PREDICTING PARTING PLANE SEPARATION AND TIE
BAR LOADS IN DIE CASTING USING COMPUTER
MODELING AND DIMENSIONAL ANALYSIS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree
of Doctor of Philosophy in the Graduate School of The Ohio State
University
By
Karthik S Murugesan M.S.
The Ohio State University
2008
*****
Dissertation Committee:
Approved by:
Dr. R. Allen Miller, Adviser
Dr. Jerald Brevick
Dr. Khalil Kabiri-Bamoradian
Adviser
Graduate Program in Industrial and
Systems Engineering
ABSTRACT
Die Casting dies and machines are high performance products that are subjected to
clamp load, cavity pressure loads and thermal loads during normal operation and the
dies and machine deflect under the action of these loads. The ability of the dies to
withstand loads and preserve the integrity of the cavity dimensions depends on the
structural design of the dies. Die castings dies are expensive products with long
production lead times and the structural behavior of the dies has to be predicted at the
design stage. The other common problem in die casting is the tie bar load imbalance.
The machine clamp load is distributed among the four tie bars depending upon the
location of the dies and the location of the cavity center of pressure on the platen. Tie
bar load imbalance causes the die parting surface to close unevenly and leads to
problems such as flash and premature tie bar failure. The problem is over come by
adjusting the length of the tie bars between the machine platens until all the tie bars
carry equal loads. Tie bar load predictions are necessary to determine the individual
length adjustments needed on each tie bars.
Numerical methods such as the finite element method are the most effective way to
predict the distortion of the dies and the machine at the design stage. Performing a
full FEA during the initial stages of the die design is time consuming and it is not cost
effective. So off the shelf design tools such as closed form expressions, design charts
ii
and guidelines are needed to make design improvements during the initial stages of
the design.
In this dissertation research work the relative contribution of the major structural
design variables of the die casting die and machine to the mechanical performance of
the dies and machines was investigated using computational (FEA) experiments. The
maximum parting plane separation was chosen as the performance measure for the
structural behavior of the dies and the machine. The computational experiments were
designed using Design of Experiments approach and closed form power law models
were developed to predict the maximum cover and ejector side parting plane
separation. The functional form for the power law model was obtained using
dimensional analysis based on Pi-theorem. These power law models were then used
to explain the sensitivity of maximum parting plane separation to the design
variables. The power law models can also be used to compare the performance of
different dies and machines and make structural design improvements of the die. In
addition a methodology to characterize the stiffness of the machine platens is also
developed.
In the second part of the research power law models were developed based on
dimensional analysis to predict the loads on the tie bars of the die casting machine as
a function of the die location, the location of the cavity center of pressure, clamp load
and the magnitude of cavity pressure. The power law model to predict die bar loads
can be used to determine the length adjustments needed on the tie bar to balance the
ii
tie bar loads. The relative contributions of the die location and cavity location on tie
bar load imbalance were also studied using the exponents and coefficients of the
power law model. The adequacy of the model was also studied by using tie bar load
measurements from a die casting machine.
iii
Dedicated to my parents and my brother
iv
ACKNOWLEDGMENT
My first and foremost thanks go to Dr. R. Allen Miller for providing me this
wonderful opportunity to pursue my PhD under his guidance and support. I thank him
for guiding my thought process by his constant questioning and constructive
criticism. I thank him for his trust, patience and enthusiasm. Working with Dr. Miller
was my most valuable academic and research experience.
I am very thankful to Dr. Khalil Kabiri-Bamoradian for guiding me in the
development of finite element models and for helping me with dimensional analysis
and power law model development. I also thank him for teaching me the best finite
element modeling practices and research methods. I cannot overemphasis the
enormous amount of time that he spent for helping me during this research work.
I would like to thank Dr. Jerald Brevick for his contribution and support as a member
of my dissertation committee. I also thank him for sharing his scientific knowledge
and expertise in die casting throughout the course of my graduate study.
I thank Dr. Theodore Allen for being a member of my general exam committee and
for the development of experimental design used in the tie bar load prediction model.
My special thanks to my former colleague Adham Ragab, who helped me with his
valuable suggestions during the initial period of this research. I am grateful to my
v
colleague and friend Abelardo Garza for constantly motivating and encouraging me
during my toughest times.
I am extremely grateful to Cedric-Size and Shih Kwang-Chen for their laboratory
support. The timely completion of this project would not have been possible without
Shih-Kwang’s help and support. My thanks to all the IWSE staff, particularly Darline
Wine for their administrative support.
I thank the US Department of Energy for the financial support they have provided for
this project. I also thank the NADCA Computer Modeling Task force for their
feedback and for their encouragement for this project.
vi
VITA
April 8, 1979……………..Born, Coimbatore, India
2000 ……………………..B.E. Mechanical Engineering, Bharathiar University, India
2003 …………………….M.S. Engineering, Purdue University, Indianapolis, Indiana
2003-2008 ……………….Graduate Research Associate, The Ohio State University,
Columbus, Ohio
PUBLICATIONS
1. K. Murugesan, A. Ragab, K. Kabiri-Bamoradian, R. A. Miller, “Effect of Die,
Cavity and Toggle Locations on Tie bar Forces, Toggle Forces and Parting Plane
Separation”, NADCA Proceedings, April 2005
2. K. Murugesan, A. Ragab, K. Kabiri-Bamoradian, R. A. Miller, “A Model to
Predict Tie Bar Load Imbalance”, NADCA Proceedings, April 2006
3. K. Murugesan, A. Ragab, K. Kabiri-Bamoradian, R. A. Miller, “An Experimental
Verification of the effect of Die Location on Tie Bar Load Imbalance”, NADCA
Proceedings, May 2007
4. K. Murugesan, K. Kabiri-Bamoradian, R. A. Miller, “Effect of support pillar
patterns on Mechanical Performance of Ejector Side Dies”, NADCA Proceedings,
May 2008
FIELDS OF STUDY
Major Field: Industrial and Systems Engineering (Manufacturing)
Minor Fields: Applied Statistics and Mathematics
vii
TABLE OF CONTENTS
Page
ABSTRACT ................................................................................................................ II
ACKNOWLEDGMENT............................................................................................ V
VITA......................................................................................................................... VII
PUBLICATIONS .................................................................................................... VII
FIELDS OF STUDY ............................................................................................... VII
LIST OF FIGURES ................................................................................................ XII
LIST OF TABLES .................................................................................................. XV
CHAPTER 1 ................................................................................................................ 1
1.1 DIE CASTING DIES AND MACHINES ...................................................................... 2
1.2 MECHANICAL AND THERMAL LOADS INVOLVED IN DIE CASTING ........................ 6
1.3 DIE DEFLECTIONS ................................................................................................ 8
1.4 IMBALANCED LOADS ON MACHINE TIE BARS ...................................................... 11
1.5 PROBLEM STATEMENT........................................................................................ 13
1.6 RESEARCH OBJECTIVES....................................................................................... 15
1.7 RESEARCH CONTRIBUTIONS ............................................................................... 16
viii
1.8 DISSERTATION OUTLINE..................................................................................... 18
CHAPTER 2 .............................................................................................................. 20
2.1 MATHEMATICAL MODELING AND ANALYSIS OF DIE CASTING PROCESS ............ 20
2.1.1 Fluid Flow, Thermal and Solidification Analysis ...................................... 21
2.1.2 Thermal Stresses and Distortion Analysis in Casting and Die................... 28
2.1.3 Models accounting for Structural Loads .................................................... 35
2.2 PARAMETRIC DIE DESIGN STUDIES .................................................................... 49
2.3 RELEVANT RESEARCH IN INJECTION MOLDING .................................................. 51
2.4 SUMMARY .......................................................................................................... 58
CHAPTER 3 .............................................................................................................. 59
3.1 INTRODUCTION ................................................................................................... 59
3.2 FINITE ELEMENT MODELING .............................................................................. 60
3.2.1 Boundary Conditions and Constraints ....................................................... 62
3.2.2 Loads and Assumptions ............................................................................. 65
3.2.3 Material Properties ..................................................................................... 65
3.2.4 Finite Element Model Predictions.............................................................. 66
3.2.5 Effect of Element Types and Cover Platen Constraint on Model Predictions
............................................................................................................................. 70
CHAPTER 4 .............................................................................................................. 75
4.1 INTRODUCTION ................................................................................................... 75
4.2 DESIGN OF EXPERIMENTS ................................................................................... 76
4.3 DIMENSIONAL ANALYSIS AND EMPIRICAL CORRELATIONS ................................ 81
4.3.1 Determination of the Model form and Non-Dimensional Parameters for
Predicting Parting Plane Separation.................................................................... 83
4.3.2 Empirical Correlation to Predict Ejector Side Parting Surface Separation 89
4.3.3 Empirical Correlation to Predict Cover Side Parting Surface Separation.. 95
ix
4.4 SENSITIVITY OF PARTING PLANE SEPARATION TO VARIATIONS IN STRUCTURAL
DESIGN PARAMETERS .............................................................................................. 97
4.4.1 Response Surface Plots of Non-Dimensional Parameters.......................... 97
4.4.2 Response Surface Plots of Explicit Design Variables.............................. 103
4.5 MODEL ADEQUACY .......................................................................................... 107
4.5.1 Rules to Characterize the Spans between Pillars and the Spans between
Pillars and Rails................................................................................................. 112
4.6 PLATEN STIFFNESS CHARACTERIZATION AND DETERMINATION OF PLATEN
THICKNESS PARAMETER TO BE USED IN POWER LAW MODELS ............................. 114
4.6.1 Methodology to Determine Equivalent Cover Platen Thickness ............. 117
4.6.2 Methodology to Determine Equivalent Ejector Platen Thickness ........... 120
4.6.3 Methodology to Determine Equivalent Thicknesses for Platens with
Different Toggle Locations ............................................................................... 122
4.7 DETERMINATION OF EQUIVALENT STIFFNESS OF A DIE CASTING MACHINE USING
A LUMPED ELEMENT MODEL ................................................................................. 125
4.8 SUMMARY ........................................................................................................ 131
CHAPTER 5 ............................................................................................................ 133
5.1 INTRODUCTION ................................................................................................. 133
5.2 DESIGN OF EXPERIMENTS ................................................................................. 134
5.3 DIMENSIONAL ANALYSIS AND EMPIRICAL CORRELATION TO PREDICT TIE BAR
LOADS .................................................................................................................... 138
5.4 MODEL ADEQUACY .......................................................................................... 146
5.4.1 Model Adequacy Study using Experimental Measurements ................... 150
5.4.2 Comparison of Experimental Measurements and Model Predictions...... 154
5.5 RESPONSE SURFACE PLOTS FOR THE EFFECT OF DIE LOCATION AND CAVITY
LOCATION ON TIE BAR LOADS ............................................................................... 157
5.6 SUMMARY AND CONCLUSIONS ......................................................................... 161
x
CHAPTER 6 ............................................................................................................ 163
6.1 CONCLUSIONS FROM THE POWER LAWS TO PREDICT PARTING PLANE
SEPARATION ........................................................................................................... 163
6.2 CONCLUSIONS FROM THE MACHINE CHARACTERIZATION STUDY ..................... 165
6.3 CONCLUSIONS FROM THE POWER LAWS TO PREDICT TIE BAR LOADS .............. 166
6.4 FUTURE WORK ................................................................................................. 168
REFERENCES........................................................................................................ 170
APPENDIX A .......................................................................................................... 176
A.1. PROCEDURE TO SELECT SAMPLE NODES AND PREDICT PURE DISTORTION OF
THE PARTING PLANE FROM FINITE ELEMENT MODELS........................................... 179
A.2. ALTERNATE METHOD TO REMOVE PSEUDO RIGID BODY MOTION USING A
LOCAL COORDINATE SYSTEM IN ABAQUS........................................................... 181
APPENDIX B .......................................................................................................... 183
B.1
LINEAR MODEL FOR TOP TIE BAR .............................................................. 183
B.2
LINEAR MODEL FOR BOTTOM TIE BAR ...................................................... 183
xi
LIST OF FIGURES
Figure
Page
Figure 1.1: Schematic of an open-close die [5]............................................................. 3
Figure 1.2: Schematic of a Hot Chamber Die Casting Machine [66] ........................... 5
Figure 1.3: Schematic of a Cold Chamber Die Casting Machine [66] ......................... 5
Figure 1.4: Free Body Diagram of Cover and Ejector Dies........................................ 10
Figure 1.5: Free Body Diagram of Die Casting Machine and the Die........................ 12
Figure 2.1: Schematic of Toggle Spring Platen Model [1] ......................................... 38
Figure 2.2: Schematic of the Models Considered in [30] ........................................... 41
Figure 2.3: Schematic of Pistons and Connecting Rods of Injection Molding Machine
Clamping Mechanism [50].......................................................................................... 53
Figure 2.4: Representation of the Clamping Mechanism in Multi-body Dynamics
Simulation [50]............................................................................................................ 54
Figure 2.5: Schematic of the procedure to estimate mold deflection [52] .................. 56
Figure 2.6: Mold Spring Diagram used to Estimate Gap Formation [52]................... 57
Figure 3.1: Geometry of the Part used in the Study .................................................... 61
Figure 3.2: Schematic of the Finite Element Model ................................................... 62
Figure 3.3: Boundary Conditions used between Cover Platen and Base .................... 64
Figure 3.4: Schematic of the Finite Element Model showing the Location of Tie Bar
Load Prediction ........................................................................................................... 67
Figure 3.5: Illustration of Ejector and Cover Side Parting Plane Separation.............. 68
Figure 3.6: Pseudo Rigid Body Movement Caused by Stretching of Tie Bars........... 69
Figure 3.7: Deflection Plots of the Cover Platen ........................................................ 73
xii
Figure 3.8: Deformed Plot of Cover Platen Superimposed on the Undeformed Plot . 73
Figure 4.1: Side View of an Ejector Die used in the Study ........................................ 77
Figure 4.2: Schematic of Pillar Patterns used in the Study ......................................... 78
Figure 4.3: Length Scales Representing the Unsupported Span behind the ............... 91
Figure 4.4: Non-Dimensional Cover Separation vs. Non-Dimensional Die Length
(П4) and Distance between Tie Bars (П1) ................................................................... 98
Figure 4.5: Non-Dimensional Cover Separation vs. Non-Dimensional Die Length
(П4) and Platen Thickness (П2) ................................................................................... 99
Figure 4.6: Non-Dimensional Ejector Separation vs. ............................................... 100
Figure 4.7: Non Dimensional Ejector Separation vs. Non Dimensional Platen
Thickness (П1-2) and Weighted Average of Spans (П4a+ 1.6П4b)............................. 101
Figure 4.8: Non-Dimensional Ejector Separation vs. ............................................... 102
Figure 4.9: Maximum Cover Separation vs. Die Thickness & Die Length .............. 104
Figure 4.10: Maximum Cover Separation vs. Die Thickness & Platen Thickness... 104
Figure 4.11: Maximum Ejector Separation vs. Die Thickness & Pillar Diameter.... 105
Figure 4.12: Maximum Ejector Separation vs. Die Thickness & Die Length .......... 106
Figure 4.13: Maximum Ejector Separation vs. Die Thickness & Die Length .......... 107
Figure 4.14: Pillar Arrangement Patterns in the three Test Cases Used to Study the
Adequacy of the Power Law Models ........................................................................ 108
Figure 4.15: Illustration of Rules for Characterizing the Spans behind the Ejector Die
................................................................................................................................... 113
Figure 4.16: Schematic of the Platen Design Chosen to Demonstrate Stiffness
Characterization Methodology.................................................................................. 116
Figure 4.17: Dimensions of the Platen Design Chosen to Demonstrate Stiffness
Characterization Methodology (All Dimensions in Inches) ..................................... 117
Figure 4.18: Schematic of Finite Element Model Used to Determine the Cover Platen
Stiffness..................................................................................................................... 119
Figure 4.19: Deflection Vs Load Curves for Cover Platen ....................................... 119
xiii
Figure 4.20: Schematic of the Finite Element Model Used to Determine the Ejector
Platen Stiffness.......................................................................................................... 121
Figure 4.21: Deflection Vs Load Curves for Ejector Platens.................................... 122
Figure 4.22: Schematic of Finite Element Models used to determine the Stiffness of
Four Toggle (Right) and Two Toggle (Left) Ejector Platens.................................... 123
Figure 4.23: Deflection Vs Load Curves for Two Toggle and Four Toggle Platens 124
Figure 4.24: Spring Stiffness Diagram for the Die and the Machine under Clamp
Load........................................................................................................................... 126
Figure 4.25: Spring Diagram with Clamp and Pressure Loads................................. 129
Figure 5.1: Coordinate System and Tie bar Labels viewed from inside face of Cover
Platen......................................................................................................................... 135
Figure 5.2: Schematic of the Finite Element Model of the 1000 Ton Machine and 250
Ton Machine Used for Model Adequacy Study [1].................................................. 148
Figure 5.3: Schematic of the test die on the machine platens .................................. 152
Figure 5.4: Schematic of the locations of tie bars, strain gauges and coordinate
system, viewed from front of cover platen................................................................ 152
Figure 5.5: Tie bar Load Measurements vs. Predictions from the Regression Model
................................................................................................................................... 156
Figure 5.6: Effect of Cavity Location on Tie Bar Load ............................................ 158
Figure 5.7: Effect of Cavity Location on Tie Bar Load ............................................ 159
Figure 5.8: Effect of Die Location on Tie Bar Load................................................. 160
Figure 5.9: Effect of Die Location on Tie Bar Load................................................. 160
Figure A.1: Contact Pressure Plot ............................................................................. 180
Figure A.2: Ejector Side Separation Obtained by Sampling Nodes in
Contact Regions Only ............................................................................................... 181
Figure A.3: Separation With Respect to a Local Coordinate System on the Parting
Surface....................................................................................................................... 182
xiv
LIST OF TABLES
Table
Page
Table 3.1: Specifications of the Machine used in the Study ....................................... 61
Table 3.2: Material Properties for ST4140.................................................................. 66
Table 3.3: Die and Machine Parameters used in the Simulations to Test the Effect of
Cover Platen Constraint Type and Element Type....................................................... 71
Table 3.4: Effect of Element Type and Cover Platen Boundary Condition on Parting
Surface Separation Prediction ..................................................................................... 72
Table 3.5: Effect of Element Type and Cover Platen Boundary Condition on Tie Bar
Load Prediction ........................................................................................................... 74
Table 4.1: Factors used in Design of Experiments...................................................... 76
Table 4.2: Response Surface Experimental Array ..................................................... 79
Table 4.3: Summary of Element Types and Constraints used in Computational
Experiments................................................................................................................. 81
Table 4.4: Non-Dimensional Structural Design Parameters ....................................... 87
Table 4.5: Parameter Estimates for Ejector Side Fit ................................................... 90
Table 4.6: Parameter Estimates for Cover Side Fit..................................................... 95
Table 4.7: Summary of the Parameter Values used for Test Cases .......................... 109
Table 4.8: Comparison of FEA and Power Law Model Predictions for Ejector Side
for the Test Cases ...................................................................................................... 110
Table 4.9: Comparison of FEA and Power Law Model Predictions for Ejector Side
for the Test Cases ...................................................................................................... 111
xv
Table 4.10: Stiffness Values to be used in Lumped Element Model for the Dies and
Machine Parts............................................................................................................ 127
Table 5.1: Description of Variables used for Tie Bar Load Prediction Model
Development ............................................................................................................. 135
Table 5.2: Experimental Array Used for Tie bar Load Prediction Model Development
................................................................................................................................... 138
Table 5.3: Non Dimensional Parameters used in Tie Bar Load Prediction Model... 142
Table 5.4: Parameter Estimates for Top Tie Bar Model Fit...................................... 144
Table 5.5: Parameter Estimates for Bottom Tie Bar Model Fit ................................ 144
Table 5.6: Summary of Finite Element Models used for Model Adequacy Study ... 147
Table 5.7: Comparison of Model Predictions for a 3500 Ton Machine ................... 148
Table 5.8: Comparison of Model Predictions for a 1000 Ton Machine ................... 150
Table 5.9: Comparison of Model Predictions for a 250 Ton Machine
(DPX=0”, DPY=-3.14”, CPX=0”, CPY=-0.423”, CPR=10000 PSI) ....................... 150
Table 5.10: Experimental Array................................................................................ 153
Table 5.11: Comparison of Tie bar Load Measurements and Tie bar Load Predictions
from the Regression Model ....................................................................................... 155
Table 5.12: Difference between Measurements and Model Predictions................... 157
xvi
CHAPTER 1
INTRODUCTION
Die casting is a net shape manufacturing process in which parts with complex
geometries are produced by injecting molten metal into steel molds/dies under high
pressure. The molten metal is held in the die cavity until it solidifies and the final part
is ejected out of the dies and the process is repeated over thousands of cycle. Die
casting offers competitive advantage over other net-shape manufacturing process
such as forging and stamping with its ability to produce parts with complex geometric
features, high surface finish and tight dimensional tolerances. A casting that is
distorted and fails to meet the specified dimensional requirements is scrapped and
remelted resulting in a decrease in process yield, loss of production time, labor and
increase in cost and energy consumption associated with remelting and rework.
