] used COPRA RF [15] coupled with the nite elementmodule MSC.Marc to simulate the 3D roll-forming process. The main objec-tive was to study how the exible roll forming process can be modelled andsimulated. The simulated U-channel was modelled with a number of 1274 fullintegrated 8-noded hexahedral volume elements. An elastic-plastic materialmodel where the swift extended power law describes the material hardeningwas used. In addition, Von Misses yield criteria and the associated ow rulewere applied. The surface of the tools was modelled using rigid shells. Tocontrol the rotating tools the load control option in the MSC.Marc was used.The simulation time was only 3-4 CPU hours since the model was relativelyrough. The result showed that the dimensions of the simulated part were closeto the dimensions of the desired prole.

Chapter 4

Experimental setup

The experimental equipment developed and built for the work presented inthe appended papers 4 to 7, is described below.

4.1 The roll forming experiment

Section 4.1 describes the experimental equipment used in the study presentedin paper 4 and 5. The aim of these experiments was to measure strain history,roll load, roll torque and springback during the roll forming of a U-channeland to compare the results with nite element simulations.

4.1.1 The roll forming machine and data acquisition equip-ment

A universal, spindle type roll-forming mill, Eckhardt [13], with six roll formingstands and universal joint driven shafts, was used in the experiment, see Figure4.1. Only the lower shafts were driven. The mill has a three-phase frequencycontrolled motor. The horizontal distance between the roll forming standsis 450 mm and, in this experiment, the vertical distance was 150 mm plusmaterial thickness. The rst and last forming stands were used as feeder rolls.The data acquisition equipment came from NATIONAL INSTRUMENTTM

and the software used for measurement was LabVIEWTM .

4.1.2 The tools

The investigated prole was a U-channel where the forming steps had the bendangles: 20o, 40o, 60o and 80o, see Figure 4.2. The bending method used toform the prole was the constant arc length method, Chaing [9], which meansthat the radius of the neutral layer of the bend decreases in every forming step.For the bend angles listed above the radii of the neutral layer of material with

35

36 Experimental setup

Figure 4.1. A universal, spindle type roll-forming mill where the lower tools aredriven with universal joint driven shafts.

a thickness of 1mm are 19.8, 9.9, 6.6 and 4.95 mm respectively. The toolswere designed for materials with a thickness of 1 mm but they were used formaterials with thicknesses up to 1.47 mm when roll torque and roll load weremeasured. The results imply that the material in the ange will be squeezedwhen the bend angle increases. To avoid this phenomenon, with materialthicker than 1 mm the upper tool for a bend angle of 80o was also used as anupper tool for a bend angle of 60o. The roll pitch radius for the upper andlower tools was 75 mm.The lower tools in the two rst forming steps had guides, see Figure 4.2, toalign the material with the machine. These guides inuence the longitudinaland transversal strain in the strip.

The surface nish of the tools is RZ = 1m, and an abundance of lubrica-tion was applied to have a similar low friction coecient between the dierentmaterials and the tools.

4.1.3 Strain measurement

Strain gages were used [25] to measure the biaxial strain history on the topsurface of the ange during the forming process. The strain gage was a45o/90o/135o rosette, relative to longitudinal direction, with a measuring gridof 1.5 mm. It was bonded close to the edge of the ange and the grid wasallocated about 2.5 mm from the edge, see Figure 4.3.

4.1 The roll forming experiment 37

Figure 4.2. Geometry of the tools used in the experiment. The tools have a bendangle of 20o, 40o, 60o and 80o.

Figure 4.3. This gure shows only half of the U-channel because the symmetry andthe rosette strain gage is bonded on the top surface close to the edge of one ange.The longitudinal direction is the same as x-direction and the transverse direction issame as y-direction.

The grids, 45o/90o/135o, were placed with 2.4 mm in longitudinal direction(x-direction) between each, to make them follow the same path during theforming process. The position of the sheet was measured with an encoder,and with the help of the position the strain measurement results from thedierent grids were adjusted in time to make them measure the same point inthe process.

The strain in longitudinal and transverse direction was determined withstrain transformation, Dahlberg [10], equation (4.1).


() = x cos2() + y sin2() + xy cos() sin(). (4.1)

The equation (4.1) and the measurements in 45o/90o/135o give the equation(4.2) and the equation (4.3) for the longitudinal and transverse strain.

x(longitudinal) = 45 + 135 90. (4.2)

y(transverse) = 90. (4.3)

4.1.4 Roll load and roll torque

The roll load was measured with two donut load cells from Honeywell SensotecSensors. They were mounted in the forming stands on both sides of the uppertool, see Figure 4.4, and were able to measure a total roll load of 17800 N inz-direction, Figure 4.3.

Figure 4.4. The roll load in z-direction was measured with two donut cells. Thetorque sensor was used to measure the roll torque and it was mounted between thelower tool and power transmission. The equipment measured the forces of each form-ing step, one by one.

