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Analysis of Material Deformation and Wrinkling

Failure in Conventional Metal Spinning Process
WANG, LIN

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2
 Analysis of Material
Deformation and Wrinkling
Failure in Conventional Metal
Spinning Process

Lin Wang

Doctor of Philosophy

School of Engineering and Computing Sciences

Durham University

2012
 Analysis of Material Deformation and Wrinkling Failure
in Conventional Metal Spinning Process

Lin Wang

Abstract
Sheet metal spinning is one of the metal forming processes, where a flat metal blank is

rotated at a high speed and formed into an axisymmetric part by a roller which gradually

forces the blank onto a mandrel, bearing the final shape of the spun part. Over the last

few decades, sheet metal spinning has developed significantly and spun products have

been widely used in various industries. Although the spinning process has already been

known for centuries, the process design still highly relies on experienced spinners using

trial-and-error. Challenges remain to achieve high product dimensional accuracy and

prevent material failures. This PhD project aims to gain insight into the material

deformation and wrinkling failure mechanics in the conventional spinning process by

employing experimental and numerical methods. In this study, a tool compensation

technique has been proposed and used to develop CNC multiple roller path (passes).

3-D elastic-plastic Finite Element (FE) models have been developed to analyse the

material deformation and wrinkling failure of the spinning process. By combining these

two techniques in the process design, the time and materials wasted by using the

trial-and-error could be decreased significantly. In addition, it may provide a practical

approach of standardised operation for the spinning industry and thus improve the

product quality, process repeatability and production efficiency. Furthermore, effects of

process parameters, e.g. roller path profiles, feed rate and spindle speed, on the

variations of tool forces, stresses, strains, wall thickness and wrinkling failures have also

been investigated. Using a concave roller path produces high tool forces, stresses and

reduction of wall thickness. Conversely, low tool forces, stresses and wall thinning have

been obtained in the FE model which uses the convex roller path. High feed ratios help

to maintain original blank thickness but also lead to material failures and rough surface

finish. Thus it is necessary to find a “trade off” feed ratio for a spinning process design.

i
Declaration

I hereby declare that this thesis is my own work, which is based on research
carried out in the School of Engineering and Computing Sciences, Durham
University, UK. No portion of the work in the thesis has been submitted in
support of an application for another degree or qualification of any other
university or institute of learning.

Copyright © 2012 by Lin Wang

The copyright of this thesis rests with the author. No quotation from it should be
published without the prior written consent and information derived from it should
be acknowledged.

ii
Acknowledgements

My deepest gratitude goes first and foremost to my supervisor, Dr. Hui Long, for her

constant encouragement and guidance through all the stages of my PhD research and

thesis writing. Her tremendous effort and tireless support made this work possible and

was critical to the successful completion of this thesis.

Secondly, I would like to acknowledge the consistent support and valuable advice from

Mr. David Ashley, Mr. Martyn Roberts, Mr. Peter White, Mr. Fred Hoye, Mr. Paul

Johnson, and Mr. Kris Carter of Metal Spinners Group Ltd, where I worked as a KTP

associate for two years.

I appreciate the contribution to this research by former students of Durham University,

Mr. Seth Hamilton, Mr. Stephen Pell, and Mr. Paul Jagger for their suggestion and

assistance on FE modelling and the experimental design of metal spinning.

My sincere thanks also go to Miss. Rachel Ashworth for sharing literatures and

discussing theoretical analysis of the metal spinning process. I gratefully acknowledge

Dr. Xiaoying Zhuang and Mr. Xing Tan for their precious advice on the thesis writing.

Finally, I should like to express my heartfelt gratitude to my beloved parents. This thesis

is by all means devoted to them because they have assisted, supported and cared for

me all of my life.

The first two years of this PhD study were financially supported by UK Technology

Strategy Board and Metal Spinners Group Ltd, Project No. 6590. The final year of study

was sponsored by School of Engineering and Computing Sciences, Durham University.

iii
Publications

Aspects of the work presented in this thesis have been published in the following journal

papers and conference proceedings:

1. Wang, L., Long, H., 2011. Investigation of material deformation in multi-pass

conventional metal spinning. Materials & Design, 32, 2891-2899.

2. Wang, L., Long, H., Ashley, D., Roberts, M., White, P., 2011. Effects of roller feed
ratio on wrinkling failure in conventional spinning of a cylindrical cup.
Proceedings of IMechE: Part B: Journal of Engineering Manufacture, 225,
1991-2006.

3. Wang, L., Long, H., 2011. A study of effects of roller path profiles on tool forces
and part wall thickness variation in conventional metal spinning, Journal of
Materials Processing Technology, 211, 2140-2151.

4. Wang, L., Long, H., Ashley, D., Roberts, M., White, P., 2010. Analysis of
single-pass conventional spinning by Taguchi and Finite Element methods, Steel
Research International, 81, 974-977.

5. Wang, L., Long, H., 2010. Stress analysis of multi-pass conventional spinning,
Proc. of 8th International Conference on Manufacturing Research, Durham, UK.

6. Wang, L., Long, H., 2011. Investigation of Effects of Roller Path Profiles on
Wrinkling in Conventional Spinning, Proc. of 10th International Conference on
Technology of Plasticity, Aachen, Germany.

7. Long, H., Wang, L., Jagger, P., 2011. Roller Force Analysis in Multi-pass
Conventional Spinning by Finite Element Simulation and Experimental
Measurement, Proc. of 10th International Conference on Technology of Plasticity,
Aachen, Germany.

iv
 Table of Contents

Abstract ............................................................................................................................i
Declaration ......................................................................................................................ii
Acknowledgements ........................................................................................................ iii
Publications ....................................................................................................................iv
Table of Contents.............................................................................................................v
List of Figures ............................................................................................................... viii
List of Tables....................................................................................................................x
List of Abbreviations .......................................................................................................xi
Nomenclature ................................................................................................................ xii
Terminology in Spinning ............................................................................................... xvi
1. Introduction .............................................................................................................. 1
1.1 Background........................................................................................................... 1
1.2 Scope of Research ............................................................................................... 5
1.3 Structure of Thesis ................................................................................................ 8
2. Literature Review ................................................................................................... 10
2.1 Investigation Techniques..................................................................................... 10
2.1.1 Theoretical Study.......................................................................................... 10
2.1.1.1 Analysis of Tool Forces .......................................................................... 10
2.1.1.2 Prediction of Strains ................................................................................11
2.1.1.3 Investigation of Wrinkling Failures ..........................................................11
2.1.2 Experimental Investigation ........................................................................... 12
2.1.2.1 Measurement of Tool Forces.................................................................. 12
2.1.2.2 Investigation of Strains and Material Deformation.................................. 14
2.1.2.3 Study of Material Failures....................................................................... 15
2.1.2.4 Design of Experiments ........................................................................... 16
2.1.3 Finite Element Analysis ................................................................................ 17
2.1.3.1 Finite Element Solution Methods ........................................................... 17
2.1.3.2 Material Constitutive Model.................................................................... 18
2.1.3.3 Element Selection .................................................................................. 20
2.1.3.4 Meshing Strategy ................................................................................... 21
2.1.3.5 Contact Treatment.................................................................................. 22
2.2 Material Deformation and Wrinkling Failure ........................................................ 23
2.2.1 Tool forces .................................................................................................... 23
2.2.2 Stresses........................................................................................................ 24
2.2.3 Strains........................................................................................................... 25
2.2.4 Wrinkling Failure ........................................................................................... 26
2.3 Key Process Parameters .................................................................................... 27
2.3.1 Feed Ratio .................................................................................................... 27
2.3.2 Roller Path and Passes ................................................................................ 28
2.3.3 Roller Profile ................................................................................................. 29
2.3.4 Clearance between Roller and Mandrel ....................................................... 30

v
 2.4 Summary............................................................................................................. 31
3. Fundamentals of Finite Element Method ............................................................. 32
3.1 Hamilton’s Principle............................................................................................. 32
3.2 Basic Analysis Procedure of FEM ....................................................................... 33
3.2.1 Domain Discretisation................................................................................... 33
3.2.2 Displacement Interpolation ........................................................................... 34
3.2.3 Construction of Shape Function ................................................................... 35
3.2.4 Formation of Local FE Equations ................................................................. 38
3.2.5 Assembly of Global FE Equations ................................................................ 40
3.3 Different Type of Finite Elements ........................................................................ 41
3.3.1 3-D Solid Element......................................................................................... 41
3.3.2 2-D Plane Stress/Strain Element .................................................................. 45
3.3.3 Plate Element ............................................................................................... 47
3.3.4 Shell Element ............................................................................................... 49
3.4 Non-linear Solution Method ................................................................................ 51
3.4.1 Implicit Method ............................................................................................. 51
3.4.2 Explicit Method ............................................................................................. 54
3.5 Material Constitutive Model................................................................................. 55
3.5.1 von Mises Yield Criterion .............................................................................. 55
3.5.2 Strain Hardening........................................................................................... 57
3.6 Contact algorithms .............................................................................................. 59
3.6.1 Contact Surface Weighting ........................................................................... 59
3.6.2 Tracking Approach........................................................................................ 59
3.6.3 Constraint Enforcement Method................................................................... 60
3.6.4 Frictional Model ............................................................................................ 61
3.7 Summary............................................................................................................. 61
4. Effects of Roller Path Profiles on Material Deformation..................................... 62
4.1 Experimental Investigation .................................................................................. 62
4.1.1 Experimental Setup ...................................................................................... 62
4.1.2 Design of Various Roller Path Profiles.......................................................... 64
4.2 Finite Element Simulation ................................................................................... 68
4.2.1 Development of Finite Element Models ........................................................ 68
4.2.2 Verification of Finite Element Models ........................................................... 70
4.2.2.1 Mesh Convergence Study ...................................................................... 70
4.2.2.2 Assessment of Scaling Methods ............................................................ 74
4.2.2.3 Comparison of Dimensional Results ...................................................... 74
4.3 Results and Discussion....................................................................................... 76
4.3.1 Tool Forces ................................................................................................... 76
4.3.2 Wall Thickness.............................................................................................. 79
4.3.3 Stresses........................................................................................................ 82
4.3.4 Strains........................................................................................................... 85
4.4 Summary and Conclusion................................................................................... 89
5. Analysis of Material Deformation in Multi-pass Conventional Spinning........... 90
5.1 Experimental Investigation .................................................................................. 90
5.1.1 Tool Compensation in CNC Programming.................................................... 90

vi
 5.1.2 Experimental Design by Taguchi Method ..................................................... 94
5.2 Experimental Results and Discussion................................................................. 95
5.2.1 Diameter of Spun Part .................................................................................. 96
5.2.2 Thickness of Spun Part................................................................................. 97
5.2.3 Depth of Spun Part ....................................................................................... 99
5.3 Finite Element Simulation ................................................................................. 100
5.3.1 Development of Finite Element Models ...................................................... 100
5.3.2 Verification of Finite Element Models ......................................................... 101
5.4 Finite Element Analysis Results and Discussion............................................... 104
5.4.1 Tool Forces ................................................................................................. 104
5.4.2 Stresses...................................................................................................... 105
5.4.3 Wall Thickness............................................................................................ 109
5.4.4 Strains..........................................................................................................111
5.5 Summary and Conclusion..................................................................................113
6. Study on Wrinkling Failures .................................................................................115
6.1 Theoretical Analysis ...........................................................................................115
6.1.1 Energy Method ............................................................................................116
6.1.2 Theoretical Model ........................................................................................117
6.2 Experimental Investigation .................................................................................119
6.2.1 Experimental Setup .....................................................................................119
6.2.2 Process Parameters ................................................................................... 120
6.3 Finite Element Simulation ................................................................................. 122
6.3.1 Element Selection....................................................................................... 123
6.3.2 Verification of FE Models............................................................................ 126
6.4 Results and Discussion..................................................................................... 127
6.4.1 Severity of Wrinkle...................................................................................... 128
6.4.2 Forming Limit of Wrinkling .......................................................................... 129
6.4.3 Tool Forces ................................................................................................. 131
6.4.4 Stresses...................................................................................................... 134
6.4.5 Thickness ................................................................................................... 138
6.5 Summary and Conclusion................................................................................. 139
7. Conclusion and Future Work .............................................................................. 141
7.1 Conclusion ........................................................................................................ 141
7.2 Future Work ...................................................................................................... 144
Reference................................................................................................................... 146
Appendix .................................................................................................................... 152
Appendix 1 Roller Path information of Multiple Pass Spinning Study ..................... 152
Appendix 2 Roller Path information of Wrinkling Failure Study............................... 163

vii
List of Figures

Figure 1.1 Setup of metal spinning process, adapted from Runge (1994).................................1
Figure 1.2 Applications of spun parts (http://www.metal-spinners.co.uk) ..................................2
Figure 1.3 Conventional spinning and shear forming, adapted from Music et al. (2010) ............2
Figure 1.4 Stress distributions of roller working zone during conventional spinning ...................3
Figure 1.5 Typical material failure modes in metal spinning (Wong et al., 2003) ........................4
Figure 1.6 System of conventional spinning process, adapted from Runge (1994)....................6
Figure 2.1 Definitions of tool force components .....................................................................13
Figure 2.2 Force measurement system, adapted from Jagger (2010) ................................14
Figure 2.3 Methods for studying strains and material deformation......................................15
Figure 2.4 Propagation of wrinkles in spinning (Kleiner et al., 2002)...................................16
Figure 2.5 Material hardening models (Dunne and Petrinic, 2005) .....................................19
Figure 2.6 Deformation of a reduced integration linear solid element subjected to bending .....21
Figure 2.7 Mesh strategy, adapted from Sebastiani et al. (2006) ............................................22
Figure 2.8 Various roller path profiles ....................................................................................29
Figure 2.9 Various shapes of roller (Avitzur et al., 1959).........................................................30
Figure 2.10 Deviation from sine law in shear forming, adapted from Music et al. (2010)..........31
Figure 3.1 Finite Element Meshing (Wang, 2005) ...............................................................34
Figure 3.2 Pascal triangle of monomials (Liu and Quek, 2003)...........................................36
Figure 3.3 Pascal pyramid of monomials (Liu and Quek, 2003)..........................................37
Figure 3.4 Hexahedron element and coordinate system (Liu and Quek, 2003) ..................42
Figure 3.5 Rectangular 2-D plane stress/strain element (Liu and Quek, 2003)...................45
Figure 3.6 Rectangular shell element (Liu and Quek, 2003) ...............................................49
Figure 3.7 First iteration in an increment (Abaqus analysis user’s manual, 2008) ..............53
Figure 3.8 Second iteration in an increment (Abaqus analysis user’s manual, 2008) .........53
Figure 3.9 Bi-linear stress-strain curve (Dunne and Petrinic, 2005) ....................................56
Figure 3.10 Isotropic strain hardening (Dunne and Petrinic, 2005) .....................................57
Figure 3.11 Stress-strain curve of linear strain hardening (Dunne and Petrinic, 2005)........58
Figure 4.1 Spinning experiment ..........................................................................................63
Figure 4.2 Roller path profile design ...................................................................................66
Figure 4.3 Experimentally spun samples by using different CNC roller paths .....................66
Figure 4.4 Concave roller path profiles using different curvatures ......................................67
Figure 4.5 True stress-strain curves of Mild steel (DC01) ...................................................69
Figure 4.6 Variations of von Mises stress in 1st forward pass of FE model .........................73
Figure 4.7 Comparison of wall thickness between FE analysis and experimental results ...76
Figure 4.8 Comparison of tool forces using various roller path profiles...............................78
Figure 4.9 Wall thickness variations using various roller path profiles.................................80
Figure 4.10 Wall thickness variations using concave path with different curvatures ...........81
Figure 4.11 Radial stress variations after 1st forward pass ..................................................83
Figure 4.12 Tangential stress variations after 1st forward pass............................................84
Figure 4.13 Maximum in-plane principal strain (radial strain) after 1st forward pass............86
Figure 4.14 Minimum in-plane principal strain (tangential strain) after 1st forward pass ......87

viii
Figure 4.15 Out-of-plane principal strain (thickness strain) after 1st forward pass...............88
Figure 5.1 Tool compensation ..............................................................................................91
Figure 5.2 Multi-pass design and spun sample without tool compensation.............................92
Figure 5.3 Roller passes design using tool compensation .....................................................93
Figure 5.4 Spinning experiment in progress ..........................................................................93
Figure 5.5 Experimental measurements ...............................................................................95
Figure 5.6 Experimental spun parts ......................................................................................96
Figure 5.7 Main effects plot for diameter .............................................................................97
Figure 5.8 Main effects plot for thickness ..............................................................................98
Figure 5.9 Main effects plot for depth....................................................................................99
Figure 5.10 Spinning process using off-line designed roller passes .....................................101
Figure 5.11 Comparison of experimental and FE analysis results of wall thickness ..............102
Figure 5.12 Evaluation of energy ratios in FE model ...........................................................103
Figure 5.13 Comparison of experimental and FE analysis results of axial forces (Long et al., 2011) ....103
Figure 5.14 History of tool forces of FE simulation ..............................................................104
Figure 5.15 Variations of stresses at the beginning of 1st forward pass ................................107
Figure 5.16 Variations of stresses at the beginning of 1st backward pass .............................108
Figure 5.17 Variations of wall thickness .............................................................................. 111
Figure 5.18 Variations of strains at the beginning of 1st pass ...............................................113
Figure 6.1 Schematic of a buckled plate in flange region.....................................................116
Figure 6.2 Spinning experiment of wrinkling investigation....................................................120
Figure 6.3 Roller passes used in the experiment.................................................................121
Figure 6.4 Experimental samples......................................................................................122
Figure 6.5 Comparison of deformed workpiece using different types and numbers of elements .124
Figure 6.6 Force comparisons of wrinkle-free models using different types and numbers of elements .126
Figure 6.7 Ratio of artificial strain energy to internal energy of the wrinkle-free models ...127
Figure 6.8 Effects of roller feed ratio on wrinkling ................................................................128
Figure 6.9 Severity of wrinkles of FE models ......................................................................128
Figure 6.10 Forming limit diagram for wrinkling...................................................................130
Figure 6.11 Force histories of wrinkle-free models (Model 5 and 7)......................................132
Figure 6.12 Force histories of wrinkling model (Model 4).....................................................134
Figure 6.13 Tangential stress distribution of wrinkling model (Model 4) ................................135
Figure 6.14 Tangential stress distribution of wrinkle-free model (Model 5)............................135
Figure 6.15 Stress distributions in flange at wrinkling zone (Model 4)...................................137
Figure 6.16 Wall thickness distributions at different feed ratios.........................................138
Figure 6.17 Effects of feed ratio in blank metal spinning ......................................................139

ix
List of Tables
Table 4.1 Mesh convergence study.....................................................................................71
Table 4.2 Scaling method study – Trial 1.............................................................................74
Table 4.3 Ratios of maximum force components using various roller path profiles .............78
Table 5.1 Experimental input factors and levels.....................................................................95
Table 5.2 Experimental runs and dimensional results ............................................................96
Table 5.3 Comparison of depth and diameter FEA vs. experimental results..........................101
Table 5.4 Ratios of maximum tool forces of FE model .........................................................105
Table 6.1 Factor α for deflection equation (Timoshenko and Woinowsky-Krieger, 1959) .. 118
Table 6.2 Process parameters of experimental runs............................................................121
Table 6.3 FE analysis process parameters and flange state of spun part .............................123
Table 6.4 FE models using different types and numbers of elements ...................................123
Table 6.5 Standard deviations of wrinkle amplitudes............................................................129
Table 6.6 Feed ratio limits of various thicknesses of blanks .................................................131

x
List of Abbreviations

2-D Two Dimension

3-D Three Dimension

ANOM Analysis of Means

CAM Computer Aided Manufacturing

CNC Computer Numerical Control

CPU Central Processing Unit

DoE Design of Experiment

DOF Degree of Freedom

FE Finite Element

FEM Finite Element Method

OFAT One-Factor-At-a-Time

PNC Playback Numerical Control

RAM Random-Access Memory

RSM Response Surface Methodology

S/N Signal to Noise ratio

xi
Nomenclature

a Length of a half-wave wrinkle in the tangential direction of flange

b Width of wrinkled plate in the radial direction of flange

B Strain matrix

cd Wave speed of the material

D Flexural rigidity of plate

D Matrix of material constants

D0 Original diameter of the blank

D1 Final diameter of the blank

dε Element strain increments

E Young's Modulus

E0 Reduced Modulus

Eb Energy due to the bending in the wrinkled flange

El Energy due to the lateral concentrated loading from the roller

Slope of the stress strain curve at a particular value of strain in the plastic
Ep
region

Energy due to the radial elongation of the flange under tensile radial
Er
stresses

Energy due to the circumferential shortening of the flange under

Et
compressive tangential stresses

fb Body force

fs Surface force

F External applied force vector / Feed rate

Fa Axial tool force

Fr Radial tool force

Ft Tangential tool force

G Shear modulus

h Plastic hardening modulus

I Internal element force vector

xii
J Jacobian matrix

K Stiffness matrix

L Matrix of partial differential operator

Le Characteristic element length

M Mass matrix / Material type

n The number of the time increment / Sample number of the experiment

nd Number of nodes forming the element

nf Number of Degree of Freedom

N Number of the nodes along the edge of workpiece

N Shape function

p& Effective plastic strain rate

P Lateral concentrated load

Radial direction in a cylindrical coordinate system of the mandrel / Roller

r
nose radius

R The radius of the round part of the mandrel

S Spindle speed

Sf Domain of area

T Time of the analysed process / Kinetic energy

T Transformation matrix

Tangential direction in a cylindrical coordinate system of the mandrel / Wall

t
thickness

t0 Original thickness of the blank

t1 Final thickness of the blank

u Vector of displacement

Ui Radial coordinate of element node i along the edge of workpiece

U Displacement vector

V Domain of volume

v Poisson’s ratio

w Buckled deflection surface

Wf Work done by external force

xiii
X x-coordinate of the global coordinate system

x Coordinate in transverse direction of local coordinate system

Y y-coordinate of the global coordinate system

yi Outputs of different samples

Coordinate in longitudinal direction of local coordinate system / Axial

z
direction in a cylindrical coordinate system of the mandrel

α Inclined angle of the mandrel in shear forming

Δt Time increment

θ Angle between local coordinate system and the global coordinate system

λ Lamé’s constant

μ Lamé’s constant

ρ Mass density of material

σ Stress

σ1 Principal stress

σ2 Principal stress

σr Radial stress

σt Tangential stress

σy Yielding stress

σe Effective stress

ε Strain

εe Elastic strain

εp Plastic strain

u& Vector of velocity

u&& Vector of acceleration

y Mean outputs of different samples

γ Maximum deflection of the buckling surface
U Mean value of radial coordinate of element nodes on the workpiece edge

∏ Strain energy

ξ Natural coordinate of an element

xiv
η Natural coordinate of an element

ξ Natural coordinate of an element

τ Shear stress
χ Curvature of a plate

xv
Terminology in Spinning

There are currently no universally agreed terminologies of the metal spinning process.

Different researchers and engineers may use different terms referring to the same

technique. The spinning terms used in this thesis, corresponding alternatives and

explanations are shown below.

Terms Alternative Explanation

Mandrel Former, chunk Rigid tool which bears the final profile of the
desired spun product.
Backplate Tailstock Circular disk which clamps the blank onto the
mandrel
Roller nose Roller round-off Blending radius between the two flat surfaces on
radius radius the outer surface of the roller (Music et al., 2010).
Roller path Tool path The trace of roller movement, e.g. linear, convex,
concave, etc.
Forward Rim-directed Roller feeds towards the edge of the blank
path movement
Backward Centre-directed Roller feeds towards the centre of the blank
path movement
Feed rate Feed Feeding speed of the roller (unit: mm/min)
Spindle Mandrel speed, Rotational speed of the mandrel (unit: rpm)
speed rotational speed
Feed ratio Feed per Ratio of feed rate to spindle speed (unit: mm/rev)
revolution, feed
Conventional Multiple-pass Spinning process which deliberately reduces the
spinning spinning, diameter of the workpiece but without changing the
manual wall thickness by using multiple roller passes
spinning
Shear Shear spinning, Spinning process which maintains the diameter of
forming power spinning the workpiece and deliberately decreases the wall
thickness by a single roller pass
Spinnability Formability The ability of a sheet metal to undergo deformation
by spinning without wrinkling or cracking failures

xvi
 Chapter 1 Introduction

1. Introduction

1.1 Background

Sheet metal spinning is one of the metal forming processes, where a flat metal blank is

formed into an axisymmetric part by a roller which gradually forces the blank onto a

mandrel, bearing the final shape of the spun part. As shown in Figure 1.1, during the

spinning process, the blank is clamped between the mandrel and backplate; these three

components rotate synchronously at a specified spindle speed. Materials used in the

spinning process include non-alloyed carbon steels, heat-resistant and stainless steels,

non-ferrous heavy metals and light alloys (Runge, 1994). The process is capable of

forming a workpiece with a thickness of 0.5 mm to 30 mm and diameter of 10 mm - 5 m.

Figure 1.1 Setup of metal spinning process, adapted from Runge (1994)

Due to its incremental forming feature, metal spinning has some unique advantages

over other sheet metal forming processes. These include process flexibility,

non-dedicated tooling, low forming load, good surface finish and improved mechanical

properties of the spun part (Wick et al., 1984). Hence, the sheet metal spinning process

has been frequently used to produce components for the automotive, aerospace,

medical, construction and defence industries, as shown in Figure 1.2.

1
 Chapter 1 Introduction

Figure 1.2 Applications of spun parts (http://www.metal-spinners.co.uk)

There are two types of sheet metal spinning: in conventional spinning, as shown in

Figure 1.3(a), a blank is formed into the desired shape by multiple roller passes to

maintain the original wall thickness (t0); however, the diameter of the spun part (D1) has

been reduced from the original diameter (D0). Conversely, during shear forming, the

roller deforms the blank by one single pass as shown in Figure 1.3(b). The diameter of

the spun part (D1) remains unchanged but the wall thickness of the spun part is reduced

deliberately. The final thickness of the spun part, t1, can be determined by the sine law:

t1 = t0 ⋅ sin α (1)

where t0 is the original thickness of the blank, α is the inclined angle of the mandrel.

(a) Conventional spinning: spun part (left), (b) Shear forming: spun part (left),
blank (right) blank (right)

Figure 1.3 Conventional spinning and shear forming, adapted from Music et al. (2010)

2
 Chapter 1 Introduction

During the conventional spinning process, a local plastic deformation zone is generated

at the roller contact area. The stress patterns of this zone depend on the roller feeding

direction (Runge, 1994). In the forward pass (the roller feeds towards the edge of the

blank), tensile radial stresses and compressive tangential stresses are induced, as

shown in Figure 1.4(a). The tensile radial stresses lead to a material flow towards the

edge of the blank causing thinning of the blank, which is balanced by the thickening

effects of the compressive tangential stresses, maintaining an almost constant

thickness. In the backward pass (the roller feeds towards the mandrel), however, the

material builds up in front of the roller, generating compressive radial stresses and

compressive tangential stresses, as shown in Figure 1.4(b).

(a) Forward pass (b) Backward pass

Figure 1.4 Stress distributions of roller working zone during conventional spinning

There are three types of common material failures in the sheet metal spinning process

(Wong et al., 2003): wrinkling, circumferential cracking and radial cracking, as shown in

Figure 1.5. Wrinkling is caused by buckling effects of the unsupported flange of the

metal sheet during spinning. Once the compressive tangential stress in the workpiece

exceeds a buckling stability limit, wrinkling will occur. Therefore, multiple roller passes

are generally required in order to keep the compressive tangential stress below the

buckling limit. In the sheet metal spinning process, excessive stresses in either radial or

tangential direction of the spun part are undesirable. High tensile radial stresses lead to

the circumferential cracking failure, mainly in the area close to the mandrel, as

3
 Chapter 1 Introduction

illustrated in Figure 1.5(b). The radial cracking shown in Figure 1.5(c) is normally

caused by the bending effects over existing severe wrinkles.

(a) Wrinkling (b) Circumferential cracking (c) Radial cracking

Figure 1.5 Typical material failure modes in metal spinning (Wong et al., 2003)

Up to now, research on the sheet metal spinning process has been carried out by using

three techniques, i.e. theoretical study, experimental investigation and FE simulation.

Each technique has its own advantages and disadvantages. For instance, theoretical

study is the least expensive method used when analysing the metal spinning process

and it has the potential to assist process design and predict material failures. However,

due to the complex nature of metal spinning, theoretical study has to be developed on

certain simplified assumptions. It is therefore almost impossible to obtain detailed and

reliable results, such as stress and strain, by applying the theoretical analysis alone. On

the other hand, accurate tool forces, strains and material failures can be obtained via

experimental investigation. Nevertheless, carrying out experiments with various

parameters at different levels costs a significant amount of time and material; thoroughly

analysing their effects on material deformation is extremely difficult. FE simulation has

the potential to provide in-depth understanding of the material deformation and failure

mechanics, and can therefore develop guidance in determining process parameters and

improve product quality. However, FE simulation of the spinning process involves three

areas of non-linearity: material non-linearity, geometry non-linearity and boundary

non-linearity. It generally takes extremely long computational time due to the nature of

incremental forming and complex contact conditions.

4
 Chapter 1 Introduction

The shear forming process has been investigated intensely by many researchers who have

been using both experimental and numerical approaches since 1960. On the other hand,

limited publications on conventional spinning mainly focus on one-pass deep drawing

conventional spinning and simple multi-pass conventional spinning (less than three passes,

linear path profile). The process design of conventional spinning thus still remains a

challenging task and material failures significantly affect production efficiency and

product quality. In the present industrial practice, the trial-and-error approach is commonly

used in the process design (Hagan and Jeswiet, 2003). With the aid of Playback Numerical

Control (PNC) of the spinning machine, all the processing commands developed by

experienced spinners are recorded and used in the subsequent spinning productions

(Pollitt, 1982). Nevertheless, the process design inevitably results in significant variations

and discrepancies in product quality and geometrical dimensions (Hamilton and Long,

2008). Furthermore, the procedure of the PNC process development and validation unduly

wastes a considerable amount of time and materials. It is therefore essential to study the

material deformation and failure mechanics in the multi-pass conventional spinning process

and to analyse the effects of process parameters on the quality of spun products.

1.2 Scope of Research

The aim of this research is to gain in-depth understanding of material deformation and

wrinkling failures, and thus to provide guidance on process design for the conventional

spinning process. A schematic diagram of the system in the conventional spinning

process is shown in Figure 1.6, where the underlined texts indicate the parameters and

outputs that have been investigated by this study.