One of the major factors that contribute to the dimensional inaccuracy of the casting
is the elastic deformations of the die cavity caused by the thermo mechanical loads
the dies are subjected to during normal operation. Deformations caused by repeated
loading and unloading of the die might also cause fatigue failure of the die. Die
casting dies are expected to run several million cycles during their life time. High
manufacturing costs prohibit prototyping and any serious die deformation or die
1
failure problems are not noticed until the first production run. A die can cost
anywhere between $50000 to $1000000 and the delivery times ranges from 3-12
months depending upon the complexity of the dies. Therefore it is extremely
important that the die distortion be predicted and controlled at the design stage.
Numerical modeling of the die casting process is the most efficient way to predict the
distortion of the dies. However, the die designer should have a thorough
understanding of the physical phenomenon involved in die distortion and the
mathematical theory employed in the numerical models to efficiently model the die
distortion phenomenon. During the initial stages of the die design, where many
primitive design iterations have to be performed quickly, numerical modeling
techniques become time consuming and they are not cost effective. Therefore the goal
of this research work is to develop off the shelf tools and guidelines for structural
design of die casting dies.
1.1 Die Casting Dies and Machines
Die casting dies are complex high performance structures with several individual
components and mechanisms that are assembled together to meet its functional
requirements. The dies are usually made of at least two halves, the fixed half or the
cover half and the moving half or the ejector half to facilitate the removal of the
solidified casting. The surface where the two halves of the dies meet is called the die
parting surface. The dies can also have moving mechanisms such as slides and cores
2
to produce undercuts and holes in the casting. Dies with these moving mechanisms
are called non-open close dies and dies without any of these moving mechanisms are
called open close dies. A schematic of an open-close die is shown in Figure 1.1. A
metal feeding system composed of runners and gates are cut out on the die to evenly
distribute the molten metal into the die cavity at required velocity and pressure.
Cooling lines are provided to extract the heat from the solidifying metal and the
location and design of cooling lines depend on the geometry of the part.
Figure 1.1: Schematic of an open-close die [5]
Die casting machines provide the clamping mechanism to hold the die halves together
when they are subjected to high pressure loads from the incoming molten metal. They
also provide the injection mechanism that injects the molten metal into the die cavity
at required velocity and pressure. Die casting machines are typically rated based on
the clamping force it can generate and the machine rating ranges from 50 tons to 4000
3
tons. Regardless of their size, die casting machines are usually classified as hot
chamber machines or cold chamber machines depending on the method used to inject
moltenmetal into a die. A schematic of a hot chamber machine is shown in Figure 1.2.
The injection mechanism of a hot chamber machine is immersed in the molten metal
bath of a metal holding furnace. The furnace is attached to the machine by a metal
feed system called a gooseneck. As the injection cylinder plunger rises, a port in the
injection cylinder opens, allowing molten metal to fill the cylinder. As the plunger
moves downward it seals the port and forces molten metal through the gooseneck and
nozzle into the die cavity. Hot chamber machines are used primarily for low melting
point alloys such as zinc and copper that do not readily attack and erode the
components in the machine’s injection system. In a cold chamber machine, the
molten metal is poured into a cylindrical sleeve and a hydraulically operated plunger
seals the cold chamber port and forces metal into the die at high pressures. A
schematic of the cold chamber machine is shown in Figure 1.3. Cold chamber
machines are used for alloys such as aluminum, magnesium and other alloys that
chemically attack steel.
4
Figure 1.2: Schematic of a Hot Chamber Die Casting Machine [66]
Figure 1.3: Schematic of a Cold Chamber Die Casting Machine [66]
5
A typical die casting machine consists of a cover/fixed platen, a movable/ejector
platen, a rear platen and four tie bars stretching between the cover and rear platens.
The tie bars are secured to the cover and rear platens and the ejector platen is free to
slide on the tie bars. The cover and ejector halves of the dies are mounted on the
cover and ejector platens of the die casting machine respectively. The toggle
mechanism that clamps the dies together is provided between the cover and rear
platens and the injection mechanism is provided behind the cover platen.
1.2 Mechanical and Thermal Loads involved in Die Casting
The major thermal and mechanical loads that act on a die casting die during normal
operation are (1) The clamp load (2) Injection pressure (3) the impact load caused by
the sudden deceleration of the plunger mechanism at end of fill (4) Intensification
pressure and (5) the heat released by the molten metal during filling and
solidification.
The clamp load applied behind the ejector platen causes the tie bars to stretch
between the cover and rear platens and tensile forces are developed on the tie bars.
Thus the clamp load is transmitted to the die parting surfaces through the machine tie
bars and the cover platen. The clamp load holds the two die halves together and
prevents the dies from opening up during the metal injection stage. The clamp load is
kept constant through out the casting cycle.
The injection pressure is applied to the molten metal by the hydraulic plunger
mechanism to fill the cavity in fractions of seconds before the metal solidifies. Die
6
castings are characterized by thin wall sections and the fill time is usually in the order
of milliseconds. If premature solidification occurs along the metal flow front, it
results in an incompletely filled cavity and leads to defective parts. Therefore to avoid
premature solidification the entire filling process is completed in a fraction of seconds
by injecting the metal at pressures as high as 10000 PSI.
After the injection stage a pressure spike is given to the metal to force the metal into
far ends of the cavity. This pressure spike also helps to reduce shrinkage defects by
forcing the remaining liquid metal to solidifying areas. It is typical to have
intensification pressure values twice as high as the injection pressure.
The molten metal is pushed into the die cavity by the plunger mechanism which is
driven by a hydraulic piston. Initially the metal is pushed at a lower velocity to avoid
gas entrapment and this phase is referred to as slow shot phase. When a certain
volume of the shot sleeve is filled, the plunger velocity is increased to achieve the
required metal pressure at the gates. This phase is called fast shot phase. After the fast
shot phase, the plunger and the hydraulic piston are brought to a sudden stop which
causes an impact loading in the die cavity.
Finally heat is released by the molten metal during the filling and solidification. The
loading sequence is repeated over thousands of cycles during the entire lifetime of the
die.
7
1.3 Die Deflections
The cyclic thermal and mechanical loads described in previous section, cause
deflections of the dies during each cycle. The elastic deflections could cause
dimensional variations in the die cavity and result in parts that are out of dimensional
tolerances. The deflections along the edges of the cavity could cause the molten metal
to escape out of the cavity and this phenomenon is called flash. Die flashing is a
major problem in die casting resulting in increased cycle time and increased cost
associated with the removal of flash from the final casting. Die flashing could also
affect the across parting plane dimensions of the casting and it could cause uneven
loading of various machine components. Flash might also get into the clearance holes
between the moving mechanisms of the dies, causing process problems. As a worst
case scenario, the repeated loading and unloading of the die could also lead to fatigue
failure of the die.
The effect of mechanical loads on the dies is illustrated in the free body diagram
shown in Figure 1.4. Ideally the parting surfaces of the die casting dies are machined
flat to mate perfectly in the absence of thermal and mechanical loads. In the first stage
of casting cycle (Figure 1.4a) when the clamp load is applied behind the dies, the die
parting surfaces are subjected to distributed compressive forces that are equivalent to
the applied clamp load. The clamp load applied between the rear and ejector platens
is transmitted to the cover die through the cover platen, whereas on the ejector side
the load path is through the pillars and/or rail support. The pillars and rails behind the
8
ejector die are shown in Figure 1.4. In the next stage of the casting cycle (Figure
1.4b), when the molten metal is injected at high pressure, the pressure load on the
cavity relieves some of the compressive forces at the parting surface. The clamp force
in excess of the cavity pressure load remains at the parting surface as shown in Figure
1.4c. In Figure 1.4, the clamp and pressure loads are assumed to be static.
During the metal injection and solidification stages the dies are also subjected to
thermal loads from the solidifying metal. Due to the heat flux from the metal, the
regions of dies that are close to the cavity attains higher temperatures than the regions
far away from the cavity causing uneven thermal expansion of the die. The uneven
thermal expansion could increase and redistribute the compressive stresses at the
parting surface. However the uneven growth and thermal distortion of the die depends
on the geometry of the casting and this issue has to be addressed by ensuring a good
part design and by proper placement of cooling lines.
9
Figure 1.4: Free Body Diagram of Cover and Ejector Dies
If the dies were perfectly rigid as shown in Figure 1.4, the pressure load would only
tend to relieve the compressive forces at the parting surface. Since the dies are not
perfectly rigid, they will undergo elastic deformation due to the clamp and pressure
loads. The magnitude of elastic deflections of the dies under clamp and pressure loads
will depend on the structural design of the dies and the rigidity of the support
provided by the machine platens. The dies are similar to flat plate structures resting
on elastic supports. The degree of deflection of these elastic supports will also add up
10
to the deflection of the dies. On the cover side, the die is mounted directly on the
cover platen and the cover platen serves as the elastic support for the die under clamp
and pressure loads. A stiffer cover platen will be subjected to less bending and hence
the support area available for the cover die will be more. In other words the stiffness
of the cover platen and the die foot print determines the unsupported span behind the
cover die. Therefore the deflection on the cover side is mostly dependent on the
stiffness of the cover platen and the die foot print. But on the ejector side, due to the
presence of ejection mechanisms the rails and support pillars provide the necessary
support between the ejector platen and the die. Therefore the deflection on the ejector
side will largely be a function of the thickness of the die and the insert and the
location, size and number of support pillars. The deflections due to mechanical loads
can be controlled by ensuring a good structural design of the dies and selecting a
suitable machine with appropriate stiffness characteristics.
1.4 Imbalanced loads on Machine tie bars
The other problem that is often encountered during die casting operation is
unbalanced loads on the tie bars of the die casting machine. The machine tie bars are
constrained to the cover and rear platens and the ejector platen is free to slide on the
tie bars. When the clamp load is applied the tie bars stretch between the cover and
rear platen and tensile forces are developed on the tie bars. This is illustrated in
Figure 1.5 which shows the free body diagram of a die casting die and machine with
the clamp and pressure loads acting on them. Figure 1.5 shows a case where the
11
geometric center of the dies and cavity center of pressure coincide with geometric
center of the platen. If the die is centered on the platen the moments about all four tie
bars will be equal and hence all four tie bars will carry equal loads.
Figure 1.5: Free Body Diagram of Die Casting Machine and the Die
However if the die is located off center on the platen it results in unequal moments
and loads on the four tie bars. As mentioned before if the dies and machine are
perfectly rigid the pressure load will only tend to reduce the compressive forces on
the die parting surface and it will not alter the tie bar loads. Since the machine is not
rigid the tie bars will be stretched further slightly after injection stage and the tie bar
loads increase further. The increase in tie bar loads after the injection stage will be
proportional to the stiffness of the tie bars and the location of the cavity center of
pressure with respect to the platen center.
12
If the four tie bars do not carry equal loads, the dies close unevenly at the die parting
surface and die flashing may occur. In extreme cases, poorly balanced tie bar loads
could also lead to tie bar failure. The common practice to overcome the tie bar load
imbalance problem is to adjust the length of the tie bars between the platens so that all
the four tie bars carry equal loads. In such a case the minimum clamp load required to
hold the dies together will be higher than the one that would be needed if the dies
were centered on the platen thus limiting the capacity of the machine.
1.5 Problem Statement
Numerical modeling of the die casting process using techniques such as the finite
element method is the most efficient way to predict die deflections and problems such
as flash at the design stage and to enable suitable design modifications. Die distortion
modeling using finite element method has been an active area of research at the
Center for Die Casting at Ohio State University. Numerous literatures have been
published by this die casting research group on realistic approximations of
die/machine geometry, loads and boundary conditions that can be used in the finite
element models to predict die distortion.
However, to make efficient design improvements based on the numerical model
predictions, the die designer should have a thorough understanding of how the
various structural design variables contribute to the die deflection. Therefore design
guidelines explaining the sensitivity of die deflection to various design parameters are
essential. Moreover, some die designers lack FEA support and they are solely
13
dependent on tools such as closed form analytical expressions, design guidelines,
charts etc. An initial set of guidelines was developed in previous research studies [14]. Computational experiments were conducted to study the sensitivity of the parting
plane separation to the major structural die/machine design variables viz., platen
thickness, die thickness, insert thickness and the location of the die with respect to the
platen center. Functional relationships describing the relative contribution of these
variables to the parting plane separation on cover and ejector sides were established
using polynomial approximations. One of the major conclusions from those studies
was that the cover platen thickness is the dominant factor on the cover side that
contributes to the parting plane separation. The variables that characterize the
unsupported span behind the ejector die such as the size, number and location of
pillars supports were not controlled in those computational experiments. Hence the
contribution of the ejector side design variables to the parting plane separation could
not be confirmed from those studies and it remained an open issue.
The other issue that needs to be investigated is the relative contributions of the
location of the dies on the platen and the location of cavity center of pressure to the
tie bar load distribution. To determine the amount of length adjustments needed on
the tie bars to balance the loads, Die Casting process engineers usually need a prior
estimate of the loads on the tie bars. Herman [5] presents a methodology based on
moment equilibrium calculations to estimate the loads on the tie bars. This method
assumes that the cavity load is equivalent to the machine clamp load. The location of
cavity center of pressure is considered for the moment calculations and location of the
14
die on the platen is completely ignored. This method also assumes the dies and the
machine as perfectly rigid bodies. The mathematical formulation of the tie bar load
prediction problem makes it a statically indeterminate system with four unknown
loads and three equations viz., a force balance equation in tie bar direction and two
moment balance equations in directions perpendicular to the tie bar direction. Herman
[5]
approaches this problem by applying a moment balance on the operator and
helper side tie bars and estimates the sum of the top and bottom tie bars on both sides.
The total load on each side of the machine are then split between the top and bottom
tie bars by assuming that the sum of the loads on top and bottom tie bars on either
side of the machine are equal to the cavity load. This approach clearly violates the
force equilibrium constraint. Due to these drawbacks this approach generally will
produce inaccurate predictions of the tie bar loads.
1.6 Research objectives
One of the main objectives of this research is to study the relative contributions of the
cover and ejector side structural design variables to die deflection and develop design
guidelines and tools to aid in structural design of the die casting dies. It was decided
to develop closed form approximations of the relationship between the structural
design parameters and the maximum distortion on the cover and ejector side parting
surfaces. These expressions can also be used to make a quick estimate of the
magnitude of the die deflections at the initial design stage. The closed form
expressions can be used to obtain the best estimate of die design parameters that
15
would result in minimal parting plane separation. The other purpose of developing
these closed form expressions is to obtain the sensitivity of the maximum cover and
ejector parting surfaces to variations in the structural design parameters and gain an
understanding of the relative contributions of these design variables to die distortion.
The second objective of the research is to develop a closed form expression to predict
the loads on the tie bars of the die casting machine by taking into account the location
of the dies on the platens, location of cavity center of pressure with respect to the
platen center, the clamp load, the cavity pressure load and the rigidity of the die and
the machine.
1.7 Research Contributions
One of the major contributions of this research work is the development of non-linear
power law models to predict the maximum deflection on the cover and ejector side
parting surfaces. Though linear polynomial models have been developed in previous
research studies to predict the maximum deflection on the cover and ejector parting
surfaces, these polynomial models produced inaccurate predictions in certain regions
of the design space. Power law models were obtained using dimensional analysis
based on Buckingham Pi-theorem. The use of non dimensional parameters and power
law models gives a better understanding of the relationship among the various
physical and geometric parameters that affect the parting plane separation and it also
increases the degree of confidence in the model predictions even outside the design
space considered in this study. The power law models were also used to generate
16
response surface plots describing the behavior of the function in response to
variations in design parameters. These plots will be useful to gain a better
understanding of the relative effects of the design variables on maximum parting
plane separation. Since the power law models represent the stiffness characteristics of
the die/machine system at the specified level of parameters, these models can be used
to evaluate and compare the structural performance of different die/machine systems.
The second major contribution of this work is a methodology to characterize the
stiffness of machine platens of different sizes and designs. The power law models to
predict parting plane separation were developed based on an 8.9 MN (1000 ton) four
toggle machine. The stiffness of the machine was represented in the power law
models by the platen thickness and the distance between the tie bars. Using the platen
thickness characterization method, an equivalent platen thickness parameter can be
estimated for cast platens with ribs and also for two toggle platens. The equivalent
platen thickness parameter can be used in the power law models to predict maximum
parting plane separation and understand the performance of the die/machine. A one
degree of freedom lumped stiffness model for the die/machine is also developed to
determine the total stiffness of the die and machine. The lumped model also provides
an understanding of how the interaction between various machine components and
the dies contribute to the total stiffness of the die.
The other major contribution of this research work is the development of a closed
power law model to predict the loads on the tie bars of the die casting machine. The
17
power law model form to predict the tie bar loads were also obtained using
dimensional analysis. Unlike Herman’s approach this model takes into account the
location of the dies on the platen, the location of the cavity center of pressure, the
magnitude of the cavity pressure and the clamp load. The model adequacy has also
been verified using tie bar load measurements form a die casting machine.
1.8 Dissertation Outline
The dissertation is organized into 6 Chapters. In the second chapter some of the
recent scientific literature on die casting die design is reviewed. A review of relevant
research studies in other casting processes and injection molding are also included in
chapter 2. Chapter 3 presents the methodology adopted in this research. The finite
element modeling procedure that was used to predict parting plane separation and tie
bar loads are presented. The assumptions on loads and boundary conditions used in
the finite element model are discussed.
The power law models that were developed to predict the maximum parting plane
separation on the cover and ejector side are presented in Chapter 4. The experimental
design, dimensional analysis, assumptions on the model form and the model
development procedure are all presented in this chapter. The sensitivity of parting
plane separation to variations in the design variables are described using the power
law models and response surface plots. Chapter 5 describes the computational
experiments that were conducted to develop the power law model to predict the loads
on the tie bars of the die casting machine. Dimensional analysis and the non-linear
18
model fitting procedures are also presented. Comparison between the power law
predictions and FEA predictions for machines of different designs and sizes is also
presented and the experimental work conducted to test the adequacy of the tie bar
load prediction model is also discussed in Chapter 5. The results and conclusions of
this dissertation work are summarized in chapter 6 and recommendations for
structural die design are presented in this chapter.
19
CHAPTER 2
LITERATURE REVIEW
This chapter provides an overview of recent developments in numerical modeling of
die casting and related processes and the application of these methods in die design.
Some of the relevant analytical and experimental work in die design and analysis is
also included in the review. Finally some related studies on the mechanical
performance of injection molding die and machine are presented.
2.1 Mathematical Modeling and Analysis of Die Casting Process
A die casting die has to meet various functional requirements: (1) Distribute the
molten metal uniformly into the cavity, (2) Should allow efficient removal of heat
from the molten metal and (3) Preserve the dimensional integrity of the die cavity.
The die design task is decomposed into these functional elements and design
solutions that meet each of these functional requirements are sought based on the
physical principles that govern these functions. There are a variety of physical
phenomena that takes place during the die casting process such as metal flow, heat
transfer, solidification and thermo mechanical distortion of the die, the casting and the
machine. Decomposing the design tasks establishes a clear boundary between these
phenomena and the analyst solves the mathematical governing equations of either one
20
or more of the phenomenon to obtain a design solution. Numerical techniques such as
finite element method, finite difference method and boundary element method are
being extensively used to model and solve these governing equations.
The focus of the die casting research community has been mostly on developing
numerical/computational methods to solve heat transfer, fluid flow, solidification and
thermal distortion related problems. Very few researchers have paid attention to the
role of mechanical loads such as clamping force and cavity pressure in die and casting
distortion. Even in the models where mechanical loads are considered, not all of them
account for the stiffness of the machine parts such as the platens, tie bars and the
toggle mechanism. Some of the recent improvements in the numerical modeling
methods in die casting and other similar processes in heat transfer, fluid flow and
structural distortion are reviewed here.
2.1.1 Fluid Flow, Thermal and Solidification Analysis
The analysis of fluid flow, heat transfer and solidification phenomenon can predict
potential problems and defects in the casting such as gas porosity, shrinkage porosity,
hot spots, hot tears, etc and these types of analysis helps in proper design of cooling
lines, metal feeding systems and the casting geometry. Die filling analysis is usually
carried out to track the advancing flow front and predict possibilities for gas
entrapment in the cavity. Fluid flow analysis coupled with heat transfer helps to
identify other flow related problems such as a cold shut. The temperature profile
obtained at the end of filling can be used for further solidification analysis to identify
21
potential problems such as hot spots and uneven cooling. To model the fluid flow in
casting processes the fluid is usually assumed to be newtonian with laminar or
turbulent behavior. A complete flow and solidification model should solve for the
Navier-Stokes equation, the continuity equation and energy equation simultaneously.
The continuity or mass conservation equation, the Navier stokes equation and the
energy equation are given as
∇⋅u = 0
ρ
(2-1)
(
)
∂u
+ u ⋅ ∇u = −∇p + ∇ ⋅ µ ∇u + (∇u )T + ρb
∂t