4.1 The roll forming experiment 39

The roll torque was measured with a torque sensor from Burster, which canmeasure up to 75 Nm. It was mounted in the universal joint shaft betweenthe lower tool and the power transmission, see Figure 4.4.

The measurement of the load and torque was made in one forming step ata time. For example the bend angle 20o was measured when the rst feederroll and the roll for a bend angle of 40o also were engaged with the strip, formeasurement of 40o are the rolls for bend angle 20o and 60o engaged at thesame time.

The load and torque are dependent on the tolerance between the thicknessof the roll formed material and the roll gap between upper and lower tools.The roll gap was adjusted to be 10 % to 15 % wider than the thickness of thematerial in order to avoid the material to be squeezed between tools causingvery high load and torque. The roll gap was checked with a feeler gage. Thismethod also takes into account shaft deection and bearing slackness.

4.1.5 Spring back

The prole will experience some spring back after each forming step and thendeform more and more as it approaches the next step. However, betweenspringback and deformation, the prole will have a shape that does not changeprovided the distance between the forming steps is large enough. In the presentexperiments this occurred about 100 mm to 150 mm before and after theforming step depending on material properties and geometry conditions of theprole.

The width (y-direction) and depth (z-direction) of the prole were mea-sured, see Figure 4.5, between the forming steps in the area where the shapedoes not change. This was done with a sliding caliper, and a measurementtolerance of about 0.1 mm.

Figure 4.5. The spring back, width and depth between the dierent forming stepswere measured with a sliding caliper.


4.2 Partial heating experiment

Section 4.2 describes the experimental equipment used in the study describedin paper 6, Lindgren et al. [38]. The aim of this study was to partiallyheat up the bending area on cold rolled AISI 301 stainless steel and then rollform the strip. The heating reduced the amount of martensite in this area.The plastic forming increases the martensite fraction again restoring the yieldstrength of the material. The material properties in the bending zone werethen investigated before and after the roll forming process and compared withheat treated samples from furnace experiments.

4.2.1 Furnace experiment

Samples, with a size of 0.7 15 84 mm, of the cold rolled stainless steel:AISI 301, were heated in an electrical batch furnace and then quenched inwater. The heat treatment times were 15, 30, 60, 90 and 120 s, respectively,with temperatures increasing from 500oC to 1200oC, in steps of 100oC each.

Figure 4.6. The picture shows the prototype resistance heating machine and thesteel strip which was fed through the machine by wheels made of copper.

4.2 Partial heating experiment 41

4.2.2 Prototype resistance heating machine

The prototype resistance heating machine and a sketch of the electrical cir-cuit are shown in Figure 4.6 and Figure 4.7. The machine consists of a pro-grammable logical control (PLC) system where the rotation speed of the heat-ing wheels can be adjusted. The power used to heat up the material was con-trolled by a thyristor that was controlled by a trigger signal from the PLC.The transformer increases the current in the material. The voltage over thematerial can be varied between 5 and 7 V (voltage) by adjusting the trans-former. For all experiments 7 V, a speed of 1 m/min and a contact pressure of5 Bar between the wheels and the sheet were used. The diameter of the wheelswas 122.5 mm and the width of the contact zone was 4 mm. Three dierentpowers called P10, P20 and P30 adjusted by the thyristor were investigated;the results described Paper 6, Lindgren et al. [38].

Figure 4.7. A sketch of the electrical circuit of the prototype resistance heatingmachine. Wikstrom [56].

4.2.3 Roll forming machine and tools

The roll forming machine in the experiment was the same as that described insection 4.1.1, see Figure 4.1. A V-section was roll formed in six forming steps:15o, 30o, 60o, 90o, 105o and 120o. The prole had an inner radius of 0.4 mmand the material was 0.7 mm thick, see Figure 4.8. The forming velocity was3.5 m/min and the forming load about 500 N. The diameter of the upper toolswas 140 mm and of the lower tools 100 mm. The tools were designed with theconstant radius method , Chaing [9], which means all forming steps had thesame inner radius.


Figure 4.8. The tools and the ower pattern. The V-section was roll formed in 6forming steps, from 15o to 120o.

4.2.4 Sample preparation

Cross-sections of all samples from the furnace heat treatment as well as par-tially heated samples from before and after the roll forming, were examinedwith light optical microscope and with Vickers microhardness tests with anapplied load of 500 g. The samples were hot mounted using a thermoset-ting resin followed by mechanical grinding and polishing with 1 m diamondsuspension. To reveal the microstructure of the samples from the furnace ex-periment, an etching solution of 60 ml deionised water, 62 ml concentratedHCl, 3 ml concentrated HNO3 and 5 ml concentrated H2SO4 was applied.The etching time was approximately 60 s in room temperature. To etch theheated zone of the partially heated samples, a solution consisting of 25 mlconcentrated HCl and 5 ml 10 wt% chrome acid solution in deionised waterwas used. In these cases etching time was 5-10 s.