5
 Chapter 1 Introduction

Workpiece Parameters

z Blank Thickness

z Blank Diameter
Outputs
z Blank Material
z Geometrical Accuracy

Tooling Parameters z Wall Thickness

z Roller Diameter z Tool Forces

z Roller Nose Radius z Production Time

z Mandrel Diameter z Stress

z Blank Support Unit z Forming Temperature

z Hardness
Process Parameters
z Strain
z Feed Rate
z Surface Finish
z Spindle Speed
z Wrinkling Failures
z Feed Ratio
z Cracking Failures
z Roller Path (Passes)

z Temperature

z Lubricant

Figure 1.6 System of conventional spinning process, adapted from Runge (1994)

In this project contributions have been made on six areas of research work on the

conventional spinning process:

1) Finite Element Simulation

3-D elastic-plastic models of metal spinning have been developed using commercial FE

software Abaqus. The explicit FE solution method has been chosen to simulate the

spinning process, because it is more robust and efficient to model 3-D problems that

involve highly nonlinearities. The computing performance of different types of elements

and different scaling methods has been evaluated respectively.

6
 Chapter 1 Introduction

2) Experimental Investigation

CNC programming has been used to develop roller path (passes) in this study by using

spinning Computer Aided Manufacturing (CAM) software - OPUS. To make the

workpiece successfully conform to the non-linear profile of the mandrel, the tool

compensation techniques have been proposed and employed in the multiple roller

passes design. In addition, the Taguchi method has been used to design an experiment

and to analyse the dimensional variation of spun samples. Experimental investigation

has also been conducted to study the wrinkling failures.

3) Theoretical Analysis of Wrinkling

Energy methods and two-directional plate buckling theory have been used to predict the

critical condition of wrinkling failure in conventional spinning. To predict wrinkling failures,

a theoretical model involving the radial stress, tangential stress, flange dimension and

material property has been developed.

4) Material Deformation

Based on the FE simulation, the variations of tool forces, stresses, strains and wall

thickness have been investigated numerically. Axial force dominates at the beginning of

the conventional spinning; radial force increases gradually over the process; tangential

force is the smallest and remains almost constant. Stress analysis shows that high

tensile and compressive radial stresses take place behind and in front of the roller

contact. Two pairs of oppositely directed radial bending effects have been observed in

the workpiece. The dominated in-plane tensile radial strains of the workpiece are

believed to be the main reason behind the wall thinning.

5) Wrinkling Failures

In order to understand the wrinkling failure mechanics, FE analysis results of tool forces

and stresses of a wrinkle-free model and a wrinkling model have been compared. It is

believed that sudden changes and fluctuations in the tool forces could be used to

determine the approximate moment that wrinkling occurs. If compressive tangential

7
 Chapter 1 Introduction

stresses at the flange area near the local forming zone do not fully “recover” to tensile

tangential stresses after leaving roller contact, wrinkling failure will take place.

6) Effects of Parameters
Using a concave roller path produces high tool forces, stresses and reduction of wall

thickness. Conversely, low tool forces, stresses and wall thinning have been obtained in

the FE model which uses the convex roller path. Moreover, results of an experiment

show that the type of material has the most significant effects on the dimensional

variations of spun parts, followed by the effects from feed rate and spindle speed. It has

been shown that high feed ratios help to maintain original blank thickness. However,

high feed ratios also lead to material failures and rough surface finish.

1.3 Structure of Thesis

This thesis consists of seven main chapters, the contents of which are detailed as

below:

Chapter 2 gives a systematic review on the published literature of research on the sheet

metal spinning. Three main investigation techniques used in the research of the sheet

metal spinning process are reviewed, i.e. theoretical study, experimental investigation

and FE analysis. Additionally, research on the material deformation and wrinkling failure

mechanics is presented. Effects of four key process parameters, namely, feed ratio,

roller path (passes), roller profile, and clearance between roller and mandrel, on the

quality of the spun parts are also discussed.

The fundamental theory of Finite Element Method has been discussed in Chapter 3,

such as Hamilton’s Principle and basic analysis procedure of FEM. Moreover, the

formulations of four different types of finite elements, i.e. 3-D solid element, 2-D plane

stress/strain element, plate element and shell element, are presented. Two commonly

used non-linear FE solution methods, implicit method and explicit method, are

compared. Additionally, the elastic-plastic material constitutive model and contact

8
 Chapter 1 Introduction

algorithms of FE simulation have been briefly outlined.

Chapter 4 presents an investigation of the effects of roller path profiles on material

deformation. Four roller path profiles are designed and developed to carry out

experimental investigation and FE simulations. The techniques of developing 3-D FE

models of metal spinning are explained in detail. These FE models are verified by

conducting a mesh convergence study, assessing scaling methods and comparing

dimensional results.

In Chapter 5, material deformation in a multi-pass conventional spinning process is

investigated experimentally and numerically. The tool compensation technique is

studied and used in the CNC multiple roller passes design. The Taguchi method is

applied to design the experiment and to analyse the effects of process parameters on

the dimensional variations of spun parts. In addition, FE simulation is conducted to

investigate the variations of tool forces, stresses, wall thickness, and strains in this

multi-pass conventional spinning process.

Theoretical analysis, experimental investigation and FE simulation of the wrinkling

failures in conventional spinning are carried out in Chapter 6. The theory of

two-direction plate buckling and the energy method are employed to determine the

critical condition of wrinkling in the conventional spinning. The severity of wrinkles is

quantified and a forming limit study is carried out by conducting FE simulations.

Furthermore, the computational performance of the solid and shell elements in

simulating the spinning process is examined. Stresses and tool forces are also

investigated in order to gain insight into the wrinkling failure mechanics.

Chapter 7 summarises key conclusions of this study on material deformation and

wrinkling failures in conventional spinning. Future research trends of sheet metal

spinning processes are also outlined.

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 Chapter 2 Literature Review

2. Literature Review

This chapter consists of three main sections which review the published literature of

studies on sheet metal spinning. In Section 2.1, three main investigation techniques in

the research of metal spinning, i.e. theoretical study, experimental investigation and FE

analysis are presented. Section 2.2 outlines research on the material deformation and

wrinkling failure mechanics in the sheet metal spinning process. Section 2.3 discusses

the effects of four key process parameters on the material deformation and failure of the

sheet metal spinning process. The end of this chapter gives a brief summary and

discusses the knowledge gap identified.

2.1 Investigation Techniques

In this section, the methodology of theoretical analysis and experimental investigation

on the tool forces, strains and material failures of the spinning process are reviewed.

Moreover, the key factors in the FE simulation, such as FE solution methods, material

constitutive model, element selection, meshing strategy and contact treatment, are

discussed in detail.

2.1.1 Theoretical Study

Compared with the limited theoretical studies on the strain and wrinkling failure, most of

the research work focuses on the theoretical analysis of tool forces, where eight

analytical force models are identified in this literature review. However, all of these

analytical force models are developed for the shear forming but not for conventional

spinning.

2.1.1.1 Analysis of Tool Forces

In those eight published papers, the deformation energy method has been used to

predict the tool forces, i.e. the work done by the external force is assumed to be equal to

10
 Chapter 2 Literature Review

the deformation energy of the workpiece. Most of the analytical models developed in

1960s only took the tangential force component into account (Avitzur and Yang, 1960,

Kalpakcioglu, 1961a, Sortais et al., 1963). This is because the tangential force

consumes most of the power in the spinning, and it is thus significantly important for the

design of spinning machines. Researchers (Avitzur and Yang, 1960, Kim et al., 2003,

Kobayashi et al., 1961) calculated the tool force based on the assumption that the

deformation mode in spinning is a combination of bending and shearing. Moreover, by

assuming uniform roller contact pressure, Kobayashi et al. (1961) estimated the radial

and axial forces from the projected contact areas. A similar approach has also been

employed by Chen et al. (2005a), Kim et al. (2006) and Zhang et al. (2010).

2.1.1.2 Prediction of Strains

By assuming hoop strain to be zero in shear forming and neglecting the thickness strain

in conventional spinning on a spherical mandrel, Quigley and Monaghan (2000)

proposed a theoretical analytical method to predict the strains using the constancy of

volume. The verifying experiment indicated that the theoretical strain results only agreed

well in the middle section of workpiece along its radial direction. Beni et al. (2011) also

applied this method and compared the theoretical results with their experimental results.

The authors reported that the theoretical strain models could not predict the strain

values accurately due to unrealistic assumption of zero hoop strain in shear forming and

zero thickness strain in conventional spinning.

2.1.1.3 Investigation of Wrinkling Failures

In general there are two methods to analyse the wrinkling failures of engineering

problems (Senior, 1956): (1) Equilibrium method, where the differential equations for the

system in equilibrium are set up and solved to obtain the critical condition of wrinkling,

such as Euler’s solution for the buckling of a longitudinally loaded column (Gere, 2001).

(2) Energy method, where a deflected form of the part is assumed and the potential

energy related to this small deflection is evaluated. When the total energy which tends

11
 Chapter 2 Literature Review

to restore the equilibrium is higher than the energy due to forces displacing it, the

system remains stable (Senior, 1956). The critical condition of wrinkling is given by

equating the two energy values. Until now, very limited theoretical analyses have been

reported on the wrinkling failure of metal spinning processes. Reitmann and Kantz

(2001) used the equilibrium method to analyse various conditions of buckling. They

reported that wrinkling in spinning processes could result from static buckling or

dynamic buckling or both ways. By modifying the instability theory of the deep-drawing

process (Senior, 1956) and using the energy method, Kobayashi (1963) proposed a

theoretical model to determine the critical condition of the flange wrinkling in

conventional spinning on a conical mandrel, with spinning ratio and cone angle as

variables. Nevertheless, this theoretical model was based on an assumption of

neglecting the radial stresses, i.e. one directional beam buckling theory (Chu and Xu,

2001). Therefore, Senior (1956)’s theoretical work may not be accurate in determining

the critical condition of wrinkling failure in metal spinning.

2.1.2 Experimental Investigation

Experimental investigation has been applied to analyse the material deformation and

failure in the sheet metal spinning process since the 1950s. In this literature review, 39

papers of experimental investigation in sheet metal spinning are included. The

methodologies of measuring tool forces, investigating strains and material deformation,

and analysing material failures are presented in this section. The statistical experimental

design methods that have been used in spinning research are also discussed.

2.1.2.1 Measurement of Tool Forces

Experimental investigations into tool forces have been carried out on both shear forming

and conventional spinning. Tool force in the spinning process is normally resolved into

three orthogonal components, e.g. axial force - Fa, radial force - Fr and tangential force -

Ft. However, the definition of force components in the shear forming study is generally

different from that in the conventional spinning study. As shown in Figure 2.1(a), in the

12
 Chapter 2 Literature Review

analysis of shear forming, the axial force is normally defined as the force in line with the

roller path in shear forming, also termed the feeding force. Radial force is defined as the

force normal to the surface of the mandrel. By contrast, as shown in Figure 2.1(b), in the

analysis of conventional spinning, the axial force is defined as the force in line with the

mandrel’s axis and the radial force is defined as the force parallel to the mandrel’s radial

direction. The definitions of tangential force in both shear forming and conventional

spinning are the same, i.e. perpendicular to the axial and radial forces.

Roller
Roller
Fa Fr
Ft
Fa
Fr Ft

Mandrel Mandrel

Blank Blank

(a) Shear forming (b) Conventional spinning

Figure 2.1 Definitions of tool force components

Piezoelectric force transducers and strain gage sensors are two types of sensor which

are commonly used to measure the force (Wilson, 2005). El-Khabeery et al. (1991) and

Wang et al. (1989) measured the tool forces by using strain gauges mounted on the

shafts supporting the roller. Nevertheless, since the spinning process is a dynamic

process, the measurement by using strain gauges may be not able to capture the high

frequency of tool force variations (Long et al., 2011). For this reason, many researchers

have applied the piezoelectric force transducer to measure the dynamic force histories

in the spinning process (Arai, 2006, Chen et al., 2001, Jagger, 2010). Figure 2.2 shows

a typical force measurement system in the spinning study. During the spinning process,

electrical signals measured from the force transducer are amplified, converted and then

recorded by a computer with a data acquisition card and software.

13
 Chapter 2 Literature Review

Force transducer Amplifier Data acquisition card Computer

Figure 2.2 Force measurement system, adapted from Jagger (2010)

2.1.2.2 Investigation of Strains and Material Deformation

By using the grid marking method, strain analyses have been carried out on the

conventional spinning process (Beni et al., 2011, Quigley and Monaghan, 2000, Razavi

et al., 2005) and shear forming process (Shimizu, 2010). A pattern of circles is etched

on the blank before forming (Joshi, 2002), as shown in Figure 2.3(a). After deformation

the circles are transferred into ellipses with different sizes, which can be measured by

optical projectors to obtain accurate strain results. To study the material deformation

during the shear forming process, the plugged holes method has been employed by

Avitzur et al. (1959). As shown in Figure 2.3(b), holes are drilled and plugged with metal

material. After the spinning experiment, the workpeice is cut until the holes are revealed

and used to study the material deformation. In addition, Kalpakcioglu (1961a) applied

the grid line method to analyse the material deformation during the shear forming

process. As illustrated in Figure 2.3(c), a blank is cut in the middle and the grid lines are

inscribed on the interface surfaces. The two parts are soldered together to be used in

the experiment, after which the two parts are separated by melting the solder. Then the

interface surfaces are cleaned and polished to study the material deformation. The

application of gird line method has also been extended to the study of deformation

mechanics in incremental forming (Jackson and Allwood, 2009).

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 Chapter 2 Literature Review

Grid pattern Plugged holes

Grid lines
Metal blank
Metal blank Metal blank

(a) Grid marking method (b) Plugged holes method (c) Grid line method

Figure 2.3 Methods for studying strains and material deformation

2.1.2.3 Study of Material Failures

Both wrinkling and cracking failures in the sheet metal spinning process have been

studied through experiments. However, most of the experimental investigations on

material failures focus on shear forming rather than conventional spinning. The early

failure studies in shear forming have been carried out by investigating the spinnability

(Kalpakcioglu, 1961b). Kegg (1961) defined it as the ability of metal to undergo shear

forming deformation without fracture. In order to predict the fracture in shear forming,

Kegg (1961) carried out a series of spinnability tests of various materials on a half

ellipsoidal mandrel. Moreover, the author proposed a method to predict the spinnability

of a given material, by correlating the maximum thickness reduction of the blank in the

spinning with the reduction of area at fracture of the test sample in the tensile test.

Hayama and Tago (1968) claimed that Kegg (1961)’s experimental results based on the

half ellipsoidal mandrel may not be valid in the case with the conical mandrel. Hayama

and Tago (1968) also expanded the term of spinnability as the ability of a sheet metal to

undergo deformation by spinning without the wrinkles in the flange and no fractures on

the wall. Furthermore, they divided the cracks into three types and analysed the cause

of each type of crack. Most recently, Kawai et al. (2007) carried out spinnability studies

of “die-less” shear forming on both conical and hemispherical parts by using a cylindrical

mandrel for general purposes.

In order to study the deformation modes and wrinkling failure, Hayama et al. (1966)

15
 Chapter 2 Literature Review

measured the radial and circumferential strains as well as the periodic variations of

curvatures on the flange, by attaching strain gauges on both sides of the flange before

spinning. In a later study, Hayama (1981) used the sudden change of the vibration of the

axial force (feeding force) to determine the exact moment when the wrinkling occurs in

the shear forming. By applying a laser range sensor to monitor the height of the flange

of the rotating workpiece, Arai (2003) experimentally measured the development of the

wrinkles at different stages of forming. Based on a one-pass deep drawing conventional

spinning experiment, Xia et al. (2005) carried out spinnability studies on blanks made by

aluminium and mild steel. Kleiner et al. (2002) experimentally monitored the

development of wrinkling failures in the first pass of a conventional spinning process.

The authors divided the propagation of wrinkles into five stages: onset of wrinkling,

expansion of wrinkling, first complete circle, increasing amplitude, and collapse of

wrinkles, as shown in Figure 2.4.

Figure 2.4 Propagation of wrinkles in spinning (Kleiner et al., 2002)

2.1.2.4 Design of Experiments

Early experimental studies (Hayama et al., 1965, Wang et al., 1989) focused on the

effects of each process parameter (factor) on the spinning process, by varying a single

factor while keeping other factors constant, i.e. the One-Factor-at-a-Time (OFAT)

method. The major disadvantage of OFAT method is that the interactions between

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 Chapter 2 Literature Review

different factors cannot be evaluated (Montgomery, 2009). For this reason, Design of

Experiment (DoE) methods, in which several factors are varied simultaneously, have

been employed to analyse the effects of process parameters on the dimensional

variations of spun parts (Auer et al., 2004, Henkenjohann et al., 2005, Kleiner et al.,

2005). Response surface methodology (RSM) has also been applied in the

experimental investigations of tool forces and surface finish (Chen et al., 2001, Chen et

al., 2005b, Jagger, 2010) in the sheet metal spinning process. RSM is a collection of

mathematical and statistical techniques for establishing relations among various

process parameters and optimising the response (Montgomery, 2009). It uses

regression analysis to fit an equation to correlate the response with the process

parameters.

2.1.3 Finite Element Analysis

Since the 1990s, FE analysis of sheet metal spinning process has seen significant

development. In this section, 31 papers on FE analysis of spinning have been reviewed.

Due to the nature of incremental forming, in the early studies, to reduce the computing

time, certain simplifications had to be made. For instance, 2-D FE models (Alberti et al.,

1989, Liu et al., 2002) or simplified 3-D FE models with axisymmetric modeling were

used where the roller was approximated as a virtual ringed tool with variable diameters

(Mori and Nonaka, 2005). More recently, with the development of computing hardware,

3-D FE models have been commonly applied to study the material deformation and

failure mechanics in the spinning process. In this section, five key factors of the FE

simulation are discussed, i.e. the FE solution method, material constitutive model,

element selection, meshing strategy, and contact treatment.

2.1.3.1 Finite Element Solution Methods

Finite Element solution methods are generally resolved into the implicit method and the

explicit method (Harewood and McHugh, 2007). The implicit FE analysis method

iterates to find the approximate static equilibrium at the end of each load increment. It

17
 Chapter 2 Literature Review

controls the increment by a convergence criterion throughout the simulation. Because of

the complex contact conditions and high non-linearity in the metal forming problems, a

large number of iterations have to be carried out before finding the equilibrium; the

global stiffness matrix thus has to be assembled and inverted many times during the

analysis. Therefore, the computation is extremely expensive and memory requirements

are also very high (Tekkaya, 2000). Additionally, the implicit method is unable to carry on

the analysis if shape defects, e.g. wrinkling, occur in the sheet metal simulation (Alberti

and Fratini, 2004). It is difficult to predict how long it will take to solve the problem or

even if convergence can be achieved (Harewood and McHugh, 2007). Thus the implicit

method is preferable to analyse some small 2-D problems and 3-D problems under

simple loading conditions, for instance, modelling the springback after spinning (Bai et

al., 2008, Zhan et al., 2008). On the other hand, the explicit FE analysis method

determines a solution by advancing the kinematic state from one time increment to the

next, without iteration. The explicit solution method uses a diagonal mass matrix to

solve for the accelerations and there are no convergence checks. Therefore it is more

robust and efficient for complicated problems, such as dynamic events, nonlinear

behaviors, and complex contact conditions. Hence the explicit FE analysis method has

been chosen by most researchers to analyse the metal spinning process.

2.1.3.2 Material Constitutive Model

The most commonly used yield criterion in engineering application, particularly for

computational analysis, is von Mises criterion (Dunne and Petrinic, 2005). Figure 2.5 (a)

shows the von Mises yield surface of isotropic hardening in 2-D space of principal

stresses (σ1, σ2). In the isotropic material hardening, if the load is reversed at the load

point (1), the material behaves elastically until reaching the load point (2), which is still

on the yield surface. Any stress increase beyond this point will lead to plastic

deformation. Clearly, the isotropic hardening leads to a very large elastic region in the

reversed loading process. However, in reality, a much smaller elastic region is expected

in the reversed loading process. This phenomenon is called the Bauschinger effect

(also known as work softening), i.e. when a metal material is subjected to tension into

18
 Chapter 2 Literature Review

the plastic range, after the load is released and compression is applied, the yield stress

in the compression is lower than that in the tension (Kalpakjian and Schmid, 2001).

Kinematic hardening model takes the Bauschinger effect into account, where the yield

surface translates in the stress space rather than expanding, as shown in Figure 2.5(b).

During the spinning process, the roller induces constant tensile and compressive

loadings on the workpiece. Hence, Klimmek et al. (2002) and Pell (2009) suggested that

the Bauschinger effect should not be neglected in the FE simulation of spinning.

However, due to the lack of specific material test data, none of those researchers have

considered the Bauschinger effect.

(a) Isotropic hardening

(b) Kinematic hardening

Figure 2.5 Material hardening models (Dunne and Petrinic, 2005)

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 Chapter 2 Literature Review

2.1.3.3 Element Selection

The accuracy of any FE simulation is highly dependent on the type of element used in

the simulation. Solid elements and shell elements are two types of the most commonly

used elements in metal spinning simulation. Quigley and Monaghan (2002a) suggested

that 8 noded hexahedral solid elements should be used, because a blank modelled by

2-D shell elements may not be able to handle the contact with the roller and mandrel at

the same time. To solve this problem, Zhao et al. (2007) applied an offset of one-half of

the blank thickness from the middle plane to both sides of the 2-D shell element. By

comparing FE results obtained from solid and shell elements, Hamilton and Long (2008)

concluded that wrinkling failure may be exaggerated if using shell elements. Most

recently, Long et al. (2011) reported that the use of continuum shell elements produced

axial force and thickness results which were in good agreement with the experiment.

The FE models using solid elements produced considerably different tool force results

in comparison with the experimentally measured axial and radial force values.

During the metal spinning process, the material undergoes a complicated loading

process that includes bending effects (Sebastiani et al., 2007), which may cause the

“hourglassing” problem as a result of using reduced integration linear solid elements. As

shown in Figure 2.6, the bending of a reduced integration linear solid element presents

a zero-energy deformation mode, as no strain energy is generated by the element

distortion (Abaqus analysis user’s manual, 2008). Moreover, this “hourglassing”

problem can propagate through the elements and produce meaningless numerical

results. On the other hand, unlike the reduced integration linear solid element, which

only uses one integration point along the thickness direction, multiple integration points

are used through the thickness of a reduced integration linear shell element. Stresses

and strains at each integration point of the shell element are calculated independently.

This may be the reason why reduced integration linear shell elements can produce

more accurate results of wrinkling (Wang et al., 2011) and tool forces (Long et al., 2011)

than reduced integration linear solid elements.

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 Chapter 2 Literature Review

Integration
point

Figure 2.6 Deformation of a reduced integration linear solid element subjected to bending

2.1.3.4 Meshing Strategy

Various researchers have employed different meshing strategies, which affect the

accuracy and efficiency of FE simulation in spinning. Four commonly used meshing

strategies, as shown in Figure 2.7, are discussed in this section. The structured

meshing technique (Mesh A) and the free meshing technique (Mesh C) solve the

problem where triangular prism elements have to be used in the centre of the circular

blank. However, the irregular mesh in Mesh A and C may result in local stress peaks

due to an inhomogeneous mass distribution in the rotating blank (Sebastiani et al.,

2006). In order to achieve a regular mesh, the central area of the blank which is

clamped and does not undergo deformation, is neglected in Mesh B and Mesh D.

Numerous researchers have used the sweep meshing technique shown in Mesh B (e.g.

Quigley and Monaghan, 1999, Zhan et al., 2007). Correlations have been achieved

between FE analysis and experimental results by using Mesh B (Awiszus and Härtel,

2011, Wang et al., 2011). However, Sebastiani et al. (2006) suggested that the small

element size around the inner region may limit the increment size of the FE explicit

solution method and thus decrease the computing efficiency. They proposed a meshing

strategy - Mesh D which provides a solution to this problem by using more element

seeds along the outer circle of the blank, while using less seeds along the inside circle.

Nevertheless, triangular prism elements have to be applied to connect the inner region

and outer region of the blank. Because of its constant bending and membrane strain

approximations, high mesh density of triangular prism elements is generally required to

accurately capture the bending deformation and the solution involves high strain

gradients (Abaqus analysis user’s manual, 2008).

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 Chapter 2 Literature Review

Mesh A Mesh B

Mesh D Mesh C

Figure 2.7 Mesh strategy, adapted from Sebastiani et al. (2006)

2.1.3.5 Contact Treatment

During the spinning process, the contact between the roller and blank is dynamic and

complex. The penalty contact method has been applied to model the contact in the

normal direction between the tool and blank surfaces. It has been shown to provide

good results in the FE simulations of metal spinning (Bai et al., 2008, Liu, 2007, Zhao et

al., 2007). Contact forces, which are calculated as the penetration distance multiplies

the penalty stiffness, are applied to the slave nodes to oppose the penetration (Abaqus

analysis user’s manual, 2008). At the same time, reaction forces act opposite on the

master surface at the penetration point.

The contact between roller and blank represents both the sliding and the rolling frictional

behaviour (Bai et al., 2008). In the early spinning simulation studies, the rotation of the

roller is simplified by neglecting the friction between the roller and the blank (Quigley

and Monaghan, 2002a, Sebastiani et al., 2006, Zhao et al., 2007). However, this

simplification cannot represent the actual spinning process, where the friction force

makes the roller rotate along its axis. Consequently, many researchers such as Liu

(2007), Wang and Long (2011c), and Zhan et al. (2008) assumed a small Coulomb

frictional coefficient (0.01-0.05) between the roller and the blank.

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 Chapter 2 Literature Review

2.2 Material Deformation and Wrinkling Failure

In this section, published literature regarding the tool forces, stresses and strains, and

wrinkling failures in both shear forming process and conventional spinning process are

reviewed. Analysing tool forces in the sheet metal spinning process is important to

select process parameters, design spinning machines, and improve the quality of spun

products (Wang et al., 1989). Moreover, investigating stresses, strains and material

failures is essential to understand the deformation and failure mechanics and thus

optimise the process design.

2.2.1 Tool forces

Until now, tool forces in shear forming, one-pass deep drawing conventional spinning,

and multi-pass conventional spinning processes have been investigated intensively.

Among the three force components generated in shear forming, as defined in Figure

2.1(a), the radial force is the highest, because the roller squeezes the material onto the

mandrel during the whole process. The axial force ranks the second and the tangential

force is the smallest (Arai, 2003, Hayama et al., 1965, Zhan et al., 2007). As reported in

an experimental investigation of shear forming (Lee and Noble, 1982), both axial force

and radial force decrease when increasing the angle between the roller axis and

mandrel axis from 0°to approximately 60°with a slight increase thereafter. The tool

forces would also decrease if a thinner blank were used (Chen et al., 2001, Hayama et

al., 1965).

According to the experimental investigations of one-pass deep drawing conventional

spinning (Hayama and Murota, 1963, Jurkovic et al., 2006, Xia et al., 2005), the peak

values of the axial force are obtained at the middle stage of spinning. On the other hand,

the maximum values of the radial force occur at the end of the process. The tangential

force is again the smallest force among these three force components and it remains

almost constant during the one-pass deep drawing spinning process.

Tool forces of the multi-pass conventional spinning process have been experimentally

23
 Chapter 2 Literature Review

investigated (Jagger, 2010, Wang et al., 1989) and analysed by FE simulation (Essa

and Hartley, 2009, Liu, 2007, Pell, 2009). It has been reported that in the conventional

spinning process, among three force components as defined in Figure 2.1(b), the axial

force is the largest, while the tangential force is the lowest. In a 3-pass conventional

spinning experiment, Wang et al. (1989) reported that when the thickness of the blank

increases, tool forces increase accordingly. Wang et al. (1989) also gave an

approximately proportional relationship among the peak values of the force components,

where

Fa : Fr : Ft = 20 : 10 : 1 (2)

The maximum force ratios obtained in an experiment (Jagger, 2010) and FE simulation

(Wang and Long, 2011b) of 3-pass conventional spinning processes show a similar

trend, as illustrated in equation (3) and (4), respectively.

Fa : Fr : Ft = 17 : 5 : 1 (3)

Fa : Fr : Ft = 35 : 17 : 1 (4)

Recently, Wang and Long (2011a) numerically compared the force histories when the

workpieces were made of three material types in a multi-pass conventional spinning

process. The authors reported that a FE model using stainless steel produced the

highest tool forces, followed by a model using mild steel, while the lowest tool forces

were obtained in a model using aluminium. Wang and Long (2011a) also compared the

maximum tool forces of a wrinkle-free model and a wrinkling model; the tool forces in

the wrinkle-free models were approximately one third of the corresponding forces in the

wrinkling models.

2.2.2 Stresses
Klimmek et al. (2002) suggested that investigating the stress distribution of the

workpiece, especially within the local forming zone, is essential to understand the cause

of the wrinkling failure. The authors investigated the stress variations of two distinct

elements during a forward roller pass. The first element is close to the centre of the

blank, while the second element is located at the rim of the blank. They reported that

24
 Chapter 2 Literature Review

once the first element gets into the local forming zone, the tensile radial stress steadily

increases to a maximum value. By contrast, as the roller gets to the edge of the blank,

the compressive tangential stress of the second element increases dramatically.

Sebastiani et al. (2007) developed a spinning model which uses three linear roller

passes, including both forward passes and backward passes. The authors report that

local radial bending effects exist in the workpiece during a forward pass, where the inner

surface (mandrel facing) of the workpiece is subjected to compressive radial stresses

while its outer surface (roller facing) is under tensile radial stresses, as also observed by

Pell (2009) and Wang and Long (2010). Moreover, after a backward roller pass, a

toothed stress pattern in the flange region has been observed by Sebastiani et al.

(2007), who believe that it may be the pre-state of the wrinkling failure. A similar toothed

stress pattern has also been noticed during a non-linear backward pass by Wang and

Long (2011b). However, no correlations have been found between the toothed stress

pattern and wrinkling failure.

2.2.3 Strains
Theoretically, in the conventional spinning process, the compressive hoop strain should

balance the tensile radial strain. The thickness strain thus remains zero. However, in an

experimental investigation of a one-pass deep drawing conventional spinning process

(Hayama and Murota, 1963), tensile radial and hoop strains resulting in compressive

thickness strains, have been observed at the bottom of a spun part. However, in the

opening of a spun part, the unbalanced compressive hoop strains and tensile radial

strains lead to a certain amount of tensile thickness strains. Thinning at the bottom and

thickening in the opening of the spun parts in one-pass deep drawing conventional

spinning have also been reported by Hamilton and Long (2008) and Xia et al. (2005).

The experimental results of the multi-pass conventional spinning on the spherical

mandrel (Beni et al., 2011, Quigley and Monaghan, 2000, Razavi et al., 2005) show that

the tensile radial strain and compressive hoop strain do not mirror each other, thus

indicating that a certain amount of thickness strain exists in the workpiece, as confirmed

25
 Chapter 2 Literature Review

in the FE simulations by Wang and Long (2011b). In addition, Quigley and Monaghan

(2000) reported that there was a certain degree of shear forming involved in the first

roller pass of conventional spinning.

In the shear forming process, considerably high tensile radial strain and compressive

thickness strain have been observed by Shimizu (2010), who experimentally

investigated the strain distributions along the radial direction of the workpiece. By

contrast, tangential strain keeps almost constant, only a slightly increase is observed in

the opening of the spun part. Bending strains have been measured by strain gauges

that mounted on both sides of the workpiece (Hayama et al., 1966), where bending and

unbending are performed repeatedly in the local forming zone of the shear forming.