 ∂T
ρC p 
+ u ⋅ ∇T  = ∇.(k∇T ) + Qgen

 ∂t
(2-2)
(2-3)
Where, ρ is the density, u is the velocity vector, t is time, P is pressure, µ is viscosity,
b is the body force, Cp is the specific heat, T is the temperature, K is the thermal
conductivity and Qgen is the heat source term.
22
For the solidification problem, the heat source term in the energy equation, Qgen
represents the latent heat of fusion evolved during solidification and it is given by
Qgen = L
df s
dt
(2-4)
Where, L is the latent heat, fs is the fraction of solid. The evolution of the fraction of
solids for a particular alloy is usually determined by microscopic models such as
Lever or Scheil equations. The release of latent heat causes an increase in the
enthalpy and hence the latent heat can be accounted for in the model using the
following relation
Cp =
H(t + ∆t) − H (t )
[T (t + ∆t ) − T (t )]
(2-5)
This model is often referred to in the literature as macroscopic specific heat model.
Since the specific heat capacity, Cp is temperature dependent the energy equation
becomes a second order non-linear partial differential equation and analytical
solutions can be obtained only by using further simplifying assumptions.
Nevertheless various numerical methods have been proposed in literature to obtain
approximate solutions for solidification.
Hetu et al [6], developed a 2.5D shell finite element model to simulate both laminar
and turbulent flow regimens in thin walled die castings. They also proposed 3D finite
element model for thick walled castings. The 2.5D model used a shell mesh to model
the thin wall sections with a thickness dimension for the shell elements and the model
23
does not account for the energy equation. In the 3D model both the part and the mold
were included in the computational domain and a laminar flow was assumed. The
Navier-stokes equations and energy equations representing the gas in the cavity and
the liquid metal were solved over the entire computational domain to obtain the
pressure, velocity and temperature distributions. The 3D model also accounted for the
latent heat of fusion released during phase change. Once the velocity profile was
obtained, the flow front was tracked using a fill factor F. A function F (x, t) denoting
the distance from the interface between the liquid and the gas was defined over the
entire computational domain to track the flow front. The flow front tracking equation
is then given by
∂F
+ U .∇F = 0
∂t
(2-6)
Where, U is the velocity field obtained during each time step. The flow rate
information based on the velocity field was used to evaluate the fill factor for each
element using the mass conservation equation. When the value of F exceeded a
critical value at any portion of the cavity, that portion was assumed to be filled.
Usually the fill factor ranges between 0 and 1, where zero indicates an empty element
and 1 indicates a completely filled element. The iterative procedure was repeated
until the filling of the die is completed.
Barone and Caulk [7], proposed a method in which the governing equations of fluid
flow were integrated through the cavity thickness instead of modeling the entire three
24
dimensional flow region and the fluid flow was described in terms of bulk velocity
and pressure resultant. The fill fraction approach shown in equation (2-6) was used to
track the motion of the flow front. The cavity gas was modeled as an ideal gas that
compresses adiabatically as the flow front advances and finite element method was
used to solve the governing equations. But the energy equation was not taken into
consideration in this method and isothermal fluid flow was assumed.
Jia [8], used a turbulent incompressible fluid flow model to obtain the pressure,
temperature and velocity profiles during cavity filling and the governing equations
were solved using a finite difference approach. Their method also employs the fill
factor tracking approach to track the flow front. The gas in the cavity was modeled as
ideal gas and the pressure obtained from the ideal gas equation was related to the
pressure at the free surface of the flow front.
Kulasegaram et al [9], proposed a mesh free Lagrangian particle method to model the
cavity fill in die casting processes. In this method the continuum is represented by a
large set of particles where each particle is described by its mass, position vector and
velocity vector. This method approximates the given function and its gradient in
terms of the values of the function at number of neighboring particles and a kernel
function. But this was a two dimensional model and the thermal effects were not
accounted for in their model. The authors argue that this method is computationally
less intensive than a finite element method.
25
Cleary et al [10] also suggested a similar particle hydrodynamics method to simulate
the molten flow. This method solved the ordinary differential equations as opposed to
solving the variational form described by Kulasegaram et al [9].
Bounds et al [11] described a hybrid numerical model in their paper to predict die
temperature. This method utilized a boundary element formulation for the die blocks
and finite element formulation for the casting. The temperature in the die blocks (Td)
was represented as two additive components; a steady state component shown in
equation (2-7) and a transient part shown in (2-8)
∇ 2 Td = 0
(
(2-7)
)
α d ∇ 2 Td =
∂Td
∂t
(2-8)
Applying appropriate boundary conditions the first part was solved to obtain the
steady state temperature. The second part was considered to be a transient
perturbation about the steady state temperature. The steady state boundary conditions
are taken as the time averaged boundary conditions as shown in (2-9) and the
perturbed boundary condition is given by (2-10)
qd = h( x,t ) [Td ( x,t ) − T0(x,t)]
pert
qd
= qd − q d
(2-9)
(2-10)
26
The governing equation for heat transfer in casting is given by
 ∂T 
∇ (kc ∇ Tc ) = Cc  c 
 ∂t 
(2-11)
Cc is the effective heat capacitance that accounts for the latent heat of solidification
of the casting. The first step to perform an analysis using this method is to perform a
steady state analysis using time averaged boundary condition. In the next step a
transient analysis over a single casting cycle is performed using perturbed boundary
conditions. Using the data from the transient analysis the boundary condition for the
next cycle of steady state analysis is updated. A number of transient cycles were
analyzed until a convergent die and casting temperature were obtained. The model
predictions showed good correlation with experimental thermocouple measurements.
However due to the updating of boundary conditions at the end of each cycle and the
coupling between the FE and BE models, the computational cost and time were
expensive.
Xiong et al [12] described three different methods to improve the computational
efficiency and stability of the finite difference scheme used for the thermal analysis of
the die casting. The temperature distribution in the casting and the die obtained from
a filling analysis at the end of solidification was used as initial conditions. In their
first method called a component wise splitting method, a simpler approximation in
each dimension was used for the Laplacian operator in the heat conduction equation.
The method was shown to be unconditionally stable. In their second method, an
27
irregular mesh was used on the casting and the dies, wherein a finer mesh was used in
the cavity areas with thin sections and coarser meshes were used in the areas far from
the cavity. In the third method, the computational domain was divided into a transient
area and steady state area. The transient area consists of the casting and the surface
layer in the die beneath the cavity where the temperature variations within a cycle
was faster. A smaller time step was used within the transient layer and a larger time
stepping which was a multiple of the time step size in the transient area was used in
the steady state area thus reducing the computational effort needed to solve the
problem.
2.1.2 Thermal Stresses and Distortion Analysis in Casting and Die
The temperature predictions at the end of a thermal and solidification analysis can
also be used as an input for subsequent stress analysis in the casting and the die.
Koric and Thomas [13] developed a computational algorithm to solve for the thermal
stresses, strains and displacements during the solidification process in continuous
casting. During each time step, the heat conduction equation including the latent heat
term was solved to obtain the temperature history in the casting. Then the temperature
values were used to predict stresses in the casting during the same time step. A small
strain assumption was used and an elastic-viscoplastic constitutive relation was used
to predict the stresses in the casting. The total strain in the elastic visco- plastic
models is given by
ε& total = ε& elastic + ε& inelastic + ε& thermal
28
(2-12)
The inelastic stress consists of the strain rate independent plasticity and the time
dependent creep and is defined by single variable, equivalent inelastic strain, ε& inelastic
given by the relation,
(ε& inelastic )ij = 3 ε& inelastic
2
σ′ij
σ
(2-13)
where σ′ij is the deviatoric stress component and σ is the average stress. To model
the stresses in the mushy zone two different approaches were taken. In the first
approach, an isotropic elastic-perfectly plastic rate-independent model was chosen for
elements where the temperature was greater than the solidus temperature for at least
one material point. A small yield strength value of 0.03Mpa was chosen to eliminate
the stresses in the mushy zone. In the second approach a viscoplastic relation was
implemented as a penalty function to generate inelastic strain in the mushy zone in
proportion to the difference between the average stress in the casting and a small
yield stress.
Song et al [14] used a non-linear coupled thermo mechanical finite element model to
predict the distortion of the casting in rapid tooling. In their study both the ceramic
matrix and the casting were treated as elastic-plastic deformable bodies. The material
properties of the casting were assumed to be temperature dependent and the material
properties of the matrix were assumed to be temperature independent. First the heat
transfer problem was solved to obtain the temperature distribution using the current
boundary conditions. A thermal contact was allowed between the casting and the
29
matrix and the heat generated due to plastic thermal work dissipation, friction and the
latent heat of solidification were accounted for in the thermal model. Then the
mechanical problem was solved using the temperature distribution obtained from the
heat transfer analysis. A mechanical contact boundary condition was also enabled
between the casting and the die. The displacement and temperature fields were
updated after the first increment to serve as a new configuration for the next
increment and the procedure was repeated until convergence was achieved.
A
displacement checking criteria was used to check the convergence and an adaptive
time stepping based on load control was used.
Dour [15], described a normalized approach to predict thermal stresses and distortion
in die casting dies. The different variables involved in the thermo-mechanical
problem was reduced to three non dimensional numbers viz., Biot Number (hL/K),
Normalized time (kt/L2) and Normalized stress (-σ/[αETa/(1-υ)]); where, h is the heat
transfer coefficient, L is the characteristic length of the slab/die, K is the thermal
conductivity of die material, k is the thermal diffusivity, t is the time spent between
filling and air gap formation, Ta is the temperature difference between the molten
metal bath and the die, E is the young’s modulus, α is the coefficient of thermal
expansion and υ is the Poisson ratio. These normalized parameters were used in the
thermal and mechanical governing equations and analytical expressions for
temperature normalized stress and radius of curvature was derived for one
dimensional case assuming appropriate thermal and mechanical boundary conditions.
The stresses and radius of curvature predicted by the analytical expressions were
30
summarized in plots and graphs for various values of normalized time, Biot number
and temperature. Since the graphs are obtained from the non dimensional form of the
equations it is claimed that these charts can be used to diagnose the thermal stresses
and distortion for any real casting condition.
Broucaret [16] used an inverse method based on Laplace transform of heat
conduction equation to determine the heat flux exchanged between the die and the
casting in gravity die casting process. One dimensional heat conduction was assumed
and a time varying heat flux and a time varying temperature were chosen as the
boundary conditions at the two ends of the die respectively. The heat conduction
problem was solved in the frequency domain and an expression for temperature as a
function of the heat flux at the front end of the slab and temperature at the back end of
the slab are obtained. The expression for temperature is then transferred back to the
time domain using inverse Laplace transform based on a numerical algorithm. The
heat flux in the temperature expression is estimated using the temperature
measurements and applying an inverse problem numerical algorithm. The
temperature measurements were also used to estimate the thermal stresses in the die.
31
The die material was considered elastic and the thermal stresses were estimated using
the following expression:
σ( x, t ) =

3( x + h) 2h
1 2h
1  
(
)
(
)
α
−
+
+
+
x
h
T
x
,
t
dx
E
T
x
t
T
(
x
,
t
)
dx
(
,
)
 
 (2-14)
∫
∫
2h 0
1 − υ  
2h 3 0

Where α is the coefficient of thermal expansion, E is the elastic modulus of the die
material, T(x, t) is the die temperature and h is the distance from the die surface.
Based on the one dimensional stress model, the effect of initial temperature of the die
and the die coating on the thermal stresses of the die was studied. It was concluded
that the more conductive the coating and the lower the initial temperature of the die,
higher the heat transfer and higher the stresses.
Lin [17] used finite element modeling and simulated annealing optimization method
to find the optimal cooling system design parameters to minimize the thermal
distortion of the dies. In this methodology a finite element software MARC/Mentantsolid was used to solve the coupled thermo-mechanical problem and predict die
distortion. A case study with fifteen different cooling system designs were analyzed
and the parameters that were varied in the case study were the distance between the
cooling lines and the cavity, the distance between the cooling lines and the diameter
of the cooling lines. The results of these analyses were used to construct a neural
network describing the relationship between the cooling system parameters and die
deformation. The polynomial equations of the neural network were used as objective
32
functions for the constrained optimization problem to minimize the die deformation.
A simulated annealing method was then used to solve the optimization problem.
Srivastava et al [18] showed that the stress and strain predictions from finite element
analysis can be utilized to predict the thermal fatigue life of a die casting dies. The
model predictions were correlated with laboratory fatigue tests. In the laboratory
fatigue tests, the fatigue loads were applied to test coupons by repeatedly immersing
them in molten aluminum bath and then quenching them in water at room
temperature. The dip times were varied and different loading conditions were studied.
The finite element model used to simulate the different test conditions and the finite
element model was solved using DEFORM 3D package. The principal stresses were
obtained from the finite element analysis. The directions of principal planes were
determined and the laboratory fatigue tests showed that the direction of cracking was
always perpendicular to the direction of the principal stress.
The stress values obtained from the finite element analysis was used to predict the
fatigue life of the die using a modified coffin mason equation which is given by
Cε f



N F = 

(
)
(
)
α∆T
1
υ
∆σ/
E
ε
−
−
−
T 

1
η
(2-15)
Where Nf is the number of cycles to failure, α is the coefficient of thermal expansion,
υ is the poison ration, εf is the fatigue strength coefficient, εT is the net thermal strain
and σ is the principal stress. The fatigue life predicted from the FEA model showed
good correlation with the laboratory fatigue life tests that were conducted.
33
Sakhuja and Brevick [19] conducted a study to compare the thermal fatigue life of the
die materials Haynes 230 and Inconel 617 that were used for copper die casting. A
sequentially coupled 2D elastic thermo mechanical finite element model was used to
predict the thermal stresses on the die. In this method a thermal analysis was
performed first to obtain the die temperatures. The nodal die temperatures obtained
from thermal analysis at appropriate sub steps were then used in the subsequent
structural analysis. The method of universal slope was then used to estimate the
thermal fatigue life of the dies. The expression used in universal slope method is
given by
( )z + 3.5 σEu (N f )γ
∆ε
= D 0.6 N f
2