4.3 3D Roll forming experiment

Section 4.3 describes the experimental equipment used in the study presentedin paper 7, Lindgren and Ingmarsson [39]. The aim of the study was toinvestigate a new tooling concept for a production of hat-channels with variablecross-sections.

4.3.1 The 3D roll forming machine

An experimental 3D roll forming machine was built in this study, see Figure4.9. The machine was equipped with slitter heads to t the metal sheet,and six forming stands with four units each, see Figure 4.10. The units have

4.3 3D Roll forming experiment 43

servo control axis, two translations and two rotations axis. A unit can bemoved up and down, in and out and rotate, and the speed of the tools canbe controlled individually. To control the servo-axis the position of the sheetmetal is measured by encoders.

The desired cross-section (depth and width) can be created in a CAD-program and read by the control system. Depending on the tooling the ma-chine can produce for example U-proles, C-proles, hat-proles etc. withvariable cross-sections.

Figure 4.9. The experimental equipment used for research and prototyping of 3Droll formed proles.

4.3.2 The tooling concept

In the study described in paper 7, a hat-section with variable depth and widthwas investigated and a special tooling concept was used. This concept demandstwo forming stands per pass, see Figure 4.11. Forming stands number 1, 3 and5 form the left side of a channel and forming stands number 2, 4, and 6 theright side. For example, a channel is roll formed in six forming steps withthe bend angles 15o , 30o , 45o , 60o, 75o and 85o and therefore the sectiondemands two laps in the machine. The rst lap forms 15o - 45o, and then thecontrol program of the machine switches to the next lap, 60o - 85o.

The tools are the same for all stands, for example Figure 4.12 shows that


Figure 4.10. The experimental equipment has 6 forming stands. Each stand has 4units (tool holders) that can rotate and translate.

Figure 4.11. View from the top: The prole is roll formed in six passes and to dothat the prole must go through the machine two times. Forming stands number 1,3 and 5 form the left side and forming stands number 2, 4, and 6 form the right side.

the tools for the left and right side are the same for both the 30o and 60o. Thedierence between the passes is that the tools that hold the ange have movedup closer to the tools that hold the web. The ower pattern is a function ofthe length of the section. This exibility makes it possible to use as manypasses as the cross-section requires without making more tools. It also makesit possible to produce a section not only of variable width but also of variable

4.3 3D Roll forming experiment 45

depth, in material of varying thickness and with dierent properties.

Figure 4.12. View from the back. The geometry of the tools is the same for allforming stands. The only dierence is that the tools for the ange are moving up andcloser to the tools that hold the web, for example bend angles 30o and 60o. Toolswith a constant radius have been used, Chaing [9].

4.3.3 Measuring equipment

An optical scanner based on the principle of triangulation was used to measurethe channels. The accuracy of the measurement equipment and the xturewere tested and showed a variation less than 0.07 mm.

A xture positioned the hat-prole at the bottom, z-direction, with 3 ref-erence points and 1 support point. The distance between the points in thelongitudinal direction was 1300 mm. In y-direction the section was positionedwith 2 reference points and 2 support points and in the x-direction one refer-ence point was used, see Figure 4.13.


Figure 4.13. The xture used.

Chapter 5

Computational model

Details about the nite element models described in paper 1-3 and paper 5are presented in the following chapter. The roll forming process was modelledwith the implicit FE package MARC/MENTAT [43], see Figure 5.1. Largedisplacement and large additive strains were taken into account in an Updated-Lagrangian approach. Nonlinear plastic material behaviour as well as contactswith friction contributed to the nonlinearities in the models.

Figure 5.1. The nite element model described in paper 5, Lindgren [37].

47

48 Computational model

5.1 Nonlinear solution procedure and convergencecriteria

To solve the nonlinear equilibrium equations the full Newton-Raphson methodwas used. Convergence is assumed when the relative unbalance force normbecomes less than 0.05 or if the relative displacement norm is less than 0.01.The analysis was performed with a direct sparse solver. The models describedin the appended papers dier with regard to the time step. For the model inpaper 5 a constant time step of 0.006 s was used whereas for the other modelsthe constant time step was between 0.002 and 0.008 s.

5.2 The geometry

The geometry of the model in paper 5The geometry of the roll forming tools, see Figure 5.1 and 5.2, was generatedwith the ORTIC CAE system [27], and the model comprised four formingsteps. The tools were modelled as analytically described rigid surfaces thatrotated with an angular speed of 0.75 rad/s which gave the prole a speed of3.375 m/min. The horizontal distance between the tools was 450 mm. Lengthand width of the unformed strip were 950 mm and 41.9 mm respectively. Thenumber of elements was 3840 and the size of the elements was 4 1 mm to4 2 mm in the bending zone. In other areas the element size was 4 5 mm.Five layers of integration points were used in the thickness direction

Figure 5.2. The geometry of the tools and the corresponding ower pattern describedin paper 5.