2.2.4 Wrinkling Failure

It is believed that in sheet metal spinning wrinkling takes place when the tangential

compressive stress in the flange exceeds the buckling stability limit (Kleiner et al., 2002,

Klimmek et al., 2002, Runge, 1994). However, it is still unknown how to determine the

buckling limit which could be used to predict and prevent wrinkling failure. Kleiner et al.

(2002) suggested that the wrinkling in the spinning process is not only caused by static

buckling, but also influenced or even triggered by the dynamic effects from the feeding

of roller and rotation of the workpiece.

The experimental results of Kleiner et al. (2002) indicate that the diameter and thickness

have the most significant effects on wrinkling failure, followed by the feed rate, the

spindle speed, the roller path and the material of the workpiece. Hayama et al. (1966)

also reported that the feed ratio, blank thickness and blank diameter are very important

contributing factors to wrinkling failure. Hayama (1981) suggested that the feed ratio is

the most important parameter affecting spinnability. In general, wrinkles tend to occur

when increasing the feed ratio and blank diameter or decreasing the blank thickness

(Hayama et al., 1966, Kleiner et al., 2002, Xia et al., 2005).

26
 Chapter 2 Literature Review

Satoh and Yanagimoto (1982) reported that wrinkling resistance would be enhanced if a

material with higher yielding stress was used, as they believe wrinkling occurs due to

elastic buckling. Wang and Long (2011a) proposed that there may be a limit to the roller

feed depth, beyond which wrinkling failures may take place. Furthermore, both Kawai

and Hayama (1987) and Wang et al. (2011) reported that wrinkles may be smoothed out

in subsequent passes of conventional spinning processes. Although several

investigations have been conducted on the wrinkling failure of metal spinning, further

investigation is still required to gain an in-depth understanding of wrinkling failure

mechanics, the effects of process parameters, and to predict and prevent wrinkling

failures.

2.3 Key Process Parameters

The process parameters play a decisive role in the design and optimisation of sheet

metal spinning process. The effects of four key process parameters, i.e. feed ratio, roller

path (passes), roller profile, and clearance between roller and mandrel, on the material

deformation and failures of the spun parts are discussed in this section.

2.3.1 Feed Ratio

Feed ratio is defined as the ratio of the roller feed rate to the spindle speed. As long as

the feed ratio remains constant, by changing the feed rate and the spindle speed

proportionally, there would be no significant effects on the tool forces (Hayama et al.,

1965, Jagger, 2010, Lee and Noble, 1982, Wang et al., 2011, Wang et al., 1989, Xia et

al., 2005), wall thickness (Xia et al., 2005, Zhan et al., 2008), wrinkling failures (Hayama

et al., 1966, Wang et al., 2011), cracking failures (Hayama and Tago, 1968) and surface

finish (Ma et al., 2010) of the spun part.

Nevertheless, it has been reported that variation of feed ratio has considerable effects

on the tool force, wall thickness, spinnability, surface finish and springback of the metal

spinning process. When a higher feed ratio is applied, tool forces will increase

27
 Chapter 2 Literature Review

accordingly (El-Khabeery et al., 1991, Essa and Hartley, 2009, Hayama et al., 1965,

Jagger, 2010, Liu, 2007, Ozer and Arai, 2009, Pell, 2009, Perez et al., 1984, Xia et al.,

2005, Zhan et al., 2007). In addition, material failures tend to take place if a high feed

ratio is used (Hayama, 1981, Hayama et al., 1966, Kawai et al., 2007, Ozer and Arai,

2009, Wang et al., 2011, Xia et al., 2005, Zhan et al., 2007).

On the other hand, Wong et al. (2003) suggested that a low feed ratio would result in

excessive material flow in the outward direction, which unduly thins the blank, as Pell

(2009), Wang et al. (2011) and Zhan et al. (2007) also reported. However, a lower feed

ratio would result in a better surface finish of the spun part (Chen et al., 2001,

El-Khabeery et al., 1991, Kleiner et al., 2005, Ma et al., 2010, Slater, 1979a, Wang et al.,

2011, Slater, 1979a). Both El-Khabeery et al. (1991) and Essa and Hartley (2010)

reported that springback would increase when a higher feed ratio is used.

2.3.2 Roller Path and Passes

By using multiple roller passes in conventional spinning, the tensile radial and

compressive tangential stresses are induced gradually; hence material failures can be

prevented (Runge, 1994, Sebastiani et al., 2007). Up to now, most investigations of

roller path and passes in conventional spinning have focused on single pass or

multi-pass with no more than 3 passes (mainly linear path). Consequently, there is still a

huge knowledge gap between academic research and industrial production, where a

considerable amount of roller passes have to be used to successfully produce

complicated spinning parts (Filip and Neago, 2010). Three types of the roller path

shown in Figure 2.8, straight line, concave curve and convex curve, have been

experimentally studied by Kang et al. (1999) and Hayama et al. (1970). The difference

between these two experiments was that only the single pass was studied by Kang et al.;

whereas multiple passes were investigated by Hayama et al. Kang et al. (1999)

concluded that the first pass in the conventional spinning plays a decisive role in the

final blank thickness. After comparing four different types of roller path designs, Hayama

et al. (1970) proposed that the involute curve path, which is a special type of concave

28
 Chapter 2 Literature Review

curve, gives the highest spinning ratio without material failures. This was supported by

Liu et al. (2002), who numerically analysed the distributions of stresses and strains

obtained from three different roller paths, i.e. straight line, involute curve and quadratic

curve. Their FE analysis results illustrate that the stresses and strains obtained under

the involute roller path are the smallest. Furthermore, Kawai and Hayama (1987)

applied an involute curve in each pass in their experiments and studied the first pass

and the remaining passes separately. They suggested that the angle of the first pass

plays a dominant role in the generation of wrinkles and cracks.

Roller passes

Mandrel Mandrel Mandrel

(a) Linear path (b) Concave path (c) Convex path

Figure 2.8 Various roller path profiles

2.3.3 Roller Profile

Figure 2.9 shows examples of various shapes of spinning roller (Avitzur et al., 1959).

Roller diameter and nose radius are two key parameters that have been investigated by

a few researchers. According to the literature review by Wong et al. (2003), the roller

diameter has little effect on the final product quality, but a small roller nose radius would

result in poor thickness uniformity. This is supported by El-Khabeery et al. (1991), who

reported that an increase in roller nose radius would lead to a smaller reduction of wall

thickness. Younger (1979) claimed that increasing the roller nose radius resulted in a

decline of the axial force and had almost no effects on the variations of tangential force

in a shear forming experiment. Both Essa and Hartley (2010) and El-Khabeery et al.

(1991) pointed out that as the roller nose radius decreased, tool forces would go down

accordingly in one-pass deep drawing conventional spinning. In a multi-pass

conventional spinning experiment, Wang et al. (1989) also reported that all of the three

29
 Chapter 2 Literature Review

tool force components decrease when applying a smaller roller nose radius. According

to an experimental wrinkling study in the shear forming, Hayama et al (1966) reported

that the roller nose radius had little effects on the wrinkling failures in the shear forming

process. Chen et al. (2001), El-Khabeery et al. (1991) and Kleiner et al. (2005) pointed

out that increasing the roller nose radius would improve the surface finish of the spun

part. Chen et al. (2001) suggested that a larger roller nose radius resulted in a larger

contact area between roller and blank, thus producing a smoother material deformation.

Figure 2.9 Various shapes of roller (Avitzur et al., 1959)

2.3.4 Clearance between Roller and Mandrel

In the single pass conventional spinning and shear forming process, the parts are spun

within one pass, where the roller forms the workpiece onto the mandrel with a specified

clearance. Therefore, the clearance between roller and mandrel is one of the key

parameters in those processes. In the one-pass deep drawing conventional spinning

process, radial and axial tool forces increase when the clearance decreases (Xia et al.,

2005). This is because a smaller clearance would result in more thinning of the

workpiece; thus the severe material deformation leads to high tool forces. According to

Essa and Hartley (2010), if the clearance is smaller than the initial wall thickness,

material builds up in front of the roller and causes wrinkles. Conversely, if the clearance

is larger than the initial wall thickness, material escapes beneath the roller, resulting in

dimensional inaccuracy.

As shown in Figure 2.10, in the shear forming process, the flange keeps straight when

the clearance is equal to the theoretical value defined by the sine law ( t0 ⋅ sin α )

30
 Chapter 2 Literature Review

(Hayama et al., 1965). This may be because that when the sine law is closely followed,

stresses are confined within the zone under the roller and thus the flange remains

virtually stress-free. However, when the clearance is smaller than the theoretical value

( t1 < t0 ⋅ sin α , over-spinning), the material builds up in front of the roller; the flange is

bent forward and away from the roller. Conversely, when the clearance is larger than the

theoretical value ( t1 > t0 ⋅ sin α , under-spinning), the material in the flange will be pulled

inward, bending the flange towards the roller and potentially causing wrinkles. These

findings are also reported by Lu et al. (2006), Slater (1979b) and Zhan et al. (2007).

t1 = t0 sin α t1 < t0 sin α t1 > t0 sin α

α α α

t0 t0 t0

(a) True shear forming (b) Over-spinning (c) Under-spinning

Figure 2.10 Deviation from sine law in shear forming, adapted from Music et al. (2010)

2.4 Summary

In this chapter, published research regarding the investigation techniques, material

deformation, wrinkling failure, and key process parameters of the sheet metal spinning

process have been reviewed. Two major knowledge gaps have been identified: (1) Most

research focus on shear forming rather than conventional spinning. Additionally, current

studies in conventional spinning are limited to single-pass spinning or multi-pass

spinning with linear path. Thus it is essential to investigate material deformation

mechanics of multi-pass conventional spinning with non-linear path design. (2) The

cause of wrinkling failures in the sheet metal spinning process is only partially

understood. Moreover, it is also worth investigating how to determine the critical

condition of wrinkling failure in order to predict and prevent it from occurring.

31
 Chapter 3 Fundamentals of Finite Element Method

3. Fundamentals of Finite Element Method

The Finite Element Method (FEM) is a numerical method seeking an approximated

solution of a complex engineering problem which is difficult to obtain analytically. It is

realised by dividing the complicated analysis body into a finite number of elements. The

behaviour of each element, which is in regular shape, may be predicted by certain

mathematical equations. The summation of the individual element behaviours produces

the expected behaviour of the whole analysis body. FEM was originally developed for

solving complex elasticity and structural analysis problems in civil and aeronautical

engineering in 1940s. By the early 70s, applications of FEM were limited in aeronautics,

automotive, defence and nuclear industries, executed on expensive mainframe

computers. Over the past decades, due to the rapid developing of computing capability,

FEM has been widely used for analysing the problems involving solid mechanics, fluid

mechanics, heat transfer, vibrations, electrical and magnetic fields, etc. The most

common applications of FEM in industry are to design new products and processes,

optimise existing products and processes so as to improve their performance. In this

chapter, the fundamentals of Finite Element method, such as, Hamilton’s Principle, the

basic analysis procedure of FEM, and four types of finite elements are discussed in

detail. Furthermore, the non-linear solution method, material constitutive model and

contact algorithm are briefly outlined.

3.1 Hamilton’s Principle

To obtain a solution of any solid mechanics FE problems, it is essential to satisfy the

requirements of equilibrium, compatibility, constitutive equations and boundary

conditions. However, obtaining the exact solution (known as strong form) of the

equilibrium is usually extremely difficult for practical engineering problems. Instead, a

weak form of equilibrium for the body as a whole is imposed even though it does not

ensure pointwise equilibrium (Dunne and Petrinic, 2005). Hamilton’s Principal, resulting

from the Conservation of Energy, is one of the most commonly used weak form of

32
 Chapter 3 Fundamentals of Finite Element Method

equilibrium in FEM. It states that the variation of the kinetic and strain energy plus the

variation of the work done by external forces acting during any time interval from t1 to t2

must be zero. Mathematically, Hamilton’s Principle states (Liu and Quek, 2003):
t2
δ ∫ Ldt = 0 (5)
t1

The Lagrangian functional, L, is defined as

L = T − Π + Wf (6)

where T is the kinetic energy, Π is the strain energy and Wf is the work done by

external forces. These components of the Lagrangian functional can be expressed as

follows
1
2∫
T= & TU
ρU & dV (7)

& is the velocity.

where ρ is the density and U

1 T 1
Π=
2 ∫ ε σdV = ∫ ε T DεdV
2
(8)

where ε and σ represent the strain and stress vectors, respectively; D the is matrix of

material constants and will be discussed in Section 3.3.

W f = ∫ UT f b dV + ∫ UT f s dS f (9)

where U is the displacement vector, the body force vector is expressed as fb and surface

force vector is defined as fs. Sf and V represent the domains of area and volume,

respectively.

3.2 Basic Analysis Procedure of FEM

The solution process of general continuum problems normally follows a step-by-step

procedure. In this section, the basic procedure of FEM is briefly summarised.

3.2.1 Domain Discretisation

The analysed body is represented as an assembly of subdivisions known as finite

elements. This procedure is called meshing, which is usually carried out by using so

called pre-processors. It is especially useful for meshing complicated geometries, such

33
 Chapter 3 Fundamentals of Finite Element Method

as the example shown in Figure 3.1, where a cargo train body is meshed with

thousands of finite elements. The elements are considered to be interconnected at

specified joints called nodes. The selection of the type, number and size of finite

elements are decided based on the characteristic of the analysed problem.

Figure 3.1 Finite Element Meshing (Wang, 2005)

3.2.2 Displacement Interpolation

Because the displacement solution in a complex structure under specified loading

conditions cannot be predicted exactly, in FEM the displacement within an element is

simply assumed by polynomial interpolation using the displacements at its nodes (Liu

and Quek, 2003):

nd
U ( x, y , z ) = ∑ N i ( x , y , z ) d i = N ( x, y , z ) d e (10)
i =1

where nd is the number of nodes forming the element; di is the nodal displacement at the

i th node and can be expressed in a general form of

34
 Chapter 3 Fundamentals of Finite Element Method

⎧d1 ⎫
⎪d ⎪
⎪ 2 ⎪
di = ⎨ ⎬ (11)
⎪M ⎪
⎪d n f ⎪
⎩ ⎭
where nf is the number of Degree Of Freedom (DOF) at a node.

The vector de in equation (10) represents the displacement vector of the entire element

and it has the form of

⎧d1 ⎫
⎪d ⎪
⎪ 2 ⎪
de = ⎨ ⎬ (12)
⎪M ⎪
⎪d n ⎪
⎩ d⎭
Thus the total DOF of the entire element is nd × nf.

In equation (10), N is the matrix defining shape functions for the nodes in an element. It

is predefined to assume the variations of displacement with respect to the coordinates

of the element.

N( x, y, z ) = [ N1 ( x, y, z ) N 2 ( x, y, z ) ... N nd ( x, y, z )] (13)

where Ni is a sub-matrix of the shape functions for displacement components, which is

arranged as

⎡ N i1 0 0 0 ⎤
⎢0 Ni 2 0 0 ⎥
Ni = ⎢ ⎥ (14)
⎢0 0 O 0 ⎥
⎢ ⎥
⎣⎢ 0 0 0 N in f ⎦⎥
where Nik is the shape function for the kth displacement component (DOF) at the i th

node.

3.2.3 Construction of Shape Function

For an element with nd nodes at space coordinates xi (i = 1, 2,....,nd), where xT = {x} for

one-dimensional problems, xT = {x, y} for 2-D problems, and xT = {x, y, z} for 3-D

problems, its displacement component u is approximated as (Liu and Quek, 2003)

35
 Chapter 3 Fundamentals of Finite Element Method

nd
u ( x ) = ∑ pi ( x )α i = pT ( x )α (15)
i =1

in which pi(x) is the basic function of monomials and α is the coefficient for the monomial

pi(x). Vector α is defined as

α T = {α1 ,α 2 ,α 3 ,K,α n } d
(16)

A basic of complete order of pi(x) in the one-dimensional problem can be written as

pT ( x) = {1, x, x 2 , x 3 , x 4 ,K, x p } (17)

pi(x) in the 2-D problem has the form

pT ( x ) = pT ( x, y ) = {1, x, y, xy, x 2 , y 2 ,K, x p , y p } (18)

and that in the 3-D problem is expressed as

pT ( x ) = pT ( x, y, z ) = {1, x, y, z , xy, yz, zx, x 2 , y 2 , z 2 K, x p , y p , z p } (19)

The nd terms of pi(x) used in the basic should be selected from the constant term to

higher orders symmetrically from the Pascal triangle (2-D problem) shown in Figure 3.2

or from the Pascal pyramid (3-D problem) shown in Figure 3.3. Moreover, the total

number of terms involved in pi(x) should be equal to the number of nodal DOFs of an

element (Rao, 2005).

Figure 3.2 Pascal triangle of monomials (Liu and Quek, 2003)

36
 Chapter 3 Fundamentals of Finite Element Method

Figure 3.3 Pascal pyramid of monomials (Liu and Quek, 2003)

The coefficient vector α can be determined by

α = P -1d e (20)

where de is the displacement vector at all the nd nodes in the element, as given in

equation (12). Matrix P can be written as

⎡ pT ( x 1 ) ⎤
⎢ T ⎥
⎢p ( x 2 ) ⎥
P=⎢ ⎥ (21)
M
⎢ ⎥
⎢⎣pT ( x nd )⎥⎦

Substituting equation (20) into equation (15),

u ( x ) = N ( x )d e (22)

Thus the matrix of shape function N(x) can be calculated as

⎡ ⎤
N( x ) = pT ( x )P −1 = ⎢pT ( x )P1 L pT ( x )Pn ⎥
−1 −1 −1
pT ( x )P2
⎢ 14N2 4 3 1424 3 1424 3⎥ (23)
⎣ 1 (x) N 2 (x) Nn (x) ⎦

−1
where Pi is the ith column of matrix P −1 .

37
 Chapter 3 Fundamentals of Finite Element Method

3.2.4 Formation of Local FE Equations

Once the shape functions are constructed, the strain vector can be calculated as

ε = Bd e (24)

where B is the strain matrix and defined by

B = LN (25)

in which L is a matrix of partial differential operator.

Substituting equation (24) into the strain energy term in (8)

ε DεdV = ∫ d e T BT DBd e dV = d e T ⎛⎜ ∫ BT DBdV ⎞⎟d e

1 T 1 1
Π=
2 ∫ 2 2 ⎝ ⎠
(26)

By denoting

k e = ∫ BT DBdV (27)

which is called the element stiffness matrix, equation (26) can be rewritten as
1 T
Π= d e k ed e (28)
2

Similarly, by substituting equation (10) into the kinetic energy term in (7)
1 & dV = 1 ρd& T NT Nd& dV = 1 d& T ⎛⎜ ρ NT NdV ⎞⎟d&
T=
2 ∫ ρU
& TU
2∫
e e
2 ⎝∫
e
⎠ e
(29)

By denoting

m e = ∫ ρ NT NdV (30)

which is known as mass matrix and the kinetic energy can be rewritten as
1& T &
T= d e med e (31)
2

Finally, by substituting equation (10) into equation (9), the work done by external forces

can be expressed as

∫U fbdV + ∫ U T f sdS f = d e (∫ N T f bdV ) + d e (∫ N T f sdS f )

T T
Wf = T
(32)

By denoting

Fb = ∫ NT f b dV (33)

Fs = ∫ NT f s dS f (34)

38
 Chapter 3 Fundamentals of Finite Element Method

Then equation (32) can be rewritten as

T T T
W f = d e Fb + d e Fs = d e f e (35)

Fb and Fs are the forces acting on nodes of the elements. In terms of the work done by

these nodal forces, it is equivalent to the body forces and surface forces on the

elements. Hence, those nodal forces can be added up to form the total nodal force

f e = Fb + Fs (36)

Substituting equations (28), (31) and (35) into (6)

1& T & 1 T T
L= d e m ed e − d e k ed e + d e f e (37)
2 2

Applying Hamilton’s Principle (5)

t ⎛1 (38)
1 ⎞
δ ∫ ⎜ d& eT m e d& e − d eT k e d e + d eT f e ⎟dt = 0
2

t1 ⎝2 2 ⎠
Since the variation and integration operators are interchangeable

∫ (δd& )
t2
m e d& e − δd e k e d e + δd e f e dt = 0
T T T
e
(39)
t1

The deriving from (38) to (39) can be found in Liu and Quek (2003). In equation (39), the

variation and differentiation with time are also interchangeable

⎛ dd T ⎞
δd& eT = δ ⎜⎜ e ⎟⎟ = δd eT
d
( ) (40)
⎝ dt ⎠ dt
Substituting equation (40) into (39) and integrating the first term
t2 t2 t2 (41)
∫ δd& eT m e d& e dt =δd eT m e d& e − ∫ δd e m e&d&e dt = − ∫ δd e m e&d& e dt
t2 T T
t1
t1 14243 t1 t1
=0

Because the initial condition at t1 and the final condition at t2 have to be satisfied for any

de, and no variation at t1 and t2 is allowed (Liu and Quek, 2003), δd e = 0 , the first term on

the right side of equation (41) vanishes.

Substituting equation (41) into (39) leads to

δd eT (− m e&d& e − k e d e + f e )dt = 0
t2 (42)
∫t1

39
 Chapter 3 Fundamentals of Finite Element Method

In order to have the integration in equation (42) as zero for an arbitrary integrand, the

integrand itself has to be zero

δd eT (− m e&d&e − k ed e + f e ) = 0 (43)

Because of the arbitrary nature of the variation of the displacements, to satisfy equation

(43), we have

k e d e + m e&d&e = f e (44)

3.2.5 Assembly of Global FE Equations

The element equation in (44) is formulated based on the local coordinate system of the

element. To assemble all the element equations to form the global system equations, a

coordinate transformation has to be carried out. By using the coordinate transformation,

the displacement vector de can be transferred into a displacement vector Ue oriented in

the global coordinate system.

d e = TU e (45)

where T is the transformation matrix. It can also be applied to transfer the force vector fe

in local coordinate system to force vector in global coordinate system.

f e = TFe (46)

Substituting equations (45) and (46) into (44), we have

&& = F
K e Ue + Me U (47)
e e

where

K e = TT k eT (48)

M e = TT m e T (49)

Fe = TT f e (50)

In the end, the FE equations of individual elements are assembled to form the global FE

system equation:

40
 Chapter 3 Fundamentals of Finite Element Method

&& = F
KU + MU (51)

where K and M are the global stiffness and mass matrix respectively, U is a vector of all

the displacements at all nodes, and F is a vector of all external nodal forces.

3.3 Different Types of Finite Elements

Since 3-D solid and 2-D structural elements are the most commonly used elements in

the FE analysis of metal forming process, the formulations of a 3-D solid element, a

shell element, as well as a 2-D plane stress/strain element and a plate element which

are used to generate the formulations of the shell element are discussed in this section.

3.3.1 3-D Solid Element

The 3-D solid element is the most general of all solid finite elements, as all its field

variables are dependent of x, y and z. A 3-D solid element can be a tetrahedron or

hexahedron in shape, where each node has three translational DOFs. Here, the 8

noded hexahedron element is taken as an example to explain the formulations of shape

function, strain matrix, stiffness matrix and mass matrix in the 3-D solid element.

As shown in Figure 3.4, a hexahedron element has eight nodes, numbered 1, 2, 3, 4

and 5, 6, 7, 8 in a counter-clockwise manner. Since each node has three DOFs, there

are total 24 DOFs in a hexahedron element. By using a natural coordinate system

( ξ ,η , ζ ) with its origin at the centre of the element, it is easier to construct the shape

functions and to evaluate the matrix integration than using the physical coordinate

system ( x , y , z ). Assuming that the dimensions of the hexahedron element

is 2a × 2b × 2c , the relationship between local natural coordinate system and physical

coordinate system is given by

ξ = x a, η = y b, ζ = z c (52)

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 Chapter 3 Fundamentals of Finite Element Method

Figure 3.4 Hexahedron element and coordinate system (Liu and Quek, 2003)

The shape function of a hexahedron element is used to interpolate the coordinate from

the nodal coordinates:

8
x = ∑ N i (ξ ,η , ζ )xi (53)
i =1

8
y = ∑ N i (ξ ,η , ζ ) yi (54)
i =1

8
z = ∑ N i (ξ ,η , ζ )zi (55)
i =1

The shape functions are given as

1
N i = (1 + ξξi )(1 + ηηi )(1 + ζζ i ) (56)
8
where ( ξi ,ηi , ζ i ) denotes the natural coordinate of node i.

In a hexahedron element, the displacement vector U, which is a function of the

coordinate x, y and z, is interpolated using the shape function

U = Nd e (57)

where the nodal displacement vector de is given by

42
 Chapter 3 Fundamentals of Finite Element Method

⎧d e1 ⎫
⎪ ⎪
⎪d e 2 ⎪
⎪d e 3 ⎪
⎪ ⎪
⎪d ⎪
de = ⎨ e4 ⎬ (58)
⎪d e 5 ⎪
⎪d e 6 ⎪
⎪ ⎪
⎪d e 7 ⎪
⎪d ⎪
⎩ e8 ⎭

In which d ei is the displacement of the node i, where u, v, and w are its displacement

components in x, y and z direction.

⎧ui ⎫
⎪ ⎪
d ei = ⎨vi ⎬ ( i = 1,2,...8 ) (59)
⎪w ⎪
⎩ i⎭
The matrix of shape function is given as

N = [N1 N2 N3 N4 N5 N6 N7 N8 ] (60)

where each sub-matrix is given by

⎡Ni 0 0 ⎤
Ni = ⎢⎢ 0 Ni 0 ⎥⎥ (61)
⎢⎣ 0 0 N i ⎥⎦
The strain matrix can be written as

B = [B1 B2 B3 B4 B5 B6 B7 B8 ] (62)

where

⎡∂N i ∂x 0 0 ⎤
⎢ 0 ∂N i ∂y 0 ⎥⎥
⎢
⎢ 0 0 ∂N i ∂z ⎥
Bi = LNi = ⎢ ⎥ (63)
⎢ 0 ∂N i ∂z ∂N i ∂y ⎥
⎢ ∂N i ∂z 0 ∂N i ∂x ⎥
⎢ ⎥
⎢⎣∂N i ∂y ∂N i ∂x 0 ⎥⎦

Because the shape functions are defined in the natural coordinates ( ξ ,η , ζ ), to obtain

the derivatives with respective to x, y and z in the stain matrix, equation (64) needs to be

used:

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 Chapter 3 Fundamentals of Finite Element Method

∂N i ∂N i ∂x ∂N i ∂y ∂N i ∂z
= + +
∂ξ ∂x ∂ξ ∂y ∂ξ ∂z ∂ξ
∂N i ∂N i ∂x ∂N i ∂y ∂N i ∂z
= + + (64)
∂η ∂x ∂η ∂y ∂η ∂z ∂η
∂N i ∂N i ∂x ∂N i ∂y ∂N i ∂z
= + +
∂ζ ∂x ∂ζ ∂y ∂ζ ∂z ∂ζ
which can be expressed in the matrix form

⎧∂N i ∂ξ ⎫ ⎧∂N i ∂x ⎫
⎪ ⎪ ⎪ ⎪
⎨∂N i ∂η ⎬ = J ⎨∂N i ∂y ⎬ (65)
⎪∂N ∂ζ ⎪ ⎪∂N ∂z ⎪
⎩ i ⎭ ⎩ i ⎭
J is called Jacobian matrix

⎡ ∂x ∂y ∂z ⎤
⎢ ∂ξ ∂ξ ∂ξ ⎥
⎢ ⎥
∂x ∂y ∂z ⎥
J=⎢ (66)
⎢ ∂η ∂η ∂η ⎥
⎢ ∂x ∂y ∂z ⎥
⎢ ⎥
⎣ ∂ζ ∂ζ ∂ζ ⎦

By inverting equation (65), we obtain

⎧∂N i ∂x ⎫ ⎧∂N i ∂ξ ⎫
⎪ ⎪ -1 ⎪ ⎪
⎨∂N i ∂y ⎬ = J ⎨∂N i ∂η ⎬ (67)
⎪∂N ∂z ⎪ ⎪∂N ∂ζ ⎪
⎩ i ⎭ ⎩ i ⎭
from which the strain matrix B is evaluated.

Substituting the strain matrix B into equation (27), the stiffness matrix can be obtained
+1 +1 +1
k e = ∫ BT DBdV = ∫ ∫ ∫ BT DBdet[J ]dξdηdζ (68)
−1 −1 −1

where the matrix of material constants, D is given by

⎡1 − v v v 0 0 0 ⎤
⎢ v 1− v v 0 0 0 ⎥
⎢ ⎥
⎢ v v 1− v 0 0 0 ⎥
E ⎢ 1 − 2v ⎥
D= ⎢ 0 0 0 0 0 ⎥ (69)
(1 + v)(1 − 2v) ⎢ 2 ⎥
1 − 2v
⎢ 0 0 0 0 0 ⎥
⎢ 2 ⎥
⎢ 0 1 − 2v ⎥
0 0 0 0
⎢⎣ 2 ⎥⎦
in which E is the Young’s Modulus and v is the Poisson’s ratio

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 Chapter 3 Fundamentals of Finite Element Method

The determinate of the Jacobian matrix det[J ] in equation (68) is given by

det[J] = abc (70)

The mass matrix can be obtained by substituting the shape function matrix equation (60)

into (30)
+1 +1 +1
m e = ∫ ρ NT NdV = ∫ ∫ ∫ ρNT Ndet[J]dξdηdζ (71)
−1 −1 −1

3.3.2 2-D Plane Stress/Strain Element

The 2-D plane stress/strain element is used to analyse structural problems where the

loading and hence the deformation takes place within the plane. As shown in Figure 3.5,

a rectangular 2-D plane stress/strain element has four nodes, numbered 1, 2, 3, 4 in a

counter-clockwise manner. Each node has two DOFs, thus there are total eight DOFs in

a rectangular 2-D plane stress/strain element. Again the formulation of the 2-D plane

stress/strain element is based on a local natural coordinate system with its origin at the

centre of the element. Figure 3.5 shows the mapping between the local natural

coordinate system ( ξ ,η ) and physical coordinate system ( x , y ) for the rectangular

element, the dimension of which is defined as 2a × 2b × h .