(2-16)
Where D=- ln (1-RA) is the logarithmic ductility, RA is the percentage reduction in
area and σu is the ultimate tensile strength. Since the RA for the die materials were not
available they were determined using constancy of volume relationship for plastic
deformation. Park et al [20] used a elastic-visco-plastic model to investigate the
thermal distortion of copper mold used for thin slab steel castings. The mold wall
temperatures measured from the plant were used to obtain the heat flux profiles using
an inverse heat conduction model and this data was used in the die distortion analysis.
The model assumed isotropic hardening with a temperature dependent yield stress
function. A good correlation was observed between the model predictions and die
distortion measurements.
34
2.1.3 Models accounting for Structural Loads
Ahuett-Garza [21] and [22], conducted an elaborate study of the loads involved in die
casting process and he concluded that die deflection simulations with reasonable
resolution can be carried out by accounting for the clamping load, heat released
during solidification, the intensification pressure, and the heat removed during
lubricant spray. His study showed that the heat released during fill, the momentum
during filling and the pressure surge at the end of fill can be ignored in the die
deflection simulations and still results with reasonable accuracy and resolution can be
achieved. By an order of magnitude analysis it was shown that the heat released
during fill can be ignored when the ratio between half the thickness of the part and the
fill time is greater than or equal to one seventh. This corresponds to a case where the
solidification time is at least an order of magnitude greater than the fill time. The
details of the scale analysis are also provided elsewhere [23]
Based on the results of his study an initial finite element modeling procedure was
developed and tested [21], [24] and [25]. The preliminary model consisted of only the
cover die, ejector die and the ejector support block. First a thermal analysis was
carried out to obtain the nodal temperature values on the dies, which were later used
in the stress analysis.
35
The heat flux from the part during solidification was calculated using
q ' (t ) = K .10 mt
(2-17)
Where q’ (t) is the heat flux from the molten metal, t is the time between injection and
ejection and K and m are empirical constants determined based on the injection and
ejection temperature of the part, the surface area of the die in contact with the casting
and the total heat released during solidification. The ejection temperature of the part
was obtained from a solidification analysis using MAGMA software. The cavity
pressure was modeled as a hydrostatic pressure with magnitude equal to that of
intensification pressure. The clamp load was modeled as a pressure boundary
condition behind the ejector support block. A rigid support was assumed behind the
cover die and nodes on the back surface of the die were constrained in all directions.
The stiffness of the machine was not accounted for in this model. In a subsequent
study Dedhia [26] compared the parting plane separation predictions of a model with
rigid support behind the cover die versus the parting plane separation predictions
from a model that accounted for the machine stiffness. Spring elements were used to
account for the stiffness of the platens and the toggle mechanism. The clamp load was
modeled by applying appropriate displacement boundary conditions to the spring
elements that represented the toggle mechanism. The separation values at several
locations along the edges of the cavity were used as a measure of die deflection. The
maximum separation value was about 12% to 20% higher in the model with spring
36
elements than the values from the model with rigid support, depending upon the
design features of the die.
Choudhary [27] developed a finite element model in which the three machine platens,
the C-frame and the tie bars were modeled explicitly. The die was a dummy structure
that consisted of two parallel plates connected by pillars on the four corners of the
plates. A roller support was modeled at the bottom of cover and ejector platens. A
support block was modeled at the bottom of the rear platen to prevent displacement in
the vertical direction. A small sliding contact was defined at all interfaces. The nodes
on the either end of the tie bars were tied to corresponding nodes on the ejector and
rear platen. The clamp load was applied as a pressure boundary condition on the
toggle blocks on the cover and rear platens. Thermal loads and cavity pressure were
ignored in the model. The deflection of the cover platen was predicted at eight
different locations behind the cover platen and the results were compared with
corresponding values from the field data. The deflection pattern from the simulations
was similar to the pattern observed on the field data. But the individual deflection
values fell in the range of 10% to 15% of the observed field data. This model was
fairly accurate given the various approximations to the boundary conditions in the
model and the procedure followed to model the clamp load.
In another die distortion modeling study, Chayapathi [28] used a finite element model
in which the tie bars were explicitly modeled and the toggle mechanism was
represented by linear spring elements. The nodes on one end of the tie bar that are in
37
contact with the nodes in the cover platen were tied to the corresponding nodes on the
cover platen. The nodes on the other end of the tie bar were constrained in all six
degrees of freedom. The corner nodes on the bottom of the cover platen were
constrained in vertical direction to prevent rigid body motion. The clamp load was
applied by specifying displacements on the free end of the spring elements. The
intensification pressure load was applied as a pressure boundary condition on the
cavity surfaces. A schematic of the finite element model used in this study is shown
in Figure 2.1
Figure 2.1: Schematic of Toggle Spring Platen Model [1]
Ragab et al [29] experimentally verified the adequacy of this finite element model
shown in Figure 2.1 in predicting the contact loads between the dies and platens on
the cover and ejector sides. The contact load between the platens and the dies were
38
measured using a total of 35 load cells, 18 load cells on the cover side and 17 load
cells on the ejector side. The contact load was measured under two different loading
conditions, under clamp load only and during actual casting operation. The load cells
and the fixtures used in the experiments were also explicitly included in the finite
element model. The summation of cover side load cell measurements decreased by
7% after intensification whereas the summation of cover side load cell predictions
from simulation remained constant. On the ejector side the summation of load cell
measurements and predictions remained constant. The difference between model
predictions and measurements on the cover side were attributed to the fact that the
model is stiffer than the die/machine actually is.
To address these observed differences between the model predictions and
measurements on the cover side further modeling improvements were tested by
Arrambide [30]. Various machine components were included in the finite element
models and the predictions were compared again to the experimental load cell
measurements. Four different models were tested. The first model included the cover
and ejector platen, dies, inserts, the load cells and fixtures all of them modeled using
quadratic tetrahedral elements. The tie bars were modeled using beam elements, with
the one end of the beam elements constrained to the cover platen and the other end
was fixed in space. Several nodes on the bottom of the cover platen was constrained
in vertical and tie bar directions. The clamp load was applied as a pressure boundary
condition behind the ejector platen. A schematic of this simple model is shown in
Figure 2.2a. In the second model the rear platen was also included and the two ends
39
of the tie bars were constrained to the cover and rear platens. The toggle mechanism
was represented by beam elements and the clamp load was applied by specifying
appropriate temperature on these beam elements. This model is also shown in Figure
2.2b. In the third and fourth models the front support frame with the support frame
was added to the previous two models as shown in Figure 2.2c and Figure 2.2d.
Comparison between load cell measurements and load cell predictions between
simulations showed that the front support frame did not have any effect on the contact
load between the dies and the platens. Including the rear platen and toggle mechanism
in the model altered the load distribution on various load cell predictions by 2- 34%,
with an average of 11%. Also the full model showed good correlation with the
experimental measurements. To test the adequacy of the full model to predict parting
plane separation, a simplified model with dies, inserts and load cells only was built.
The clamp load was applied behind the load cells directly using the predictions from
the full model and also the loads from the experimental measurements. The maximum
separation showed a difference of 0.001” which falls within the resolution of the
numerical simulation.
40
Figure 2.2: Schematic of the Models Considered in [30]
In all of the die distortion modeling studies discusses above, the intensification
pressure was assumed to be hydrostatic and it was modeled as a pressure boundary
condition on the cavity surfaces in the finite element models. In reality the liquid
metal carries the hydrostatic pressure from the plunger mechanism and transfers it to
the cavity surfaces. But the solid elements used in the structural finite element models
cannot carry this hydrostatic pressure to the cavity.
Garza-Delgado [31] developed a two dimensional fluid structure interaction (FSI)
finite element model using ADINA to predict die distortion. It was a fully coupled
41
thermo mechanical test model that was developed to gain an understanding of the
capability of the fluid structure interaction model to predict die distortion. An FSI
boundary condition was defined at the interface between the solid and liquid domain.
A solid domain was used to represent the dies and a liquid domain was used to
represent the casting and the pressure load was simulated by specifying a nodal
pressure boundary condition in the gate area of the fluid domain. The FSI method
uses a conjugate heat transfer to calculate the heat fluxes across the interface and
hence no interfacial heat transfer coefficients had to be defined between the liquid
and the solid domain. Latent heat effects were included in the model by specifying
temperature dependent specific heat curve. This model was developed for
demonstration purposes and it is yet to be implemented in complex die distortion
simulations.
Another important dynamic load that has been ignored in die distortion simulations is
the dynamics impact load caused by the sudden deceleration of the plunger
mechanism at the end of fill. Xue et al [32], attempted to predict the pressure
distribution in the die cavity due to this impact loading using a CFD model. The goal
was to use the pressure predictions from this CFD model to approximate the dynamic
cavity pressure in structural die distortion simulations. FLOW-3D was used to
simulate the metal flow in the shot sleeve, runner and the dies. The molten aluminum
was treated as slightly compressible fluid with temperature independent material
properties and a K-ε turbulent model was used in the simulation. Heat transfer
between the metal and the dies was included in the model and the heat conduction in
42
the die was neglected. Pressure history at different cavity locations was investigated.
The pressure spike was found to be more than twice of the intensification pressure
that is normally used in the production of this experimental cast part used in this
study. It was also observed that the pressure within the cavity was almost uniform and
the maximum pressure difference in the cavity was also very small (about 50 PSI) at
the instant the impact occurs. It was also observed from the predictions that the
pressure in the cavity was zero during the slow shot phase and it reached the peak at
different locations at different instants of time during the fast shot. The maximum
pressure occurred during the deceleration phase through out the cavity.
Miller at al [33], developed a finite element model to predict the deflection of the
slides in non-open close dies and the results from the model were compared with field
data. The field data consisted of the slide blowback and tilt values from nominal
position under different pressure loads for different slide design. The simple finite
element model assumed a rigid support behind the cover die. The model predictions
showed a good correlation with the experimental data except for the high pressure
cases. This was due to the assumption of rigid support behind the cover die.
Vashist [34] studied the effect of different support structures for the die on the parting
plane separation. The goal was to study the parting plane separation patterns on a
production die that flashed severely after it was moved from a 1000 ton machine to a
2500 ton machine. The die had to be mounted lower on the 2500 ton machine due to
the location of the shot hole. Therefore, to evenly spread out the clamp load, the die
43
foot print was increased on the cover on the machine platen by adding support
structures to the dies. This study analyzed the effect of different types of support
structures and different clamp loads. The results showed that the added supports
stiffened the die/machine structure and increased the platen coverage area, but they
did not aid in transferring the clamp load to the die faces. Flynn et al [35] measured
the deflection of the same production die at various locations of the die using LVDT’s
and they reported a good match between the model predictions by Vashist [34] and
the experimental measurements.
Garza-Delgado [36] studied the failure of tie bolts that occurred on a hot chamber
machine, using a sequentially coupled thermo mechanical model. This case study
showed that non-uniform heat growth on the parting surface of the die resulted in
unequal distribution of loads on the tie bolts and resulted in tie bolts failure. The
machine frame, shank and bracket were modeled explicitly in the structural model.
The toggle mechanism and the tie bolts were modeled using 3D beam elements.
Milroy et al [37] used a boundary element method to predict the die deformation at
the die parting surface or interface. The two die blocks were analyzed individually.
The clamp load was represented by a uniform pressure load at the die parting surface
and a static pressure load was applied on the cavity surface. Few nodes on the rear
surface of the die blocks were constrained to prevent rigid body motion. The steady
state die temperature distribution was obtained by the boundary element method
described earlier in [11].
44
The heat flux on the die surfaces were then approximated by
K
(
∂T
= heff Tcon − T K
∂n
)
(2-18)
Triangular elements were used to discretize the die surface and pipe elements were
used to model the cooling lines. Though triangular elements are geometrically linear,
quadratic variations of displacement and traction were used in the model and these
elements with the order of geometric variation lower than that of field variables are
called sub parametric elements. The temperature distribution within an element that is
obtained from the thermal model is linear. Hence a linear variation of temperature
coupled with quadratic variation of displacement and traction was used in the first set
of analysis. Another two sets of analysis with isoparametric elements (variation of
displacement and traction linear over the triangular element). Experiments were
performed to evaluate the deformed profile of the die interface using transducers. The
deflection of the interface from fixed datum points was measured at different
locations of the die interface. The summation of the difference between the model
predictions and measurements at these locations were taken as the measure of error.
The authors argue that using the deformation profiles the inverse of the deformation
can be machined into the cold die so that the dies will mate perfectly under operating
conditions. But this model did not account for the stiffness of the machine platens, tie
bars and the toggle mechanism and the validity of the model predictions for cases
with die and cavity off-center from the platen center were not verified.
45
Rasgado et al [38], used the BE model described in [11], [37] to predict the thermo
mechanical stresses in copper based dies. The mechanical loads and boundary
conditions are same as in [37] and the thermal analysis procedure used in [11] was
used to obtain the temperature history in the die. The effect of temperature on the
elastic body was represented by adding a body force, bk=γTk to the corresponding
nodes denoted by ‘k’ and increasing the nodal traction to γTnk, where γ=2µα (1+υ) (12υ)-1, µ is a Lame’s constant, α is the coefficient of thermal expansion of the die
material and υ is the Poisson ratio. An experimental rig with a die block mounted on
it was used to simulate the actual thermal loads in the die casting process and strain
measurements were obtained at various locations on the die. The measurements were
compared with model predictions. The model consistently over predicted the stresses
on the die until ejection and it under predicted the stresses on the die after the casting
was ejected.
Barone and Caulk [39] presented a method to predict the ultimate distortion of both
the casting and the die due to thermal and mechanical loads in the die casting process.
They formulated the die distortion problem as a nonlinear thermo elastic contact
problem solved by iterative boundary element method. But the casting distortion was
analyzed as an unconstrained thermo elastic shrinkage using finite element method.
Their model included the dies, the ejector support, and cover and ejector platens. The
tie bars and toggle mechanisms were represented by spring elements behind the
ejector and cover platen respectively with appropriate stiffness values. Suitable
displacement constraints were provided on selected nodes on the bottom of the cover
46
platen to prevent rigid body motion of the entire structure. This boundary condition is
an approximation of anchoring the machine to the base. The uneven contact on the
parting surface was handled using contact/gap elements whose formulation provides
for load transfer between the die components only when the mutual surface traction is
compressive. Friction between the contact surfaces were ignored in this model. The
cavity pressure was modeled as a hydrostatic pressure with a magnitude equal to that
of the intensification pressure. The modeling approach was tested on a front drive
transmission case die and the results were presented. The advantage of this method as
claimed by the authors is that the casting and die distortion can be analyzed
simultaneously and the shrinkage allowance for the die cavity can be estimated.
Ragab et al [40] studied the effect of casting material constitutive model on the
deflection and residual stress predictions on the casting. The cover and ejector platens
were included in the model and the tie bars were represented using spring elements.
The clamp load was applied by specifying displacement boundary condition on the
spring elements representing the toggle mechanism. The cavity pressure was modeled
as a pressure boundary condition. A contact constraint was used between the die and
the casting. A fully coupled thermo mechanical analysis was conducted. Three
different material models were considered for the casting, viz., elastic, elastic-plastic
and elastic-viscoplastic. The residual stress predictions were affected significantly by
the material models were as the distortion predictions were less affected. The elastic
model over estimated the stresses and the visco-plastic model lacked the required
material property data. Therefore in a further study Ragab, [41] used the elastic47
plastic model to predict the effect of key modeling factors on casting distortion
predictions. The factors considered were the yield strength and strain hardening of the
casting material, the heat transfer coefficient between the die and the casting and the
injection temperature of the metal. The study concluded that the yield strength had a
major effect on residual stresses at ejection while the injection temperature and the
heat transfer coefficient had a major effect on the residual stresses at room
temperature. The disadvantage of the model used in Ragab’s study was that the solid
casting could not follow the distorted shape of the cavity due to the use of solid
elements for the castings and hence it might affect the casting distortion prediction.
Garza-Delgado [42] addressed this issue by using a shell mesh representing the
casting surface in the sequentially coupled thermo mechanical model that included
the clamp loads, intensification pressure load and the thermal load. The nodal
distortion values of the shell mesh were then mapped on to the surface nodes of a
solid mesh for the casting. Then the solid cavity was tied to the distorted shape of the
die using tied contact and the cooling stages of the casting were simulated using a
fully coupled elastic-plastic thermo mechanical model. Modeling the tie bars
explicitly and applying the clamp loads through spring elements caused problems in
establishing contact between the die and the casting. Therefore the clamp force was
modeled as a pressure boundary condition behind the ejector platen in this work.
48
2.2 Parametric Die Design Studies
Based on the die distortion modeling methodologies reviewed in the previous section
numerous parametric die design studies has been conducted at the Center for Die
Casting to understand the role of structural die design parameters on die deflection.
Some of these parametric die design studies are reviewed here.
Jayaraman [43], conducted a parametric study of the slide design variables for an
inboard lock design and the work was continued by Chakravarti [44, 45] using a
refined model. The variables considered were the preload, the angle of the locking
surface and the pivot. Results suggested that the preload had no effect on the
blowback and tilt values but it affected the fatigue life of the slides. The trends also
showed that the tip separation increased with locking face angle and slides with high
pivot were better supported by the ejector platen. There was negligible effect on the
parting plane separation within the range of design variables.
The effect of proud inserts on the parting plane separation was first analyzed by
Dedhia [26], [46]. The study concluded that using proud inserts resulted in a lower
separation during the initial cycles but at the later stages as the die warms up and
grows the separation values were similar to the cases with flush inserts (insert parting
surface in line with the die parting surface). A much refined model was later used by
Tewari [3], [45] to study the effect of proud inserts on the parting plane separation.
The contact pressure between the dies and the platens and the compressive stresses in
the die pocket were also studied. It was observed that the parting plane separation was
49
reduced around the cavity by having a proud insert and the contact pressure between
the dies and the platens was not affected due to the use of proud inserts. The proud
insert had negligible effect on the compressive stresses in the die pocket.
Tewari [3], [45], analyzed the effect of adding a back plate behind the ejector support
box. Results from his study indicate that the back plate has negligible effect on both
the parting plane separation and the contact pressure between the ejector die and the
platen.
A parametric study conducted by Dedhia et al [26], [46], showed that using proud
pillars had no effect on the parting plane separation. The difference in parting plane
separation values between the cases with proud pillars and flush pillars was less than
5%.
A series of parametric studies [1-3], [28] were conducted to gain understanding of the
effect of important structural variables of the dies and the machine on the die
distortion. The summary of the work was published in [4], [45].
The variables investigated were die size (% of platen area covered), insert thickness,
thickness of die steel behind the insert and die location on the platen. Response
surface models based on design of experiments were used to study the interaction
between the variables and their effect of parting plane separation. The approach for
the sensitivity analysis was developed by Chayapathi [1], [28] and the initial
experimental design array consisted of 15 experimental runs. An additional 16 runs
were further added by Kulkarni and Tewari [2], [3] to ascertain the results. The study
50
showed that the dominant factor that affects the die distortion on the cover side is the
cover platen thickness. Thin dies performed better than thick dies. The more the steel
behind the dies the lesser was the parting plane separation observed. Small or medium
sized dies (covering 40% to 50% of platen area) performed better. Kulkarni [2]
attempted to study the cover and ejector side performances separately. But the ejector
side design variables such as the rail size, number, size and location of the support
pillars were not controlled in the computational experiments. Therefore the study was
inconclusive about the contribution of ejector side design to the ejector side
separation.
2.3 Relevant Research in Injection Molding
Isayev[47] proposed an approach to simulate the cavity filling, packing stage and
flash formation in injection molding using a finite difference method. The flow
during flash formation was calculated using a power law fluid model. The mass
balance equation during flash formation was given as
Q
∂ ln ρ
∂  ∂P S 
=− ∗
−
+
∂t
∂x  ∂x b 
b w
(2-19)
Where Q is the volumetric flow rate due to leakage calculated at each nodal point on
the cavity edges, ρ is the density of the melt, P is the packing pressure, S is the melt
fluidity, b* is the variable gap thickness, b and W are the gap thickness and width
during flash formation respectively. The variable gap thickness b* was obtained by
assuming the mold as a beam under uniform pressure loading with pin supports at the
51
location of the leader pins. Chen et al [48] developed an analytical solution to predict
flash length on straight and curved parting surfaces of injection molds. The power law
fluid model was assumed for the melt flow in this study.
Carpenter et al [49] studied the effect of injection molding machine stiffness on the
mold deflection predictions of a finite element model. The cavity pressure distribution
at the end of filling and at the instant of maximum packing pressure was obtained
from coupled thermal fluid flow simulation using MOLDFLOW. The pressure
predictions were used in a structural finite element models to predict the mold
deflection. Three different loading scenarios were analyzed in the structural model. In
the first case, only the clamp loads were modeled. In the second and third cases, in
addition to the clamp load, the cavity pressure at end of filling and the maximum
cavity pressure were included respectively. The three loading cases were repeated on
a model that included only the mold halves and also on a model that included the
machine platens and the tie bars. The strain predictions from these two types of
models were compared to experimental measurements and it was shown that the
predictions from a mold and machine model showed better agreement with the
experimental measurements than the predictions from the mold only model.
Significant reductions in mold opening were predicted from the mold and machine
model as compared to the predicted mold opening in the mold only model.
Hostert [50] demonstrated a multi body dynamics method to simulate the dynamic
characteristics of a two platen HUSKY injection molding machine. ADAMS a multi
52
body simulation software were used to model the mechanical structure and DSH a
fluid power analysis software was used to estimate the hydraulic force that drives the
clamping mechanism. The four clamp pistons that engage the tie bar threads were
modeled as rigid bodies and the three connecting bars were modeled as flexible
bodies. A schematic of the clamp pistons and connecting rods and their representation
in ADAMS along with the constraints and boundary conditions are shown in Figure
2.3 and Figure 2.4 respectively.
Figure 2.3: Schematic of Pistons and Connecting Rods of Injection Molding
Machine Clamping Mechanism [50]
53
Figure 2.4: Representation of the Clamping Mechanism in Multi-body Dynamics
Simulation [50]
The fluid power analysis package reads the position and velocity of the connecting
rod from ADAMS, calculates the hydraulic force and returns the value back to
ADAMS. The connecting rods were originally modeled in ANSYS and their mode
shapes were obtained using finite element method. The mode shape predictions were
read into ADAMS using a transformation procedure that reduces the number of
degrees of freedom for the multi body simulation as compared to the original number
of degrees of freedom in the finite element model. The stress time histories of the
connecting rod were then predicted in ADAMS. The advantage of the multi body
dynamics simulation over FEA is that large non linear motions can be modeled with
less computational time and cost.
Beiter et al [51] developed a decision support system that takes into account
mechanical requirements, manufacturing costs and material selection for design of
54
injection molded parts. The mechanical requirements of the part were estimated by
approximating the part geometry as a flat plate and the maximum deflection was
estimated using classical plate deflection equation. The support system can also take
into account deformation of the part due to creep and impact loadings. A filling
analysis capability was also included to estimate the minimum number of gates and
minimum wall thickness based on a defined flow length. Provisions were also
provided for estimating the cycle time required for the part. Finally a cost model was
included that predicts the manufacturing costs based on the material and part design
selection.
Menges [52] describes methods to predict mold deflection in injection molding. The
structural features are assumed as linear springs with certain stiffness. The deflection
for each of the structural feature is estimated using standard design formulas. The
individual deflections are then summed up to obtain the total deflection in the mold.
A schematic of the procedure is described in Figure 2.5
55
Figure 2.5: Schematic of the procedure to estimate mold deflection [52]
Menges [52] et al also described the effect of the relative stiffness of the mold and the
machine on gap or flash formation using a force deformation diagram. In this method
the mold, the parting surface, the machine clamp unit and tie bars are represented by
springs with equivalent stiffness. Then the deformation of the springs representing the
molds and the machine were obtained under different loads and a load deformation
curve was obtained as shown in Figure 2.6. It can be observed from the figure that
when the clamp load is applied the tensile forces on the press/machine are equivalent
to the compressive forces on the mold faces. When the cavity load is applied, the
compressive force on the mold faces are relieved causing gap formation and the
machine/press is stretched further and hence the load on the machine increases. The
stretching of the machine and the increase in locking force after application of cavity
56
load will depend on the stiffness of the machine. If the machine is perfectly rigid
there will not be any increase in the locking force. This method provides a rather
simple and realistic explanation of the effect of machine stiffness on mold
deformation.
Figure 2.6: Mold Spring Diagram used to Estimate Gap Formation [52]
Sasikumar et al [53], investigated the premature failure of a tie bar in an injection
molding machine using experimental failure investigation methods. The failed surface
and samples cut from failed tie bar was subjected various visual and macro
examination methods such as chemical analysis and fractography. Mechanical
property measurements such as yield strength, tensile strength, elongation and impact
toughness measurements were also carried out. The analysis revealed fatigue at the
first thread of the tie bar as the major root cause of failure. It was concluded that the
improper process conditions caused a tie bar load imbalance resulting in a torsional
component of stress, misalignment of thread and consequent gouging at the threads. It
was also found that material defects such as inclusions, untransformed phases and
57
fine cracks at the root of the thread caused the initiation of the crack and the crack
propagated due to the pulsating tensile and torsional stresses.
2.4 Summary
Thus there is a vast amount of work done in mathematical modeling of the casting
and similar processes. As mentioned earlier the complexity of the models and the
assumptions on the physical phenomenon depends on the goals of the analyst. The
goal of this research work is to study the effect of structural design variables of the
die and the machine on the mechanical performance of the dies and machine and this
work addresses the structural die design issues that were untouched by Chapyapathi
and others [1], [2], [3] and [4]. Improved finite element modeling methods developed
by Ragab [29] and Arrambide [30] were utilized in this dissertation. Non-Linear
power law models to predict parting plane separation are developed in this research
using dimensional analysis as opposed to the linear polynomial models developed in
previous research [1], [2], [3] and [4]. A non-linear model to predict tie bar loads on
the die casting machine is also developed in this research. To the author’s knowledge
no other work has been done to develop power law models to predict parting plane
separation and tie bar loads which make this dissertation work unique.
58
CHAPTER 3
RESEARCH METHODOLOGY
3.1 Introduction
Empirical correlations to predict the tie bar loads and the maximum parting plane
separation were developed by conducting computational experiments and fitting
regression models to the data obtained from the computational experiments. The
computational experiments refer to the finite element analysis conducted at each
design point in the chosen experimental design. The regression models then serve as
surrogate functional approximations of the finite element model and they are also
commonly referred to as response surface models or meta-models [54]. The surrogate
response surface models can be used to find the optimal design parameters that would
result in the desirable output of the system and these surrogate approximations has
been widely used in structural design optimization [55]. The other objective in
constructing these functional approximations is to predict the output of the system at
an untried design point [56]. The accuracy of the predictions will not only depend on
the functional form and also on the resolution and accuracy of the underlying
numerical models.
59
In this research dimensional analysis based on Pi-theorem [58], [62] was used to
determine the functional relationship and non-dimensional parameters for predicting
the tie bar loads and parting plane separation. The functional forms obtained from the
dimensional analysis for the parting plane separation prediction problem and the tie
bar load prediction problem were imposed on the parting plane separation predictions
and tie bar load predictions from the respective computational experiments. This
chapter describes the finite element modeling methodology that was adopted in the
computational experiments to predict the tie bar loads and parting plane separation.
3.2 Finite Element Modeling
A hypothetical box shaped part and a nominal 8.9 MN (1000 ton) four toggle die
casting machine that have been used in previous research studies is chosen for this
research. A schematic of the part geometry is shown in Figure 3.1 and the
specifications of the die casting machine are shown in Table 3.1.
60
Figure 3.1: Geometry of the Part used in the Study
Machine Parameters
Specifications
Machine Tonnage
1000 tons
Tie bar diameter
7.5”
Tie bar length
115”
Space between tie bars
44”
Cover Platen-Width×Height
60”×70”
Ejector Platen-Width×Height
60”×60”
Rear Platen-Width×Height
60”×60”
Table 3.1: Specifications of the Machine used in the Study
61
The schematic of the finite element model used in this study is shown in Figure 3.2.
The dies, inserts, machine platens, the machine base and the tie bars are modeled
explicitly using 3D solid elements. Ten node tetrahedron elements and eight or six
node brick elements that are available in ABAQUS have been used for the 3D solid
elements as will be explained in the next chapter. The toggle mechanism was
modeled using 2D beam elements.
Figure 3.2: Schematic of the Finite Element Model
3.2.1 Boundary Conditions and Constraints
The machine base is constrained in all directions at six nodes on the bottom edges to
prevent rigid body motion. A small sliding contact was defined between the ejector
62
platen and the base, between the ejector platen and the tie bars and also between the
rear platen and the base. The nodes on the tie bars are tied to the corresponding nodes
on the rear and cover platens using a tied contact formulation in ABAQUS. Similarly
a tied contact constraint is defined between the nodes on the rear side of the insert and
the corresponding contact nodes on the die shoes. The corner nodes of the dies were
tied to the neighboring nodes on the platens using tied multi point constraint. A
coulomb friction coefficient of 0.3 was assumed for all contacting surfaces.
The boundary condition that has a significant effect on the tie bar load prediction and
parting plane separation prediction is the one used between the cover platen and the
machine base. In actual die casting machine, the cover platen is bolted to the base and
keyways are provided to prevent platen movement in the tie bar direction. The
keyways are also provided with clearances. The stiffness of this constraint will affect
the bending of the cover platen and hence the support available behind the cover die
and the stretching of the tie bars will also be affected. Three different approaches
were considered to model this constraint between the cover platen and the machine
base. In the first approach the degrees of freedom of the edge nodes on cover platen
and machine base were tied using a multipoint constraint formulation. A schematic of
this constraint is shown in Figure 3.3a. In the second approach a tied contact
formulation was used to ties the degrees of freedom of all the nodes on the contacting
surfaces of the cover platen and the base. The tied contact formulation in ABAQUS
ties the degrees of freedom of the slave nodes to the corresponding degrees of
freedom of the neighboring master surface node [57]. A schematic of this boundary
63
conditions is shown in Figure 3.3b . However both of these constraint types might
lead to an over constrained model. A better methodology to represent the bolted joint
in FEA is to explicitly model the bolted joints. This achieved by using truss elements
connecting nodes on the cover platen and the base as shown in Figure 3.3c. The head
of the bolt is modeled using rigid elements connecting the reference node (node on
the truss element) to the neighboring nodes on the platen or the base. The rigid
elements are also shown in Figure 3.3c. A preload of 400 MegaPascals was applied
on the truss elements by specifying appropriate temperature values on the reference
nodes. This causes the truss elements to stretch and generate the required preload.
64
Figure 3.3: Boundary Conditions used between Cover Platen and Base
3.2.2 Loads and Assumptions
The clamp load was applied in the first step of the model by specifying appropriate
temperature at the beam element nodes. This causes an increase in the length of the
beam elements thus producing the required clamping force. The contact force
between the cover die and cover platen was chosen as a measure of the clamp force.
The temperature on the beam element nodes were adjusted until the desired clamp
65
load was obtained. Following the same assumption proposed by Garza [21], the
pressure load was assumed to be hydrostatic with a magnitude equal to that of
intensification pressure. The pressure load is modeled as a pressure boundary
condition on the cavity surfaces. The thermal loads are cavity geometry specific and
the thermal distortion issues have to be addressed by appropriate design of cooling
lines and part geometry. Therefore thermal loads are intentionally ignored in this
study.
3.2.3 Material Properties
A linear elastic material model was used in the analysis. The platens, tie bars,
machine base, toggle mechanism, the dies and inserts are all assumed to be made of
ST4140 steel and the material properties that are used for the ST4140 steel in the
simulations are given in Table 3.2.
Material Property
Value
Young’s modulus
2.068×1011 N/m2
Poisson ratio
0.29
Coefficient of thermal Expansion
1.170 ×10-5 /οC
Table 3.2: Material Properties for ST4140
3.2.4 Finite Element Model Predictions
The predictions obtained from the finite element models were the loads on the tie bars
and the maximum separation of the cover and ejector parting surfaces from their
66
nominal locations. To obtain the tie bar loads, the tie bars were partitioned into two
different volumes at a location between the cover and ejector platens and the
contacting surfaces of the two volumes were tied together using a tied contact
formulation. At the end of the analysis the contact loads between these two contacting
surfaces were obtained and this contact load is taken as a measure of the tie bar load.
The location were the volume partition was done and tie bar loads were predicted is
shown in Figure 3.4.
Figure 3.4: Schematic of the Finite Element Model showing the Location of Tie
Bar Load Prediction
The other prediction that was obtained from the finite element model was the
separation of the cover and ejector parting surfaces from the nominal parting plane
location. This phenomenon is illustrated in Figure 3.5. The black lines represent the
undistorted nominal parting plane. The blue lines represent the distorted ejector
67
parting surface and the red lines show the distorted cover parting surface. The
distance between the nominal location of the parting surfaces and their final location
after the application of loads was obtained from ABAQUS. The nominal parting
plane represented by the black lines is a plane that passes through all the nodes that
on the parting surface that are in perfect contact after the application of clamp and
pressure loads.
Figure 3.5: Illustration of Ejector and Cover Side Parting Plane Separation
The default nodal displacement output obtained from ABAQUS is given with respect
to a global coordinate system. The dies and inserts undergo a pseudo rigid body
motion due to the stretching of the tie bars as shown in Figure 3.6. The black lines in
68
the figure show the undeformed shape and the green lines show the deformed shape
of the dies and the machine.
Figure 3.6: Pseudo Rigid Body Movement Caused by Stretching of Tie Bars
Therefore the default displacement output from ABAQUS includes this pseudo rigid
body motion and the rigid body component should be removed to predict the
displacement of cover and ejector parting surfaces from their nominal location which
is the plane that passes through the parting surface nodes that are in contact. A
methodology that was developed in previous research [2], [3] and [4], was used to
remove this rigid body translation and rotation component from the displacement
outputs of ABAQUS.
69
The equation used to estimate this transformation component is as follows:
R 0
 + [E 0]
T 1 
[X s 1] = [X f 1] 
(3-1)
Where Xs and Xf are the starting and final coordinates of the sample nodes
respectively, R is the rotation matrix, T is the translation vector and E is the distortion
component. The initial coordinates Xs and final coordinates Xf of a few sample nodes
on the parting plane were obtained from ABAQUS and a least squares method that
minimizes the sum of squares 'trace(ET E)' was employed to estimate the
transformation matrix in equation (3-1). Then the best estimate of the pure distortion
component is given by
R∗
[E 0] = [X s 1] − [Xf 1]  ∗
T
0