5.2 The geometry 49

The geometry of the model in the paper 2 and 3The nite element model consisted of two forming stands and two stands whereno forming took place, see Figure 5.3. The latter were used as a belt feeder tothe other two, where the forming was done. The element size was 4 1 mmin the bending zone and in other areas the element size was 4 4 mm. Threelayers of integration points were used in the thickness direction. The numberof thick shell elements ranged from 1600 to 2400. Due to its symmetry onlyhalf the geometry had to be modelled. The initial angular speed of the toolswas 10 rad/s, which gave the strip a speed of 0.6 m/s.

Figure 5.3. Four forming stands were used. The two rst stands were used as a beltfeeder, the other two forming stands have the same bend angle.

The geometry of the model in the paper 1This model comprised six forming stands and two stands where no formingtook place. The latter were used as a belt feeder to the six forming stands.The forming began with an undeformed strip and ended with the nishedsymmetrical U-prole. The distance between each of the forming stands was330 mm.

The strip was 1.502 mm thick; the width was 31.8 mm and the length1000 mm. The strip was modelled by the use of 2750 thick shell elements.The element size was 4 1 mm in the bending zone whereas in other areas itwas 4 4 mm. Three layers of integration points were used in the thicknessdirection. Due to its symmetry, only one half of the geometry was modelled.The initial angular speed of the tools was 10 rad/s which gave the strip aspeed of 0.6 m/s.


5.3 Shell element

The strip was modelled with a bilinear thick shell element type number 75,MARC elements Library [42]. This is a four node element that calculatesmembrane strains and the curvatures at the middle surface. The element canbe used in a curved shell analysis and thanks to the simple formulation theelement is not so expensive. A nite element model, with 3, 5, 7, 9 and 11integrations points in the thickness direction, was evaluated for the work inpaper 5. It was found that at least ve integration points must be used topredict the springback decently. The integrations points were spaced equallythroughout the thickness with one point both on the top and bottom surface.To calculate the stress and strain MARC uses numerical integration throughthe thickness.

5.4 Material model

An elastic-plastic material model with the von Mises yield criterion and isotropichardening was assumed together with the associated ow rule. The yield cri-terion is written as

f = y =

32sijsij y. (5.1)

where sij is the deviatoric Cauchy stress giving the eective stress . Thedeformation is elastic as long as this is lower than the yield limit y.

The associated ow rule is written as

pij =

ij=

sij. (5.2)

where is a plastic parameter to be determined by the consistency conditionthat f 0 during plastic ow.

The tensile test data for the materials used in paper 5, see Figure 5.4. Thestress versus plastic strain was implemented as a table in the nite elementprogram. The work hardening slope is described in Marc with a piecewise lin-ear representation in a table. Heating was ignored in all models and thereforethe ow stress at room temperature was sucient. In the studies presented inpaper 1-3 material data were implemented in same way as described above.

5.5 Contact 51

Figure 5.4. The tensile test data for the materials used in paper 5.

5.5 Contact

The friction between the prole and the tools was modelled as a type ofCoulomb friction. MARC uses

Ft = Fn2arctan(

vr

C). (5.3)

wherevr = Sliding velocityC = Relative sliding velocity, in the study is C = 0.01Fn = Normal forceFt = Tangential force = Friction coecient, in the study = 0.085

The tools were modelled as rigid bodies and the material was deformable.MARC uses a method called direct constraint method for this contact situa-tion. During the procedure, the node seldom hits the surface of the rigid bodyexactly. In order to make this situation easier a contact tolerance around therigid surface is given. MARC detects contact if a node in the deformable ma-terial comes inside this contact tolerance area (D) and it is default 25 % ofthe thickness of the smallest shell element used, see Figure 5.5. The contacttolerance is also default symmetrically (D1 = D2) placed around the surfaceof the rigid body. Greater tolerance gives less accuracy when calculating and


a smaller tolerance means that MARC will have more problems discoveringcontact, which causes higher computational costs. In this thesis a toleranceof 4 - 5 % of the shell thickness was used. As a result of rotating tools, manynodes are in almost contact in the roll forming process. This means that manycontact separations can occur thereby causing an extended calculation time.In order to avoid this it is possible to displace the tolerance distance closer tothe surface of the rigid body using a bias factor which means that D1 < D2.The default is a bias factor of 0 which means that D1 = D2 but in this thesisa bias factor of 0.25 was used.

Figure 5.5. The used tolerance distance D1 and D2 in paper 5 is 5 %

Experimental and Computational Investigation of the Roll Forming Process

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