Figure 3.5 Rectangular 2-D plane stress/strain element (Liu and Quek, 2003)

The displacement vector U is assumed to have the form

U ( x, y ) = N ( x , y ) d e (72)

where the nodal displacement vector d e is given as

45
 Chapter 3 Fundamentals of Finite Element Method

⎧ u1 ⎫
⎪v ⎪
⎪ 1⎪
⎪u2 ⎪
⎪ ⎪
⎪v ⎪
de = ⎨ 2 ⎬ (73)
⎪u3 ⎪
⎪ v3 ⎪
⎪ ⎪
⎪u4 ⎪
⎪v ⎪
⎩ 4⎭

The matrix of a shape function has the form of

⎡N 0 N2 0 N3 0 N4 0⎤
N=⎢ 1
N 4 ⎥⎦
(74)
⎣0 N1 0 N2 0 N3 0

The shape function is given as

1
N j = (1 + ξξ j )(1 + ηη j ) (75)
4
The strain matrix

⎡∂ ∂x 0 ⎤
B = LN = ⎢⎢ 0 ∂ ∂y ⎥⎥ N (76)
⎢⎣∂ ∂y ∂ ∂x ⎥⎦

Having obtained the shape function and the strain matrix, the stiffness matrix can be

calculated as
+1 +1
k e = ∫ BT DBdV = ∫ ∫ abhBT DBdξdη (77)
−1 −1

where the matrix of material constants, D is given by

⎡1 v 0 ⎤
E ⎢ ⎥ (for plane stress)
D= v 1 0 (78)
1 − v2 ⎢ ⎥
⎢⎣0 0 (1 − v) 2⎥⎦

⎡ 1 v (1 − v) 0 ⎤
E (1 − v) ⎢ ⎥ (for plane strain) (79)
D= ⎢ v (1 − v) 1 0 ⎥
(1 + v)(1 − 2v)
⎢⎣ 0 0 (1 − 2v) (2(1 − v))⎥⎦

The element mass matrix can be obtained by

+1 +1
m e = ∫ ρ NT NdV = ∫ ∫ abhρNT Ndξdη (80)
−1 −1

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 Chapter 3 Fundamentals of Finite Element Method

3.3.3 Plate Element

A plate element is geometrically similar to a 2-D plane stress/strain element, as shown

in Figure 3.5. However the plate element only carries transversal loads and the bending

deformation in the plate. The deformation resulted from the transverse loading on a

plate is represented by the deflection and rotation of the normal of the middle plane of

the plate, which are functions of x and y but independent of z. Considering a 4 noded

plate element in the x-y plane, each node has three DOFs, i.e. the deflection w in z axis,

the rotation of the normal of the middle plane about x axis θ x the rotation of the normal

of the middle plane about y axis θ y . Hence the total DOFs of a 4 noded plate element is

12.

For a 4 noded plate element, the deflections and rotations can be written as

⎧w⎫
⎪ ⎪
⎨θ x ⎬ = Nd e (81)
⎪θ ⎪
⎩ y⎭
The shape functions of 4 noded plate element are the same as that of 4 noded 2-D

plane stress/strain element, as given in equation (75).

Based on the Reissner-Mindlin plate theory, which does not require the cross-section to

be perpendicular to the axial axes after deformation (Liu and Quek, 2003), the two

displacement components parallel to the underformed middle surface at a distance z

from the centroidal axis can be expressed by

u ( x , y , z ) = zθ y ( x , y ) (82)

v ( x , y , z ) = − zθ x ( x , y ) (83)

The in-plane strain is given as

ε = − zχ (84)

where χ is the curvature, given as

⎧ − ∂θ y ∂x ⎫
⎪ ⎪
χ = Lθ = ⎨ ∂θ x ∂y ⎬ (85)
⎪∂θ ∂x − ∂θ ∂y ⎪
⎩ x y ⎭

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 Chapter 3 Fundamentals of Finite Element Method

The off-plane shear strain is given as

⎧ ∂w ⎫
⎧ξ xz ⎫ ⎪⎪ θ +
∂x ⎪⎪
y
γ =⎨ ⎬=⎨ ∂w ⎬ (86)
⎩ξ yz ⎭ ⎪− θ x + ⎪
⎪⎩ ∂y ⎪⎭

Hamilton’s Principle is used to derive the equation of motion. The strain energy of a 4

noded plate element can be obtained as

1 h 1 h T
2 ∫A ∫0 2 ∫A ∫0
Π= ε T
σ dz dA + τ γdzdA (87)

where τ is the average shear stresses related to the shear strain in the form

⎧τ xz ⎫ ⎡G 0 ⎤
τ = ⎨ ⎬ =κ⎢ ⎥γ = κD sγ (88)
⎩τ yz ⎭ ⎣ 0 G⎦

in which G is the shear modulus, and κ is a constant that is usually taken to be π 2 12

(Liu and Quek, 2003). Substituting equation (84) and (88) into (87), the strain energy

becomes

1 h3 T 1
Π= ∫ χ DχdA + ∫ κhγ T D s γdA (89)
A
2 12 2 A

The kinetic energy of the plate element is given by

T=
1
2∫
( )
ρ u& 2 + v& 2 + w& 2 dV (90)

which is a summation of the contributions of three velocity components in

the x , y and z directions. Substituting equation (82) and (83) into (90), the kinetic energy

can be obtained (Liu and Quek, 2003)

1 h3 & 2 h3 & 2 1
T= ∫ ρ hw + θ x + θ y )dA = ∫ (dT Id)dA
2
( & (91)
2 12 12 2

where

⎡ ρh 0 0 ⎤
⎢
I = ⎢ 0 ρh 12
3
0 ⎥⎥ (92)
⎢⎣ 0 0 ρh3 12⎥⎦

To obtain the shape function, substituting equation (81) into (87) will lead to

ke = ∫
A 12
[ ]
h3 I T T
[ ]
B DBI dA + ∫ κh BO D s BO dA
A
(93)

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 Chapter 3 Fundamentals of Finite Element Method

where

⎡0 0 − ∂N j ∂x ⎤
⎢ ⎥
BIj = ⎢0 ∂N j ∂y 0 ⎥ (94)
⎢0 ∂N j ∂x − ∂N j ∂y ⎥
⎣ ⎦
⎡∂N j ∂x 0 Nj⎤
BOj = ⎢
0 ⎥⎦
(95)
⎣∂N j ∂y − N j

The mass matrix can be derived from the kinetic energy, i.e. equation (91)

m e = ∫ N T INdA (96)
A

3.3.4 Shell Element

A shell element is able to carry loads in all direction, i.e. bending and twisting, as well as

in-plane deformation. The simplest but widely used shell element can be formulated by

combining the 2-D plane stress/strain element and the plate element, where the 2-D

plane stress/strain element handles the membrane or in-plane effects, while the plate

element deals with the bending or off-plane effects. In this section, the derivation of a 4

noded rectangular shell element is presented. There are 6 DOFs at a node of a shell

element, i.e. three translational displacements and three rotational deformations. Figure

3.6 shows a rectangular shell element and the DOFs at each node.

Figure 3.6 Rectangular shell element (Liu and Quek, 2003)

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 Chapter 3 Fundamentals of Finite Element Method

The nodal displacement vector of a rectangular shell element can be written as

⎧d e1 ⎫
⎪ ⎪
⎪d ⎪
de = ⎨ e2 ⎬ (97)
⎪d e 3 ⎪
⎪⎩d e 4 ⎪⎭

where d ei are the displacement vector at node i

⎧ui ⎫
⎪v ⎪
⎪ i ⎪
⎪⎪wi ⎪⎪
d ei = ⎨ ⎬ (98)
⎪θ xi ⎪
⎪θ yi ⎪
⎪ ⎪
⎪⎩θ zi ⎪⎭

The stiffness matrix of a 2-D solid rectangular element is used to deal with the

membrane effects of a rectangular shell element

⎡ k11
m
k12m m
k13 k14m
⎤
⎢ m ⎥
k k m22 k m23 k m24 ⎥
k e = ⎢ 21
m
(99)
⎢ k 31
m m
k 32 m
k 33 m ⎥
k 34
⎢ m ⎥
⎢⎣k 41 k 42 k m44 ⎥⎦
m
k m43

where the superscript m represents the membrane matrix. Each sub-matrix has a

dimension of 2 × 2, as it corresponds to the two DOFs u and v at each node.

The stiffness matrix of a rectangular plate element is used to take account the bending

effects in a rectangular shell element

⎡ k11
b b
k12 b
k13 k14b
⎤
⎢ b b ⎥
k k b22 k b23 k 24 ⎥
k eb = ⎢ 21 (100)
⎢ k 31
b b
k 32 b
k 33 b ⎥
k 34
⎢ b ⎥
⎣⎢k 41 k b42 k b43 k b44 ⎦⎥

where the superscript b stands for the bending matrix. Each sub-matrix has a dimension

50
 Chapter 3 Fundamentals of Finite Element Method

of 3 × 3, corresponding to three DOFs, i.e. w, θ x and θ y .

The stiffness matrix of a rectangular shell element in the local coordinate system is then

formulated by combining equation (99) and (100):

⎡ k 11
m
0 m
0 k 12 0 m
0 k13 0 m
0 k14 0 0⎤
⎢ b b b b ⎥
⎢ 0 k11 0 0 k12 0 0 k13 0 0 k14 0⎥
⎢0 0 0 0 0 0 0 0 0 0 0 0⎥
⎢ m ⎥
⎢k 21 0 0 k m22 0 0 k m23 0 0 k m24 0 0⎥
⎢ 0 kb 0 0 k b22 0 0 k b23 0 0 k b24 0⎥
⎢ 21
⎥
⎢0 0 0 0 0 0 0 0 0 0 0 0⎥
ke = ⎢ m (101)
k 0 0 k m
0 0 k m
0 0 k m
0 0⎥
⎢ 31 32 33 34 ⎥
⎢ 0 k 31
b
0 0 b
k 32 0 0 b
k 33 0 0 b
k 34 0⎥
⎢ ⎥
⎢0 0 0 0 0 0 0 0 0 0 0 0⎥
⎢k m41 0 0 k m42 0 0 k m43 0 0 k m44 0 0⎥
⎢ b b b ⎥
⎢ 0 k 41 0 0 k 42 0 0 k 43 0 0 k b44 0⎥
⎢0
⎣ 0 0 0 0 0 0 0 0 0 0 0⎥⎦

Similarly, the mass matrix of a rectangular shell element can be obtained by combining

the mass matrices of a 2-D rectangular solid element and a rectangular plate element.

3.4 Non-linear Solution Method

In this section, two non-linear FE solution method, i.e. static implicit method and

dynamic explicit method are summarised.

3.4.1 Implicit Method

Newton-Raphson method is one of the most commonly used solution procedures in

non-linear implicit analysis (Harewood and McHugh, 2007). Unlike solving a linear

problem, the solution cannot be obtained by solving a single system of equations for the

non-linear problem. Hence, in the non-linear implicit analysis, the loading is applied

gradually and incrementally. Normally, the simulation breaks into a number of load

increments and finds the approximate equilibrium at the end of the each increment. It

51
 Chapter 3 Fundamentals of Finite Element Method

often takes several iterations to determine an acceptable solution for a given increment.

The equilibrium equation in an iteration of the implicit method is defined as (Abaqus

analysis user’s manual, 2008):

F − I = Kc (102)

where F is the external force vector, I is the internal force vector, K is the stiffness matrix

of the structure, and c is the displacement correction vector.

Figure 3.7 shows the non-linear response of a structure to a load increment ΔF. Based

on structure’s configuration at the initial displacement of U0, the initial stiffness matrix of

the structure K0, is determined. A displacement correction ca, is calculated from

−1
ca = K 0 ΔF (103)

The configuration of the structure is now updated to a new displacement of Ua and a

new stiffness of the structure Ka, is formed based on Ua. Internal force Ia is also

calculated at this stage, where

Ia = K a U a (104)

The force residual of this iteration, Ra, can be calculated as:

R a = F − Ia (105)

In order to ensure the accuracy of the solution, certain convergence criterion is applied.

If both Ra and ca are smaller than the defined tolerance values, it means that the solution

has converged. If any of the convergence check does not satisfy the criterion, further

iterations will be performed in order to bring the external force and internal force into

balance.

52
 Chapter 3 Fundamentals of Finite Element Method

Figure 3.7 First iteration in an increment (Abaqus analysis user’s manual, 2008)

Assuming the first iteration does not achieve convergence, the second iteration uses the

stiffness Ka and force residual Ra to determine another displacement correction cb, which

brings the system closer to equilibrium, as shown in Figure 3.8. Using the internal force

calculated from structure’s new configuration, the force residual of the second iteration,

Rb, is calculated. Both Rb and cb are compared against the tolerance of force residual

and displacement increment (Ub-U0) at this iteration. If necessary, further iterations will

be performed.

Figure 3.8 Second iteration in an increment (Abaqus analysis user’s manual, 2008)

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 Chapter 3 Fundamentals of Finite Element Method

3.4.2 Explicit Method

Similar to the implicit method, the non-linear response in the explicit method is also

obtained incrementally. At the beginning of an increment (t) in the explicit analysis,

based on the dynamic equilibrium equation:

..
F − I = Mu (106)
..
The nodal accelerations ( u ) can be calculated by:
..
u | ( t ) = M −1 ( F − I ) | ( t ) (107)

where M is the nodal mass matrix. The acceleration of any node is completely

determined by the mass and the net force acting on it.

The accelerations are integrated through time using the central difference rule, whereby

the change of velocity is calculated from equation (108), assuming that the acceleration

is constant over the time increment:

. . Δt ( t + Δt ) + Δt ( Δt ) ..
u | (t + Δt / 2 ) = u | (t − Δt / 2 ) + u | (t ) (108)
2

The velocities are integrated through time and added to the displacement ( u ) at the

beginning of the increment to calculate the displacements at the end of the increment:
.
u | ( t + Δt ) = u | ( t ) + Δt | ( t + Δt ) u | (t + Δt / 2) (109)

In order to obtain accurate results from the explicit method, the time increment has to be

extremely small which ensures that the acceleration through the time increment is

nearly constant. Therefore an explicit analysis typically requires many thousands of

increments. The size of the increment is determined by the stability limit:

Le
Δt = min( d ) (110)
c
d
where Le is the characteristic element length, c is the wave speed of the material:

λ + 2μ
cd = (111)
ρ

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 Chapter 3 Fundamentals of Finite Element Method

where λ and μ are Lamé’s elastic constants. If T is the actual time of the analysed

process, the number of time increments required n can be obtained by:

T T
n= = (112)
Δt ρ
min( L ×
e
)
λ + 2μ

Normally it is unfeasible to run a quasi-static analysis with its real time scale, as the

computing time is extremely long (Harewood and McHugh, 2007). Two techniques have

been used to speed up the analysis: the first method is by artificially increasing the

loading or deformation rate, known as load rate scaling; the second method is to

increase the density of the material, known as mass scaling. According to equations

(110) - (112), mass scaling by a factor of q2 should have the same speeding up effects

as load rate scaling by a factor of q. Mass scaling is preferable as it does not affect the

mechanical response of rate-dependent material (Tekkaya, 2000). Nevertheless, if a

very large speed-up factor is applied, the corresponding inertial forces will affect the

mechanical response and produce unrealistic dynamic results. The general rule to

control the inertial effects resulting from mass scaling is to ensure that the kinetic energy

of the material should not exceed a small portion (typically 5%-10%) of its internal

energy during the majority of the duration of the process (Abaqus analysis user’s

manual, 2008).

3.5 Material Constitutive Model

3.5.1 von Mises Yield Criterion

Figure 3.9 shows a idealised bi-linear stress-strain curve, where the material behaves

elastically until the initial yielding stress σ y is reached (Dunne and Petrinic, 2005). If at a

strain of ε the load is reversed (unloading), the stress-strain response is assumed to

follow the elastic law, i.e. the gradient of the unloading stress-strain curve is again the

Young’s modulus E, as it is in the elastic loading process. Once a stress of zero is

obtained, the remaining strain is the plastic strain ε p and the recovered strain is the

55
 Chapter 3 Fundamentals of Finite Element Method

elastic strain ε e . Therefore, the total strain ε can be decomposed into elastic and plastic

parts (Belytschko et al., 2000)

ε =εe +εp (113)

which is known as the classical additive decomposition of strain.

Figure 3.9 Bi-linear stress-strain curve (Dunne and Petrinic, 2005)

As shown in Figure 3.9, for a bilinear stress-strain relationship, the stress achieved at a

strain of ε can be calculated by

σ = Eε e = E (ε − ε p ) (114)

As a commonly used yield criterion for isotropic, non-porous metal materials, the von

Mises yield function is defined as (Dunne and Petrinic, 2005)

f = σe −σy (115)

where σ y is the yield stress measured from the uni-axial material tensile test, while σ e is

known as the effective stress or von Mises equivalent stress. The effective stress

in terms of principal stresses is defined as,

[ ]
1
σe =
1
(σ 1 − σ 2 )2 + (σ 2 − σ 3 )2 + (σ 3 − σ 1 )2 2 (116)
2

or in terms of direct and shear stresses,

1
⎡3
( ⎤2
σ e = ⎢ σ 112 + σ 22 2 + σ 33 2 + 2σ 12 2 + 2σ 232 + 2σ 312 ⎥ ) (117)
⎣2 ⎦

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 Chapter 3 Fundamentals of Finite Element Method

where the subscripts 1, 2, and 3 represent the X, Y, and Z directions.

The yield criterion shown in equation (115) is given by

f < 0 : Elastic deformation (118)

f = 0 : Plastic deformation (119)

3.5.2 Strain Hardening

Figure 3.10 shows the isotropic strain hardening under a uni-axial stress σ 2 , as

discussed in Section 2.1.3.2. In order to ensure the hardening to take place and the load

point to stay on the yield surface, the yield surface must expand as the stress increases.

The amount of the expansion can be expressed as a function of accumulated plastic

strain p , which is given as

p = ∫ dp = ∫ p& dt (120)

where p& is the effective plastic strain rate

[( ) ( ) ( )]
1
2 p 2 2 2 2
p& = ε&1 − ε&2 p + ε&2 p − ε&3p + ε&3p − ε&1p (121)
3

Figure 3.10 Isotropic strain hardening (Dunne and Petrinic, 2005)

The yield function becomes

f (σ , p ) = σ e − σ y ( p) = 0 (122)

57
 Chapter 3 Fundamentals of Finite Element Method

As shown in Figure 3.10,

σ y ( p) = σ y0 + r ( p) (123)

in which σ y 0 is the initial yield stress and r ( p) is called isotropic hardening function.

Figure 3.11 shows a stress-strain curve of linear strain hardening and the linear isotropic

hardening function can be written as

dr ( p) = hdp (124)

where h is known as the plastic hardening modulus.

Figure 3.11 Stress-strain curve of linear strain hardening (Dunne and Petrinic, 2005)

For uni-axial conditions, dp = dε p and the stress increase due to isotropic hardening is

just dr , therefore
dσ
dε p = (125)
h
The increment in elastic strain
dσ
dε e = (126)
E
The total strain can be expressed as
dσ dσ E+h
dε = + = dσ ( ) (127)
E h Eh
Thus
E
dσ = E (1 − )dε (128)
E+h

58
 Chapter 3 Fundamentals of Finite Element Method

3.6 Contact algorithms

Contact simulation, which is among the most difficult aspects for non-linear FE problems,

is also implemented in an incremental manner (Belytschko et al., 2000). In each time

increment of a metal forming problem simulation, there are three main aspects of the

contact modellling: identifying the area on the surfaces that are in contact; calculating

the contact force in the normal direction of the surfaces due to penetrations; thereafter

calculating the tangential force caused by friction. Four corresponding contact

formulations, i.e. contact surface weighting, tracking approach, constraint enforcement

method, and friction models, are discussed in this section.

3.6.1 Contact Surface Weighting

To analyse the contact problem, master and slave surfaces have to be defined. The

master surface is the surface with “hard” material, for instance, the tools in the metal

forming simulation; whereas the slave surface is the surface with relatively “soft”

material such as the deformable blank. In general, there are two kinds of contact

weighting methods, i.e. a pure master-slave surface weighting and a balanced

master-slave surface weighting. In the pure master-slave surface weighting method,

only the penetrations of slave nodes into master surface will be resisted. There is no

restriction for the master surface to penetrate into the slave surface. Conversely, the

balanced master-surface weighting applies the pure master-slave weighting twice,

reversing the surfaces on the second pass, which minimises the penetrations of the

contact bodies. Therefore it provides more accurate results than the pure master-slave

surface in most cases and should be used whenever possible. The pure master-slave

surface weighting is used in the case such as when a rigid surface contacts a

deformable surface.

3.6.2 Tracking Approach

In the simulation of the contact / impact problem, a node on a contact surface is possible

to contact any facet on the opposite contact surface. There are two commonly used

59
 Chapter 3 Fundamentals of Finite Element Method

contact tracking approaches, i.e. global search and local search (ABAQUS analysis

user’s manual, 2008). At the beginning of the simulation, an exhaustive global search is

carried out to determine the closet master surface facet for each slave node of each

contact pair. As the computing cost of global search is relatively high, a global search is

only conducted every few hundred time increments in a contact / impact model.

Conversely, a less expensive local search is performed in most time increments, where

a given slave node only searches the facets which are attached to the previous tracked

master surface node to determine the closest facet. Because the time increments are

small in most situations, the movement amongst contacting bodies is very limited from

one increment to the next and the local search is adequate to track the motion of the

contact surfaces.

3.6.3 Constraint Enforcement Method

In the simulation of contact problem, two methods to enforce contact conditions have

been commonly used, i.e. a kinematic predictor/corrector method and a penalty method

(ABAQUS analysis user’s manual, 2008). By using the kinematic contact method, the

kinematic state of the model is first advanced without considering the contact conditions

in each increment of the analysis. If the slave nodes penetrate into the master surface,

the time increment, the depth of the penetration of each node, the mass associated with

it will be used to calculate the resisting force to oppose the penetration. This resisting

force will make the slave nodes to exactly contact with the master surface. In other

words, no penetration of slave nodes into the master surface is allowed in the kinematic

contact method. The penalty contact applies less stringent enforcement of contact

constraints than the kinematic contact method. It searches for the slave node

penetration in the current configuration. Contact forces, which calculated as the

penetration distance times the penalty stiffness, are applied to the slave nodes to

oppose the penetration. At the same time, opposite forces act on the master surface at

the penetration point.

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 Chapter 3 Fundamentals of Finite Element Method

3.6.4 Frictional Model

Apart from the normal forces, when surfaces are in contact, they normally transmit

shear forces across their interface as well. Coulomb friction is a common friction model

used to describe the interaction between two contacting surfaces. The tangential motion

will not start until the frictional shear stress reaches a critical value ( τ crit ), which is

defined by:

τ crit = μ ⋅ p (129)

where μ is the coefficient of friction and p is the normal contact pressure. If the shear

stress is below τ crit , there will no relative motion between the contact surfaces (sticking).

While when the frictional shear stress reaches its critical value relative motion (slipping)

occurs.

3.7 Summary

In this chapter, the fundamental theory of Finite Element Method, such as Hamilton’s

Principle and basic analysis procedure of FEM, has been discussed. The formulations

of four different types of finite elements, i.e. 3-D solid element, 2-D plane stress/strain

element, plate element and shell element, are presented. Two commonly used

non-linear FE solution methods, implicit method and explicit method, are compared.

Additionally, the elastic-plastic material constitutive model and contact algorithms of FE

simulation have been briefly outlined. Furthermore, the application of FEM to the metal

spinning simulation, e.g. development and verification of the metal spinning FE models,

are given in Section 4.2, 5.3 and 6.3 of the following chapters.

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4. Effects of Roller Path Profiles on Material

Deformation

Until recently, the limited research work on the roller path profiles have been mainly

carried out by means of experimental investigation alone (Hayama et al., 1970) or by

simplified 2-D FE simulation (Liu et al., 2002). Possible causes of wall thinning in the

workpiece during conventional spinning are still not fully understood. For this reason,

four representative roller path profiles, i.e. combined concave and convex, convex,

linear, and concave curves, have been designed and used to carry out experimental

investigation in this study. 3-D elastic-plastic FE models of metal spinning have been

developed by using commercial FE software - Abaqus. The FE models have been

verified by conducting mesh convergence study, evaluating scaling techniques, and

comparing dimensional results. Finally, effects of these roller path profiles on the

variations of tool force, stress, wall thickness, and strain in the conventional spinning

have been analysed numerically. The results show that using a concave path produces

relatively high tool forces, stresses and reduction in wall thickness, comparing with the

corresponding results obtained from other roller paths. Conversely, low tool forces,

stresses and wall thinning are obtained in the FE model using a convex roller path.

4.1 Experimental Investigation

4.1.1 Experimental Setup

By using the commercial spinning Computer-Aided Manufacturing (CAM) software -

OPUS, CNC programs of various roller passes have been developed. Figure 4.1

illustrates the setup and the schematic diagram of the spinning experiment, where the

angle between the roller axis and the mandrel axis is 45°. The blank is made of mild

steel (DC01) and its diameter and thickness are 240 mm and 2 mm, respectively. A feed

rate of 800 mm/min and a spindle speed of 400 rpm are chosen for all the experimental

62
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

runs, as these parameters are commonly used in the production of Metal Spinners

Group Ltd and have been verified by FE simulation before conducting the experiments.

(a) Experimental setup

(b) Schematic diagram

Figure 4.1 Spinning experiment

63
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4.1.2 Design of Various Roller Path Profiles

Based on the results of FE simulations and corresponding verifying experiments using a

range of CNC roller path designs, four representative roller path profiles have been

selected for analysing their effects on the material deformation of the conventional

spinning process. As shown in Figure 4.2, these roller path profiles include combined

concave and convex roller path (Trial 1), convex roller path (Trial 2), linear roller path

(Trial 3) and concave roller path (Trial 4). It is clear that all roller path designs consist of

two forward passes and one backward pass between them. The equations of trendlines

of the first forward pass of each path design have been given in Figure 4.2. The

positions of the final points of the corresponding roller passes in these path designs are

approximately the same, where the coordinates of the final points are also shown in

Figure 4.2.

60
y = 3E-06x 6 + 0.0002x 5 + 0.0028x 4 + 0.0123x 3 - 0.19x 2 - 4.2467x + 0.2081

50
-35.607, 45.512

40
Y (mm)

30

20

10

0
-60 -50 -40 -30 -20 -10 0
X (mm)

(a) Roller path - Trial 1

64
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

60
y = -9E-08x 6 - 9E-06x 5 - 0.0003x 4 - 0.0079x 3 - 0.1458x 2 - 3.7673x + 0.0047

-35.531, 48.38
50

40

Y (mm) 30

20

10

0
-60 -50 -40 -30 -20 -10 0
X (mm)

(b) Roller path - Trial 2

60
y = -6E-07x 6 - 6E-05x 5 - 0.0024x 4 - 0.0476x 3 - 0.525x 2 - 4.9312x + 0.081

50
-35.804, 47.834

40
Y (mm)

30

20

10

0
-60 -50 -40 -30 -20 -10 0
X (mm)

(c) Roller path - Trial 3

65
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

60
y = 2E-06x 6 + 0.0001x 5 + 0.002x 4 - 0.0007x 3 - 0.3219x 2 - 4.0501x + 0.4564

50
-35.571, 45.395

40

Y (mm) 30

20

10

0
-60 -50 -40 -30 -20 -10 0
X (mm)

(d) Roller path - Trial 4

Figure 4.2 Roller path profile design

Verifying experiments have been carried out by using roller path Trials 1, 2 and 3. Figure

4.3 illustrates the experimentally spun samples; their wall thickness variations have

been measured and used to evaluate the FE analysis results. In addition, the roller path

Trials 5 and 6, shown in Figure 4.4, are developed to study the effects of roller path

curvature on the wall thickness variation. Clearly, both of the roller path designs use

concave curves, but a greater curvature is used in roller path Trial 6.

(a) Trial 1 (b) Trial 2 (c) Trial 3

Figure 4.3 Experimentally spun samples by using different CNC roller paths

66
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

60
y = 6E-06x 6 + 0.0003x 5 + 0.0074x 4 + 0.0729x 3 + 0.2419x 2 - 2.5452x + 0.1283

50

40
Y (mm)

30

20

10

0
-60 -50 -40 -30 -20 -10 0
X (mm)

(a) Roller path – Trial 5

60
y = 2E-06x6 + 3E-05x5 - 0.0012x4 - 0.0396x3 - 0.4869x2 - 4.2998x + 0.1098

50

40
Y (mm)

30

20

10

0
-60 -50 -40 -30 -20 -10 0
X (mm)

(b) Roller path – Trial 6

Figure 4.4 Concave roller path profiles using different curvatures

67
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4.2 Finite Element Simulation

4.2.1 Development of Finite Element Models

The explicit FE solution method has been chosen to analyse the spinning process.

Comparing with the implicit FE method, the explicit FE method determines a solution by

advancing the kinematic state from one time increment to the next, without iteration. It is

more robust and efficient for analysing the metal spinning process, which can be

considered as a quasi-static problem including large membrane deformation and

complex contact conditions.

In order to improve the computational efficiency, the spinning tools – roller, mandrel, and

backplate, are modeled as 3-D analytical rigid bodies, leaving the blank as the only

deformable body. 3-D eight noded reduced integration linear continuum shell elements

(SC8R), which provide a better capability to model two–side contact behaviour and

transverse shear deformation than 2-D conventional shell elements, are used to mesh

the blank. Moreover, it has been observed that the SC8R element can produce accurate

numerical results, based on a force measurement experiment and the corresponding FE

simulation (Long et al., 2011).

In this study, nine integration points along the element thickness direction have been

used in order to accurately compute the state of stress through the blank thickness, as

also suggested by Klimmek et al. (2002). Enhanced hourglass control, which provides

an increased resistance to the hourglassing problem and more accurate displacement

solutions than the default hourglass control (Abaqus analysis user’s manual, 2008), is

used in the FE analysis models. A central area (radius of 50 mm) of the blank is

neglected, since it is clamped between the mandrel and the backplate and almost no

deformation takes place. The sweep meshing technique is employed to mesh the blank,

in which 4320 elements and 9000 nodes are generated.

The material of the blank is assumed to be homogeneous and isotropic. The elastic

behaviour of the material is defined by Young’s modulus of 198.2 GPa, Poisson’s ratio

68
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

of 0.3, and mass density of 7861 kg/m3. The von Mises yielding criterion and isotropic

hardening have been used to model the material plastic response. Figure 4.5 shows the

true stress-strain curve of the material obtained from a uni-axial tensile test, where a flat

sample was tested at room temperature under ASTM E8M (Standard Test Methods for

Tension Testing of Metallic Materials) by Westmoreland Mechanical Testing & Research,

Ltd.

450

400

350
True Stress (MPa)

300

250

200

150

100

50

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

True plastic strain

Figure 4.5 True stress-strain curves of Mild steel (DC01)

The penalty enforcement method and Coulomb’s friction law have been used to

simulate the normal and sliding contact behaviors between the tools and blank,

respectively. The frictional coefficient between the roller and the blank is set to be low,

since the roller rotates along its own axis during the spinning process. Three Coulomb

frictional coefficients have been assigned to three contact pairs of tools and blank:

mandrel-blank 0.2, backplate-blank 0.5, and roller-blank 0.02 (Razavi et al., 2005).

Three steps have been applied in the spinning simulation. At the first step, a

compressive force of 150 kN is applied on the backplate, which accounts for the blank

being clamped between the backplate and mandrel. The second step involves applying

69
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

a rotational boundary condition to the backplate and mandrel, in order to model the

synchronous rotation of the blank, backplate and mandrel. Finally, to realise the

complex nonlinear roller path (passes), two displacement boundary conditions in the

local x and z directions, as shown in Figure 4.1(b), are applied to the roller.