1 
(3-2)
Where R* and T* are the estimates of the rotation and translation components
obtained from the least squares method. The matrix least squares method and the
procedure to select the sample nodes are described in APPENDIX A. An alternate
method to predict the displacement of the parting plane from their nominal location
using a local coordinate system in ABAQUS is also presented in APPENDIX A.
3.2.5 Effect of Element Types and Cover Platen Constraint on Model Predictions
As mentioned earlier three different types of constraint between the cover platen and
the machine base were considered in this research. The finite element model
predictions are also sensitive to the element type used to model the platens, dies,
inserts and tie bars. A nominal 8.9 MN (1000 ton) four toggle machine and a nominal
70
die and insert were chosen to test the effect of the cover platen constraint and the
element types on the model predictions. The dimensions of the machine and the die
and the magnitude of the loads used in these test simulations are shown in Table 3.3.
Table 3.4 shows the effect of the element types and the cover platen constraint on the
total parting plane separation prediction of the finite element model. Total parting
plane separation in the finite element model is obtained as the normal distance from
the slave node to the master node of the contacting parting surfaces. It can also be
viewed as the sum of the cover and ejector side parting surface separation. It can be
observed from Table 3.4 that the cover platen boundary condition and the element
types have a negligible effect on the parting plane separation prediction from the
finite element model.
Parameter
Platen Thickness
Die Length
Die Width
Die Thickness
Insert Thickness
Rail Width
Rail Height
Clamp Load
Cavity Pressure
Value
11"
31"
31"
7.5"
4.125"
3"
6"
722 tons
10000 PSI
Table 3.3: Die and Machine Parameters used in the Simulations to Test the
Effect of Cover Platen Constraint Type and Element Type
71
Element type
used for
platens, base
and tie bars
Element type
used for die
shoes and
inserts
10 node
tetrahedron
10 node
tetrahedron
8/6 node brick
10 node
tetrahedron
8/6 node brick
8/6 node brick
8/6 node brick
Constraint
type between
the cover
platen and
base
surface to
surface tied
constraint
Maximum
Separation
(mm)
Maximum
Separation
(thousands
of inch)
0.273
10.763
Edge nodes tied
0.269
10.604
10 node
tetrahedron
surface to
surface tied
constraint
0.269
10.589
10 node
tetrahedron
Truss elements
used to model
bolted joint
0.269
10.604
8/6 node brick
Edge nodes tied
0.262
10.310
Table 3.4: Effect of Element Type and Cover Platen Boundary Condition on
Parting Surface Separation Prediction
Figure 3.7 shows the deflection plots of cover platen in the tie bar direction for cases
with bolt constraint and multi point constraint. It can be seen that the deflection of the
platen in the multi point constrain case is higher (0.38 mm) than the case in which the
bolts were modeled explicitly (0.258 mm). The multi point constraint is stiffer than
the bolt constraint and hence the bending of the cover platen is higher in this case.
The constraint is less stiff and hence the bottom of the cover platen moves slightly
towards the toggle side. This trend is shown in Figure 3.8, where the deformed shapes
of the platens are superimposed on the undeformed shape of the platen.
72
Figure 3.7: Deflection Plots of the Cover Platen
Figure 3.8: Deformed Plot of Cover Platen Superimposed on the Undeformed
Plot
73
Table 3.5 shows the effect of cover platen boundary condition on the tie bar load
predictions of the finite element model. In Table 3.5, the tie bar loads are shown as a
percentage of the nominal load which is one fourth of the total clamp load.
Tie bar Load/Nominal Load*100
Constraint
Top tie
bar-1 (T1)
Top tie
bar-2 (T2)
Bottom tie
bar-1 (B1)
Bottom tie
bar-2 (B2)
Edge nodes
tied
97.8
98.4
102.0
101.6
Surface to
surface tie
constraint
97.4
98.0
102.4
102.0
Truss
elements
99.0
99.2
101.5
100.3
Table 3.5: Effect of Element Type and Cover Platen Boundary Condition on Tie
Bar Load Prediction
It can be observed from Table 3.5 that modeling the bolts explicitly results in slightly
lower imbalance (approximately 2% less imbalance) between the top and bottom tie
bars as compared to the other two constraints. It should also be noted that the cases
shown in the Table are the ones with the dies centered between the tie bars and the
cavity center of pressure centered on the geometric center of the die.
74
CHAPTER 4
EMPIRICAL CORRELATIONS TO PREDICT PARTING
PLANE SEPARATION
4.1 Introduction
This chapter discusses the power law models that were developed to predict
maximum parting plane separation on the cover and ejector side. A design of
experiments was developed based on the major structural variables of the die casting
die and the machine. A static finite element analysis was conducted at each design
point specified in the design array using the modeling methodology described in the
previous chapter. Maximum separation of the cover and ejector side parting surfaces
from their nominal location was obtained from the finite element models. Power law
models were fit to the parting plane separation data and the non dimensional
structural design parameters. The non dimensional parameters were obtained using
dimensional analysis based on Buckingham pi-theorem. The design of experiments,
the non dimensional parameters and the power law models are presented in this
chapter.
75
4.2 Design of Experiments
The design variables included in the study are the die width, die length, die thickness,
thickness of the die steel behind the insert (die shoulder thickness), pillar diameter
and the pattern of ejector pillar supports. The description of the factors along with
their high and low values is shown in Table 4.1. The fourth factor die thickness ratio
(TR) is the ratio between the thickness of the die steel behind the insert and the total
die thickness. This factor defines the thickness of the insert used. The factor is
illustrated in Figure 4.1. In Figure 4.1, t denotes the die thickness and IT denotes the
insert thickness. The thickness ratio, TR is given by (t-IT)/t.
Factor
Pt
Lx
Ly
t
TR
PD
X
Description
High Level
Low Level
Platen thickness
9”
13”
Horizontal dimension of
24”
38”
the die of the
Vertical dimension
24”
38”
die
Die thickness
5”
10”
Die thickness ratio
0.4
0.5
Pillar diameter
1.5”
4”
Pillar Pattern (discrete
4 Levels/Patterns
factor)
Table 4.1: Factors used in Design of Experiments
76
Figure 4.1: Side View of an Ejector Die used in the Study
(Dimensions not to scale)
The sixth factor, pillar pattern, is a discrete factor. The schematic of the four different
pillar arrangements behind the ejector die that were analyzed is shown in Figure 4.2
which shows the rear view of an ejector die. The back surface of the rails is hatched
in the figure. The rails are 3 inches wide in all of the cases. The rails and pillars are 6
inches long in all of the cases. In pillar pattern-1 there are a total of nine pillars, one
directly behind the center of pressure and the outer pillars are located at a radial
distance of 6.75 inches from the center pillar. In pattern-1 the outer pillars are 45°
apart from each other. In pattern-2, there are no pillars and the ejector die has only
rail support. In pattern-3 and pattern-4 there are five pillars, one on the center and
four outer pillars each 90° apart. The difference between pattern-3 and pattern-4 is in
the orientation of the outer pillars. A 58 run central composite response surface
77
experimental design was chosen based on the five continuous factors. Then the 58runs were repeated for each level of the discrete factor. The experimental array in uncoded units is shown in Table 4.2.
Figure 4.2: Schematic of Pillar Patterns used in the Study
(a) Pattern-1, 9 pillars, (b) Pattern-2, No pillars, (c) Pattern-3, Five Pillars, (d)
Pattern-4, 5 pillars
78
RUN
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Pt
9
13
9
13
9
13
9
13
9
13
9
13
9
13
9
13
9
13
11
11
11
11
11
11
11
11
11
11
11
11
Lx"
24
24
38
38
24
24
38
38
24
24
38
38
24
24
38
38
31
31
24
38
31
31
31
31
31
31
31
31
31
31
Ly"
24
24
38
38
24
24
38
38
24
24
38
38
24
24
38
38
31
31
24
38
31
31
31
31
31
31
31
31
31
31
t"
5
5
5
5
10
10
10
10
5
5
5
5
10
10
10
10
7.5
7.5
7.5
7.5
5
10
7.5
7.5
7.5
7.5
7.5
7.5
7.5
7.5
TR
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.45
0.45
0.45
0.45
0.45
0.45
0.4
0.5
0.45
0.45
0.45
0.45
0.45
0.45
PD"
4
1.5
1.5
4
1.5
4
4
1.5
1.5
4
4
1.5
4
1.5
1.5
4
2.75
2.75
2.75
2.75
2.75
2.75
2.75
2.75
1.5
4
2.75
2.75
2.75
2.75
Continued
Table 4.2: Response Surface Experimental Array
79
Table 4.2 continued
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
11
11
13
13
9
9
9
9
13
13
9
9
13
13
13
13
9
9
11
11
11
11
31
31
24
38
24
38
24
38
24
38
24
38
24
38
24
38
24
38
24
38
31
31
31
31
38
24
38
24
38
24
38
24
38
24
38
24
38
24
38
24
31
31
24
38
7.5
7.5
5
5
10
10
5
5
10
10
5
5
10
10
5
5
10
10
7.5
7.5
7.5
7.5
0.45
0.45
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.45
0.45
0.45
0.45
2.75
2.75
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
4
4
4
4
4
4
4
4
2.75
2.75
2.75
2.75
The maximum cover and ejector side parting plane separation were predicted for each
case in the experimental array using the finite element modeling methodology
described in previous chapter. As it can be seen from, the last five runs are repeated
runs with mid level setting of each factor. The runs 27 to 32 in the experimental array
were utilized to account for these uncertainties in the mesh size, element type and the
boundary condition between the cover platen and the base. A summary of the element
80
types and cover platen constraints used among the 52 cases in the experimental array
is shown in Table 4.3.
Runs
Element type used for
platens, base and tie bars
Element type used
for die shoes and
inserts
Constraint type
between the cover
platen and base
1-27,
32-58
10 node tetrahedron
10 node tetrahedron
Edge nodes tied
28
10 node tetrahedron
10 node tetrahedron
29
8 or 6 node brick
10 node tetrahedron
30
8 or 6 node brick
10 node tetrahedron
31
8 or 6 node brick
10 node tetrahedron
32
8 or 6 node brick
8 or 6 node brick
surface to surface
tied constraint
Edge nodes tied
surface to surface
tied constraint
Truss elements
used to model
bolted joint
Edge nodes tied
Table 4.3: Summary of Element Types and Constraints used in Computational
Experiments
4.3 Dimensional Analysis and Empirical Correlations
As mentioned in the first chapter, the objective of the computational experiments is to
develop empirical correlations to predict the maximum parting plane separation as a
function of the design variables involved. Dimensional analysis was used to
determine
the
non-dimensional
parameters
for
the
empirical
correlations.
Dimensional analysis is based on the principle that any valid physical relationship
should be dimensionally homogeneous. Based on this principle Buckingham [58]
showed in his famous Pi-theorem that any equation describing the physical
81
relationship between the variables can be reformulated as a function of dimensionless
products of variables.
For example if the original relationship of n dimensional variables is written as
f ( x1 , x2 ,.... xn ) = 0
(4-1)
The Pi-theorem states that we can express this as a new function of a set of
dimensionless parameters that are conventionally represented by П’s
f (Π1 , Π 2 ,....Π n −m ) = 0
(4-2)
Equation (4-2) shows that there are only n-m variables in the relationship as
compared to the original n variables in equation (4-1). Here m represents the number
of fundamental or independent dimensions in the relationship such as mass, length,
time, temperature etc. While the number of П’s is fixed there can be any number of
sets of these products. Based on the knowledge of the governing physical principles
these products can be manipulated by multiplying or dividing them by one another.
Thus by the nature of dimensional analysis the number of variables required for
experimentation is reduced. The dimensionless parameters also capture the
fundamental non-linear relationships between the original variables. Omitting a
variable in the dimensional analysis does not invalidate the non-dimensional
parameters or the functional relationship obtained from it, but it only restricts the
generality of the solution obtained.
82
In addition to the measured variables, dimensional constants such as viscosity,
density, young’s modulus, Poisson ratio etc may enter the non-dimensional
parameters. There might be also some appropriate non-dimensional values such as
ratios or scale factors. Again the selection of these constants and ratios will depend on
the background knowledge in the physical phenomenon. Vignaux et al., [59]
demonstrated how dimensional analysis can be used to simplify regression models
and how the regression model obtained based on dimensionless parameters remains
dimensionally homogeneous regardless of the measurement units and transformations
to the original data.
4.3.1 Determination of the Model form and Non-Dimensional Parameters for
Predicting Parting Plane Separation
The choice of variables to be used in the dimensional analysis and the grouping of
variables that form the non-dimensional groups can be determined using the
knowledge of the physical phenomenon and engineering judgment. The dies and
inserts are assumed as pre-stressed flat plates that are resting on elastic supports and
subjected to uniform loading. The semi analytical empirical equation for predicting
maximum deflection of a flat plat under uniform loading is given by [60]
αWL4
Maximum deflection =
Et 3
(4-3)
W is the uniformly distributed load, l is the length of the plate, E is the young’s
modulus, t is the plate thickness and α is a constant that depends on the aspect ratio
83
and boundary conditions. The unsupported span behind the cover die is characterized
by the length and width of the die and a model for cover side separation will take a
form similar to equation (4-3). However on the ejector side the unsupported span is
characterized by the distance between the pillars and rail supports. Jofreit [61]
developed a semi empirical equation to predict the maximum deflection of concrete
flat plates supported on straight beams around the periphery and a grid of pillars in
between the beam supports. The equation to predict the maximum deflection within
any inner grid of pillars is given by
 l + 3l n 
αW  s

4 

Maximum deflection =
Et 3
4
(4-4)
Where ls and ln are the long span and the short span between the pillar supports. It can
be observed that the span variable L in equation (4-4) is replaced by the weighted
average of the shorter and longer span between the pillar supports in equation (4-4).
Therefore a model for the ejector side separation will take a form similar to the one
shown in equation (4-4). Equations (4-3) and (4-4) suggest that the empirical
correlations to predict the maximum parting plane separation on the cover and ejector
side will take a power law model form. Based on the structure of the equations (4-3)
and (4-4), the physical relation between the maximum parting plane separation and
the design variables can be written as follows:
f (δ max , Pt, L tb , t, TR, L, l n , PD, RW, Distributed Load, E) = 0
84
(4-5)
Where, δmax is the maximum parting plane separation, Ltb is the distance between the
tie bar centers, L is the length scale representing the length and width of the die or the
span between the pillars, ln is the length scale for span between pillars, PD is the
pillar diameter, RW is the width of the die and E is the young's modulus of the die
material. The term Distributed Load denotes the distributed contact load at the parting
surface and it is approximates as follows:
Distrubted Contact Load =
Total Clamp Load Cavity Pressure × Projected Area of Cavity
−
Lx × Ly
Lx × Ly
(4-6)
Both on the cover side and ejector side, the platens behave as an elastic foundation for
the dies. Therefore the deflection of the dies will also depend on the stiffness of the
platens which in turn depends on the thickness of the platen and the distance between
the tie bar centers. Therefore the distance between the tie bar centers is also included
in the functional relationship shown in equation (4-6). The scaling/repeating variables
to be used in the dimensional analysis were determined based on linear elastic stressstrain relation. By linear elastic theory, the relation between load and deflection is
given by,
F
= εE
A
(4-7)
Where, F is the total load, A is the area over which the load acts, ε is the strain and E
is the young's modulus.
85
For the problem of predicting the maximum separation, δmax , equation (4-7) can be
written as,
F δ max
=
E
A
t
(4-8)
Or
δ max E
= Constant
t × Distributed Load
(4-9)
The left hand side of Equation (4-9) is a non dimensional number which suggests the
choice of scaling variables to be used in the dimensional analysis. The variables, E,
distributed load and t were chosen as the scaling variables for the dimensional
analysis. Then the remaining variables were written as a product of the repeating
variables raised to an exponent. For example, the non-dimensional parameter
involving the distance between the tie bars, Ltb can be written as
Ea × DistLoad b × t c × L tb = ∏
(4-10)
Writing equation (4-10) in terms of the basic dimensions of Mass (M) length (L) and
Time (T) we have
[ML
] × [ML
-1 - 2 a
T
] × [L] × [L] = M L T
-1 - 2 b
T
0 0 0
c
(4-11)
Equating the exponents of Mass (M) length (L) and Time (T) on both sides of the
equation (4-11), we obtain a=0, b=0 and c=-1. Therefore the non dimensional
parameter involving the distance between the tie bar centers is given by
86
∏=
L tb
t
(4-12)
The other non-dimensional groups can also be obtained in a similar manner and
equation (4-5) reduces to
δ max E
Pt L
Lx Ly Lcx Lcy l n PD RW 

, , tb , TR,
,
,
,
f
,
,
 = 0 (4-13)
t
t
t
t
t t t
t 
 t × Distribute d Load t
The Lx and Ly in equation (4-13) is the width and length of the die respectively and
Lcx and Lcy are the characteristic dimensions of the cavity respectively. A linear
combination of the parameters, L/t, ln/t, PD/t, and RW/t represents the unsupported
span behind the dies and these parameters can be represented by a single non
dimensional number, say, ‘L/t’ for the purposes of dimensional analysis. Therefore
the non-dimensional parameters in (4-13) can be summarized as shown in Table 4.4.
П1
(Ltb/t) or (Distance between tie bar centers/Die thickness)
П2
(Pt/t ) or (Platen thickness/Die thickness)
П3
(TR) or (Die shoulder thickness/Die thickness)
П4
(L/t ) or (Span/Die thickness)
П5
(Max Separation/t)× (Young’s Modulus/Distributed contact load)
П6
(Lx/Ly) or (Aspect ratio of the die)
П7
(Lcx/Lcy) or (Aspect ratio of the cavity)
Table 4.4: Non-Dimensional Structural Design Parameters
All non dimensional parameters have some physical significance. The parameter П1,
indicates that increasing the distance between the tie bar center has the same effect as
decreasing the die thickness. If the distance between the tie bars increases for a given
87
platen thickness, the platen bends more and the support available for the dies is
reduced and the dies deflect more. Decreasing the die thickness will also result in an
increase in the deflection of the die. Though the variable Ltb was not explicitly varied
in the computational experiments, the parameter П1 varies among the experimental
cases due to the variation in die thickness among the experimental cases and the
effect of Ltb can be captured from the computational experiments.
The second parameter П2 can be combined with the first parameter П1, to obtain a
new non dimensional parameter, say, П1-2, which is given by
∏1 − 2 =
∏1 Pt
=
∏ 2 L tb
(4-14)
The parameter П1-2, in equation (4-14) shows that there is always a combination of
values of platen thickness and distance between tie bar centers for which the platen
stiffness remains a constant. The most important non dimensional parameter is П4
which represents the ratio between the unsupported span and the die thickness. This
parameter suggests that larger the span behind the die, thicker should be the die. A
right combination of the span and die thickness can always be chosen to obtain a
desired magnitude of parting plane separation. On the ejector side, the parameter П4
represents the ratio of the span between the pillar supports to the die thickness. In our
experiments the pillar locations with respect to the center of pressure were fixed. But
the thickness of the die was varied and hence the parameter П4 varies among the
different experimental cases. This enables the power law model fit to capture the
88
inherent variability in die deflection caused by changes in unsupported span and die
thickness.
4.3.2 Empirical Correlation to Predict Ejector Side Parting Surface Separation
As mentioned in the previous section, the non-dimensional parameter П4=L/t is a
linear combination of the other span variables, ln/t, PD/t, and RW/t. Therefore seven
common length scales (L1x, L1y, L2, L3, L4, RWX and RWY) that characterize the span
between pillars in each pillar pattern were identified as shown in Figure 4.3 and a
general closed form expression that can be extended to any arbitrary pillar pattern
was developed. The model form used in the regression for the ejector side data is
shown in equation (4-15). The term Liavg in equation (4-15) represents the average of
the internal spans (L2, L3 and L4) between the pillar supports in each of the four pillar
support cases. For pillar pattern-1, Liavg was taken to be the average of L2, L3 and L4.
For pillar pattern-2, there are no pillar supports and hence Liavg was taken to be zero.
For pattern-3, Liavg is the average of L2 and L4 and for pillar pattern-4 it is the average
of L3 and L4. The schematic of these length scales for the four pillar patterns are
illustrated in Figure 4.3. The variable np was introduced as a correction for the
number of pillar supports. The power law model was obtained by non linear
regression using sequential quadratic programming optimization algorithm in SPSS
[67]. The parameter estimates, the standard error values and the confidence intervals
for the parameter estimates are shown in Table 4.5. The values in Table 4.5 were
89
obtained from SPSS. The standard error values and the confidence intervals were
estimated using the linearized form of equation (4-15).

Max Ejector Sep
E
  PT
×
= c8 + c0 1 + 
t
Dist load
L
  tb



c1+
c7
1+ np

c6
  L tb 
×
  t  ×

 
 L1x + L1y RWX + RWY 
 L iavg
× 
+
 + c 2
2t
2t

 t

  Lx Lcy  
 
× 1 + 
×
  Ly Lcx  



 
c 3+
c4
1+ np
(4-15)
×
c6
Parameter Estimates
95%
Confidence
Std.
Parameter Estimate
Interval
Error
Lower Upper
Bound Bound
c0
26.586
6.25 14.261 38.91
c1
-0.954
0.1
-1.15 -0.758
c2
1.62
0.078 1.467 1.773
c3
3.785
0.113 3.563 4.008
c4
1.484
0.084 1.319 1.649
c5
-0.407 0.108 -0.62 -0.193
c6
-1.488 0.119 -1.722 -1.253
c7
3.053
0.354 2.356 3.751
c8
20.205 3.672 12.964 27.446
Adjusted R squared = 0.982.
Table 4.5: Parameter Estimates for Ejector Side Fit
It can be observed from Table 4.5 that the estimates are at least an order of magnitude
higher than the standard error. The lower and upper bounds for the 95% confidence
interval does not include zero for any of the estimates and hence it can be concluded
that all the estimates are significantly different from zero. An adjusted r-squared
90
value of 0.982 was obtained for the ejector side fit which indicates that the model
explains 98.2% of the variability in the data.
The ejector side power law model in the form of non dimensional parameters is
shown in equation (4-16) and the same model is shown in the form of explicit
variables in equation (4-17).
Figure 4.3: Length Scales Representing the Unsupported Span behind the
Ejector Die
91
3.1 

− 0.95 +
− 1.5
(
)
×
∏ 5 = 20.2 + 26.6 1 + ∏ 1- 2
1+ np  × [∏ 1 ]


1.5
× [(∏ 4a ) + 1.6(∏ 4b )]3.8 + 1+ np ×
(4-16)
× [1 + (∏ 6 × ∏ 7 )]− 0.41

Max Ejector Sep
E
  Pt
×
= 20.2 + 26.6 1 + 
t
Dist load
L
  tb



− 0.95 +
3.1
1+ np

−1.5
  L tb 
×
×
  t 



 L1x + L1y RWX + RWY 
 L iavg
× 
+
 + 1.6
2t
2t

 t

  Lx Lcy  
 
× 1 + 
×
  Ly Lcx  



 
3.8 +
1.5
1+ np
(4-17)
×
− 0.41
It can be observed from equations (4-16) and (4-17) that the term П4a is the average
of the span between the pillar supports and the rail support. This span is represented
by the explicit variables L1x and L1y in equation (4-17). Similarly the parameter П4b in
equation (4-16) is the average of the spans between the pillars. The average of the
span between the pillars is represented by the explicit variables Liavg in equation
(4-17). The relative contributions of the design variables on the parting plane
separation can be inferred from the magnitude and signs of the exponents of those
variables in equation (4-17). It can be observed from equation (4-17) that the largest
contribution to the ejector side separation comes from the unsupported span behind
the die which is characterized by the number, size and location of the support pillars
and rails. The term ‘np’ in the exponent for the average span variable suggests that as
92
the number of pillars increases, the value of the exponent decreases and the separation
becomes less sensitive to the span between the pillars as opposed to case which has
fewer pillar supports. When ‘np’ decreases the value of the exponent increases and
the model prediction becomes more sensitive to the span between the pillars.
The next major variable affecting the separation on the ejector side is the thickness of
the die shoe. It can be shown from equation (4-17) that for a case with no pillar
supports, the effect of die thickness on the maximum ejector side separation is in the
order, Ө (t
-2.8
) while all other factors are held constant. Increasing the thickness of
the die results in an increase in the stiffness of the die and hence thicker the die, less
the separation.
The dimensionless parameter representing the platen thickness, (Pt/Ltb), by itself had
no effect on the maximum ejector separation and hence a first order correction term
(1+ (Pt/Ltb)c) was included to represent the effect of platen stiffness. Thus the platen
thickness has only a first order effect on the ejector side separation. The parameters
П6 and П7 represent the aspect ratio of the die and the cavity respectively.
93
The variables Lcx and Lcy represent the characteristic horizontal and vertical
dimensions of the cavity respectively. The largest of the horizontal and vertical
dimension of the bounding box of the cavity can be chosen as fixed length scale and
the other length scale can be obtained from the projected area of the cavity as shown
below:
Lcy =
Projected Area of the Cavity
Lcx
(4-18)
It can also be noted that the parameter die thickness ratio (П3=TR) does appear in the
ejector side power law model. Though the term was included in the initial curve
fitting procedure, the error involved in the estimate for its exponent was higher than
the estimate itself. In other words the amount of variability in the maximum
separation data caused by the thickness ratio alone was negligible. This might be due
to the tied constraint used between the rear surface of the insert and the die shoe
surface in the finite element model. Due to the tied constraint the insert and the die
shoe behave as a single block and hence the insert thickness/die shoulder thickness
has a negligible effect on the maximum separation. Therefore this parameter was
dropped in the final power law model fit.
94
4.3.3 Empirical Correlation to Predict Cover Side Parting Surface Separation
The model form used for the cover side regression is shown in (4-19) and the
parameter estimates, the standard error values and the confidence intervals for the
parameter estimates of the cover side are shown in Table 4.6. The values in Table 4.6
are also obtained from SPSS.
Max Cover Sep
E
L 
×
= c0  tb 
t
Dist load
 t 
c4
 Pt 
× 
 t 
c1
 Lx 
× 
 t 
c2
 Lx Lcy 
×
×

 Ly Lcx 
c3
(4-19)
Parameter Estimates
95%
Confidence
Interval
Std. Lower Upper
Parameter Estimate Error Bound Bound
c0
0.359 0.016 0.327 0.391
c1
-1.908 0.023 -1.954 -1.862
c2
3.449 0.028 3.395 3.504
c3
-1.822 0.021 -1.863 -1.782
c4
-1.763 0.033 -1.828 -1.698
Adjusted R squared = 0.995
Table 4.6: Parameter Estimates for Cover Side Fit
The adjusted r-squared value for the cover side fit is 0.995 which shows that the
model explains 99.5% of the variability in the data obtained from the computational
experiments. Similarly none of the confidence intervals include zero which indicate
that the parameter estimates are significantly different from zero or in other words the
magnitude of the parameter estimates are larger than the standard error involved in
their estimates.
95
The power law model obtained for the cover side is presented in the form of non
dimensional parameters and in the form of explicit variables in equations (4-20) and
(4-21) respectively.
∏ 5 = 0.4 [∏ 1 ]−1.8 × [∏ 2 ]−1.9 × [∏ 4 ]3.5 × [∏ 6 × ∏ 7 ]−1.8
Max Cover Sep
E
L 
×
= 0.4  tb 
t
Dist load
 t 
− 1.81
 Pt 
× 
 t 
− 1 .9
 Lx 
× 
 t 
3 .5
(4-20)
 Lx Lcy 
×
×