These two displacement boundary conditions are calculated by transferring the CNC

programs, which are initially developed in the global coordinate system (X-Y), into the

local coordinates (x-z). The following equations have been used to calculate the local

coordinates from the global coordinates:

x = − X sin θ − Y cos θ (130)

z = X cos θ − Y sin θ (131)

where (x, z) is the local coordinates, (X, Y) represents the global coordinates, and θ is the

angle between axis X and axis z, which is 45° in this experimental setup.

Additionally, to speed-up the FE solution time, the mass scaling method with a scaling

factor of 25 has been used in all of the models. Parallel FE analyses which can

effectively reduce the computing time in the simulation of spinning process (Quigley and

Monaghan, 2002b) have been performed using a workstation of Quad-Core AMD

OpteronTM CPU 2.2GHz and RAM of 6GB.

4.2.2 Verification of Finite Element Models

In this section, FE models are verified not only by evaluating mesh convergence and

assessing scaling methods, but also by comparing the FE analysis dimensional results

with experimental measurements.

4.2.2.1 Mesh Convergence Study

A mesh convergence study has been carried out on the FE model using roller path Trial

1. Table 4.1 gives details of the meshes used in the three FE models and the

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

corresponding numerical results of maximum tool forces and minimum wall thicknesses.

The mesh density used in FE model 2 and FE model 3 is about two and three times of

that used in FE model 1, respectively.

Comparing with the axial force obtained from model 3, both model 1 and model 2

produce similar axial force results, where the relative errors are 2%. However,

significant differences in the radial force and tangential force have been observed

between model 1 and model 3, where the relative errors are 48% and 9%, respectively.

By increasing the number of elements from 1920 to 4320, the relative errors of the radial

force and tangential force between model 2 and model 3 have been decreased to 7%

and 1%, respectively. On the other hand, a minimum wall thickness of 1.945 mm is

obtained by model 1, while model 2 produces a wall thickness of 1.908 mm, which is

much closer to the corresponding value of model 3, i.e. 1.909 mm.

Table 4.1 Mesh convergence study

Number Average CPU

Radial Circumferential Fa Fr Ft Thickness
Model of aspect time*
nodes nodes (N) (N) (N) (mm)
elements ratio (h:m:s)

1 1920 17 120 2.46 112:42:55 4182 1833 265 1.945

2 4320 25 180 1.65 223:51:37 4177 1327 289 1.908

3 6380 30 220 1.41 338:39:50 4104 1239 292 1.909

Furthermore, the sensitivity of stress to the mesh density of the FE models has also

been analysed. As illustrated in Figure 4.6(a) – (c), similar von Mises stress distributions

of the three FE models listed in Table 4.1 have been observed during the 1st forward

pass. As shown in Figure 4.6(d), which compares the variations of von Mises stress

along a radial node path under the local forming zone, a good agreement between

model 2 and model 3 has been observed and the maximum difference of stresses in the

local forming zone is only 2%. Nevertheless, there is a considerable difference of

stresses in the local forming zone between model 1 and model 3, where the maximum

71
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

error is 9%. Therefore, by analysing the variations of tool forces, wall thickness and

stress of these three FE models with different mesh density, it is believed that sufficient

convergence has been achieved by using the meshing of FE model 2, which has been

used in all the FE models of this study.

Unit: MPa

Local forming
zone

(a) Model 1

Unit: MPa

Local forming
zone

(b) Model 2

72
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

Unit: MPa

Local forming
zone

(c) Model 3

300

250
Local forming
200 zone Model 1
Stress (MPa)

Model 2
150

Model 3
100

50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(d) Stress distribution along radial path

Figure 4.6 Variations of von Mises stress in 1st forward pass of FE model

73
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4.2.2.2 Assessment of Scaling Methods

The simulation accuracy and computing efficiency of two scaling methods, i.e. load rate

scaling and mass scaling, have been evaluated in this section. As shown in Table 4.2,

CPU time and FE analysis results of two models, which use mass scaling by a factor of

25 (Model 2) and load rate scaling by an equivalent factor of 5 (Model 4), are compared

with a model which does not use any scaling (Model 5). In terms of result accuracy, both

the scaling models agree well with none scaling model. The maximum errors of radial

forces in both scaling models are below 1%. The corresponding errors of axial and

tangential forces are less than 6%. There is almost no difference of wall thickness

among these models. As for the computing efficiency, both scaling methods speed up

the simulation process by a factor of 5, indicating that using a mass scaling factor of q 2

or a load rate scaling factor of q would speed up the spinning FE simulation by a factor

of q, in agreement with equation (110)-(112).

Table 4.2 Scaling method study – Trial 1

Stable
Scaling Scaling Processing CPU time Fa Fr Ft Thickness
Model increment
method factor time (s) (h:m:s) (N) (N) (N) (mm)
(s)
Mass
2 25 1.029e-6 10.29 223:51:37 4177 1327 289 1.908
scaling
Rate
4 5 2.058e-7 2.06 222:56:03 4180 1357 292 1.908
scaling
No
5 1 2.058e-7 10.29 1119:04:40 3927 1345 278 1.914
scaling

4.2.2.3 Comparison of Dimensional Results

Figure 4.7 compares the wall thickness variations of three experimentally spun samples

using different roller path profiles with the corresponding FE analysis results. The

maximum errors between experimental and FE analysis results of thickness for Trial 1,

Trial 2 and Trial 3 are 4.5%, 2.5% and 3%, respectively. Therefore it is considered that

the FE simulation shows good correlation with the experiment.

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

2.2

2.1

2
Thickness (mm)
1.9 FEA

1.8 Exp

1.7

1.6

1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(a) Trial 1

2.2

2.1

2
Thickness (mm)

1.9
FEA

1.8 Exp

1.7

1.6

1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(b) Trial 2

75
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

2.2

2.1

2
Thickness (mm)
1.9
FEA

1.8 Exp

1.7

1.6

1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(c) Trial 3

Figure 4.7 Comparison of wall thickness between FE analysis and experimental results

4.3 Results and Discussion

In this section, the effects of various roller path profiles on the tool forces, stresses, wall

thickness, and strains have been analysed numerically.

4.3.1 Tool Forces

In this study, the definition of tool force components is shown in Figure 2.1(b), where the

radial tool force – Fr and axial tool force – Fa are defined in line with the radial direction

and the axial direction of the mandrel, respectively; and the tangential tool force – Ft is

perpendicular to both the axial and radial forces. Figure 4.8 compares the maximum

radial, axial and tangential tool forces using the four roller path designs over three

passes as given in Figure 4.2. Clearly, the axial forces are the highest among three

force components, while the tangential force is the lowest. In addition, the concave path

produces the highest radial, axial and tangential forces amongst these four roller path

profiles. As shown in Figure 4.8(a), in the first forward pass, there is not much difference

76
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

amongst the four roller paths in terms of their effects on radial forces; while the convex

roller path produces the lowest force in the second forward roller pass. As illustrated in

Figure 4.8(b) and 4.8(c), in the forward passes, the lowest axial and tangential forces

are observed in the FE models which use the convex roller path. Therefore, it is clear

that a convex roller path generally produces the lowest tool forces.

7000

6000
Convex &
5000 Concave

Convex
Force (N)

4000

3000 Linear

2000 Concave

1000

0
1st forward Backward pass 2nd forward
pass pass

(a) Maximum radial force

7000

6000
Convex &
Concave
5000
Convex
Force (N)

4000

3000 Linear

2000 Concave

1000

0
1st forward Backward pass 2nd forward
pass pass

(b) Maximum axial force

77
 Chapter 4 Effects of Roller Path Profiles on Material Deformation

400

350

Force (N)
Convex &
300 Concave

250 Convex

200
Linear
150

100 Concave

50

0
1st forward Backward pass 2nd forward
pass pass

(c) Maximum tangential force

Figure 4.8 Comparison of tool forces using various roller path profiles

Table 4.3 shows the ratios of maximum force components using the four roller path

profiles. It is noticeable that the ratios between maximum radial forces to maximum

tangential forces of all the four roller path profiles remain unchanged as 5:1. However,

the ratios of maximum axial force to maximum tangential force vary between 13:1 for

the convex roller path and 17:1 for the linear roller path.

Table 4.3 Ratios of maximum force components using various roller path profiles

Roller Path Profiles Ratios of Maximum Forces

Trial 1 - Combined convex and concave path Fa : Fr : Ft = 14: 5: 1

Trial 2 - Convex path Fa : Fr : Ft = 13: 5: 1

Trial 3 - Linear path Fa : Fr : Ft = 17: 5: 1

Trial 4 - Concave path Fa : Fr : Ft = 16: 5: 1

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4.3.2 Wall Thickness

Figure 4.9 illustrates the effects of the four roller path profiles on wall thickness

variations. It is clear that the wall thickness reduces significantly in two regions: Region

A and B. Region A is located between the backplate’s clamped area and the workpiece’s

last point of contact with the mandrel. Region B is located between the workpiece’s last

contact point with the mandrel and its contact point with the roller.

As illustrated in Figure 4.9(a), after the 1st forward pass, a dramatic wall thickness

reduction is observed in Region B, especially in the FE model which uses the concave

roller path (Trial 4), where the wall thickness is reduced by 4%. On the other hand, only

0.8% of wall thinning is seen when using the convex roller path (Trial 2); while the wall

thicknesses decrease by 1% in the FE analysis models which apply combined convex

and concave roller path (Trial 1) and linear roller path (Trial 3).

According to the FE analysis results, there are almost no thickness changes during the

backward passes for all of these four roller path trials, thus the thickness variation

diagram of the backward pass is not shown here. As shown in Figure 4.9(b), after the

second forward roller pass, there is almost no thickness change for the Region A, due to

the fact that in the second forward pass the roller does not deform this region, which has

already been formed in the first forward pass.

However, at Region B, the wall thickness of the model which uses the concave roller

path further decreases by 4% and reaches a thickness of 1.84 mm, while the wall

thickness of the model using the convex roller path profile only further decreases by

0.5% and remains at 1.97 mm. It is thus clear that using the concave roller path in

spinning tends to cause the highest reduction of the wall thickness and using the convex

roller path helps to minimise wall thickness variations, as also reported by Auer et al.

(2004).

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

2.01
A B
1.99
Convex &
1.97
Concave

Thickness (mm)
1.95
Convex
1.93

1.91 Linear
1.89

1.87 Concave

1.85

1.83
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(a) After 1st forward pass

2.01
A B
1.99
Convex &
1.97 Concave
Thickness (mm)

1.95
Convex
1.93

1.91 Linear

1.89

1.87 Concave

1.85

1.83
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(b) After 2nd forward pass

Figure 4.9 Wall thickness variations using various roller path profiles

Figure 4.10 illustrates the wall thickness variations of FE models using concave roller

path profiles with different curvatures - Trial 5 and 6, as shown in Figure 4.4. Clearly, the

model using roller path Trial 6, which has a greater curvature of the concave curve than

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

that of Trial 5, produces a greater wall thickness reduction. Thus it is believed that a

greater curvature used in the concave path would result in a greater thinning in wall

thickness of a spun part formed by the conventional spinning process.

2.01

1.99

1.97
Thickness (mm)

1.95
Trial 5
1.93

1.91 Trial 6
1.89

1.87

1.85

1.83
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(a) After 1st forward pass

2.01

1.99

1.97
Thickness (mm)

1.95
Trial 5
1.93

1.91 Trial 6
1.89

1.87

1.85

1.83
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(b) After 2nd forward pass

Figure 4.10 Wall thickness variations using concave path with different curvatures

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4.3.3 Stresses
A global cylindrical coordinate system has been used to output the stress values. The

radial stress is defined in line with the radial-axis of the mandrel; while the tangential

normal stress is in the direction perpendicular to the radial-axis and axial-axis of the

mandrel. Figure 4.11 illustrates radial stress (σr) variations on both the outer (roller

facing) and inner surface (mandrel facing) of a cross-section away from the roller

contact position after the first forward pass in the FE models using the four roller path

profiles.

As shown in Figure 4.11(a) and (b), the distributions of these radial stresses by using

the four roller path profiles are similar. However, much higher radial stresses are

induced using the concave roller path in comparison with the corresponding stresses of

FE models generated using other roller paths. Furthermore, it is clear that the outer

surface of Region A is subjected to high compressive radial stresses; while its inner

surface is under tensile radial stresses. Conversely, around Region B, higher tensile

and compressive radial stresses are observed on its outer and inner surfaces,

respectively. Therefore, Region A and B are subjected to radial bending effects in

opposite directions.

Tangential stress (σt) variations on both the outer and inner surface of a cross-section

away from the roller contact position after the first roller pass have been plotted in

Figure 4.12. It is noticeable that the outer surface of Region A is under compressive

tangential stresses, as shown in Figure 4.12(a). Moreover, tensile and compressive

tangential stresses are observed on the outer and inner surfaces of Region B,

respectively. Much higher tangential stresses are shown on the inner surface of

workpiece when the concave roller path is applied.

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

250
200
150 Convex &
100 Concave

Stress (MPa) 50 Convex

0
-50 A B Linear
-100
-150 Concave
-200
-250
-300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(a) Outer surface of workpiece

250
200
150 Convex &
Concave
100
Stress (MPa)

50 Convex
0 A B
-50 Linear
-100
-150 Concave
-200
-250
-300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(b) Inner surface of workpiece

Figure 4.11 Radial stress variations after 1st forward pass

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

250
200
150 Convex &
100 Concave
A B
Stress (MPa) 50 Convex
0
-50 Linear
-100
-150 Concave
-200
-250
-300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(a) Outer surface of workpiece

250
200
150 Convex &
100 Concave
Stress (MPa)

50 Convex
0
-50 A B Linear
-100
-150 Concave
-200
-250
-300
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(b) Inner surface of workpiece

Figure 4.12 Tangential stress variations after 1st forward pass

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4.3.4 Strains
In order to analyse the cause of wall thinning in the conventional spinning process,

variations of the maximum in-plane principal strain (radial strain) and minimum in-plane

principal strain (tangential strain) after the first forward pass have been plotted in Figure

4.13 and 4.14, respectively. As shown in Figure 4.13(a), high tensile radial strains are

observed in both Region A and B in the outer surface of workpiece; whilst in the inner

surface high tensile radial strains take place mainly around Region B, as demonstrated

in Figure 4.13(b). These high in-plane tensile radial strains are believed to be the reason

of the significant wall thinning in Regions A and B shown in Figure 4.9. In addition, the

FE model which uses a concave roller path profile produces much higher tensile radial

strains in Region B than FE models which use other roller path profiles, resulting in a

higher amount of wall thinning when the concave roller path is applied.

As shown in Figure 4.14, except for Region A in the outer surface of the workpece, both

the inner and outer surfaces are subjected to in-plane compressive tangential strains,

which would lead to a certain degree of compensation to the wall thinning. However, the

dominant high in-plane tensile radial strains shown in Figure 4.13 play a decisive role in

wall thinning. It is also noticeable that unlike the variations of in-plane radial strains

amongst the models using various roller path profiles, there is not much difference of

in-plane tangential strains.

Figure 4.15(a) illustrates the variations of out-of-plane principal strain (thickness strain)

in the outer surface of the workpiece. It is clear that both Region A and B of the outer

surface are subjected to high compressive thickness strains, indicating the decreasing

of wall thickness in those regions. The variations of the thickness strain in the inner

surface of the workpiece have been plotted in Figure 4.15(b). In Region A of the inner

surface, the low tensile thickness strain is cancelled out by the relatively high

compressive thickness strain of the outer surface. Conversely, the compressive

thickness strain in the inner surface of Region B is enhanced by the compressive

thickness strain of its outer surface. These combined effects of the strains result in less

wall thinning in Region A than Region B, as shown in Figure 4.9.

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

0.09
0.08
Convex &
0.07 A B Concave
0.06
Strain Convex
0.05
0.04
Linear
0.03
0.02
Concave
0.01
0
-0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(a) Outer surface of workpiece

0.09
0.08
Convex &
0.07
Concave
0.06
Convex
Strain

0.05
A B
0.04
Linear
0.03
0.02
Concave
0.01
0
-0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(b) Inner surface of workpiece

Figure 4.13 Maximum in-plane principal strain (radial strain) after 1st forward pass

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

0.01
0
A B Convex &
-0.01
Concave
-0.02
Strain Convex
-0.03
-0.04
-0.05 Linear

-0.06
-0.07 Concave

-0.08
-0.09
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(a) Outer surface of workpiece

0.01
0
A B Convex &
-0.01
Concave
-0.02
Convex
Strain

-0.03
-0.04
-0.05 Linear

-0.06
-0.07 Concave

-0.08
-0.09
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(b) Inner surface of workpiece

Figure 4.14 Minimum in-plane principal strain (tangential strain) after 1st forward pass

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

0.03
0.02
0.01 Convex &
Concave
0
Strain -0.01 A B Convex

-0.02
-0.03 Linear
-0.04
-0.05 Concave
-0.06
-0.07
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(a) Outer surface of workpiece

0.03
0.02
0.01 Convex &
Concave
0
A B
Strain

-0.01 Convex

-0.02
-0.03 Linear
-0.04
-0.05 Concave
-0.06
-0.07
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised Radial Distance

(b) Inner surface of workpiece

Figure 4.15 Out-of-plane principal strain (thickness strain) after 1st forward pass

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 Chapter 4 Effects of Roller Path Profiles on Material Deformation

4.4 Summary and Conclusion

In this chapter, four different roller path profiles, i.e. combined concave and convex,

convex, linear and concave curves, have been designed by using spinning CAM

software - OPUS. The 3-D elastic-plastic FE models of metal spinning are developed by

using FE software - Abaqus. The FE models have been verified by carrying out mesh

convergence study, assessing scaling methods, and also by comparing the dimensional

results. Based on the experimental investigation and FE analysis of the conventional

spinning process using the various roller path profiles, the following conclusions may be

drawn:

a) Both mass scaling and load scaling methods are able to significantly speed up the

FE simulation of a conventional spinning process. Using a mass scaling factor of f 2

or a load rate scaling factor of f would speed up the spinning FE simulation by a

factor of f.

b) FE analysis results indicate that the concave roller path produces the highest tool

forces amongst the four different roller path designs. The lowest tool forces are

generally observed when the convex roller path is used.

c) Using the concave roller path tends to cause the highest reduction of the wall

thickness of the spun part, while the convex roller path helps to maintain the

original wall thickness. A greater curvature of the concave path would result in a

higher amount of wall thinning of the spun part.

d) High tensile radial strains and low compressive tangential strains have been

observed in the FE models. In addition, if a concave roller path is applied, much

higher tensile radial strains would be obtained, resulting in greater wall thinning.

e) Two pairs of oppositely directed radial bending effects have been observed in the

workpiece during the conventional spinning process.

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

5. Analysis of Material Deformation in Multi-pass

Conventional Spinning

In the current academic research, most published papers on conventional spinning are

based on mandrels with a simple linear profile, e.g. conical and cylindrical shapes.

Investigations on spinning are mainly limited to no more than three passes (mainly a

linear path). Nevertheless, in the industry application, a considerable amount of roller

passes have to be used in order to successfully spin a complicated product. Therefore,

it is essential to develop a method of generating multiple roller passes for a mandrel with

a nonlinear profile and to investigate the material deformation in the multi-pass spinning

process. In this chapter, to make the workpiece conform to the non-linear profile of a

mandrel, the tool compensation technique has been proposed and employed in the

CNC multiple roller passes design. The Taguchi method is used to design the

experimental runs and analyse the effects of three process parameters on the

dimensional variations of the spun parts. Furthermore, FE simulation is conducted to

investigate the variations of tool forces, stresses, wall thickness and strains in this

multi-pass spinning process.

5.1 Experimental Investigation

Based on the same experimental setup of Chapter 4, i.e. Figure 4.1, multiple roller

passes which can successfully complete the spun part are developed and used to

conduct the experiment. The blanks are again made of mild steel (DC01) with diameter

of 240 mm and thickness of 2 mm. In this section, the methodologies of tool

compensation and Taguchi experimental design are discussed.

5.1.1 Tool Compensation in CNC Programming

To make the workpiece conform to the nonlinear profile of the mandrel, tool

compensation has to be taken into account in the process of designing multiple roller

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

passes. By studying the case that the roller traces the contour of the mandrel, how to

set the tool compensation in the multi-pass design is investigated. As shown in Figure

5.1, Point A, i.e. the intersection point of tangential lines of the roller nose, represents

the point on the roller path generated by CNC programming. R is the radius of the round

part of the mandrel; r is the roller nose radius. Point o is the center of the round part of

the mandrel, which is also assigned to be the origin of the coordinate system. Assuming

that the coordinate of point A is (x, y), certain geometrical relationships among r, R, x and

y, can be obtained by analysing the right-angled triangle (oBC), where

oC = oF + FC = x + r (132)

CB = CE + EB = y + r (133)

oB = oD + DB = R + r (134)

Thus, according to the Pythagoras Theorem, we have

( x + r )2 + ( y + r )2 = ( R + r )2 (x>0, y>0) (135)

which is the CNC roller path when the roller traces the contour of mandrel. Therefore the

design of roller passes should be based on the modified mandrel geometry given in

equation (135) as shown in the blue dotted curve of Figure 5.1. This tool compensation

technique can also be extended to the design of roller passes for other mandrel profiles,

such as linear and concave curve, etc.

Figure 5.1 Tool compensation

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

By contrast, if the roller passes are developed based on the mandrel profile, rather than

the modified mandrel profile as shown in Figure 5.1, the workpiece cannot fully contact

with the mandrel, resulting in wrinkling failures due to unsupported flange. Figure 5.2

illustrates a multi-pass design which does not take the tool compensation into account;

and a severe wrinkling failure is observed on the corresponding experimental spun part.

Roller
passes Blank

Mandrel
profile

Figure 5.2 Multi-pass design and spun sample without tool compensation

As shown in Figure 5.3, by offsetting a desired clearance on the modified mandrel

profile, multiple CNC roller passes are designed, where six pairs of roller passes have

been used. Since the roller is not a perfect rigid body, to make the workpiece fully

conform to the mandrel, a clearance of 1.5 mm, which is slightly lower than the blank

thickness (2 mm), has been used in this study. The detailed information (coordinates,

time) of the roller passes is given in Appendix 1. Figure 5.4 shows four different stages

of this multi-pass spinning experiment in progress. Clearly, the roller progressively forms

the workpiece onto the mandrel.

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

Roller
passes

Blank

Mandrel
profile

Offset
mandrel
Clearance profile
between Modified
mandrel mandrel
and roller profile

Figure 5.3 Roller passes design using tool compensation

Figure 5.4 Spinning experiment in progress

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

5.1.2 Experimental Design by Taguchi Method

An L4 (23) orthogonal array of the Taguchi method has been used to generate the

experimental runs. The orthogonal array is a technique that only requires a fraction of

the full factorial experiment and provides sufficient information to determine the effects

of input factors by using Analysis of Means (ANOM). ANOM is an analytical process

which quantifies the mean response for each level of the input factor (Fowlkes and

Creveling, 1995). The term “orthogonal” refers to the balance of various combinations of

input factors so that no one factor contributes more or less weight in the experiment

than other factors.

The Taguchi method concerns the variation as well as the ability of a system to meet a

target. A term called Signal to Noise ratio (S/N) is used to measure the variability. In the

case where the aim is to achieve the larger the better value but with minimum variability,

the Signal to Noise ratio is defined as (Fowlkes and Creveling, 1995):

1 n 1
S / N = −10 log( ∑ )
n i =1 yi 2
(136)

Similarly when the aim is to achieve the smaller the better values but with minimum

variability, the ratio is defined as:

1 n 2
S / N = −10 log( ∑ yi )
n i =1
(137)

where n is the sample number used in the experiment and yi are the outputs from

different samples.

Three variables, i.e. spindle speed, feed rate and type of material, are considered as the

experimental input factors. Each input factor has two levels, as shown in Table 5.1. The

values of input factors are chosen based on the process design experience in Metal

Spinners Group Ltd. Two samples have been used for each experimental run, to

minimise the experimental errors. The depths, inside diameters and wall thickness

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

variations of the spun parts have been measured as the output factors. Figure 5.5

shows the measurement of the thickness and depth variations by using a probe

indicator and a depth gauge, respectively.

Table 5.1 Experimental input factors and levels

Level
Factor Code
1 2
Feed Rate (mm/min) F 300 900
Spindle Speed (rpm) S 500 1000
Material M Aluminum (1050H14) Mild Steel (DC01)

Figure 5.5 Experimental measurements

5.2 Experimental Results and Discussion

Table 5.2 illustrates the experimental runs and the corresponding dimensional results,

i.e. the average values of the depths, inside diameters and the minimum wall thickness

of the experimental spun parts. Figure 5.6 illustrates one spun part from each

experimental run, where crackling failure is observed near the opening of the cup in

experimental run 4. By comparing the experimental run 1 and 4, in which the blanks are

made of aluminum, it is clear that higher feed ratio (ratio of feed rate to spindle speed)

results in crackling failures, as the feed ratio in run 1 and run 4 are 0.6 mm/rev and 0.9

mm/rev, respectively. In addition, mild steel has much stronger ability to stand cracking

failures in metal spinning, since the feed ratio applied on mild steel (run 3) is as twice as

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

that used on aluminum (run 4), and yet no cracks take place in experimental run 3.

Software Minitab has been used to analyse the effects of input factors on the

dimensional output factors in this experiment.

Table 5.2 Experimental runs and dimensional results

Feed Wall
Sample Diameter Depth Material
Run F S M Ratio Thickness
Number (mm) (mm) Failure
(mm/rev) (mm)
1.1 173.7 1.47 66.90
1 1 1 1 0.6 None
1.2 173.6 1.45 67.13

2.1 173.6 1.50 63.32

2 1 2 2 0.3 None
2.2 173.7 1.55 63.24

3.1 173.3 1.60 61.08

3 2 1 2 1.8 None
3.2 173.4 1.65 61.28

4.1 173.9 0.60 75.69

4 2 2 1 0.9 Cracking
4.2 173.9 0.98 73.79

(a) Run 1 (b) Run 2 (c) Run 3 (d) Run 4

Figure 5.6 Experimental spun parts

5.2.1 Diameter of Spun Part

Figure 5.7 shows the main effects plot for diameter means and S/N ratios, respectively.

It has shown that in this experiment both the spindle speed (S) and material type (M)

have relatively strong impacts on the diameter means and variability of spun part, while

only minor effects have been found from the feed rate (F). Nevertheless, it is noticeable

that the maximum differences of the diameter values at different input levels are only 0.3

mm, indicating none of these three input factors have significant effects on the diameter

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

of the spun part in this experiment.

Main Effects Plot for Diameter Means

Data Means

F S
173.8
Mean of Diameter Means (mm)
173.7

173.6

173.5
1 2 1 2
M
173.8

173.7

173.6

173.5
1 2

Main Effects Plot for Diameter SN ratios

Data Means

F S
-44.785

-44.790
Mean of SN ratios

-44.795

-44.800
1 2 1 2
M
-44.785

-44.790

-44.795

-44.800
1 2
Signal-to-noise: Smaller is better

Figure 5.7 Main effects plot for diameter

5.2.2 Thickness of Spun Part

As can be seen from Figure 5.8, the type of material (M) has slightly higher effects on

the mean value and variability of wall thickness than the effects from spindle speed (S)

and feed rate (F). It is clear that using a relatively “soft” material, i.e. aluminum,

produces a lower wall thickness value. In addition, a thinner workpiece is also obtained

when a high spindle speed is used, as agreed with Wang et al. (2010). Nevertheless, it

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

has been shown that in this experiment a higher level of feed rate leads to a thinner

workpiece according to the response table of the mean thickness. This may be

explained by the fact that severe thinning take places near the crack of sample 4,

resulting in a low mean value of thickness. However, it has been reported that in the

case of crack-free spinning process, using a higher feed ratio can help to maintain the

original wall thickness (Pell, 2009). This conclusion is also confirmed by comparing the

thickness results of experimental run 2 and 3 shown in Table 5.2, where a greater wall

thickness is achieved when a higher feed ratio is used on the mild steel samples.

Main Effects Plot for Thickness Means

Data Means
F S
Mean of Thickness Means (mm)

1.5
1.4
1.3
1.2
1.1
1 2 1 2
M
1.5
1.4
1.3
1.2
1.1
1 2

Main Effects Plot for Thickness SN ratios

Data Means

F S
4
3
Mean of SN ratios

2
1
0
1 2 1 2
M
4
3
2
1
0
1 2
Signal-to-noise: Larger is better

Figure 5.8 Main effects plot for thickness

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

5.2.3 Depth of Spun Part

As illustrated in Figure 5.9, the type of the material (M) has the most significant effects

on the mean value as well as the variability of the depth of the spun part, followed by the

spindle speed (S), and the feed rate (F). Due to the volume constancy of the material

during the spinning process, the depth has a converse change in relation to the

thickness change. Hence, the input factors have the opposite effects on the depth than

the thickness.

Main Effects Plot for Depth Means

Data Means
F S
70
Mean of Depth Means (mm)

68
66
64
62
1 2 1 2
M
70
68
66
64
62
1 2

Main Effects Plot for Depth SN ratios

Data Means
F S
-36.00
-36.25
-36.50
Mean of SN ratios

-36.75
-37.00
1 2 1 2
M
-36.00
-36.25
-36.50
-36.75
-37.00
1 2
Signal-to-noise: Smaller is better

Figure 5.9 Main effects plot for depth

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

5.3 Finite Element Simulation

5.3.1 Development of Finite Element Models

By using the same techniques of FE simulation discussed in Section 4.2, a FE model of

the experimental run 3 has been developed and used to analyse the material

deformation in multi-pass conventional spinning. Figure 5.10 shows eight stages of the

spinning process in FE simulation. Clearly, the roller forms the workpiece progressively

onto the mandrel by six pairs of roller passes.

(a) Beginning of the spinning process (b) End of the 1st forward pass

(c) Beginning of 2nd forward pass (d) End of the 2nd forward pass

(e) End of 3rd forward pass (f) End of the 4th forward pass

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

(g) End of 5th forward pass (h) End of the 6th forward pass

Figure 5.10 Spinning process using off-line designed roller passes

5.3.2 Verification of Finite Element Models

In this chapter, FE models have been verified by three approaches: at first, the

dimensional results of the FE model have been compared with the corresponding

experimental measurements. As shown in Table 5.3, the errors of the depth and

diameter are 3.41% and 0.38%, respectively, indicating that the FE analysis results are

in agreement with the experimental results. According to Figure 5.11, the maximum

difference in thickness between the FE simulation and experimental results is 0.3 mm

(15%). The relatively high error in thickness may be the result of: 1) thickness variations

of the raw metal blank; 2) errors in measurement of the thin workpiece; 3) wear of the

roller nose causing errors in the tool compensation of the CNC roller passes design; 4)

in the current FE simulation it is assumed that the coefficient of friction between roller

and workpiece is constant, while in reality the coefficient of friction may decay from the

assumed constant value after “sliding” takes place (Abaqus analysis user’s manual,

2008), resulting in errors accumulating through the FE simulation of the multi-pass

spinning process.

Table 5.3 Comparison of depth and diameter FEA vs. experimental results

Mean experimental results FEA results Errors

Depth (mm) 61.18 59.09 3.41%

Diameter (mm) 173.35 174.01 0.38%

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

2.40

2.00

Thickness (mm)
1.60

1.20 FEA

Exp

0.80

0.40

0.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

Figure 5.11 Comparison of experimental and FE analysis results of wall thickness

Secondly, FE models have also been verified by investigating the energy ratios of the

spinning process (Abaqus analysis user’s manual, 2008). As shown in Figure 5.12, at

the beginning of the spinning process, the rotation of the workpiece dominates, thus the

ratio of the kinetic energy to the internal energy of the workpiece is extremely high.

However, this ratio decreases gradually throughout the spinning process, resulting from

the increasing degree of the plastic deformation of the workpiece. Clearly, during more

than 2/3 time period of the spinning process, the ratio of the kinetic energy to the

internal energy of the workpiece is below 10%, indicating that the inertia effects due to

mass scaling do not significantly affect the simulation results. In addition, the ratio of the

artificial strain energy to the internal energy of the workpiece is below 1% throughout the

spinning process. Thus it is believed that the “hourglassing” problem is well controlled

and would not affect the simulation accuracy.

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

100%
90%
Kinetic
80% Energy
to
70%
Internal

Energy Ratio
60% Energy
50%
Artificial
40% Strain
Energy
30%
to
20% Internal
Energy
10%
0%
0 5 10 15 20 25 30
Time (s)

Figure 5.12 Evaluation of energy ratios in FE model

Thirdly, it has been shown that the FE simulation technique used in this study is able to

produce accurate results of tool forces (Long et al., 2011), by comparing the tool forces

from the FE analysis with the corresponding tool forces measured in a 3-pass spinning

experiment. As shown in Figure 5.13, a good agreement of axial forces is achieved

between the experiment and FE simulation.

1000

-1000
Exp
Force (N)

-2000

-3000 FEA

-4000

-5000
0.0 5.0 10.0 15.0 20.0 25.0

Time (s)

Figure 5.13 Comparison of experimental and FE analysis results of axial forces (Long et al., 2011)

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

5.4 Finite Element Analysis Results and Discussion

In this section, to gain insight into the material deformation of the multi-pass

conventional spinning process, the variations of tool forces, stresses, wall thickness and

strains in the FE model of experimental run 3 have been analysed.

5.4.1 Tool Forces

The history of three tool force components has been plotted in Figure 5.14, which

clearly demonstrates six stages, representing the six pairs of roller passes in the

spinning process. Figure 5.14 shows that the axial force dominates and increases

gradually in the first 3 forward passes, since at the beginning of the spinning process

the workpiece is mainly subjected to bending effects, as demonstrated in Figure

5.10(a)-(e). By contrast, the radial force increases dramatically over the six roller passes,

as a result of that the roller gradually compresses the workpiece onto the mandrel,

especially during the last 3 passes shown in Figure 5.10(f)-(h). Furthermore, the

tangential force is much lower than the axial and radial forces during the spinning

process, as also reported in the experimental investigations of multi-pass spinning

(Jagger, 2010, Wang et al., 1989).

1000

-1000

-2000 Radial
Force
Force (N)

-3000 Axial
Force
-4000
Tangential
Force
-5000

-6000

-7000

-8000
0 5 10 15 20 25 30

Time (s)

Figure 5.14 History of tool forces of FE simulation

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

Table 5.4 shows the maximum tool forces and their ratios of this FE model over six roller

passes. It is noticeable that the ratios of the maximum axial force to the maximum

tangential force decrease from 25 to 11 through this multi-pass spinning process. By

contrast, the ratios of the maximum radial force to the maximum tangential force

increase from 4 to 18 from the first to the sixth roller pass.

Table 5.4 Ratios of maximum tool forces of FE model

Fa Fr Ft
Pass Number Ratios of Force
(N) (N) (N)

1 3661.7 559.0 146.1 Fa : Fr : Ft = 25: 4: 1

2 5369.1 1635.4 249.6 Fa : Fr : Ft = 22: 7: 1

3 6133.2 2727.7 339.7 Fa : Fr : Ft = 18: 8: 1

4 4918.5 4359.2 334.0 Fa : Fr : Ft = 15: 13: 1

5 3757.5 4571.5 355.8 Fa : Fr : Ft = 11: 13: 1

6 2940.6 4954.3 277.2 Fa : Fr : Ft = 11: 18: 1

5.4.2 Stresses
The contours of radial and tangential stresses of the workpiece at the beginning of the

first forward roller pass have been plotted in Figure 5.15. Apparently, the stress

distribution is much more complicated than the theoretical stress distribution shown in

Figure 1.4.

As illustrated in Figure 5.15(a), Region A, the area between the roller contact and the

backplate clamped area, is under high tensile radial stress. Conversely, Region B,

which is between the roller contact point and the rim of the workpiece, is under

compressive radial stress. In addition, a ring zone of high compressive radial stresses

has been observed at Region C of the workpiece. It indicates that when Region A

rotates away from the local forming zone and enters Region C, the high tensile radial

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

stresses turn into high compressive radial stresses.

Figure 5.15(b) shows the distribution of tangential stress of workpiece at the beginning of

the first forward roller pass. Although the roller does not contact Region D yet, it has been

subjected to high compressive tangential stresses. As soon as Region D rotates away

from the local forming zone, the high compressive tangential stresses “recover” to low

tensile tangential stresses, as shown in the Region E of Figure 5.15(b). Moreover, the

areas close to the roller contact point along the circumferential direction of workpiece,

i.e. Region F, are under high tensile tangential stresses.

The distribution of radial and tangential stresses in the workpiece at the beginning of the

first backward roller pass is shown in Figure 5.16. It is clear that the area under roller

contact is subjected to compressive stresses in both radial and tangential directions, as

agreed with theoretical stress pattern shown in Figure 1.4. Region G, which is located

between the backplate clamped region and roller contact zone, as shown in Figure

5.16(a), is under high tensile radial stress.

As shown in Figure 5.16(b), a toothed pattern of tangential stresses is observed along

flange area of the workpiece, as also reported by Sebastiani et al. (2007), who claimed

that the toothed pattern may be a pre-state to wrinkling. However, in this study no

correlations have been observed between this toothed stress pattern and the wrinkling

failure. It may be explained by two reasons: at first, this toothed stress pattern is

originated from the high tensile tangential stresses. No high compressive tangential

stresses, which normally result in wrinkling failures, are observed except at the roller

contact area. Secondly, wrinkling generally occurs in forward passes when there is no

support to the flange and it is thus rare to have wrinkles in backward passes (Runge,

1994).

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

C
Unit: MPa

A B

Roller
Contact

(a) Radial stress

E
Unit: MPa

F
D

F
Roller
Contact

(b) Tangential stress

Figure 5.15 Variations of stresses at the beginning of 1st forward pass

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

Unit: MPa

Roller Contact

(a) Radial stress

Toothed
stress pattern

Unit: MPa

Roller Contact

(b) Tangential stress

Figure 5.16 Variations of stresses at the beginning of 1st backward pass

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

5.4.3 Wall Thickness

According to the FE analysis results, there is almost no thickness change after the

backward pass in the multi-pass conventional spinning. Figure 5.17(a) illustrates the

wall thickness variations after each forward roller pass of the FE model. Clearly, in this

conventional spinning process, the workpiece has been thinned gradually over these six

roller passes. From the first roller pass to the fifth roller pass, the workpiece is thinned

by approximately 3% after each pass, while in the last pass the wall thickness is

reduced by 4%. The total reduction of the wall thickness after six roller passes of this

spinning process is 19%. In addition, it has been observed that the highest thinning

zone of the workpiece has been shifted from the bottom of the cup to the cup opening

during this spinning process, as shown in Region R, S, and T of Figure 5.17(b), (c) and

(d), which are the wall thickness contours after the first, third and sixth roller pass,

respectively. This may indicate that materials flow towards the rim of blank through the

forward passes in the conventional spinning process.

2.05

2.00

1.95
R
Thickness (mm)

1.90 1st pass

2nd pass
1.85 S 3rd pass
4th pass
1.80
5th pass
1.75 6th pass

1.70
T
1.65

1.60
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised Radial Distance

(a) Wall thickness variations after each pass

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

Unit: mm

(b) Contour of wall thickness after 1st pass

Unit: mm

(c) Contour of wall thickness after 3rd pass

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

Unit: mm

(d) Contour of final wall thickness

Figure 5.17 Variations of wall thickness

5.4.4 Strains
Figure 5.18 shows the radial, tangential and thickness strains of the workpiece at

beginning of the first roller pass. It is noticeable that these strains distribute almost

uniformly in the circumferential direction but vary along the radial direction of the

workpiece. As shown in the Region J of Figure 5.18(a) and (b), a ring zone of

compressive radial and tangential strains has been observed between the roller contact

and the rim of the workpiece. Those compressive strains result in tensile thickness

strains in the corresponding ring zone, as shown in the Region J of Figure 5.18(c). This

slight thickening of the workpiece may result from the fact that material builds up in front

of the roller at the beginning of the first forward pass.

Conversely, tensile radial and tangential strains take place in a ring zone between the

backplate clamped area and roller contact area, as illustrated in the Region K of Figure

5.18(a) and (b). These tensile strains lead to compressive thickness strains in the

corresponding zone shown in Figure 5.18(c). Thus, it is clear that wall thickness

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

increases in Region J, while decreases in Region K. Nevertheless, according to the

thickness strain contour shown in Figure 5.18(c), the thickness strain of the thinning in

Region K is 3.6 times as the thickening in Region J.

K
J

z r
Roller Contact

(a) Radial strain

K
J

z r
Roller Contact

(b) Tangential strain

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

K
J

z r
Roller Contact

(c) Thickness strain

Figure 5.18 Variations of strains at the beginning of 1st pass

5.5 Summary and Conclusion

In this chapter, to make the workpiece conform to the non-linear profile of the mandrel,

the tool compensation technique is proposed and used in the CNC multiple roller

passes design. The Taguchi method has been employed to analyse the effects of

process parameters on the dimensional variations of the experimental samples. In order

to gain insight into the material deformation of the multi-pass conventional spinning

process, history of tool forces, distributions of stress and strains, and wall thickness

variations of a FE model have been analysed numerically. According to the results of the

experiment investigation and FE simulation, within the range of the process parameters

used in this chapter, the following conclusions may be drawn:

a) Experimental results indicate that the type of material has the most significant

effects on the variations of thickness and depth of the spun parts, followed by the

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 Chapter 5 Analysis of Material Deformation in Multi-pass Conventional Spinning

spindle speed and the feed rate. However, none of these input factors show

significant effects on the diameter variations of spun parts in this experiment.

b) A high feed ratio can help to maintain the original wall thickness but cracking

failures may take place if a large feed ratio is used. Comparing with aluminum, mild

steel is much “stronger” to withstand cracking failures in the conventional spinning

process, since a feed ratio of 1.8 mm/rev is applied on mild steel blanks while a

feed ratio of 0.9 mm/rev is used on aluminum blanks, and yet no cracks take place

on the mild steel blanks.

c) According to the FE analysis results of this spinning process, the axial force is the

highest and the tangential force is the lowest. The axial force increases in the first

three passes and decreases gradually in the last three passes. Conversely, the

radial force continues to increase throughout the six roller passes.

d) Stress analysis by FE simulations shows that in the forward roller pass, high tensile

and compressive radial stresses take place behind and in front of the roller contact.

In the backward pass, a toothed tangential stress pattern has been noticed in the

flange area.

e) FE analysis results show that the wall thickness of the workpiece decreases

gradually after each forward roller pass. The thinnest zone on the workpiece has

been shifted from the bottom to the opening of the cup, while the spinning process

progresses.

f) Strains distribute almost uniformly in the circumferential direction but vary along the

radial direction of the workpiece. At the beginning of a forward pass, a ring zone of

tensile radial and tangential strains has been observed between the backplate

clamped area and the roller contact zone, resulting in compressive thickness

strains, i.e. wall thinning.

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 Chapter 6 Study on Wrinkling Failures

6. Study on Wrinkling Failures

In this chapter, wrinkling failure in conventional spinning has been studied by theoretical

analysis, experimental investigation and Finite Element simulation. The energy method

and two-directional plate buckling theory have been used to predict the critical condition

of wrinkling failure in conventional spinning. The severity of wrinkles is quantified by

calculating the standard deviation of the radial coordinates of element nodes on the

edge of the workpiece obtained from the FE models. A forming limit study for wrinkling

has been carried out and it shows there is a feed ratio limit beyond which wrinkling

failures will take place. It is believed that if the high compressive tangential stresses in

the local forming zone do not “recover” to tensile tangential stresses after roller contact,

wrinkling failure will occur. Furthermore, the computational performance of the solid and

shell elements in simulating the spinning process is examined.

6.1 Theoretical Analysis

It is believed that wrinkling in sheet metal forming is a local phenomenon, which

depends on the local curvatures and stress states (Hutchinson and Neale, 1985).

Hence, the region being investigated for flange wrinkling may be simplified as a

rectangular plate (Wang and Cao, 2000), as shown in Figure 6.1. The length of a

half-wave wrinkled flange in the tangential direction (x axis) is a; the width of the

wrinkled flange in the radial direction (y axis) is b. The following assumptions have been

made in order to simplify the theoretical analysis:

1) Shear stresses are neglected;

2) No dynamic effects are considered;

3) Wall thickness remains constant throughout the spinning process;

4) The loading of roller is simplified as a lateral concentrated load P, perpendicular

to the plate.

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 Chapter 6 Study on Wrinkling Failures

σt
x

t
P
a
σr

b y

Figure 6.1 Schematic of a buckled plate in flange region

6.1.1 Energy Method

According to the theoretical stress pattern in a metal spinning process under a forward

roller pass, it is assumed that the rectangular plate of the flange region is subjected to

compressive tangential stresses and tensile radial stresses. To determine the critical

condition of wrinkling failures in the spinning process, an energy method has been

employed. In the wrinkling of metal spinning due to lateral collapse, four main energies

generated in the plate of the flange region have been taken into account:

Eb, the energy due to bending in the wrinkled flange;

Et, the energy due to circumferential shortening of the flange under compressive

tangential stresses;

Er, the energy due to radial elongation of the flange under tensile radial stresses;

El, the energy due to displacement caused by lateral concentrated load from the

roller.

The critical condition reaches when

Er + Eb = Et + El (138)
where Er and Eb are the energies which tend to restore the equilibrium, while Et,

and El are the energies that lead to wrinkling failures.

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 Chapter 6 Study on Wrinkling Failures

6.1.2 Theoretical Model

Assuming that the buckled deflection surface, as illustrated by the dotted curves in

Figure 6.1, is given by the equation (Timoshenko, 1936):

πx πy
w = γ sin sin (139)
a b
where γ is the maximum deflection of the buckling surface.

The energy due to bending can be calculated by (Timoshenko, 1936):

1 b a ⎧⎪⎛ ∂ 2 w ∂ 2 w ⎞ ⎡ ∂ 2 w ∂ 2 w ⎛ ∂ 2 w ⎞ 2 ⎤ ⎫⎪
2

Eb = D ∫ ∫ ⎨⎜⎜ 2 + 2 ⎟⎟ − 2(1 − ν )⎢ 2 −⎜ ⎟ ⎥ ⎬dxdy (140)

2 0 0 ⎪⎝ ∂x ∂y ⎠ ⎢ ∂x ∂y 2 ⎜⎝ ∂x∂y ⎟⎠ ⎥ ⎪
⎩ ⎣ ⎦⎭
Substituting (139) into (140),
2
ab ⎛π 2 π 2 ⎞
Eb = Dγ 2 ⎜⎜ 2 + 2 ⎟⎟ (141)
8 ⎝a b ⎠

where D is the flexural rigidity of plate

Et 3
D=
(
12 1 − ν 2 ) (142)

where E is the Young’s Modulus, t is the wall thickness, v is the Poisson’s ratio. It has

been shown that the elastic bending theory can be extended to cover plastic bending by

replacing E with the reduced modulus E0 (Gere, 2001):

4 EE p
E0 =
( )
2
(143)
E + Ep

where Ep is the slope of the stress-strain curve at a particular value of strain in the

plastic region.

The energy due to tangential stress σ t can be calculated by

1 a b ⎛ ∂w ⎞
2

Et = ∫ ∫ σ t t ⎜ ⎟ dxdy (144)
2 0 0 ⎝ ∂x ⎠
Substituting (139) into (144), we obtain

bγ 2π 2 tσ t
Et = (145)
8a

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 Chapter 6 Study on Wrinkling Failures

The energy due to radial stress σ r can be calculated by

2
1 b a ⎛ ∂w ⎞
E r = ∫ ∫ σ r t ⎜⎜ ⎟⎟ dydx (146)
2 0 0 ⎝ ∂y ⎠
Substituting (139) into (146), we obtain

aγ 2π 2 tσ r
Er = (147)
8b

According to Timoshenko and Woinowsky-Krieger (1959), the deflection resulting from a

concentrated load P can be expressed as

Pb 2
w =α (148)
D

where α is a numerical factor, the value of which depends on the ratio of a/b and is given

in Table 6.1.

Table 6.1 Factor α for deflection equation (Timoshenko and Woinowsky-Krieger, 1959)

a/b 1.0 1.1 1.2 1.4 1.6 1.8 2.0 3 ∞

α 0.01160 0.01265 0.01353 0.01484 0.01570 0.01620 0.01651 0.01690 0.01695

The lateral energy El due to the concentrated loading P can be calculated as:
γ
Dγ 2
El = ∫ Pdw = (149)
0
2αb 2

Substituting (141), (145), (147) and (149) into (138), we have

2
⎛ π 2 π 2 ⎞ π 2t π 2t 4D
D⎜ 2 + 2 ⎟ + 2 σ r = 2 σ t +
⎜ ⎟ (150)
⎝a b ⎠ b a αab3

π 2t
Dividing Equation (150) by and introducing notations σ e and λ , where
a2

π 2D
σe = (151)
a 2t
a
λ= (152)
b

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 Chapter 6 Study on Wrinkling Failures

The critical condition of wrinkling can be expressed as:

⎡ 4λ3 ⎤
σ e ⎢(1 + λ2 ) − = σ t − σ r λ2
2

απ 4 ⎥⎦
(153)
⎣
It is clear that in this theoretical model, the critical tangential stress and radial stress of

wrinkling depend on the geometry of the half-wave wrinkled plate and material

properties of the blank.

6.2 Experimental Investigation

6.2.1 Experimental Setup

An experiment has been carried out to study the wrinkling failures in the conventional

spinning process. Figure 6.2 shows the setup of the spinning experiment and its

schematic diagram, where the angle between the roller axis and the mandrel axis is 45°.

The blank is made of mild steel (DC01). The thickness and diameter of the blank are 1.2

mm and 120 mm, respectively.

a) Experimental setup

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 Chapter 6 Study on Wrinkling Failures

b) Schematic diagram

Figure 6.2 Spinning experiment of wrinkling investigation

6.2.2 Process Parameters

Figure 6.3 illustrates the roller passes used in the experiment, clearly only the two

forward passes are effective as the backward pass does not deform the blank. Table 6.2

shows the process parameters used in four experimental runs. In an experimental run,

the feed rate in the axial direction of the roller (z-axis in Figure 6.2b) is almost constant,

while the feed rate in the radial direction of the roller (x-axis in Figure 6.2b) changes with

time. 300% and 600% of the initial feed rate used in experimental run E1 are applied to

experimental run E2 and E3, respectively. Only the first pass of experimental run E3 is

used in experimental run E4.

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 Chapter 6 Study on Wrinkling Failures

5
4
3
2
1st forward
X (mm) 1 pass
0 Backward
pass
-1
2nd forward
-2 pass
-3
X
-4
-5
-30 -25 -20 -15 -10 -5 0
Y
Y (mm)

Figure 6.3 Roller passes used in the experiment

Table 6.2 Process parameters of experimental runs

Spindle Speed Longitudinal Feed Rate (z-axis)

Sample Number
(rpm) (mm/min)
E1 400 212 (Initial feed rate)

E2 400 300% of Initial feed rate

E3 400 600% of Initial feed rate

600% of Initial feed rate
E4 400
(only the first roller pass is applied)

As shown in Figure 6.4(a), sample E1 has no wrinkles with relatively smooth surface.

Increasing the initial feed rate by 300% and 600%, the surface of the sample E2 and E3

become rough accordingly, as shown in Figure 6.4(b) and 6.4(c). In addition, by

comparing the experimental spun part E3 and E4 of Figure 6.4(c) and 6.4(d), it is clear

that wrinkling occurs in the first pass, and then smoothed out during the second pass.

However, the high feed rate of the roller pass leads to extremely rough surface finish,

which also reported by Chen et al. (2001).

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 Chapter 6 Study on Wrinkling Failures

(a) E1 (b) E2

(c) E3 (d) E4

Figure 6.4 Experimental samples

6.3 Finite Element Simulation

Since the wrinkling failures only take place in the first forward roller pass of this

experiment, the backward pass and the second forward pass are neglected in the FE

simulation. The information of the roller pass has been given in the Appendix 2. Table

6.3 presents the process parameters of some of the FE models and the corresponding

flange states. The mass scaling technique is used in these FE models to speed up the

computation. A mass scaling factor of 25 has been used in Model 1 - 6, in which the

spindle speed varies between 400 rpm to 800 rpm. Conversely, no mass scaling is used

in Model 7, which applies a significantly high spindle speed – 1800 rpm, in order to

prevent the inertial effects due to mass scaling. The first three models (Model 1 - Model

3) use the same setting as experimental sample E1, E2 and E4.

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 Chapter 6 Study on Wrinkling Failures

Table 6.3 FE analysis process parameters and flange state of spun part

Replay Feed Spindle Mass Feed Process

Flange
Model feed rate speed scaling ratio duration
state
rate (mm/min) (rpm) factor (mm/r) (s)
Model 1 100% 212 400 25 0.53 5.44 No wrinkles
Model 2 300% 636 400 25 1.59 1.81 Wrinkling
Model 3 600% 1272 400 25 3.18 0.91 Wrinkling
Model 4 800% 1696 800 25 2.12 0.68 Wrinkling
Model 5 200% 424 600 25 0.71 2.72 No wrinkles
Model 6 600% 1272 800 25 1.59 0.91 Wrinkling
Model 7 600% 1272 1800 1 0.71 0.91 No wrinkles

6.3.1 Element Selection

To evaluate the performance of the 8 noded reduced integration linear solid element

(C3D8R) and the 8 noded reduced integration linear continuum shell element (SC8R)

for the wrinkling simulation of the spinning process, three FE models have been

compared using the experimental setting of sample E4. Detailed meshing information is

shown in Table 6.4, where Model 3a and 3b applied the same process parameters as

Model 3 defined in Table 6.3, but with different element types and number of elements

through thickness direction of the blank.

Table 6.4 FE models using different types and numbers of elements

Elements Number Stable CPU

Type of Flange
Model through of increment time
element state
thickness element (s) (h:m:s)
Model No
C3D8R 1 5200 4.71e-7 08:51:12
3a wrinkles
Wrinkling Model Minor
C3D8R 4 20800 1.37e-7 48:09:38
Model 3b wrinkles
Model Severe
SC8R 1 5200 5.66e-7 14:18:07
3 wrinkles
Model No
C3D8R 1 5200 5.12e-7 13:14:48
5a wrinkles
Wrinkle
Model No
-free C3D8R 4 20800 2.12e-7 62:55:34
5b wrinkles
Model
Model No
SC8R 1 5200 5.66e-7 21:44:38
5 wrinkles

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 Chapter 6 Study on Wrinkling Failures

Figure 6.5 compares the deformed workpieces of these FE models with the

corresponding experimental sample. As shown in Figure 6.5(a), although an extremely

fine mesh has been used, Model 3a, which uses one single layer of solid elements in

the thickness direction, cannot capture the wrinkling failure occurred in the experiment.

Using four layers of elements through the thickness direction slightly improves the

results, where minor wrinkling is observed on the deformed FE workpiece, as shown in

Figure 6.5(b). However, it is still unable to represent the real severe wrinkles of the

experiment sample as shown in Figure 6.5(d).

(a) Model 3a (b) Model 3b

Sample E4

(c) Model 3 (d) Experimental sample (E4)

Figure 6.5 Comparison of deformed workpiece using different types and numbers of elements

On the other hand, Model 3, which uses a single layer of continuum shell elements in

the thickness direction with nine integration points, produces much better results, as

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 Chapter 6 Study on Wrinkling Failures

shown in Figure 6.5(c). By increasing the number of the solid element layers through the

thickness direction could improve FE results, but it is computationally unfeasible to carry

out a spinning process simulation using a FE model with several element layers in the

thickness direction. Significantly long computing time is required not only due to a large

number of elements but also an extremely small element length which significantly

decreases the stability limit value of the explicit solution, according to equation (110). In

this study, it has shown that the computing time of Model 3b is almost four times of that

of Model 3, as details given in Table 6.3.

To further investigate the computing performance of the two element types on spinning

simulation not involving wrinkling, axial and radial forces obtained from wrinkle-free

models – Model 5a, 5b and 5 have been compared. As can be seen from Figure 6.6, the

tool forces of Model 5a, which uses one layer of solid elements, is about three times

higher than the corresponding values of Model 5 where one layer of continuum shell

elements is used. This has been confirmed by Long et al. (2011) that the axial force of

FE model using one layer solid elements is approximately 2.3 times of the

experimentally measured value. Long et al. (2011) also report that using one layer

continuum shell elements can produce accurate force results.

Additionally, by using four layers of the solid element in Model 5b, the force values are

significantly decreased, as a result of using four integration points through the thickness

direction. However, the forces are still higher than the corresponding values obtained

from Model 5, where the maximum difference is approximately 50%. The overestimation

of the tool forces by Model 5a and 5b may be resulted from the artificially introduced

hourglass control stiffness to limit the propagation of the “hourglassing” deformation

mode. It is therefore evident that the reduced integration linear solid element is not

suitable for the metal spinning simulation. Conversely, the reduced integration linear

continuum shell element is able to produce accurate FE analysis results such as

wrinkling and tool forces.

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 Chapter 6 Study on Wrinkling Failures

0
-500 Model 5a:
-1000 1-layer
C3D8R
-1500 element

Force (N)
-2000 Model 5b:
-2500 4-layer
C3D8R
-3000 element
-3500 Model 5:
-4000 1-layer
SC8R
-4500 element
-5000
0 0.5 1 1.5 2 2.5

Time (s)

(a) Axial forces

0
-500 Model 5a:
-1000 1-layer
C3D8R
-1500 element
Force (N)

-2000 Model 5b:

-2500 4-layer
C3D8R
-3000 element
-3500 Model 5:
-4000 1-layer
SC8R
-4500 element
-5000
0 0.5 1 1.5 2 2.5

Time (s)

(b) Radial forces

Figure 6.6 Force comparisons of wrinkle-free models using different types and numbers of elements

6.3.2 Verification of FE Models

The number of wrinkles and the height of the experimental spun cup have been

measured to verify the FE models. By comparing Figure 6.5(c) and Figure 6.5(d), there

are 20 wrinkles according to the FE results, whereas 24 wrinkles are observed in the

experimental sample. The difference may result from the fact that some of the minor

wrinkles on the experimental sample are difficult to be captured in the FE model. The

average height of the FE sample is about 13.8 mm, while approximately 14 mm has

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 Chapter 6 Study on Wrinkling Failures

been measured from the experimental sample. Therefore it is believed that the FE

analysis results are in agreement with the experimental results.

Figure 6.7 compares the ratios of the artificial strain energies to the internal energies

from the three wrinkle-free models – Model 5a, 5b and 5. It is clear that the energy ratio

is around 1% throughout the whole spinning process in Model 5 which uses one layer

continuum shell elements, demonstrating that “hourglassing” is not an issue in this

model. However, the energy ratio is extremely high in Model 5a with one layer of solid

elements. It shows that “hourglassing” is a major problem which could significantly

affect the results. By using 4 layers of solid elements in Model 5b, this energy ratio has

decreased dramatically but is still much higher than the energy ratio obtained from

Model 5.

0.9
0.8 Model 5a:
1-layer
0.7 C3D8R
0.6 element
Energy Ratio

0.5 Model 5b:

4-layer
0.4 C3D8R
element
0.3
Model 5:
0.2 1-layer
SC8R
0.1
element
0
0 0.5 1 1.5 2 2.5

Time (s)

Figure 6.7 Ratio of artificial strain energy to internal energy of the wrinkle-free models

6.4 Results and Discussion

In this section, an approach to quantify the severity of wrinkles is proposed. A forming

limit diagram is drawn based on results of a number of FE models using various feed

rates and spindle speeds. Furthermore, the variations of tool forces, stresses and wall

thickness have been studied numerically. The theoretical analysis model of wrinkling is

verified by comparing the theoretical stress state with the FE analysis stress results.

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 Chapter 6 Study on Wrinkling Failures

6.4.1 Severity of Wrinkle

Deformed workpieces obtained from Model 2, 4 and 5 defined in Table 6.2 are shown in

Figure 6.8, which suggests that with an increasing of feed ratio, wrinkling failure tends to

take place. Moreover, the severity of wrinkles increases accordingly when applying

higher feed ratios. In order to quantify the severity of wrinkles, the radial coordinates of

element nodes located on the edge of the deformed workpiece along the circumferential

direction are plotted in Figure 6.9.

(a) Model 5 (0.71 mm/rev) (b) Model 2 (1.59 mm/rev) (c) Model 4 (2.12 mm/rev)

Figure 6.8 Effects of roller feed ratio on wrinkling

Radial nodal coordinate of flange edge (mm)

54.5

53.5

52.5
Model 5
51.5 Model 2

50.5 Model 4

49.5

48.5

47.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalised circumferential distance

Figure 6.9 Severity of wrinkles of FE models

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 Chapter 6 Study on Wrinkling Failures

The mean, maximum, minimum and standard deviations of these radial coordinate

values, which illustrate various degrees of the wrinkles, are calculated by equation (154)

and (155) and shown in Table 6.5.

∑U i
U= i =1
(154)
N

1 N
s= ∑ （U i − U ) 2
N − 1 i =1
(155)

Where U is the mean value of these radial coordinate, Ui is the radial coordinate of

node i and N is the number of the nodes along the edge of workpiece in the FE model.

Table 6.5 Standard deviations of wrinkle amplitudes

Maximum Minimum Mean

Feed Standard
radial radial radial
Model ratio deviation
coordinate coordinate coordinate
(mm/rev) (mm)
(mm) (mm) (mm)
Model 5 0.71 51.645 51.304 51.515 0.068
Model 2 1.59 52.020 49.721 50.842 0.467
Model 4 2.12 53.791 48.965 51.132 1.179

As shown in Table 6.5, it is clear that when the feed ratio is 0.71 mm/rev, the standard

deviation of the radial coordinates of the element nodes on the edge of the deformed

workpiece is 0.068 mm, indicating no wrinkling takes place in Model 5, as confirmed in

Figure 6.8(a). However, by increasing the feed ratios to 1.59 mm/rev and 2.12 mm/rev,

the standard deviation of the radial coordinates increase to 0.467 mm and 1.179 mm,

respectively. It shows that the severity of the wrinkles increases exponentially when

increasing the feed ratio.