 Ly Lcx 
−1.8
(4-21)
The last term in the equations (4-20) and (4-21) is the correction factor for the aspect
ratio of the die and the cavity geometry. It can be inferred from equations (4-20) and
(4-21) that the die length is the most important factor that contributes the separation
on the cover side. The model suggests that the larger the die with respect to the area
between the tie bar centers, the higher is the maximum separation on the cover side.
The next important factor that contributes to the separation on the cover side is the
platen thickness. It can be seen from these equations that a thicker platen will result in
lower parting plane separation on the cover side. While all other factors are held
constant the effect of platen thickness and distance between tie bars on the cover
separation are in the order of Pt~Ө (-1.9) and Ltb~Ө (-1.8) respectively. Similarly the
effect of die thickness is in the order, t~Ө (1.2). In summary, a large thick platen
performs better on a small thin die. The die thickness ratio parameter (П3=TR)
showed no effect on the cover side separation data also. Therefore the term was
dropped in the final edited cover side power law model.
96
4.4 Sensitivity of Parting Plane Separation to Variations in
Structural Design Parameters
In the previous section, the sensitivity of the parting plane separation to various
structural design variables was explained using the magnitude and direction of the
exponents of the variables. The sensitivity of the maximum separation to the design
variables can also be better understood by studying at their surface/contour plots
which are also referred to as response surface plots. These plots are presented in this
section, first in terms of the non-dimensional parameters and then in terms of the
explicit design variables. The power law models obtained for predicting the
maximum cover and ejector side parting surfaces were used to generate response
surface plots of maximum cover and ejector side parting plane separation.
4.4.1 Response Surface Plots of Non-Dimensional Parameters
The response surface plots are presented in the form of non-dimensional parameters
in this section. The plots of non-dimensional parameters not only gives a better
understanding of the interaction between the individual design variables, but the
effect of individual variables can also be inferred from fewer plots.
97
3.5
(Max Cov Sep * E)/(t * Dist Load)
4
3
3.5
3
2.5
2.5
2
2
1.5
1.5
1
0.5
8
0
9
1
7
8
7
6
6
0.5
5
5
4
Ltb/t
4
Lx/t
Figure 4.4: Non-Dimensional Cover Separation vs. Non-Dimensional Die Length
(П4) and Distance between Tie Bars (П1)
(П2=4.5, П6=1, П7=0.58)
Figure 4.4 shows the effect of the non dimensional parameter П1=Ltb/t and П4=Lx/t
on the non dimensional cover separation П5 while other parameters are held constant.
The figure shows that small dies result in less separation on the cover side. It can also
be observed that the maximum cover separation is less sensitive to the distance
between the tie bars for a small die and it is more sensitive to the distance between
the tie bars for a large die. Figure 4.5 shows the effect of the non dimensional
parameters П2=Pt/t and П4=Lx/t on the maximum cover separation. The plot shows
that the maximum cover separation decreases as the platen thickness increases and
this effect is more pronounced for a large die than for a small die. It can be concluded
98
from Figure 4.4 and Figure 4.5 that small thin dies on larger thick platens offer the
best mechanical performance on the cover side.
(Max Cov Sep / t) * (E / Dist Load)
6
8
5.5
5
6
4.5
4
4
3.5
2
3
0
2.3
2.5
2.2
2
2.1
2
1.9
Pt/t
1.8
4.5
5
6
5.5
6.5
7
7.5
1.5
1
Lx/t
Figure 4.5: Non-Dimensional Cover Separation vs. Non-Dimensional Die Length
(П4) and Platen Thickness (П2)
(П1=8.8, П6=1, П7=0.58)
Figure 4.6, Figure 4.7 and Figure 4.8 show the effect of non dimensional parameters
on the maximum non dimensional ejector separation. Figure 4.6 is a plot of the ratio
between the weighted average of the spans vs. the non dimensional ejector separation.
The plot was obtained for the pillar pattern-1 shown in Figure 4.2. A higher value of
the weighted average indicates a higher value of span between the center of pressure
and the pillars and it also indicates a lower value of span between the outer most
pillars and the rail support. In other words as the inner spans increase, the pillars
99
move closer to the rail support and the outer spans decrease. It can be inferred from
the plot that an increase in span for a given die thickness results in an increase in the
ejector side separation. The plot also shows that for any given span between the
supports, a decrease in thickness results in an increase in the maximum ejector
separation. Since the die thickness is usually determined during the cooling line
design, an appropriate arrangement of the pillar and rail supports should be chosen to
minimize the separation on the ejector side.
(Max Eje Sep / t)* (E / Dist Load)
250
200
150
100
50
3.5
4
4.5
5
Weighted Average of Span / t
5.5
Figure 4.6: Non-Dimensional Ejector Separation vs.
Non-Dimensional Weighted Average of Span (П4a+ 1.6 П4b)
(П1-2=5.9, П1=8.8, П6=1, П7=0.58)
Figure 4.7 shows the effect of the stiffness of the platen and the stiffness of the die on
the maximum ejector separation in the form of the non dimensional parameters. The
100
parameter Pt/Ltb is the ratio between the platen thickness and the distance between the
tie bar centers which represents the stiffness of the platen. The ratio between the
weighted average of the span and the die thickness governs the stiffness of the die.
The plot shows that the stiffness of the platen has a negligible effect on the ejector
separation even for thin dies with larger unsupported spans.
650
600
(Max Eje Sep / t) * (E / Dist Load)
550
700
500
600
450
500
400
400
300
350
200
300
100
5.5
250
1.35
5
1.3
4.5
1.25
4
150
1.2
3.5
Weighted Average of Spans/T
200
3
1.15
Pt/Ltb
Figure 4.7: Non Dimensional Ejector Separation vs. Non Dimensional Platen
Thickness (П1-2) and Weighted Average of Spans (П4a+ 1.6П4b)
(П1 =8.8, П6=1, П7=0.58)
101
3200
(Max Eje Sep / t) * (E / Dist Load)
3000
3500
2800
3000
2600
2500
2400
2000
2200
1500
2000
1000
1.32
1800
1.3
1600
1.28
4
1.26
5
1.24
1.22
Pt/Ltb
1400
6
1.2
7
8
9
1200
Ltb/t
Figure 4.8: Non-Dimensional Ejector Separation vs.
Non-Dimensional Distance between Tie Bars (П1-2) and Platen Thickness (П1)
(П4a+ П4b =6.4, П6=1, П7=0.58)
The interaction between the platen thickness and the distance between the tie bars and
its effect on ejector separation is shown in Figure 4.8. It can be observed from this
figure that the ejector separation decreases as the distance between the tie bar centers
increases. In other words as the distance between the tie bar centers increases, the die
foot print becomes smaller relative to the available platen area. Therefore this
observation confirms that small foot print dies result in less separation. It can be
concluded that a large thick platen provides the best support for the dies on the ejector
side.
102
4.4.2 Response Surface Plots of Explicit Design Variables
The sensitivity of parting plane separation to the explicit design variables is also
shown in Figure 4.9, Figure 4.10, Figure 4.11, Figure 4.12, Figure 4.13 and Figure
4.14. They convey the same information as the plots of non-dimensional parameters.
Figure 4.9 shows the variation of maximum cover side separation in response to
changes in die length and die thickness for various platen thicknesses while all other
variables are kept constant. The figure shows that small thin dies on thicker platens
perform better on the cover side. It can also be observed that the slopes of the surface
representing the 9” platen are much steeper than the slopes of the surfaces
representing the other two thicker platens. This shows the significance of the platen
stiffness in reducing the separation on the cover side.
Figure 4.10 shows the response surface plot of maximum cover separation as a
function of the platen thickness and die thickness for different die lengths. It can be
observed from Figure 4.10 that the slope in the platen thickness direction is steeper
than the slope in the die thickness direction which again indicates the significance of
the cover platen stiffness in reducing the cover side parting plane separation.
It can be inferred from Figure 4.9 and Figure 4.10 that small thin dies get squeezed on
to the platen surface and receives better support from the platens. Therefore thin dies
result in smaller separation on the cover side.
103
Figure 4.9: Maximum Cover Separation vs. Die Thickness & Die Length
(Ltb= 44 inches, Lx/Ly = 1, Lcx/Lcy = 0.58)
Figure 4.10: Maximum Cover Separation vs. Die Thickness & Platen Thickness
(Ltb= 44 inches, Lx/Ly = 1, Lcx/Lcy = 0.58)
104
Figure 4.11 shows the effect of die thickness and the distance between the tie bar
centers on the maximum cover side separation for three different die sizes. It can be
observed from Figure 4.11 that as the distance between the tie bar centers decreases
the maximum cover side separation increases and the effect is more pronounced for a
thick die. In other words as the distance between the tie bars decrease, the dies
become larger relative to the available platen area and results in a larger cover side
separation.
Figure 4.11: Maximum Cover Separation vs. Die Thickness & Distance between
Tie Bar Centers (Pt= 11 inches, Lx/Ly = 1, Lcx/Lcy = 0.58)
Figure 4.12 shows the response surface plot of maximum ejector separation with
respect to pillar diameter and die thickness for the four different pillar patterns
105
considered in this study. It can be observed that for pattern-2 that has no pillar
supports the parting plane separation varies non-linearly along the die thickness
direction only. Since the maximum vertical and horizontal span between pillars in
pillar pattern-1 and pattern-4 are less than the maximum horizontal and vertical span
in pattern-3, the maximum separation is always less for pattern-1 and patter-4 as
compared to pattern-3. Moreover, pattern-1 and pattern-4 behaves identically with
respect to the maximum parting plane separation.
Figure 4.12: Maximum Ejector Separation vs. Die Thickness & Pillar Diameter
Figure 4.13 and Figure 4.14 shows the variation in maximum ejector separation with
respect to changes in die length and die thickness for pillar pattern-1 and three
different platen thicknesses. In Figure 4.13, the pillar diameter was held at 4 inches
106
and in Figure 4.14 the pillar diameter was held at 1.5 inches. It can be observed from
these figures that the slope in die thickness direction is much higher than the slope in
the die length direction. Comparing the highest points (large thin die) in Figure 4.13
and Figure 4.14, it can also be seen that the platen thickness has a slightly higher
effect on the maximum separation for cases with 1.5 inch diameter pillars as
compared to cases with 4 inch diameter pillars. This indicates that the effect of platen
thickness increases as the pillar diameter decreases.
Figure 4.13: Maximum Ejector Separation vs. Die Thickness & Die Length
(Ltb= 44 inches, PD=4”, Lx/Ly = 1, Lcx/Lcy = 0.58)
107
Figure 4.14: Maximum Ejector Separation vs. Die Thickness & Die Length
(Ltb= 44 inches, PD=1.5”, Lx/Ly = 1, Lcx/Lcy = 0.58)
4.5 Model Adequacy
The power law models for predicting maximum parting plane separation were
developed using parting plane separation data from cases with pillar arrangement
patterns shown in Figure 4.2. Dimensional analysis was then used to characterize the
length scales representing the span between the pillar and rail supports. To study the
adequacy of this model for pillar arrangement patterns other than the ones shown in
Figure 4.2, three test cases with pillar arrangement patterns as shown in Figure 4.15
were chosen. The summary of the platen thickness, die thickness, die size, pillar
diameter, clamp load and cavity pressure used for the three test cases are shown in
Table 4.7
108
Figure 4.15: Pillar Arrangement Patterns in the three Test Cases Used to Study
the Adequacy of the Power Law Models
109
Variable
Value used in Test Cases
Platen Thickness
11"
Die Thickness
7.5"
Horizontal Die Dimension
31"
Vertical Die Dimension
31"
Pillar Diameter
2.75"
Clamp Load
800 tons
Cavity Pressure
10000 PSI
Table 4.7: Summary of the Parameter Values used for Test Cases
The power law models were used to predict the maximum ejector and cover parting
plane separations respectively for the three test cases. The length scales used in the
ejector side power law model to describe the span variables for the three test cases is
also shown in Figure 4.15. The span between the outer most pillars and the rail
support is denoted by L1x and L1y for all three test cases as shown in Figure 4.15. The
spans between the center pillar and the outer pillars are represented by L2 and L3.
Since there is only a center pillar in test case-1, the length scales representing the span
between the pillars is ignored in the power law model.
In test case-2, L2 is the only length scale representing the span between the pillars. To
be consistent with the length scales that were defined during model fitting procedure,
the span between the outer pillars are always defined between the edge of the center
pillar and the edge of the outer pillar when a center pillar is present. In the absence of
a center pillar, the span is defined between the center of pressure and the edge of the
110
outer pillar. These two scenarios are demonstrated in test case-2 and test case-3
respectively. Test case-3 has two length scales L2 and L3 representing the span
between the pillars. The maximum ejector side parting plane separation was also
predicted for the three test cases using finite element analysis. The comparison
between the FEA predictions and power law model predictions are shown in Table
4.8.
Test Case
FEA Prediction
(mm/inches)
Power Law Prediction
(mm/inches)
1
0.183/0.0072
0.130/0.0050
2
0.182/0.0072
0.172/0.0068
3
0.159/0.0063
0.124/0.0049
Table 4.8: Comparison of FEA and Power Law Model Predictions for Ejector
Side for the Test Cases
It can be observed from Table 4.8 that the difference between the FEA and power law
predictions ranges from 0.01 mm to 0.05mm (0.0004 in to 0.0022 in). The maximum
difference of 0.05 mm (0.0022 in) was observed in test case-1, which has a center
pillar only. The average of the absolute value of residuals from the power law model
fitting procedure was also in the order of 0.05 mm (0.002 in). Given the three test
cases lie outside the model domain, the magnitudes of these observed differences
between the FEA and power law predictions are very reasonable. The average of the
absolute value of the residuals of the power law model fit was also found to be 0.056
mm (0.002 inches).
111
The comparison between the finite element model predictions and power law model
predictions on the cover side for these three cases are summarized in Table 4.9.
Test Case
FEA Prediction
(mm/inches)
Power Law Prediction
(mm/inches)
1
0.162/0.0064
0.152/0.0059
2
0.162/0.0064
0.152/0.0059
3
0.163/0.0064
0.152/0.0059
Table 4.9: Comparison of FEA and Power Law Model Predictions for Ejector
Side for the Test Cases
It can be observed from Table 4.9 that the difference between the FEA and power law
predictions on the cover side is 0.01 mm (0.0005 in). Since the parameters on the
cover side are the same for all three test cases, the power law model predicts the same
value of cover side separation for all three test cases. The average of the residuals
from the model fitting procedure was also found to be in the same order of magnitude
as the observed difference of 0.01 mm (0.0005 in).
Therefore the predictions from the ejector side power law models can be expected to
differ from a corresponding finite element model in the range of ±0.05 mm (±0.002
inches). Similarly the predictions from the cover side power law model can be
expected to differ from a corresponding finite element model in the range of ±0.01
mm (±0.0005 inches).
112
4.5.1 Rules to Characterize the Spans between Pillars and the Spans between
Pillars and Rails
It was observed during the model adequacy study that the ejector side parting plane
separation is very sensitive to the span between the pillars and the definition of spans
should be consistent with span definitions used in the power law model development.
The following rules serve as guidelines to characterize the spans between the pillars
and between the pillars and the rails.
1. The span between the pillars and the rails should be defined from the edge of
the outermost pillar and the inner edge of the rails. This is illustrated in Figure
4.16a, where the Lo’s denote the spans between the pillars and rails.
2. If a pillar is present directly behind the center of pressure, the inner spans are
defined as the horizontal and vertical distances from the edge of this center
pillar to the edge of the outer pillars as shown in Figure 4.16a where the inner
spans are represented by Li’s.
3. If a pillar is not present behind the center of pressure, the inner spans Li
should are defined as the horizontal and vertical distances from the center of
pressure to the edge of the outer pillars as shown in Figure 4.16c.
4. If only one pillar is present then the outer spans Lo are defined between the
outer edge of the pillar to the inner edge of the rails and the inner spans Li are
defined from the center of pressure to the outer edge of the pillar. This is
illustrated in Figure 4.16d
113
Figure 4.16: Illustration of Rules for Characterizing the Spans behind the
Ejector Die
The rules summarized above can be used as a guideline to characterize the spans
between the pillars and the rail supports to be used in the power law model to predict
the ejector side separation.
114
4.6 Platen Stiffness Characterization and Determination of Platen
Thickness Parameter to be Used in Power Law Models
The power law models were developed based on finite element modeling of an 8.9
MN (1000 ton) four toggle die casting machine. These models also represent the
stiffness characteristics of the die casting die and the machine. The stiffness of the
machine is represented by the platen thickness parameter. Certain machine designs
consist of cast platens with rib like structures for enhancing the stiffness of the platen
and reducing the material simultaneously. For the power law model to be applicable
to machines with such platen designs, an equivalent platen thickness parameter has to
be developed. The equivalent platen thickness parameter will represent the constant
thickness of an 8.9 MN (1000 ton) machine platen that has the same stiffness
characteristic as that of the machine platen under consideration.
The location of the toggle mechanism behind the ejector platen could also vary
among different machine designs. The location of the toggle mechanism acts as a
constraint on the ejector platen and the stiffness of the ejector platen to resist the
clamp and pressure loads will depend on the toggle location. Two platens with same
thickness but different toggle locations could have different stiffness characteristics
due to the differences in the mechanical constraints imposed by the toggle
mechanism. For the power law models to be applicable for machines with toggle
locations other than the four toggle mechanism, an equivalent constant thickness of
an 8.9 MN (1000 ton) four toggle machine platen should be determined. In this
115
section the methodology to characterize the stiffness of the machine platens and
determine an equivalent platen thickness parameter using finite element modeling is
presented.
The support available for the dies from the platen is affected by the magnitude of the
deflection of the platens in the tie bar direction. The deflection of the platens in the
two directions perpendicular to the tie bars is negligible. Therefore the equivalent
thickness parameter to be used in the power law models are determined by comparing
the stiffness of the platen in the tie bar direction alone. The deflection of the platens
will depend on the size and geometry of the platen, constraints acting on it and the
location and magnitude of the loads. The relation between the load, deflection and
stiffness is given by
F = K∆x
(4-22)
Where F is the load, K is the stiffness and ∆x is the deflection. The clamp load is
transmitted to the machine platens through the die shoes. The stiffness of the platen
can be estimated by obtaining the deflection of the platen at a chosen location under
loads of different magnitudes. The stiffness of the platen is given by the slope of the
load deflection curve.
In the proposed methodology a static finite element analysis was used to obtain the
load deflection data for the platens. The deflection of the platen under different
magnitudes of load is obtained from a finite element models that has the exact
geometric representation of the platen under consideration and the platen stiffness is
116
determined from the load deflection curve. Then the thickness of a solid 8.9 MN
(1000 ton) machine platen that possess approximately the same stiffness value is also
determined using finite element modeling and this value of the platen thickness is
chosen as the equivalent thickness parameter. The location of the loading area in the
finite element model is the same as the location of the dies on the platen. The
constraints used in the models are different for the cover and ejector platens.
An arbitrary four toggle ribbed platen design shown in Figure 4.17 and Figure 4.18
was chosen to demonstrate the stiffness characterization methodology. The
dimensions of this platen are shown in Figure 4.18. The nominal thickness of this
platen is 6 inches and the solid rib on the center is 15 inches thick. The procedure to
determine the equivalent thickness parameter for cover and ejector side platens is
described in section 4.6.1 and section 4.6.2 respectively.
Figure 4.17: Schematic of the Platen Design Chosen to Demonstrate Stiffness
Characterization Methodology
117
Figure 4.18: Dimensions of the Platen Design Chosen to Demonstrate Stiffness
Characterization Methodology (All Dimensions in Inches)
4.6.1 Methodology to Determine Equivalent Cover Platen Thickness
The cover platen is constrained to the machine tie bars and the bottom of the cover
platen is constrained to the machine base. The other ends of the tie bars are secured to
the rear platen. In the proposed method the tie bars are modeled explicitly and one
end of the tie bars is constrained to the platen using a tied contact. The nodes on the
118
other end of the tie bar are constrained in space in all degrees of freedom. Twelve
nodes on the bottom of the platen are constrained in all directions. The constraints are
shown in Figure 4.19. Uniformly distributed load is applied on the platen center over
an area of 31” by 31” which is the size of a nominal die on a 1000 ton machine.
However this loading area can be changed depending on the size of the die in the case
for which the equivalent stiffness is being determined. The magnitudes of loads
considered are 250 tons, 500 tons, 750 tons and 1000 tons. The maximum deflection
of the ribbed platen is obtained for all of the four loading cases. The deflection is
obtained with respect to a coordinate system that was described using the three corner
nodes on the inside face of the platen. The coordinate system is also shown in Figure
4.18. Then the deflections of solid platens with thicknesses of 0.2795 meters (9
inches), 0.254 meters (10 inches) and 0.2286 meters (11 inches) are also obtained
from finite element models. The deflection values for the solid platens are obtained at
a location which is same as the location of maximum deflection on the ribbed platen
finite element model.
The load deflection curves for the ribbed platen, and solid platens with thickness of
0.2795 meters (9 inches), 0.254 meters (10 inches) and 0.2286 meters (11 inches) are
shown in Figure 4.19 along with the equations for best fit lines.
119
Figure 4.19: Schematic of Finite Element Model Used to Determine the Cover
Platen Stiffness
Deflection Vs Load
10000000
y_rib = 4470805440.54x - 17.88
y_9" = 4551731632.71x - 6.83
y_10" = 5318714310.27x - 5.32
y_11" = 6047360126.79x + 9.07
9000000
Load (Newton)
8000000
7000000
6000000
4000000
Ribbed
Solid-9"
Solid-10"
3000000
Solid-11"
5000000
2000000
1000000
0
0
0.0005
0.001
0.0015
0.002
0.0025
Deflection (meters)
Figure 4.20: Deflection Vs Load Curves for Cover Platen
It can be observed from Figure 4.20 that a solid 9 inch platen exhibits the same
stiffness characteristic as the ribbed platen shown in Figure 4.17 and Figure 4.18. It
120
can also be seen from the slopes of the equations in Figure 4.20 that the stiffness of
the 9 inch platen is approximately equal to the stiffness of the ribbed platen.
Therefore the equivalent thickness of the cover side ribbed platen can be chosen as 9
inches and this thickness parameter can be used in the power law model.
4.6.2 Methodology to Determine Equivalent Ejector Platen Thickness
The procedure used to determine the stiffness of the ejector platen is similar to the
procedure used for the cover platen. However the constraints in the finite element
model used to determine the ejector platen stiffness are different from the ones in the
finite element models used to determine the cover platen stiffness. The ejector platen
is free to slide on the tie bars and the toggle mechanism acts as a constraint behind the
ejector platen. The toggles were modeled using 3D beam elements in the finite
element models and one end of the beam elements is constrained in space in all
degrees of freedom. The schematic of the finite element model and the constraints are
shown in Figure 4.21.
The maximum deflection of the ribbed platen under loads of 250 tons, 500 tons, 750
tons and 1000 tons were obtained from the finite element model. The loads were
applied as uniformly distributed load over an area of 31” by 31” on the center of the
platen. The deflection is obtained with respect to the coordinate system that was
defined using the three corner nodes on the inside face of the platen. The coordinate
system is also shown in Figure 4.21. The deflections of solid platens at the same
121
location as that of the location of maximum deflection of ribbed platen are also
obtained under same loading conditions.
Figure 4.21: Schematic of the Finite Element Model Used to Determine the
Ejector Platen Stiffness
The load deflection curves for the ribbed platen, and solid platens with thickness of
0.2795 meters (9 inches) and 0.254 meters (10 inches) are shown in Figure 4.22. It
can be observed from Figure 4.22 that a solid 9 inch platen exhibits the same stiffness
characteristic as the ribbed platen shown in Figure 4.17 and Figure 4.18. It can also be
seen from the slopes of the equations in Figure 4.22 that the stiffness of the 9 inch
platen is approximately equal to the stiffness of the ribbed platen. Therefore the
equivalent thickness of the ejector side ribbed platen can be chosen as 9 inches.
122
Deflection Vs Load
y_rib = 3578847885.62418x
y_9" = 2852503674.45180x + 2.8
y_10" = 3793433298.8133x
10000000
9000000
8000000
Load (Newton)
7000000
6000000
Ribbed
5000000
Solid-9"
Solid-10"
4000000
3000000
2000000
1000000
0
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
Deflection (meters)
Figure 4.22: Deflection Vs Load Curves for Ejector Platens
4.6.3 Methodology to Determine Equivalent Thicknesses for Platens with
Different Toggle Locations
As mentioned in the previous section the location of the constraints on the platen
affects the stiffness of the platen and hence a two toggle platen will have a different
stiffness characteristic as compared to a four toggle platen of same thickness. While
using the power law models to predict parting plane separation for a die on a two
toggle machine an equivalent thickness has to be determined for a corresponding four
123
toggle machine. The schematic of the finite element models used to determine the
stiffness of four toggle and two toggle ejector platens are shown in Figure 4.23.
Figure 4.23: Schematic of Finite Element Models used to determine the Stiffness
of Four Toggle (Right) and Two Toggle (Left) Ejector Platens
The toggles are modeled using 3D beam elements and the node on the free ends of the
beam elements are constrained in space in all three directions. The maximum
deflection of the two toggle platen is obtained under loads of 250 tons, 500 tons, 750
tons and 1000 tons. The loads are modeled as uniformly distributed load on an area
representing the foot print of the dies (31” by 31” inches in this case used for
demonstration of the method). The deflections of the four toggle platen at the same
location are also determined under same loading conditions.
A 0.254 meters (10 inch) two toggle platen is chosen as an example. The load
deflection curve for this two toggle platen is shown in Figure 4.24. The load
124
deflection curves for four toggle platens with 0.2286 meters (9 inches) and 0.2032
meters (8 inches) thicknesses are also shown in Figure 4.24.
Deflection vs Load
y_10"_2TG = 11874368631.68x - 89.05
y_9"_4TG = 15348952508.01x - 222.54
y_8"_4TG = 10950266433.03x - 164.25
10000000
9000000
Load (Newton)
8000000
7000000
6000000
5000000
4000000
2-Toggle-10"
4-Toggle-9"
4-Toggle-8"
3000000
2000000
1000000
0
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
Deflection (meters)
Figure 4.24: Deflection Vs Load Curves for Two Toggle and Four Toggle Platens
It can be observed from Figure 4.24 that the stiffness of the 0.2032 meters (8 inch)
thick four toggle platen is approximately same as the stiffness of the 0.254 meters (10
inches) thick two toggle platen. Therefore the equivalent four toggle platen thickness
for a 0.254 meters (10 inches) thick two toggle platen is 0.2032 meters (8 inches). It
can also be observed that a two toggle platen is less stiff than a corresponding four
toggle platen with the same thickness.
125
4.7 Determination of Equivalent Stiffness of a Die Casting Machine
Using a Lumped Element Model
The method described in the previous section can be used to obtain the stiffness
values for the platens. Using the same procedure the stiffness values can also be
obtained for the dies, inserts, toggles and tie bars. The equivalent stiffness of the
die/machine system can then be determined using lumped element modeling. In the
lumped element modeling of static mechanical systems the various components of the
system are represented with springs of appropriate stiffness values. The springs are
connected in series and/or parallel depending upon the behavior of the corresponding
components under the loads acting on the system.
When the clamp load is applied by the die casting machine, the toggles are
compressed between the ejector and rear platens and the tie bars are stretched
between the cover and rear platens. The dies and inserts are compressed between the
cover and ejector platens. Therefore the die/machine system under the action of
clamp load can be represented by the lumped spring stiffness diagram as shown in
Figure 4.25.
126
Figure 4.25: Spring Stiffness Diagram for the Die and the Machine under Clamp
Load
The top portion of Figure 4.25 has three springs connected in series, KRP, KTB and
KCP, representing the stiffness of the rear platen, tie bars and cover platen
respectively. One end of the top circuit is grounded and the clamp load acts on the
other end of the series circuit. The bottom part of the diagram has five springs
connected in series, KCD, KPS, KED, KEP and KTG, representing the stiffness of the
cover die, parting surface, ejector die, ejector platen and toggle respectively. The
clamp load stretches the tie bars, cover platen and the rear platen and it compresses
the cover die, parting surface, ejector die, ejector platen and the toggle. The pressure
load applied in the next stage will stretch the parting surface spring, KPS between the
cover die and the ejector die and the displacement due to the pressure load on the top
circuit is negligible.
The behavior of the system shown in Figure 4.25 is illustrated using an example. This
example also serves as a validation for the model behavior. The stiffness of the
platens, tie bars, toggles were approximated using average displacement values
predicted from the finite element model described in Chapter 3. Since the stiffness
127
values of the dies and the machine parts are lumped in the linear springs with only
one degree of freedom, average displacements were chosen for the lumped model as
opposed to maximum displacement used in the development of power law models.
The stiffness was calculated using the relation
K=
FClamp
(4-23)
∆X
Where ∆X is the average displacement obtained from the finite element model under
clamp load only. The stiffness values thus obtained for the dies and the machine parts
are summarized in Table 4.10. The value of clamp load used in the simulations is
6.432 MN.
Part
Cover Platen
Rear Platen
Tie Bars
Cover Die
Ejector Die
Ejector Platen
Toggle
Stiffness (N/mm)
3×107
2.3×107
0.82×107
17.1×107
25×107
240×107
2.6×107
Table 4.10: Stiffness Values to be used in Lumped Element Model for the Dies
and Machine Parts
Since the tie bars, cover platen and rear platen are all in series, the equivalent stiffness
of the top circuit in Figure 4.25 is given by
1
1
1
1
=
+
+
K A K CP K RP K TB
(4-24)
128
Using the stiffness values in Table 4.10, we obtain
K A = 5.03 × 10 6 N/mm
(4-25)
Therefore the stretch due to the clamp load on the top portion of the circuit in Figure
4.25 is given by
δ Clamp
A
=
FClamp
KA
=
6.432 × 10 6
5.03 × 10 6
= 1.3 mm
(4-26)
Similarly, the stiffness of the springs to the left of the parting surface in Figure 4.25
are combined to obtain an equivalent stiffness given by
1
1
1
1
=
+
+
K 2B K ED K EP K TG
(4-27)
Using the equivalent stiffness values KA and K2B, the circuit shown in Figure 4.25 can
be reduced to one shown in Figure 4.26. In Figure 4.26, an additional load,
FClamp-Fpressure, acts on either side of the parting surface, which is equivalent to the
clamp load on the parting surface in excess of the pressure load applied during the
metal injection stage. A pressure load of 6.8948 Mpa was used in this example. Based
on the projected area of the cavity used in this research (0.0775 sq. meter), the
magnitude of the pressure load is calculated as follows:
FClamp - FPressure = FClamp - (Pressure × Projected Area of Cavity )
6
= 1.331 × 10 Newton
129
(4-28)
Figure 4.26: Spring Diagram with Clamp and Pressure Loads
The displacement of the cover die spring, KCD, between nodes 3 and 4, due to the net
load on the parting surface is given by
-P
δC
CD =
FClamp − FPressure
KCD
(4-29)
= 0.008mm
Similarly, the displacement of the spring K2B, between nodes 1 and 2 due to the net
load on the parting surface is given by
δ C2B-P =
FClamp − FPressure
K 2B
= −0.06 mm
(4-30)
Therefore the total displacement in the bottom part of the circuit, between the nodes 1
and 4 is given by
-P
-P
C-P
δC
= δC
B
CD - δ 2B = 0.068 mm
(4-31)
This total displacement of the bottom circuit due to the pressure load adds up to the
displacement on the top of the circuit, shown in equation (4-26).
130
Therefore the total displacement on the top of the circuit after clamp and pressure is
given by
-P
δ CA-P = δ Clamp
+ δC
= 1.3 + 0.068 mm = 1.37 mm
B
A
(4-32)
Therefore the total load on the top of the circuit after clamp and pressure is given by
FAC - P = K A δCA- P = 6.891 × 106 N
(4-33)
The initial applied clamp load was 6.432 MN and the clamp load in the tie bars, cover
platen and rear platen after the pressure stage is 6.891 MN as shown in equation
(4-33). Therefore there is 7% increase in the clamp load after the application of the
cavity pressure in the lumped model where as the finite element model shows a 3%
increase in the clamp load after the cavity pressure stage. Given that the stiffness of
the springs are based on the average displacements obtained from the finite element
model, a 4% difference between the finite element and a one degree of freedom
lumped model is reasonable. This also shows that the lumped element model behaves
as intended.
The equivalent stiffness of the die/machine system can be obtained from the spring
diagram shown in Figure 4.25. The equivalent stiffness of the bottom circuit is given
by
1
1
1
=
+
K B K 2B K CD
(4-34)
131
The top and bottom circuits are connected in parallel. Therefore the equivalent
stiffness of the die/machine system is given by
K eq = K A + K B
(4-35)
The methodology described in previous section to determine the stiffness of the
machine platens can be used to obtain the stiffness of the dies and the other machine
parts as well. Thus the lumped element model provides a better understanding of the
interaction between the dies and machine parts. It also provides a means to compare
the stiffness characteristics of different machines.
4.8 Summary
Power law models to predict the ejector and cover side parting surface separation
were developed using dimensional analysis and data from finite element analysis
experiments. These power law models can be used to predict maximum cover and
ejector parting plane separation and compare the performance of different die designs
and machines. The sensitivity of parting plane separation to the structural design
parameters were explained from the magnitude of the exponents and coefficients of
the power law model. Response surface plots are also provided as a visual aid in
understanding the relative contributions of the structural design variables on parting
plane separation. The accuracy of the ejector side power law model is within ±0.0022
inches and the accuracy of the cover side model is within ±0.0005 inches. Rules and
guidelines to characterize the unsupported spans on the ejector side are also provided.
132
A methodology to characterize the stiffness of platens of various designs and sizes
was described in this chapter. This method can be used to determine the equivalent
platen thickness parameter to be substituted in the power law models to predict
parting plane separation. A one degree of freedom lumped element model is also
provided to estimate the total stiffness of die casting machines and this model also
provides an understanding of the interaction between various machine parts and the
dies.
133
CHAPTER 5
EMPIRICAL CORRELATIONS TO PREDICT TIE BAR
LOADS
5.1 Introduction
As mentioned in the first chapter, the current approach used in industry to predict the
tie bar loads ignores the location of the die with respect to the platen center and it
assumes that the machine and the dies are perfectly rigid. The current approach also
violates the force equilibrium constraint thus leading to inaccurate predictions. To
address these issues, a non-linear power law model was developed to predict the tie
bar loads of the die casting machine based on the location of the die and cavity center
of pressure with respect to the tie bars. The model was obtained by curve fit to tie bar
load prediction data from computational experiments. The computational experiments
were conducted using the finite element modeling method described in chapter 3. An
experimental design was developed based on the horizontal and vertical dimension of
the die, the locations of the die and cavity center of pressure with respect to the platen
center and the magnitude of cavity center of pressure. Dimensional analysis was used
to incorporate other important scale factors and obtain the non-dimensional
134
parameters. The non-linear model was then fit to the non-dimensional form of the
location, scale and load variables. Experimental tie bar load measurements were then
compared to the power law model predictions to check the adequacy of the power law
models. The design of experiments, the dimensional analysis, the non-linear
regression model and the experimental verification are described in this chapter.
5.2 Design of Experiments
The factors that were considered are the die length, die width, location of the die with
respect to the platen center and the location of the cavity center of pressure with
respect to the platen center and the magnitude of the cavity pressure. The description
of the variables and their levels are shown in Table 5.1. The location variables in
Table 5.1 are defined with respect to a coordinate system with origin on the center of
the platen area between the tie bars. The schematic of the coordinate system and the
tie bar labels are shown in Figure 5.1.
135
Factor
Factor Description
Level
1
Level
2
Level
3
Level
4
Level
5
LX
Die Width (inches)
26.49
30.4
32.6
34.6
36.5
LY
Die Height (inches)
26.49
30.4
32.6
34.6
36.5
DPX
Die location in Xdirection (inches)
-4
-2
0
2
4
DPY
Die location in Ydirection (inches)
-4
-2
0
2
4
-4
-2
0
2
4
-4
-2
0
2
4
2
4
6
8
1
CPX
CPY
CPR
Location of center of
pressure in Xdirection (inches)
Location of center of
pressure in Ydirection (inches)
Cavity Pressure (KSI)
Table 5.1: Description of Variables used for Tie Bar Load Prediction Model
Development
136
Figure 5.1: Coordinate System and Tie bar Labels viewed from inside face of
Cover Platen
The initial goal of the study was to develop a linear polynomial model to predict the
tie bar load imbalance as compared to the nominal load (one fourth of total clamp).
Allen et al [63, 64], proposed an experimental design methodology for developing
polynomial response surface model that minimizes both the model bias and random
error. This method assumes that the true model is a third order polynomial and the fit
model is a second order polynomial. The proposed criterion is called Expected
Integrated Mean Square Error (EIMSE) criteria and it is given by
[
]
ˆ {ε, x, η(x),ξ} − η(x) 2 
EIMSE (ξ ) = E  Y