6.4.2 Forming Limit of Wrinkling

A formability study has been carried out by applying different combinations of the feed

rate and spindle speed in the FE models. A feed ratio of 0.71 mm/rev has been found to

be the forming limit of wrinkling for the spinning process considered, beyond which

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 Chapter 6 Study on Wrinkling Failures

wrinkling failures will take place. A forming limit diagram for wrinkling is illustrated in

Figure 6.10, which also indicates that the wrinkling tends to occur when using high feed

rates or low spindle speeds. In other words, using high feed ratio increases the

possibility of the wrinkling failure. This finding has also been confirmed by the

experimental investigations of both shear forming (Hayama et al., 1966) and one-pass

deep drawing spinning (Xia et al., 2005). The wrinkling zone and wrinkling-free zone is

approximately separated by a straight line, as shown in Figure 6.10. It suggests that as

long as keeping the feed ratio below the feed ratio limit, increasing the feed rate and

spindle speed proportionally, the wrinkling failure can be prevented, as agreed with

Hayama et al. (1966).

2000
1800
1600
Feed rate (mm/min)

1400
1200
1000 Wrinkling

800
No wrinkles
600
400
200
0
0 500 1000 1500 2000 2500
Spindle speed (rpm)

Figure 6.10 Forming limit diagram for wrinkling

However, the feed ratio limit obtained in Figure 6.10 is only valid for this specific

experimental setting. The feed ratio limit depends on a number of key parameters of the

spinning process, such as the blank material, blank thickness, blank diameter and roller

path etc. In this study, the effects of blank thickness have also been investigated. As

shown in Table 6.6, the feed ratio limit increases when using thicker blanks. It suggests

that the thicker the blank, the higher capability to stand the wrinkling failure, as also

reported by Kleiner et al. (2002).

130
 Chapter 6 Study on Wrinkling Failures

Table 6.6 Feed ratio limits of various thicknesses of blanks

Thickness of blank Feed ratio limit of wrinkling

(mm) (mm/rev)
1.2 0.71
1.6 1.06
2.0 1.88

6.4.3 Tool Forces

Figure 6.11 compares the tool force components of the two wrinkle-free models, i.e.

Model 5 and Model 7 as defined in Table 6.3, which use same feed ratio but with

different spindle speeds and feed rates. Clearly, the corresponding tool force

components of these two models are almost exactly the same. This supports the

assertion that as long as keeping the feed ratio constant, by changing the feed rate and

the spindle speed proportionally, there would be no significant effects on the final spun

product. Furthermore, during the spinning process the tangential force is the smallest

among three force components. In the initial stage of spinning, because the workpiece

is mainly subjected to bending effects, the axial force is greater than the radial force. At

the middle stage of the process, the axial force begins to decrease due to the remaining

flange decreases gradually; while the radial force continues to increase and peaks at

the end of the process. This may be a result of the roller forcing the blank towards the

mandrel at the final stage, as corresponding to the roller passes shown in Figure 6.3.

-200

-400
Force (N)

-600 Model 5

-800 Model 7

-1000

-1200

-1400
0 0.5 1 1.5 2 2.5
Time (s)

(a) Axial force

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 Chapter 6 Study on Wrinkling Failures

-200

-400

Force (N)
-600 Model 5

-800 Model 7

-1000

-1200

-1400
0 0.5 1 1.5 2 2.5
Time (s)

(b) Radial force

1400

1200

1000
Force (N)

800 Model 5
Model 7
600

400

200

0
0 0.5 1 1.5 2 2.5
Time (s)

(c) Tangential force

Figure 6.11 Force histories of wrinkle-free models (Model 5 and 7)

Figure 6.12 illustrates the force history of a wrinkling model – Model 4. Comparing with

the corresponding forces of wrinkle-free model shown in Figure 6.11, the magnitudes of

the forces are much higher at the second half stage of the spinning process, when

severe deformation takes place due to wrinkles appearing on the flange. In addition,

according to Figure 6.12, sudden changes and fluctuations of forces are clearly shown

around 0.4s. These sudden changes and fluctuations of tool forces may be resulted

from the existing wrinkles on the workpiece interacting with the roller. As shown in

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 Chapter 6 Study on Wrinkling Failures

Figure 6.13, wrinkling failure initiates at 0.34s and is extending to the whole flange

around 0.36s. Consequently, it is considered that these sudden changes and

fluctuations in tool forces may be used to determine the moment when wrinkling occurs,

as also reported by Arai (2003) and Hayama (1981).

-200

-400
Force (N)

-600

-800

-1000

-1200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time (s)

(a) Axial force

-500

-1000
Force (N)

-1500

-2000

-2500

-3000

-3500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time (s)

(b) Radial force

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 Chapter 6 Study on Wrinkling Failures

500

400

Force (N)
300

200

100

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time (s)

(c) Tangential force

Figure 6.12 Force histories of wrinkling model (Model 4)

6.4.4 Stresses
A distinct difference of the tangential stress distributions has been found between the

wrinkling model (Model 4) and the wrinkle-free model (Model 5). Figure 6.13 shows the

tangential stress variations through the wrinkling developing process in Model 4. Figure

6.14 shows the tangential stress contours of the wrinkle-free model – Model 5 at the

corresponding stages. According to Figure 6.13(a) and 6.14(a), at Stage 1 there is no

apparent difference of the tangential stress distributions between these two models. As

can be seen from Figure 6.13(b), wrinkles take place around processing time of 0.34s.

The compressive tangential stresses distribute not only in the roller contact zone but

also in other areas of the flange. At stage 3 shown in Figure 6.13(c), the compressive

tangential stresses locate regularly along the circumferential direction of workpiece and

more wrinkles are generated. Conversely, at the corresponding stages of the

wrinkling-free model illustrated in Figure 6.14(b) and 6.14(c), there are no compressive

tangential stresses observed on the flange area, except at the roller contact zone.

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 Chapter 6 Study on Wrinkling Failures

Unit: MPa Unit: MPa

Roller
contact

(a) Stage 1, time = 0.295 s (a) Stage 1 time=1.179s

Tangential
compressive
stress

Unit: MPa Unit: MPa

(b) Stage 2, time = 0.34s (b) Stage 2, time = 1.360s

Development of tangential
compressive stress

Unit: MPa Unit: MPa

(c) Stage 3, time = 0.363s (c) Stage 3, time = 1.451s

Figure 6.13 Tangential stress distribution Figure 6.14 Tangential stress distribution
of wrinkling model
Figure 6.13 (Model stress
Tangential 4) ofwrinkling
distribution of wrinkle-free model
model (Model
(Model 4) 5)
Figure 6.14 Tangential stress distribution of wrinkle-free model (Model 5)

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 Chapter 6 Study on Wrinkling Failures

It is believed that in the wrinkle-free model, the compressive tangential stresses

“recover” to tensile tangential stresses when the current contact area moves away from

the roller, as also observed in Figure 5.15(b). Conversely, in the wrinkling model the

compressive tangential stresses induced at the roller contact zone do not fully “recover”

to tensile tangential stresses after being deformed. This may be because the

compressive stresses at the roller contact zone are beyond the buckling stability limit,

resulting in some compressive tangential stresses remaining in the previous contact

zone thus leading to the wrinkling failure.

In order to verify the critical condition of wrinkling by theoretical analysis as given in

equation (153), stress distributions of a flange area where wrinkling initiates at have

been plotted in Figure 6.15, where the length in the tangential direction of flange area,

a=11.3 mm, while the width in the radial direction of flange area, b=10.4 mm, which

give λ =1.1 and α =0.01265 according to Table 6.1. Substituting the material and

dimensional parameters into Equations (142), (151), and (152), the left side in equation

(153) can be calculated as:

⎡ 4λ3 ⎤
σ e ⎢(1 + λ2 ) −
2
= 47.2MPa
απ 4 ⎥⎦
(156)
⎣
By using the query function of Abaqus, radial and tangential stresses in the elements at

the edges of this flange area have been output. It has been observed that the value of

tangential stress at the edge along the radial direction of the flange area varies between

72 MPa and 225 MPa, while the value of radial stress at the edge along the tangential

direction of the flange area ranges from 134 MPa to 161 MPa. Thus the maximum value

of the right side in equation (153) obtained by substituting the highest tangential stress

and the lowest radial stress:

σ t − σ r λ2 = 54.2MPa (157)

Clearly, the result of theoretical analysis obtained in (156) is in agreement with the result

of FE simulation given by (157). However, this theoretical analysis model has its

limitations: At first, since the theoretical model in (153) use the tangential and radial

stresses to predict the critical condition of wrinkling failure, it would be quite difficult to

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 Chapter 6 Study on Wrinkling Failures

use the model in the spinning industry, where the stress state of a blank cannot be

determined without conducting numerical analysis such as FE simulation. Secondly, in

the theoretical analysis the radial and tangential stresses are assumed to be uniform

along the edges of the flange area, while FE simulation indicates that the distributions of

stresses on the edges of the flange region are much more complicated as shown in

Figure 6.15.

z r Roller
Contact

(a) Radial stress

z r Roller
Contact

(b) Tangential stress

Figure 6.15 Stress distributions in flange at wrinkling zone (Model 4)

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 Chapter 6 Study on Wrinkling Failures

6.4.5 Thickness
Figure 6.16 illustrates the effects of feed ratio on the wall thickness distribution of the

spun cylindrical cup. Clearly, less thinning of the wall thickness takes place if a high feed

ratio is applied. This finding agrees with Runge (1994) who suggests that lower feed

ratios produce excessive material flow to the edge of the workpiece and unduly thin the

wall thickness. Shearing between the roller and workpiece due to frictional effects may

be one of the main reasons of the material thinning. Considering the roller feeds the

same distance during the spinning process, when using a lower feed ratio, the roller will

scan the workpiece with more revolutions, thus leading to higher shearing effects than

spinning at a high feed ratio.

1.25

Model 5
1.20
(Feed ratio
Thickness (mm)

0.71mm/r)
1.15 Model 6
(Feed ratio
1.59mm/r)
1.10 Model 4
(Feed ratio
2.12mm/r)
1.05

1.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized Radial Distance

Figure 6.16 Wall thickness distributions at different feed ratios

It is clear that in order to maintain the original blank thickness unchanged, high feed

ratios should be used. However, high feed ratios could also lead to rough surface finish

and material failures. Figure 6.17 shows the effects of feed ratio on the quality of spun

part. It may be necessary to find a “trade-off” feed ratio, which not only can help to

maintain the original blank thickness unchanged but also prevent the material failures

and produce good surface finish.

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 Chapter 6 Study on Wrinkling Failures

Feed
High Feed Ratio Zone
Rate
Wrinkling Failure
Cracking Failure “Trade-off” Feed

High Tool Forces Ratio Zone

Low Feed Ratio Zone

Poor Thickness Uniformity
Good Surface Finish
Low Production Efficiency

Spindle Speed

Figure 6.17 Effects of feed ratio in blank metal spinning

6.5 Summary and Conclusion

In this chapter, wrinkling failure in conventional spinning has been studied by theoretical

analysis, experimental investigation and Finite Element simulation. The following

conclusions may be drawn:

a) The energy method and two-directional plate buckling theory have been used to

predict the critical condition of wrinkling failure in the conventional spinning.

However, theoretical analysis has shown considerable limitations in comparing with

FE simulation as the stress state of the blank during spinning is far more

complicated than the simplified assumptions made in the theoretical analysis.

b) This study has shown that the reduced integration linear solid element is not

suitable for sheet metal spinning simulation, due to the “hourglassing” problem it

suffered from. Conversely, the reduced integration linear continuum shell element,

which can produce accurate FE analysis results such as wrinkling and tool forces,

should be used to simulate the spinning process.

c) The severity of the wrinkles is quantified by calculating the standard deviation of the

radial coordinates of element nodes on the edge of the spun cup in the FE models.

The results have shown that the severity of the wrinkles increases exponentially

when increasing the feed ratio.

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 Chapter 6 Study on Wrinkling Failures

d) A forming limit study of wrinkling indicates that there is a feed ratio limit beyond

which wrinkling failures will occur. Increasing the feed rate and spindle speed

proportionally, there won’t be much effect on the conventional spinning process.

e) If the high compressive tangential stresses in the local forming zone do not

“recover” to tensile tangential stresses after roller contact, wrinkling failure will

occur.

f) Sudden changes and fluctuations of the tool forces, resulted from existing wrinkles

on the workpiece interacting with the roller, could be used to determine the

approximate moment of wrinkling occurrence.

g) High feed ratios help to maintain the original blank thickness unchanged. However,

high feed ratios also lead to material failures and rough surface finish. It may be

necessary to find a “trade-off” feed ratio zone in the spinning process design.

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 Chapter 7 Conclusion and Future Work

7. Conclusion and Future Work

This chapter summarises the key conclusions on material deformation and wrinkling

failures of conventional spinning drawn from this study. Future research trends of sheet

metal spinning process are also outlined.

7.1 Conclusion

In this study, numerical analysis and experimental investigation have been carried out to

study the material deformation and wrinkling failures in the conventional spinning

process. Key conclusions on six main aspects of this research work are summarised in

the following:

1) Finite Element Simulation

3D elastic-plastic FE simulation of conventional spinning has been carried out under FE

software platform – Abaqus. The computing performances of scaling methods and

element types in the simulation of spinning have been evaluated, respectively. FE

analysis results indicate that both mass scaling and load rate scaling methods with a

suitable scaling factor can produce accurate numerical results, by comparing with a FE

model that does not employ any scaling method. Using a mass scaling factor of f 2 or a

load rate scaling factor of f would speed up the spinning FE simulation by a factor of f. In

addition, FE analysis results also illustrate that the reduced integration linear solid

element is not suitable for the metal spinning simulation, due to the “hourglassing”

problem it suffered from. Conversely, it is believed that the reduced integration linear

continuum shell element, which can produce accurate FE analysis results such as

wrinkling and tool forces, should be used to simulate the sheet metal spinning process.

2) Experiment Investigation

In order to make the workpiece conform to the nonlinear profile of the mandrel, the tool

compensation technique has been investigated and applied in the process of designing

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 Chapter 7 Conclusion and Future Work

multiple roller passes. The tool compensation method is developed based on the

geometry relationship between the roller and the round part of mandrel, when using a

roller to trace the mandrel contour. It has been reported that without taking tool

compensation into account, wrinkling failures may take place on the workpiece due to

unsupported flange. In addition, The Taguchi method has been used to design the

experiment and to study the effects of process parameters on the dimensional variations

of experimental samples. Experimental investigation is also employed in the analysis of

wrinkling failures in the conventional spinning process.

3) Theoretical Analysis of Wrinkling

By using the energy method and two-dimensional plate buckling theory, a theoretical

analysis model to predict the critical condition of wrinkling in the conventional spinning

process has been developed. It shows that the radial and tangential stresses at the

critical condition of wrinkling depend on the geometry of the half-wave wrinkled flange

and the material properties of the blank. Nevertheless, theoretical analysis has shown

considerable limitations in comparing with FE simulation as the stress state of the blank

during spinning is far more complicated than the simplified assumptions on the stresses

of the theoretical model.

4) Material Deformation

To gain insight into the material deformation of conventional spinning, variations of tool

forces, stresses, and strains have been analysed numerically. Axial force dominates at

the beginning of the conventional spinning when the workpiece is mainly subjected to

bending. Radial force increases gradually as roller forms workpeice onto mandrel.

Tangential force keeps almost constant during the conventional spinning process and is

the smallest forces among three force components. Stress analysis shows that in the

forward roller pass, high tensile and compressive radial stresses take place behind and

in front of the roller contact. Two pairs of oppositely directed radial bending effects have

been observed in the workpiece. High tensile radial strains and low compressive

tangential strains are also noticed in the FE models.

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 Chapter 7 Conclusion and Future Work

5) Wrinkling Failures

Wrinkling failure in the conventional spinning process has been studied by using both

numerical and experimental approaches. Compressive tangential stresses are

observed at the flange area near the local forming zone but these will change into

tensile tangential stresses after roller contact in the wrinkle-free models. However, if the

compressive tangential stresses do not fully “recover” to tensile tangential stresses,

wrinkling failure will take place. The severity of the wrinkles is quantified by calculating

the standard deviation of the radial coordinates of element nodes on the edge of spun

cup in the FE models. A forming limit study of wrinkling indicates that there is a feed

ratio limit beyond which wrinkling failures will occur. Increasing the feed rate and spindle

speed proportionally, there will not be much effect on the conventional spinning process.

Sudden changes and fluctuations of the tool forces, resulted from existing wrinkles on

the workpiece interacting with the roller, could be used to determine the approximate

moment of wrinkling occurrence.

6) Process Parameters

According to the experimental results analysis by the Taguchi method, the type of

material has the most significant effects on the thickness and depth variations of spun

parts, followed by the spindle speed and the feed rate. Nevertheless, none of these

input factors show significant effects on the diameter variations. Among four different

roller path designs, i.e. combined concave and convex, convex, linear, and concave

curves, the concave roller path produces the highest tool forces, stresses and deduction

of wall thickness. A greater curvature of the concave path, a higher amount of wall

thinning would take place. On the other hand, lowest tool forces, stresses and wall

thinning are obtained when using the convex roller path. Furthermore, it has been

shown that high feed ratios help to maintain the original blank thickness. However, high

feed ratios also lead to material failures and rough surface finish. It is thus necessary to

find a “trade-off” feed ratio zone in the spinning process design.

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 Chapter 7 Conclusion and Future Work

7.2 Future Work

In this section, four future areas of investigation on the sheet metal spinning process are

identified:

a) FE Simulation of Cracking Failures in Spinning

Up to now, there is no research being conducted on the FE simulation of cracking

failures in the conventional spinning process, although cracking failures also

significantly affects the production efficiency and product quality. Hence, it is believed

that applying FE analysis method to predict and prevent cracking failures in the

conventional spinning is beneficial to the industrial spinning process design. Developing

a suitable material damage model, e.g. defining damage initiation criteria and damage

evolution law, may be one of the key aspects in the FE simulation of cracking failures in

the sheet metal spinning process.

b) Experimental Design and Optimisation

To further analyse the effects of process parameters and their interactions on the quality

of spun part, future experimental design of metal spinning should attempt to take more

input factors at various levels into account, such as the tooling parameters shown in

Figure 1.6. In addition, future work by employing the regression modelling or response

surface methods may be essential to predict the relationships amongst process

parameters and optimise the outputs of metal spinning process.

c) Innovation of Sheet Metal Spinning

Metal spinning process is constrained by two features (Music and Allwood, 2011): it can

only produce axially symmetric parts; a dedicated mandrel is required for each part. In

order to break through these limitations, further research works on asymmetric spinning

and spinning using a general purpose mandrel may be essential. This may require to

design spinning machines with new control systems, such as, force feedback control

(Sekiguchi and Arai, 2010) and synchronous control (Shimizu, 2010).

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 Chapter 7 Conclusion and Future Work

d) Theoretical Analysis of Wrinkling Involving Process Parameters

In this study, the tangential and radial stresses in the local forming zone of workpiece

have been used to predict the critical condition of wrinkling failure. However, this

theoretical model is quite difficult to be directly used in the spinning industry, where the

stress state of workpiece cannot be analysed without carrying out numerical analysis

such as FE simulation. Thus, it is of practical importance to develop a theoretical

analysis approach to relate process parameters, for instance, feed rate, spindle speed

and roller path etc, to the critical stress state in the workpiece and thereby to the

wrinkling failure.

145
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151
 Appendix

Appendix

Appendix 1 Roller Path information of Multiple Pass Spinning Study

Global Amplitude of local

coordinate displacement
Local coordinate Time*
(calculated from boundary
CNC program) conditions
X Y x z T Amp x Amp z
0 0 0 0 0 0 0
0.943 -0.224 -0.50841 0.825194 0.072693 -0.01755 0.011344
1.881 -0.467 -0.99985 1.660287 0.145365 -0.03451 0.022823
2.813 -0.73 -1.4729 2.505279 0.217995 -0.05084 0.034439
3.74 -1.013 -1.92828 3.360879 0.290688 -0.06656 0.046201
4.661 -1.314 -2.36669 4.224963 0.363358 -0.08169 0.058079
5.576 -1.635 -2.78671 5.098947 0.436084 -0.09619 0.070093
6.483 -1.974 -3.18834 5.980002 0.508705 -0.11005 0.082205
7.383 -2.332 -3.5716 6.869542 0.581349 -0.12328 0.094433
8.276 -2.709 -3.93646 7.767568 0.654048 -0.13587 0.106778
9.161 -3.105 -4.28224 8.673372 0.726765 -0.1478 0.11923
10.037 -3.518 -4.60963 9.584832 0.7994 -0.1591 0.131759
10.904 -3.95 -4.91722 10.50336 0.87205 -0.16972 0.144386
11.763 -4.4 -5.20643 11.42897 0.94478 -0.1797 0.15711
11.662 -4.3 -5.20572 11.28684 0.95544 -0.17968 0.155156
12.61 -4.637 -5.63776 12.19547 1.030899 -0.19459 0.167647
13.559 -4.974 -6.07051 13.10481 1.106428 -0.20953 0.180147
14.507 -5.311 -6.50255 14.01344 1.181887 -0.22444 0.192638
15.456 -5.649 -6.9346 14.92349 1.257442 -0.23935 0.205148
16.404 -5.986 -7.36664 15.83212 1.332901 -0.25427 0.217639
17.353 -6.323 -7.79939 16.74146 1.40843 -0.2692 0.230139
18.301 -6.66 -8.23143 17.65009 1.483889 -0.28411 0.24263
19.249 -6.997 -8.66347 18.55872 1.559348 -0.29903 0.25512
20.198 -7.334 -9.09622 19.46806 1.634877 -0.31396 0.267621
21.146 -7.671 -9.52826 20.3767 1.710336 -0.32888 0.280111
22.095 -8.008 -9.96101 21.28604 1.785866 -0.34381 0.292612
23.043 -8.345 -10.3931 22.19467 1.861324 -0.35872 0.305102
23.992 -8.682 -10.8258 23.10401 1.936854 -0.37366 0.317603
24.94 -9.019 -11.2578 24.01264 2.012313 -0.38857 0.330093
25.889 -9.356 -11.6906 24.92198 2.087842 -0.40351 0.342594
26.837 -9.693 -12.1226 25.83061 2.163301 -0.41842 0.355084
27.785 -10.03 -12.5547 26.73924 2.23876 -0.43333 0.367575
28.734 -10.367 -12.9874 27.64858 2.314289 -0.44827 0.380075
29.682 -10.704 -13.4195 28.55721 2.389748 -0.46318 0.392566
30.631 -11.042 -13.8515 29.46726 2.465303 -0.4781 0.405076

152
 Appendix
31.579 -11.379 -14.2836 30.37589 2.540762 -0.49301 0.417567
32.528 -11.716 -14.7163 31.28523 2.616291 -0.50794 0.430067
33.476 -12.053 -15.1483 32.19386 2.69175 -0.52286 0.442558
34.424 -12.39 -15.5804 33.1025 2.767209 -0.53777 0.455048
35.373 -12.727 -16.0131 34.01184 2.842738 -0.55271 0.467549
36.321 -13.064 -16.4452 34.92047 2.918197 -0.56762 0.480039
37.27 -13.401 -16.8779 35.82981 2.993727 -0.58255 0.49254
38.218 -13.738 -17.31 36.73844 3.069186 -0.59747 0.50503
39.167 -14.075 -17.7427 37.64778 3.144715 -0.6124 0.517531
40.115 -14.412 -18.1748 38.55641 3.220174 -0.62732 0.530021
41.063 -14.749 -18.6068 39.46504 3.295633 -0.64223 0.542512
42.012 -15.086 -19.0396 40.37438 3.371162 -0.65716 0.555012
42.96 -15.423 -19.4716 41.28302 3.446621 -0.67208 0.567503
43.909 -15.76 -19.9043 42.19235 3.522151 -0.68701 0.580003
44.857 -16.097 -20.3364 43.10099 3.597609 -0.70193 0.592494
45.8 -16.435 -20.7642 44.00679 3.67274 -0.71669 0.604946
46.705 -16.772 -21.1658 44.88502 3.745168 -0.73055 0.617019
47.7 -17.109 -21.6311 45.82688 3.823958 -0.74661 0.629966
48.605 -17.446 -22.0327 46.70511 3.896386 -0.76048 0.642039
49.509 -17.783 -22.4337 47.58263 3.968744 -0.77431 0.654102
50.504 -18.12 -22.8989 48.5245 4.047533 -0.79037 0.667049
51.409 -18.457 -23.3006 49.40272 4.119961 -0.80424 0.679122
52.404 -18.794 -23.7659 50.34459 4.19875 -0.8203 0.692069
53.309 -19.131 -24.1675 51.22282 4.271178 -0.83416 0.704142
54.304 -19.468 -24.6328 52.16468 4.349967 -0.85022 0.717089
55.209 -19.805 -25.0344 53.04291 4.422395 -0.86408 0.729162
56.203 -20.142 -25.499 53.98407 4.501113 -0.88012 0.7421
57.108 -20.479 -25.9006 54.86229 4.573542 -0.89398 0.754173
58.103 -20.816 -26.3659 55.80416 4.652331 -0.91004 0.76712
59.008 -21.153 -26.7675 56.68239 4.724759 -0.9239 0.779193
60.003 -21.49 -27.2328 57.62425 4.803548 -0.93996 0.79214
59.007 -21.146 -26.7718 56.67673 4.882578 -0.92405 0.779115
58.102 -20.801 -26.3758 55.79285 4.955218 -0.91038 0.766965
57.106 -20.456 -25.9155 54.84462 5.034272 -0.89449 0.75393
56.2 -20.111 -25.5188 53.96003 5.106982 -0.8808 0.741769
55.205 -19.766 -25.0592 53.0125 5.185965 -0.86494 0.728744
54.209 -19.421 -24.5988 52.06427 5.26502 -0.84905 0.715709
53.303 -19.076 -24.2021 51.17968 5.33773 -0.83535 0.703549
52.308 -18.731 -23.7425 50.23216 5.416713 -0.81949 0.690524
51.402 -18.387 -23.3451 49.34828 5.489396 -0.80577 0.678373
50.407 -18.042 -22.8855 48.40075 5.56838 -0.78991 0.665348
49.501 -17.697 -22.4888 47.51616 5.64109 -0.77622 0.653188
48.505 -17.352 -22.0285 46.56793 5.720144 -0.76033 0.640153
47.6 -17.007 -21.6325 45.68405 5.792784 -0.74666 0.628002
46.604 -16.662 -21.1722 44.73582 5.871838 -0.73077 0.614968
45.608 -16.317 -20.7119 43.78759 5.950893 -0.71489 0.601933

153
 Appendix
44.732 -15.972 -20.3364 42.92421 6.021505 -0.70193 0.590064
43.776 -15.628 -19.9036 42.00497 6.097705 -0.68699 0.577428
42.82 -15.283 -19.4716 41.08503 6.173931 -0.67208 0.564781
41.863 -14.938 -19.0389 40.16437 6.250228 -0.65714 0.552125
40.907 -14.593 -18.6068 39.24443 6.326454 -0.64223 0.539479
39.951 -14.248 -18.1748 38.32448 6.40268 -0.62732 0.526833
38.995 -13.903 -17.7427 37.40453 6.478906 -0.6124 0.514187
38.038 -13.558 -17.31 36.48388 6.555202 -0.59747 0.501531
37.082 -13.213 -16.8779 35.56394 6.631428 -0.58255 0.488885
36.126 -12.869 -16.4452 34.6447 6.707629 -0.56762 0.476248
35.17 -12.524 -16.0131 33.72475 6.783855 -0.55271 0.463602
34.213 -12.179 -15.5804 32.8041 6.860152 -0.53777 0.450946
33.257 -11.834 -15.1483 31.88415 6.936378 -0.52286 0.4383
32.301 -11.489 -14.7163 30.96421 7.012604 -0.50794 0.425654
31.345 -11.144 -14.2843 30.04426 7.08883 -0.49303 0.413008
30.388 -10.799 -13.8515 29.12361 7.165126 -0.4781 0.400352
29.432 -10.454 -13.4195 28.20366 7.241352 -0.46318 0.387706
28.476 -10.11 -12.9867 27.28442 7.317553 -0.44825 0.375069
27.52 -9.765 -12.5547 26.36448 7.393779 -0.43333 0.362423
26.564 -9.42 -12.1226 25.44453 7.470005 -0.41842 0.349777
25.607 -9.075 -11.6899 24.52388 7.546301 -0.40349 0.337121
24.651 -8.73 -11.2578 23.60393 7.622527 -0.38857 0.324475
23.695 -8.385 -10.8258 22.68399 7.698753 -0.37366 0.311829
22.739 -8.04 -10.3938 21.76404 7.774979 -0.35875 0.299183
21.782 -7.695 -9.96101 20.84339 7.851276 -0.34381 0.286527
20.826 -7.351 -9.52826 19.92415 7.927477 -0.32888 0.27389
19.87 -7.006 -9.09622 19.0042 8.003703 -0.31396 0.261244
18.914 -6.661 -8.66418 18.08426 8.079929 -0.29905 0.248598
17.957 -6.316 -8.23143 17.1636 8.156225 -0.28411 0.235942
17.001 -5.971 -7.79939 16.24366 8.232451 -0.2692 0.223296
16.045 -5.626 -7.36735 15.32371 8.308677 -0.25429 0.21065
15.089 -5.281 -6.9353 14.40377 8.384903 -0.23938 0.198003
14.132 -4.936 -6.50255 13.48311 8.4612 -0.22444 0.185348
13.176 -4.592 -6.0698 12.56387 8.5374 -0.2095 0.172711
12.22 -4.247 -5.63776 11.64393 8.613626 -0.19459 0.160065
11.264 -3.902 -5.20572 10.72398 8.689852 -0.17968 0.147419
12.086 -4.482 -5.37684 11.71535 8.765304 -0.18559 0.161047
12.907 -5.062 -5.54725 12.706 8.840695 -0.19147 0.174665
13.728 -5.643 -5.71696 13.69737 8.916129 -0.19733 0.188293
14.548 -6.225 -5.88525 14.68873 8.991545 -0.20313 0.201921
15.368 -6.808 -6.05283 15.6808 9.067004 -0.20892 0.215558
16.188 -7.392 -6.21971 16.67358 9.142507 -0.21468 0.229206
17.006 -7.976 -6.38517 17.66494 9.217888 -0.22039 0.242834
17.824 -8.561 -6.54993 18.65701 9.293312 -0.22608 0.256471
18.642 -9.147 -6.71398 19.64979 9.36878 -0.23174 0.270119
19.459 -9.734 -6.87661 20.64257 9.444231 -0.23735 0.283766

154
 Appendix
20.276 -10.321 -7.03925 21.63535 9.519682 -0.24296 0.297413
21.092 -10.909 -7.20047 22.62812 9.595115 -0.24853 0.311061
21.907 -11.498 -7.36027 23.6209 9.670532 -0.25405 0.324708
22.722 -12.088 -7.51937 24.61439 9.745993 -0.25954 0.338365
23.536 -12.679 -7.67706 25.60787 9.821437 -0.26498 0.352022
24.35 -13.27 -7.83474 26.60136 9.896881 -0.27042 0.365679
25.163 -13.862 -7.99101 27.59484 9.972309 -0.27582 0.379337
25.975 -14.455 -8.14587 28.58833 10.04772 -0.28116 0.392994
26.787 -15.049 -8.30002 29.58252 10.12318 -0.28648 0.40666
27.599 -15.643 -8.45417 30.57671 10.19863 -0.2918 0.420327
28.41 -16.238 -8.6069 31.5709 10.27407 -0.29707 0.433994
29.22 -16.834 -8.75822 32.5651 10.34949 -0.3023 0.447661
30.03 -17.431 -8.90884 33.55999 10.42496 -0.3075 0.461337
30.839 -18.029 -9.05804 34.55489 10.50041 -0.31264 0.475014
31.647 -18.627 -9.20653 35.54909 10.5758 -0.31777 0.488681
32.455 -19.226 -9.35432 36.54399 10.65124 -0.32287 0.502357
33.263 -19.826 -9.50139 37.53959 10.72672 -0.32795 0.516044
34.07 -20.427 -9.64706 38.5352 10.80219 -0.33298 0.52973
34.876 -21.028 -9.79201 39.5301 10.87759 -0.33798 0.543406
35.682 -21.63 -9.93626 40.5257 10.95304 -0.34296 0.557093
36.487 -22.233 -10.0791 41.52131 11.02848 -0.34789 0.570779
37.291 -22.837 -10.2205 42.51692 11.1039 -0.35277 0.584465
38.095 -23.441 -10.3619 43.51252 11.17932 -0.35765 0.598151
38.899 -24.046 -10.5027 44.50884 11.25478 -0.36251 0.611847
39.702 -24.652 -10.642 45.50515 11.33023 -0.36732 0.625543
40.504 -25.259 -10.7798 46.50146 11.40567 -0.37207 0.639239
41.306 -25.867 -10.917 47.49848 11.48115 -0.37681 0.652945
42.107 -26.475 -11.0535 48.4948 11.55657 -0.38152 0.666641
42.907 -27.084 -11.1886 49.49111 11.63198 -0.38618 0.680337
43.707 -27.694 -11.3229 50.48813 11.70743 -0.39082 0.694043
44.507 -28.304 -11.4573 51.48515 11.78288 -0.39546 0.707748
45.305 -28.916 -11.5888 52.48217 11.85831 -0.4 0.721454
46.104 -29.528 -11.721 53.4799 11.93379 -0.40456 0.735169
46.901 -30.14 -11.8518 54.47621 12.00916 -0.40907 0.748865
47.698 -30.754 -11.9812 55.47394 12.08461 -0.41354 0.762581
48.495 -31.368 -12.1106 56.47167 12.16007 -0.41801 0.776296
49.29 -31.984 -12.2372 57.4694 12.2355 -0.42238 0.790012
50.086 -32.599 -12.3652 58.46712 12.31094 -0.42679 0.803727
50.88 -33.216 -12.4903 59.46485 12.38636 -0.43111 0.817442
51.675 -33.833 -12.6162 60.46329 12.46183 -0.43546 0.831167
52.468 -34.452 -12.7392 61.46172 12.53728 -0.4397 0.844893
53.261 -35.07 -12.863 62.45945 12.61268 -0.44398 0.858608
54.053 -35.69 -12.9846 63.45788 12.68812 -0.44817 0.872333
53.251 -35.076 -12.8517 62.45662 12.76387 -0.44358 0.858569
52.449 -34.462 -12.7187 61.45536 12.83963 -0.439 0.844805
51.647 -33.848 -12.5858 60.45409 12.91538 -0.43441 0.831041

155
 Appendix
50.845 -33.234 -12.4529 59.45283 12.99113 -0.42982 0.817277
50.043 -32.62 -12.3199 58.45157 13.06689 -0.42523 0.803513
49.241 -32.007 -12.1863 57.45101 13.1426 -0.42062 0.789759
48.439 -31.393 -12.0533 56.44975 13.21835 -0.41603 0.775995
47.637 -30.779 -11.9204 55.44849 13.2941 -0.41144 0.762231
46.835 -30.165 -11.7875 54.44722 13.36986 -0.40685 0.748467
46.033 -29.551 -11.6545 53.44596 13.44561 -0.40226 0.734703
45.231 -28.937 -11.5216 52.4447 13.52136 -0.39768 0.720939
44.43 -28.323 -11.3894 51.44414 13.59706 -0.39311 0.707184
43.628 -27.709 -11.2564 50.44288 13.67281 -0.38852 0.69342
42.826 -27.095 -11.1235 49.44161 13.74857 -0.38394 0.679656
42.024 -26.481 -10.9906 48.44035 13.82432 -0.37935 0.665892
41.222 -25.867 -10.8576 47.43909 13.90007 -0.37476 0.652128
40.42 -25.253 -10.7247 46.43782 13.97583 -0.37017 0.638364
39.618 -24.639 -10.5918 45.43656 14.05158 -0.36558 0.6246
38.816 -24.025 -10.4588 44.4353 14.12733 -0.36099 0.610836
38.014 -23.411 -10.3259 43.43403 14.20309 -0.35641 0.597072
37.212 -22.798 -10.1922 42.43348 14.2788 -0.35179 0.583318
36.41 -22.184 -10.0593 41.43221 14.35455 -0.3472 0.569554
35.608 -21.57 -9.92636 40.43095 14.4303 -0.34262 0.55579
34.806 -20.956 -9.79343 39.42969 14.50606 -0.33803 0.542026
34.004 -20.342 -9.66049 38.42843 14.58181 -0.33344 0.528262
33.202 -19.728 -9.52756 37.42716 14.65756 -0.32885 0.514498
32.4 -19.114 -9.39462 36.4259 14.73332 -0.32426 0.500734
31.598 -18.5 -9.26168 35.42464 14.80907 -0.31967 0.48697
30.796 -17.886 -9.12875 34.42337 14.88483 -0.31509 0.473206
29.994 -17.272 -8.99581 33.42211 14.96058 -0.3105 0.459442
29.192 -16.658 -8.86288 32.42085 15.03633 -0.30591 0.445678
28.39 -16.044 -8.72994 31.41958 15.11209 -0.30132 0.431914
27.588 -15.43 -8.597 30.41832 15.18784 -0.29673 0.41815
26.786 -14.816 -8.46407 29.41706 15.26359 -0.29214 0.404386
25.984 -14.202 -8.33113 28.41579 15.33935 -0.28756 0.390622
25.182 -13.589 -8.19749 27.41524 15.41506 -0.28294 0.376868
24.38 -12.975 -8.06455 26.41397 15.49081 -0.27835 0.363104
23.578 -12.361 -7.93162 25.41271 15.56656 -0.27377 0.34934
22.776 -11.747 -7.79868 24.41145 15.64232 -0.26918 0.335576
21.974 -11.133 -7.66574 23.41018 15.71807 -0.26459 0.321812
21.172 -10.519 -7.53281 22.40892 15.79383 -0.26 0.308048
20.37 -9.905 -7.39987 21.40766 15.86958 -0.25541 0.294284
19.568 -9.291 -7.26694 20.40639 15.94533 -0.25082 0.28052
18.766 -8.677 -7.134 19.40513 16.02109 -0.24624 0.266755
17.964 -8.063 -7.00106 18.40387 16.09684 -0.24165 0.252991
18.624 -8.805 -6.94308 19.39523 16.17132 -0.23965 0.266619
19.282 -9.549 -6.88227 20.3866 16.24581 -0.23755 0.280247
19.936 -10.297 -6.8158 21.37796 16.32033 -0.23525 0.293875
20.587 -11.047 -6.7458 22.36862 16.39481 -0.23284 0.307493

156
 Appendix
21.234 -11.8 -6.67085 23.35857 16.46927 -0.23025 0.321102
21.879 -12.555 -6.59306 24.34851 16.54375 -0.22756 0.33471
22.52 -13.313 -6.51033 25.33776 16.6182 -0.22471 0.348309
23.159 -14.074 -6.42407 26.32771 16.69273 -0.22173 0.361918
23.794 -14.838 -6.33285 27.31695 16.76724 -0.21858 0.375516
24.426 -15.604 -6.2381 28.30548 16.84172 -0.21531 0.389106
25.054 -16.373 -6.13839 29.29331 16.91618 -0.21187 0.402685
25.68 -17.144 -6.03586 30.28114 16.99067 -0.20833 0.416264
26.302 -17.918 -5.92838 31.26826 17.06514 -0.20462 0.429834
26.921 -18.695 -5.81666 32.25538 17.13964 -0.20077 0.443403
27.537 -19.474 -5.7014 33.2418 17.21413 -0.19679 0.456963
28.149 -20.256 -5.58119 34.2275 17.2886 -0.19264 0.470513
28.758 -21.04 -5.45745 35.2125 17.36306 -0.18837 0.484054
29.364 -21.827 -5.32946 36.1975 17.43756 -0.18395 0.497594
29.967 -22.616 -5.19794 37.1818 17.51203 -0.17941 0.511125
30.566 -23.408 -5.06147 38.16538 17.58651 -0.1747 0.524646
31.162 -24.203 -4.92076 39.14897 17.66103 -0.16984 0.538167
31.754 -25 -4.7758 40.13114 17.73549 -0.16484 0.551669
32.343 -25.799 -4.62731 41.1126 17.80994 -0.15971 0.56516
32.929 -26.601 -4.47457 42.09407 17.88443 -0.15444 0.578652
33.512 -27.406 -4.31759 43.07553 17.95898 -0.14902 0.592144
34.091 -28.212 -4.15708 44.05487 18.03341 -0.14348 0.605607
34.667 -29.022 -3.99162 45.03492 18.10795 -0.13777 0.619079
35.239 -29.833 -3.82262 46.01285 18.18238 -0.13194 0.632522
35.808 -30.647 -3.64938 46.99078 18.25687 -0.12596 0.645966
36.373 -31.464 -3.47119 47.968 18.33137 -0.11981 0.659399
36.935 -32.282 -3.29017 48.94381 18.40581 -0.11356 0.672813
37.494 -33.103 -3.10491 49.91962 18.4803 -0.10717 0.686227
38.049 -33.927 -2.91469 50.89472 18.55481 -0.1006 0.699632
38.601 -34.753 -2.72095 51.86911 18.62932 -0.09392 0.713026
39.149 -35.581 -2.52296 52.84209 18.70379 -0.08708 0.726402
39.694 -36.411 -2.32143 53.81436 18.77826 -0.08013 0.739767
40.235 -37.244 -2.11496 54.78593 18.85275 -0.073 0.753123
40.773 -38.079 -1.90495 55.75678 18.92725 -0.06575 0.766469
41.308 -38.916 -1.6914 56.72693 19.00176 -0.05838 0.779805
41.838 -39.755 -1.4729 57.69496 19.07618 -0.05084 0.793112
42.366 -40.597 -1.25087 58.6637 19.15072 -0.04317 0.806429
42.889 -41.44 -1.0246 59.62961 19.22513 -0.03536 0.819707
43.41 -42.286 -0.79479 60.59622 19.29964 -0.02743 0.832995
43.926 -43.134 -0.56003 61.56072 19.37409 -0.01933 0.846253
44.439 -43.985 -0.32103 62.52521 19.44862 -0.01108 0.859512
44.949 -44.837 -0.0792 63.48829 19.52309 -0.00273 0.872751
45.455 -45.692 0.167584 64.45066 19.5976 0.005784 0.885981
45.957 -46.548 0.4179 65.41091 19.67203 0.014424 0.899181
46.456 -47.407 0.672459 66.37116 19.74654 0.02321 0.912381
46.951 -48.268 0.93126 67.33 19.82102 0.032143 0.925562

157
 Appendix
46.372 -47.44 0.75519 66.3351 19.8968 0.026066 0.911885
45.793 -46.611 0.578413 65.3395 19.97264 0.019964 0.898199
45.214 -45.783 0.402344 64.3446 20.04841 0.013887 0.884522
44.635 -44.954 0.225567 63.34899 20.12425 0.007786 0.870836
44.056 -44.126 0.049497 62.35409 20.20003 0.001708 0.85716
43.477 -43.297 -0.12728 61.35848 20.27587 -0.00439 0.843473
42.898 -42.469 -0.30335 60.36358 20.35165 -0.01047 0.829797
42.319 -41.641 -0.47942 59.36869 20.42742 -0.01655 0.81612
41.74 -40.812 -0.6562 58.37308 20.50326 -0.02265 0.802434
41.16 -39.984 -0.83156 57.37747 20.57908 -0.0287 0.788748
40.581 -39.155 -1.00833 56.38187 20.65492 -0.0348 0.775062
40.002 -38.327 -1.1844 55.38697 20.7307 -0.04088 0.761385
39.423 -37.498 -1.36118 54.39136 20.80653 -0.04698 0.747699
38.844 -36.67 -1.53725 53.39646 20.88231 -0.05306 0.734022
38.265 -35.841 -1.71403 52.40086 20.95815 -0.05916 0.720336
37.686 -35.013 -1.8901 51.40596 21.03393 -0.06524 0.70666
37.107 -34.185 -2.06617 50.41106 21.1097 -0.07132 0.692983
36.528 -33.356 -2.24294 49.41545 21.18554 -0.07742 0.679297
35.949 -32.528 -2.41901 48.42055 21.26132 -0.08349 0.66562
35.37 -31.699 -2.59579 47.42494 21.33716 -0.0896 0.651934
34.791 -30.871 -2.77186 46.43005 21.41293 -0.09567 0.638257
34.212 -30.042 -2.94864 45.43444 21.48877 -0.10177 0.624571
33.633 -29.214 -3.1247 44.43954 21.56455 -0.10785 0.610895
33.054 -28.385 -3.30148 43.44393 21.64039 -0.11395 0.597208
32.474 -27.557 -3.47684 42.44833 21.71621 -0.12001 0.583522
31.895 -26.729 -3.65291 41.45343 21.79198 -0.12608 0.569846
31.316 -25.9 -3.82969 40.45782 21.86782 -0.13218 0.556159
30.737 -25.072 -4.00576 39.46292 21.9436 -0.13826 0.542483
30.158 -24.243 -4.18254 38.46732 22.01944 -0.14436 0.528797
29.579 -23.415 -4.35861 37.47242 22.09522 -0.15044 0.51512
29 -22.586 -4.53538 36.47681 22.17105 -0.15654 0.501434
28.421 -21.758 -4.71145 35.48191 22.24683 -0.16262 0.487757
27.842 -20.93 -4.88752 34.48701 22.32261 -0.1687 0.474081
27.263 -20.101 -5.0643 33.49141 22.39845 -0.1748 0.460395
26.684 -19.273 -5.24037 32.49651 22.47422 -0.18088 0.446718
26.105 -18.444 -5.41715 31.5009 22.55006 -0.18698 0.433032
26.538 -19.339 -5.09046 32.43994 22.62463 -0.1757 0.44594
26.967 -20.236 -4.75954 33.37756 22.6992 -0.16428 0.45883
27.392 -21.135 -4.42437 34.31377 22.77378 -0.15271 0.471699
27.813 -22.036 -4.08496 35.24857 22.84837 -0.141 0.48455
28.231 -22.938 -3.74272 36.18195 22.92293 -0.12918 0.49738
28.644 -23.843 -3.39482 37.11391 22.99754 -0.11717 0.510192
29.054 -24.749 -3.04409 38.04447 23.07212 -0.10507 0.522984
29.46 -25.657 -2.68913 38.9736 23.14672 -0.09282 0.535756
29.861 -26.566 -2.32992 39.89991 23.22124 -0.08042 0.54849
30.259 -27.477 -1.96717 40.82552 23.2958 -0.0679 0.561214

158
 Appendix
30.653 -28.39 -1.60018 41.74971 23.37038 -0.05523 0.573918
31.043 -29.305 -1.22895 42.67248 23.44497 -0.04242 0.586604
31.429 -30.222 -0.85348 43.59384 23.51959 -0.02946 0.599269
31.811 -31.14 -0.47447 44.51308 23.59417 -0.01638 0.611906
32.189 -32.059 -0.09192 45.4302 23.66869 -0.00317 0.624513
32.563 -32.981 0.295571 46.34661 23.74332 0.010202 0.63711
32.933 -33.904 0.686601 47.2609 23.8179 0.023699 0.649679
33.299 -34.828 1.081166 48.17306 23.89244 0.037317 0.662218
33.661 -35.754 1.479974 49.08382 23.967 0.051082 0.674738
34.019 -36.682 1.883025 49.99316 24.0416 0.064994 0.687238
34.373 -37.611 2.289612 50.90037 24.11617 0.079028 0.699709
34.723 -38.542 2.700441 51.80618 24.19076 0.093208 0.712161
35.069 -39.474 3.114805 52.70986 24.26532 0.10751 0.724584
35.411 -40.408 3.533413 53.61213 24.33992 0.121958 0.736987
35.749 -41.343 3.955555 54.51228 24.41449 0.136529 0.749361
36.083 -42.279 4.381234 55.4103 24.48902 0.151222 0.761706
36.413 -43.218 4.811862 56.30762 24.56367 0.166085 0.774041
36.739 -44.157 5.245318 57.20211 24.63822 0.181046 0.786337
37.06 -45.098 5.683724 58.09448 24.71279 0.196178 0.798604
37.378 -46.04 6.124959 58.98543 24.78735 0.211408 0.810852
37.691 -46.984 6.571143 59.87427 24.86194 0.226808 0.82307
38.001 -47.929 7.020156 60.76169 24.93654 0.242306 0.835269
38.306 -48.875 7.473412 61.64628 25.01108 0.25795 0.84743
38.607 -49.823 7.93091 62.52945 25.08568 0.273741 0.85957
38.905 -50.772 8.391236 63.41121 25.16028 0.28963 0.871692
39.198 -51.722 8.855805 64.29015 25.23484 0.305665 0.883774
39.487 -52.673 9.32391 65.16696 25.30939 0.321822 0.895827
39.771 -53.626 9.796964 66.04165 25.38397 0.33815 0.907851
40.052 -54.58 10.27285 66.91493 25.45856 0.354575 0.919856
40.329 -55.535 10.75227 67.78608 25.53314 0.371122 0.931831
40.601 -56.491 11.23593 68.65441 25.60768 0.387816 0.943768
40.244 -55.537 10.81378 67.72739 25.68408 0.373246 0.931025
39.887 -54.583 10.39164 66.80038 25.76047 0.358675 0.918281
39.529 -53.629 9.970206 65.87265 25.8369 0.344129 0.905528
39.172 -52.675 9.548063 64.94564 25.91329 0.329558 0.892785
38.815 -51.721 9.12592 64.01862 25.98969 0.314988 0.880041
38.458 -50.767 8.703777 63.0916 26.06608 0.300417 0.867298
38.1 -49.812 8.281635 62.16317 26.14258 0.285847 0.854535
37.743 -48.858 7.859492 61.23615 26.21897 0.271276 0.841792
37.386 -47.904 7.437349 60.30914 26.29537 0.256706 0.829048
37.029 -46.95 7.015206 59.38212 26.37176 0.242135 0.816305
36.672 -45.996 6.593064 58.4551 26.44816 0.227564 0.803562
36.314 -45.042 6.171628 57.52738 26.52458 0.213018 0.790809
35.957 -44.088 5.749485 56.60036 26.60098 0.198448 0.778065
35.6 -43.134 5.327342 55.67335 26.67737 0.183877 0.765322
35.243 -42.179 4.904493 54.74562 26.75384 0.169282 0.752569

159
 Appendix
34.885 -41.225 4.483057 53.8179 26.83026 0.154736 0.739816
34.528 -40.271 4.060914 52.89088 26.90666 0.140165 0.727072
34.171 -39.317 3.638771 51.96386 26.98305 0.125595 0.714329
33.814 -38.363 3.216629 51.03685 27.05945 0.111024 0.701586
33.456 -37.409 2.795193 50.10912 27.13587 0.096478 0.688832
33.099 -36.455 2.37305 49.18211 27.21226 0.081908 0.676089
32.742 -35.5 1.950201 48.25438 27.28873 0.067313 0.663336
32.385 -34.546 1.528058 47.32736 27.36513 0.052742 0.650593
32.028 -33.592 1.105915 46.40035 27.44152 0.038171 0.637849
31.67 -32.638 0.684479 45.47262 27.51794 0.023625 0.625096
31.313 -31.684 0.262337 44.54561 27.59434 0.009055 0.612353
30.956 -30.73 -0.15981 43.61859 27.67074 -0.00552 0.599609
30.599 -29.776 -0.58195 42.69157 27.74713 -0.02009 0.586866
30.241 -28.822 -1.00338 41.76385 27.82355 -0.03463 0.574113
29.884 -27.867 -1.42623 40.83612 27.90002 -0.04923 0.56136
30.111 -28.852 -0.89025 41.69314 27.97583 -0.03073 0.573141
30.334 -29.837 -0.35143 42.54732 28.05157 -0.01213 0.584883
30.554 -30.823 0.190212 43.40009 28.12734 0.006565 0.596606
30.769 -31.809 0.735391 44.24933 28.20303 0.025383 0.60828
30.981 -32.797 1.284106 45.09786 28.27882 0.044322 0.619944
31.19 -33.785 1.834942 45.94426 28.35456 0.063334 0.63158
31.394 -34.774 2.390021 46.78784 28.43029 0.082493 0.643176
31.595 -35.764 2.947928 47.63001 28.50606 0.10175 0.654753
31.792 -36.755 3.509371 48.47005 28.58184 0.121129 0.666301
31.985 -37.746 4.073642 49.30726 28.65756 0.140605 0.67781
32.175 -38.738 4.640742 50.14306 28.73331 0.160179 0.689299
32.361 -39.731 5.211377 50.97674 28.80908 0.179875 0.700759
32.543 -40.724 5.784841 51.80759 28.8848 0.199668 0.712181
32.721 -41.719 6.362547 52.63703 28.96061 0.219608 0.723583
32.896 -42.713 6.941667 53.46364 29.0363 0.239597 0.734946
33.066 -43.709 7.525737 54.28812 29.11208 0.259756 0.74628
33.233 -44.705 8.111929 55.11049 29.18783 0.279989 0.757584
33.397 -45.702 8.700949 55.93144 29.26361 0.30032 0.76887
33.556 -46.699 9.293504 56.74885 29.33933 0.320772 0.780106
33.712 -47.697 9.888888 57.56486 29.41509 0.341322 0.791324
33.864 -48.696 10.48781 58.37874 29.49087 0.361994 0.802512
34.012 -49.695 11.08956 59.18979 29.56662 0.382764 0.813661
34.156 -50.694 11.69413 59.99801 29.64232 0.403632 0.824771
34.297 -51.694 12.30154 60.80482 29.71806 0.424597 0.835862
34.433 -52.695 12.91318 61.6088 29.79382 0.445708 0.846914
34.566 -53.696 13.52695 62.41066 29.86956 0.466893 0.857937
34.696 -54.698 14.14355 63.2111 29.94534 0.488175 0.868941
34.821 -55.7 14.76368 64.00801 30.02107 0.509579 0.879896
34.943 -56.703 15.38664 64.80351 30.09685 0.531081 0.890831
35.061 -57.706 16.01243 65.59617 30.17259 0.552681 0.901727
35.175 -58.71 16.64176 66.38672 30.24838 0.574403 0.912595

160
 Appendix
35.285 -59.714 17.27391 67.17444 30.32413 0.596222 0.923423
35.391 -60.718 17.90889 67.95933 30.39985 0.618139 0.934213
35.494 -61.723 18.5467 68.7428 30.47561 0.640153 0.944983
35.279 -60.701 17.97607 67.86811 30.55394 0.620457 0.932959
35.065 -59.678 17.40402 66.99342 30.63233 0.600713 0.920935
34.85 -58.656 16.83338 66.11873 30.71066 0.581017 0.908911
34.635 -57.634 16.26275 65.24404 30.78898 0.561321 0.896887
34.421 -56.612 15.69141 64.37005 30.8673 0.541601 0.884872
34.206 -55.59 15.12077 63.49536 30.94562 0.521905 0.872848
33.991 -54.568 14.55014 62.62067 31.02395 0.502209 0.860824
33.777 -53.546 13.97879 61.74669 31.10226 0.482488 0.84881
33.562 -52.523 13.40745 60.87129 31.18067 0.462768 0.836776
33.347 -51.501 12.83682 59.9966 31.25899 0.443072 0.824752
33.132 -50.479 12.26618 59.12191 31.33732 0.423376 0.812728
32.918 -49.457 11.69484 58.24792 31.41563 0.403656 0.800714
32.703 -48.435 11.1242 57.37323 31.49396 0.38396 0.78869
32.488 -47.413 10.55357 56.49854 31.57229 0.364264 0.776665
32.274 -46.391 9.982226 55.62455 31.6506 0.344544 0.764651
32.059 -45.368 9.410884 54.74916 31.729 0.324824 0.752617
31.844 -44.346 8.840249 53.87447 31.80733 0.305128 0.740593
31.629 -43.324 8.269614 52.99977 31.88566 0.285432 0.728569
31.415 -42.302 7.698272 52.12579 31.96397 0.265712 0.716555
31.2 -41.28 7.127636 51.2511 32.0423 0.246016 0.704531
30.985 -40.258 6.557001 50.37641 32.12063 0.22632 0.692507
30.984 -41.279 7.279664 51.09766 32.1972 0.251263 0.702421
30.983 -42.301 8.003035 51.81961 32.27385 0.276231 0.712346
30.982 -43.322 8.725698 52.54086 32.35043 0.301174 0.722261
30.981 -44.344 9.449068 53.26282 32.42708 0.326142 0.732185
30.98 -45.365 10.17173 53.98407 32.50365 0.351085 0.7421
30.979 -46.387 10.8951 54.70602 32.5803 0.376053 0.752024
30.978 -47.408 11.61776 55.42727 32.65688 0.400996 0.761939
30.977 -48.43 12.34113 56.14923 32.73353 0.425963 0.771864
30.976 -49.452 13.0645 56.87118 32.81018 0.450931 0.781788
30.975 -50.473 13.78717 57.59243 32.88675 0.475874 0.791703
30.973 -51.495 14.51125 58.31368 32.9634 0.500866 0.801618
30.972 -52.516 15.23391 59.03493 33.03998 0.52581 0.811532
30.971 -53.538 15.95728 59.75689 33.11663 0.550777 0.821457
30.97 -54.559 16.67994 60.47814 33.1932 0.575721 0.831372
30.969 -55.581 17.40331 61.20009 33.26985 0.600688 0.841296
30.968 -56.602 18.12598 61.92134 33.34643 0.625632 0.851211
30.967 -57.624 18.84935 62.6433 33.42308 0.650599 0.861135
30.966 -58.645 19.57201 63.36455 33.49965 0.675542 0.87105
30.965 -59.667 20.29538 64.0865 33.5763 0.70051 0.880975
30.964 -60.688 21.01804 64.80775 33.65288 0.725453 0.890889
30.963 -61.71 21.74141 65.52971 33.72953 0.750421 0.900814
30.962 -62.732 22.46478 66.25166 33.80618 0.775389 0.910738

161
 Appendix
30.961 -63.753 23.18745 66.97291 33.88275 0.800332 0.920653
30.959 -64.775 23.91152 67.69416 33.9594 0.825324 0.930568
30.958 -65.796 24.63419 68.41541 34.03598 0.850267 0.940483
30.957 -66.818 25.35756 69.13737 34.11263 0.875235 0.950407
30.956 -67.839 26.08022 69.85861 34.1892 0.900178 0.960322
30.955 -68.861 26.80359 70.58057 34.26585 0.925146 0.970246
30.954 -69.882 27.52625 71.30182 34.34243 0.950089 0.980161
30.953 -70.904 28.24962 72.02378 34.41908 0.975057 0.990085
30.952 -71.925 28.97229 72.74502 34.49565 1 1

* Run 3 & 4 (Feed rate = 900 mm/min)

162
 Appendix

Appendix 2 Roller Path information of Wrinkling Failure Study

Global Amplitude of local

coordinate displacement
Local coordinate Time*
(calculated from boundary
CNC program) conditions
X Y x z T Amp x Amp z
0 0 0 0 0 0 0
-0.002 -0.106 0.076368 0.073539 0.16 0.008938 0.005768
-0.245 -0.645 0.629325 0.282843 0.32 0.073657 0.022184
-0.429 -1.227 1.170969 0.564271 0.48 0.137052 0.044257
-0.552 -1.794 1.658873 0.878227 0.64 0.194157 0.068881
-0.672 -2.542 2.272641 1.32229 0.8 0.265994 0.10371
-0.78 -3.335 2.909744 1.806658 0.96 0.340561 0.1417
-0.847 -4.036 3.452802 2.254964 1.12 0.404122 0.176862
-0.745 -4.538 3.735645 2.682056 1.28 0.437226 0.21036
-0.525 -4.952 3.872824 3.130362 1.44 0.453281 0.245522
-0.261 -5.377 3.986668 3.617558 1.6 0.466606 0.283734
0.027 -5.831 4.104048 4.142232 1.76 0.480344 0.324885
0.313 -6.298 4.232034 4.674683 1.92 0.495324 0.366646
0.53 -6.815 4.444166 5.193699 2.08 0.520152 0.407354
0.518 -7.505 4.940555 5.673118 2.24 0.57825 0.444956
0.471 -8.215 5.475835 6.14193 2.4 0.6409 0.481726
0.464 -8.896 5.962324 6.618519 2.56 0.69784 0.519106
0.511 -9.523 6.372446 7.095109 2.72 0.745841 0.556486
0.564 -10.149 6.777618 7.575235 2.88 0.793263 0.594144
0.62 -10.818 7.211075 8.087887 3.04 0.843996 0.634352
0.666 -11.546 7.693322 8.635188 3.2 0.900439 0.677278
0.731 -12.309 8.186882 9.220672 3.36 0.958206 0.723199
0.878 -12.998 8.570134 9.811814 3.52 1.003062 0.769564
0.982 -13.763 9.037532 10.42629 3.68 1.057767 0.817758
1.096 -14.561 9.521193 11.07117 3.84 1.114376 0.868338
1.109 -15.493 10.17102 11.73939 4 1.190433 0.920748
1.043 -16.534 10.95379 12.42882 4.16 1.282049 0.974821
0.91 -17.673 11.85323 13.14017 4.32 1.387321 1.030614
0.659 -18.923 12.9146 13.84656 4.48 1.511545 1.086019
0.346 -20.208 14.04455 14.53387 4.64 1.643797 1.139926
0.301 -20.933 14.58903 15.01471 4.8 1.707523 1.177639
0.602 -20.728 14.23123 15.08259 4.96 1.665646 1.182963
1.506 -19.289 12.57448 14.70429 5.12 1.471737 1.153292
2.357 -17.167 10.47225 13.80555 5.28 1.225689 1.082802
2.974 -15.057 8.543971 12.74984 5.44 1 1

* Model 1 of Table 6.2

163