η, x,ε
(5-1)
Where Ŷ{ε, x, η(x), ξ} and η(x) are the predicted value and true model value at the
prediction point x respectively. The experimental design is ξ and ε is a vector of
137
random errors. The symbol ‘E’ denotes the statistical expected value taken over the
variables and the EIMSE optimal method attempts to minimize this expected value
using optimization method. The optimization method searches over various candidate
experimental designs and finds the one that yields the minimum EIMSE with
minimum number of runs.
For our study a full factorial design based on seven factors with five levels was
chosen as the initial candidate set and this full factorial design comprised of 78125
runs. Some of these runs were eliminated due to the constraints imposed on the die
and cavity location by the platen size and the die size respectively. The reduced
candidate set was given as the input for EIMSE optimal design selection and the best
subset of points were chosen based on the criteria described in equation (5-1). Finally
an experimental design with fifty runs was obtained and a linear polynomial model to
predict the tie bar loads was developed [65]. The linear polynomial model is
presented in the APPENDIX B.
One of the limitations of the linear polynomial model is that the accuracy of the
model predictions is dependent on the range of the design variables within which the
experimental data are collected. The model cannot be extrapolated beyond the domain
it is intended to be used. The range of cavity loads used in the experimental design
fell within 13%-83% of the applied clamp load. The linear polynomial models that
were initially developed to predict the tie bar loads showed poor predictions when the
static cavity load approached 100% of the clamp load. Therefore it was decided to
138
develop a non-linear power law model using dimensional analysis to capture the
inherent non-linearity in the tie bar load data.
A few additional cases were also added to the experimental array to include cases
with a cavity load equal to 100% of clamp load. The initial experimental design
obtained from EIMSE criteria was an unbalanced design and hence the selection of
additional cases with high cavity load values was left as an arbitrary choice.
Therefore the best we could do without seriously violating the properties of the
experimental matrix was to repeat the cases in the experimental array with 2 KSI
cavity pressure values at 12 KSI (corresponds to 100% of clamp load). This added an
additional 10 cases to the experimental array. It was also decided to add a few cases
with no cavity pressure load. Therefore the tie bar load predictions were obtained
under clamp load only for cases in the experimental array with 10 KSI and these ten
cases represent the zero cavity pressure cases. Thus an experimental design with a
total of 70 cases was finally arrived at and the experimental array is shown in Table
5.2.
139
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
LX "
26.49
26.49
34.6
26.49
32.6
32.6
34.6
32.6
32.6
30.4
26.49
36.5
36.5
34.6
36.5
26.49
26.49
32.6
30.4
36.5
34.6
36.5
34.6
32.6
30.4
36.5
26.49
30.4
LY "
34.6
30.4
34.6
26.49
32.6
34.6
34.6
30.4
32.6
26.49
32.6
32.6
32.6
26.49
26.49
30.4
26.49
30.4
30.4
32.6
32.6
32.6
26.49
26.49
30.4
34.6
26.49
36.5
DPX "
-4
2
0
0
-2
0
0
0
2
4
4
0
0
0
0
-2
0
2
4
0
0
0
0
0
4
0
4
2
DPY "
0
-2
0
4
0
0
0
-2
2
4
0
-2
0
-4
0
-2
2
2
-2
2
2
-2
0
4
2
0
-4
0
CPX "
-4
2
-4
0
-2
2
4
0
0
4
4
-4
-4
-4
4
-2
0
2
4
0
2
2
-4
-2
4
-4
4
2
CPY "
0
0
-4
4
-2
4
-4
0
2
4
0
-2
-2
-4
0
-4
2
2
-4
0
4
0
0
4
4
4
-4
4
CPR
(KSI)
12
12
6
12
10
6
12
8
4
8
6
12
10
10
6
8
10
8
6
4
10
12
4
8
4
6
8
4
Continued
Table 5.2: Experimental Array used for Tie Bar Load Prediction Model
Development.
140
Table 5.2 continued
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
30.4
26.49
34.6
30.4
26.49
32.6
30.4
26.49
26.49
32.6
36.5
36.5
36.5
36.5
30.4
26.49
36.5
36.5
32.6
34.6
32.6
30.4
26.49
26.49
30.4
26.49
30.4
34.6
32.6
30.4
26.49
26.49
26.49
26.49
26.49
34.6
30.4
36.5
36.5
30.4
34.6
36.5
36.5
36.5
34.6
26.49
26.49
30.4
36.5
32.6
32.6
36.5
26.49
32.6
34.6
26.49
36.5
34.6
30.4
26.49
32.6
30.4
32.6
26.49
30.4
26.49
30.4
34.6
30.4
26.49
-4
-2
0
4
2
2
-4
0
0
-2
0
0
0
0
-4
-4
0
0
-2
0
2
4
-4
2
4
4
4
0
0
4
4
-2
-4
2
0
0
-2
0
0
2
0
0
0
0
0
0
-2
-2
0
-2
2
0
-4
0
0
2
0
0
-2
4
0
-2
2
4
2
-4
-2
0
-2
4
141
-4
-2
-2
4
2
0
-4
0
0
0
0
2
-4
2
-4
-4
4
2
-4
4
4
4
-4
2
4
4
4
2
-2
4
4
-2
-4
2
0
4
-2
2
-2
0
4
-4
-4
0
2
0
-2
0
-2
-4
2
4
-4
2
-4
2
-4
0
0
4
0
-4
4
4
4
-4
-2
0
0
4
6
10
6
10
4
8
10
4
10
12
4
10
4
6
6
8
6
12
12
10
12
4
0
0
0
0
0
0
0
0
0
0
2
2
2
Table 5.2 continued
64
65
66
67
68
69
70
34.6
36.5
36.5
32.6
36.5
32.6
32.6
34.6
32.6
32.6
34.6
26.49
32.6
26.49
0
0
0
-2
0
-2
2
0
-2
-2
0
-4
0
2
4
-4
2
0
2
-4
4
-4
-2
0
2
-4
2
2
2
2
2
2
2
2
2
5.3 Dimensional Analysis and Empirical Correlation to Predict Tie
Bar Loads
The functional relationship between the tie bar loads and the variables involved in the
physical phenomenon is given by
f (T, Lx, Ly, DPX, DPY, CPX, CPY, L tb , Clamp,CPR, A ) = 0
(5-2)
Where, T is the load on the tie bar, Lx and Ly are the horizontal and vertical
dimensions of the die, DPX and DPY are the horizontal and vertical location of the
dies with respect to the platen center, CPX and CPY are the horizontal and vertical
location of the cavity center of pressure, Ltb is the distance between the tie bar
centers, CPR is the pressure load and A is the projected area of the cavity. This
functional relation can be reduced to one involving fewer non-dimensional
parameters using dimensional analysis.
The load on the tie bars is essentially a percentage of the total clamp load and the
percentage of clamp load on individual tie bars depend on the location of the die and
142
the cavity center of pressure. Ideally, if the dies and cavity center of pressure were
centered on the platen, the load on each tie bar should be equal to one fourth of the
clamp load. Therefore the tie bar load T, in equation (5-2) is scaled by one fourth of
the clamp load. During the metal injection stage, the cavity pressure load relieves a
fraction of the initial clamp load depending on its magnitude. The pressure load also
redistributes the contact force on the die parting surface depending on the location of
the cavity center of pressure. Hence the moments on each tie bars are also
redistributed during the metal injection stage. Therefore the pressure load was
approximated as the product of the cavity pressure and the projected area of the cavity
and the pressure load was scaled by the total clamp load. The remaining length scale
variables were scaled by the square root of the projected area of the cavity. Therefore
equation (5-2) can be rewritten as

T
Lx Ly DPX DPY CPX CPY L tb CPR × A 
f 
 = 0 (5-3)
,
,
,
,
,
,
,
,
A Clamp 
A
A
A
A
 0.25 × Clamp A A
The moments caused by the die locations DPX, DPY and the cavity location CPX and
CPY about the tie bars will also depend on the distance between the tie bar centers.
Therefore the non-dimensional parameters involving the locations variables DPX,
DPY, CPX, CPY and the non-dimensional parameter involving the distance between
the tie bars, Ltb can be combined together and the functional relationship in equation
(5-3) can be reduced further as follows:

T
Lx Ly DPX DPY CPX CPY CPR × A 
 = 0
f 
,
,
,
,
,
,
,
 0.25 × Clamp A A L tb L tb L tb L tb Clamp 
143
(5-4)
The non-dimensional numbers in the equation (5-4) are summarized in Table 5.3
П1
Lx/√A
П2
Ly/√A
П3
DPX/Ltb
П4
DPY/Ltb
П5
CPX/Ltb
П6
CPY/Ltb
П7
(CPR × A) / Clamp
П8
T/ (0.25 × Clamp)
Table 5.3: Non Dimensional Parameters used in Tie Bar Load Prediction Model
Though the cavity geometry and the machine stiffness were not varied in the
experiments, the use of non-dimensional parameters shown in Table 5.3 ensures that
the model could be extended to machines of different sizes and tonnages.
The loads on the four tie bars were obtained for all of the 70 cases in the experimental
array and a non-linear regression model was fit to the non-dimensional parameters
using the tie bar load data from the computational experiments. Non-linear model was
fit to the load data for each individual tie bars. The same model form was obtained for
all the four tie bars and the magnitudes of the coefficients and exponents for the two
top tie bars were approximately equal with different signs. Similarly the magnitudes
of the exponents and coefficients for the two bottom tie bars were almost the same.
Therefore the tie bar load data for the top tie bars-1 & 2 were pooled together and the
data for the bottom tie bar-1 & 2 were pooled together and the same model form was
144
fit to the top tie bar data and bottom tie bar data. The models forms used for the top
and bottom tie bars in terms of the non-dimensional parameters are given by
equations (5-5) and (5-6) respectively. It can be noted that the signs for the nondimensional parameters П4 and П6 are reversed between the equations (5-5) and (5-6).
These parameters represent the vertical location of the die and center of pressure
respectively and hence their signs are positive for the top tie bars and negative for the
bottom tie bars. The sign convention is explained in detail in the following section.
c1
c2
  DPX     DPY  
Top Tie bar load
 ×
  1 + 
= c01 ± 



Nominal Load
  0.5L t b     0.5Lt b  
(5-5)

 



CPR × A 
CPX 
 1 + CPY   CPR × A    ×
+
× 1 + c3
exp   1 ±
c
4


CLAMP 
0.5L t b 
0.5Lt b   CLAMP   







 CPR × A
 CPR × A   
× 1 − c5
× exp
  
 CLAMP   
 CLAMP

c1
c2
  DPX     DPY  
Bottom Tie bar load
 ×
  1 − 
= c01 ± 



Nominal Load
  0.5Lt b     0.5Lt b  
(5-6)



CPR × A 
CPX 
CPY   CPR × A   
×
+ c4 1 −
× 1 + c3
exp   1 ±




CLAMP
0.5Lt b 
0.5L t b   CLAMP   


 




 CPR × A
 CPR × A   
× 1 − c5
× exp
  
CLAMP
 CLAMP   


SPSS was used to perform the non-linear regression and the sequential quadratic
programming algorithm in SPSS was used to solve the non-linear regression problem.
The parameter estimates, the standard error in the estimates and the confidence
intervals for the estimates for the top and bottom tie bars are provided in Table 5.4
and Table 5.5 respectively.
145
Parameter
c0
c1
c2
c3
c4
c5
Parameter Estimates
95% Confidence
Interval
Std.
Lower
Upper
Estimate
Error
Bound
Bound
1.005
0.002
1.001
1.009
0.354
0.011
0.333
0.374
0.303
0.012
0.279
0.327
0.063
0.005
0.052
0.074
0.886
0.045
0.796
0.976
-0.098
0.006
-0.109
-0.086
Adjusted R-square = 0.96
Table 5.4: Parameter Estimates for Top Tie Bar Model Fit
Parameter
c0
c1
c2
c3
c4
c5
Parameter Estimates
95% Confidence
Interval
Std.
Lower
Upper
Estimate
Error
Bound
Bound
1.04
0.002
1.036
1.044
0.294
0.012
0.271
0.318
0.256
0.014
0.228
0.284
0.062
0.005
0.051
0.073
1.025
0.052
0.922
1.128
-0.106
0.005
-0.117
-0.095
Adjusted R-Square = 0.95
Table 5.5: Parameter Estimates for Bottom Tie Bar Model Fit
146
The power law models to predict top and bottom tie bar loads are given by equation
(5-7) and (5-8) respectively.
  DPX 
Top Tie bar load

= 1.0051 ± 

Nominal Load
  0.5Lt b 
0.354
  DPY 

1 + 

  0.5Lt b 
0.303
×
(5-7)

 



CPR × A 
CPX 
 1 + CPY   CPR × A   ×
exp   1 ±
0.886
× 1 + 0.063
+


CLAMP 
0.5Lt b 
0.5Lt b   CLAMP  








CPR
A
×
 CPR × A



× 1 − 0.098
× exp
 
 CLAMP  
 CLAMP

  DPX
Bottom Tie bar load
= 1.041 ± 
Nominal Load
  0.5L t b




0.294
  DPY
1 − 
  0.5L t b






CPR × A 
CPX
× 1 + 0.062
exp   1 ±



CLAMP
0.5L

tb



0.256
×
(5-8)
 



 + 1.03 1 − CPY   CPR × A   ×
 0.5L   CLAMP  

t b 


 

 CPR × A
 CPR × A  
× 1 − 0.106
× exp
 
 CLAMP  
 CLAMP

The term nominal load on the left hand sides of the equations (5-7) and (5-8) is
defined as one fourth of the total clamp load. The ± sign before the terms involving
DPX and CPX indicates that the signs of these variables depend on the location of the
die/cavity and also on the tie bar for which the load is calculated. If the die and/or
cavity is positioned towards the tie bar for which the load is to be predicted, a positive
sign should be chosen for the corresponding variables and if the die and/or cavity is
positioned away from the tie bar for which the load is to be predicted a negative sign
should be chosen. For example, if the die is horizontally off-center towards the helper
side, a positive sign should be used before the variable DPX to predict the loads on
the helper side tie bars and a negative sign should be used before DPX to predict the
147
loads on the operator side tie bars and vice versa. The same sign convention applies to
the terms involving the horizontal cavity location CPX.
The third term in the equation represents the moments caused by the cavity pressure
load. The moment terms and the load term appear as exponential terms in the model.
This indicates that the tie bar loads increase or decrease in an exponential fashion as
the cavity center of pressure is moved towards or away from the respective tie bars.
The parameters involving the die dimensions, Lx/√A and Ly/√A were found to have a
negligible effect on the model predictions and hence they were ignored in the model.
These non-dimensional parameters were included in the initial model fitting. But the
error involved in their estimates for the corresponding exponents were much higher
than the value of the exponents itself. Therefore this model behaves as a lumped
model where the clamp and pressure loads are approximated by point loads acting on
the die center and cavity center of pressure respectively.
5.4 Model Adequacy
The power law models shown in equations (5-7) and (5-8) were obtained by curve
fitting to tie bar load data from an 8.9 MN (1000 ton) four toggle machine. It can be
seen from equations (5-7) and (5-8) that the location variables are all scaled by the
distance between the tie bar centers. The distance between tie bar centers is
determined by the machine manufacturer based on the size and clamping capacity of
the die casting machine. To study the adequacy of the model to predict the tie bar
loads on machines of other designs and clamping capacity, the power law model
148
predictions were compared against the finite element model predictions of tie bar load
on machines of other designs and tonnages. Three different machine finite element
models were considered, viz, a 3500 ton four toggle machine, 1000 ton four toggle
machine and a 250 ton two toggle machines. The die location, cavity location, the
clamp load and magnitude of cavity pressure for these three cases are summarized in
Table 5.6
Machine
Design
3500 ton-4
toggle
1000-ton-4
toggle
250-ton-2
toggle
DPX
(in)
DPY
(in)
CPX
(in)
CPY
(in)
CPR
(PSI)
Ltb
(in)
Cavity
Load
(tons)
Clamp
Load
(tons)
4
0
4
0
0
84.25
0
3500
0
1.25
0
3.63
10000
44
602
722
0
-3.14
0
-0.423
10000
21.75
135
250
Table 5.6: Summary of Finite Element Models used for Model Adequacy Study
The boundary conditions used in the finite element model of 3500 ton machine is
same as the boundary conditions in the computational experiments that yielded the
data for power law model development. However the finite element models of the
1000 ton machine and 250 ton machine are different from the finite element models
in the computational experiments. A schematic of the finite element model used for
the 1000 ton and 250 ton machines are shown in Figure 5.2. In this model the corner
nodes on the bottom of the cover platen are constrained in vertical direction only. One
end of the tie bars is constrained to the cover platen and the other end is constrained
in space. The rear platen is not included in the model and the toggle mechanism was
149
modeled using spring elements of appropriate stiffness. The clamp load was applied
by specifying displacements on the nodes of the free ends of the spring elements.
Since the tie bars are constrained in space and the compliance of the rear platen is not
included in the model shown in Figure 5.2, the model is stiffer than the finite element
models used for power law model development. Therefore comparison of the power
law model predictions and predictions from the finite element model shown in Figure
5.2 will provide insight into the sensitivity of the tie bar load predictions to the
boundary conditions used in the finite element models.
Figure 5.2: Schematic of the Finite Element Model of the 1000 Ton Machine and
250 Ton Machine Used for Model Adequacy Study [1]
The comparison between the FEA predictions and power law predictions for the 3500
ton machine, 1000 ton machine and 250 ton machine are shown in Table 5.7, Table
5.8 and Table 5.9 respectively.
150
Top Tie Bar-1
Top Tie Bar-2
Bottom Tie Bar-1
Bottom Tie Bar-2
Tie Bar Load/Nominal Load Prediction
FEA
Power Law
-3.8%
-3.0%
2.8%
3.8%
-2.9%
-2.6%
4.0%
3.1%
Table 5.7: Comparison of Model Predictions for a 3500 Ton Machine
(DPX=4”, DPY=0”, CPX=4”, CPY=0”, CPR=0 PSI)
For the 3500 ton machine shown in Table 5.7, the die is off-center to the right,
towards the top tie bar-2 and bottom tie bar-2 and hence these two tie bars carry
higher loads than the nominal. Top tie bar-1 and bottom tie bar-2 carry less than the
nominal load. It can be observed from Table 5.7 that the differences between the FEA
predictions for the top tie bar-1 and bottom tie bar-1 of the 3500 ton machine is 1.1%
and the differences between the FEA predictions for top tie bar-2 and bottom tie bar-2
is 1.2%. Since the same types of constraints were used in both the FEA and power
law models, the difference between the predictions is only about 1% which is
negligible. This shows that the non dimensional power law model is adequate to
predict the tie bar loads well independent of the size of the machine. Though the die
is not off-center vertically there are differences in the FEA predictions between the
top and bottom tie bars. This is due to the constraint imposed on the bottom of the
cover platen by tying the nodes on the platen to the machine base.
Table 5.8 shows the FEA and power law predictions for a 1000 ton four toggle
machine. Since the die and cavity are both off-center towards the top tie bars, the top
tie bars carry loads higher than the nominal in this case and the bottom tie bar loads
are lower than the nominal. In the FEA, two nodes on the bottom of the cover platen
151
were constrained in vertical direction alone. Therefore the FEA predictions for the top
and bottom tie bars are symmetric about the nominal load. However the power law
model shows that the top tie bar loads are 6.7% higher than the nominal while the
bottom tie bar loads are only 2.1% lower than the nominal load. The constraint used
between the nodes of the cover platen and the base in the computational experiments
causes this asymmetry in the tie bar load predictions between the top and bottom tie
bars. The same trend is observed on the tie bar load predictions for the 250 ton two
toggle machine shown in Table 5.9. The die and cavity are off-center vertically
towards the bottom tie bars in the 250-ton machine and hence the top tie bars loads
are lower than the nominal and the bottom tie bar loads are higher than the nominal.
Both the FEA and power law model are able to capture this trend. However the power
law predictions for the bottom tie bar are higher than the FEA predictions. The
boundary conditions used in the FEA for the 250 ton machine are same as those
shown in Figure 5.2 and hence the observed symmetry in FEA prediction between the
top and bottom tie bars. The higher predictions of the power law model on the bottom
tie bars are caused due to the constraint on the bottom of the cover platen.
Top Tie Bar-1
Top Tie Bar-2
Bottom Tie Bar-1
Bottom Tie Bar-2
Tie Bar Load/Nominal Load Prediction
FEA
Power Law
4.1%
6.7%
4.1%
6.7%
-4.1%
-2.1%
-4.1%
-2.1%
Table 5.8: Comparison of Model Predictions for a 1000 Ton Machine
(DPX=0”, DPY=1.25”, CPX=0”, CPY=3.63”, CPR=10000 PSI)
152
Top Tie Bar-1
Top Tie Bar-2
Bottom Tie Bar-1
Bottom Tie Bar-2
Tie Bar Load/Nominal Load Prediction
FEA
Power Law
-8.2%
-8.3%
-8.2%
-8.3%
8.2%
14%
8.2%
14%
Table 5.9: Comparison of Model Predictions for a 250 Ton Machine
(DPX=0”, DPY=-3.14”, CPX=0”, CPY=-0.423”, CPR=10000 PSI)
5.4.1 Model Adequacy Study using Experimental Measurements
The predictions from the power law model are not expected to match exactly the tie
bar load measurements from a die casting machine due to various approximating
boundary conditions in the finite element model. Comparison with measurements on
a die casting machine provide further insight into the adequacy of this model and also
its limitations.
Therefore experiments were conducted on a 250 ton two-toggle machine by varying
the die location and obtaining the tie bar loads under clamp load only. The schematic
of the test die on the machine platens is shown in Figure 5.3. The test die measures
13.38 inches by 18 inches and the distance between the tie bar centers is 21.75 inches.
Four uniaxial strain gauges were attached to each tie bar to measure the longitudinal
strains. The strain gauges were attached to the tie bars at a distance of 267 mm (10.5
in) from the inside face of the stationary platen so that the strain gauges are half-way
between the stationary and movable platens when the test die is closed. The schematic
of the strain gauge locations is shown in Figure 5.4. The four strain gauges on each tie
bar are 90ο apart. Thirteen different die setups were studied, one with a die centered
153
on the platen, four cases with vertically off center dies, four cases with horizontally
off center dies and four cases with diagonally off center dies. The thirteen cases are
summarized in Table 5.10. Not all combinations of diagonally off center dies could
be studied due to the limitation of the space available between the tie bars.
Figure 5.3: Schematic of the test die on the machine platens
154
Figure 5.4: Schematic of the locations of tie bars, strain gauges and coordinate
system, viewed from front of cover platen
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
DPX
(inches)
0
0
0
2
4
0
0
-2
-4
2
4
-2
-4
DPY (inches)
0
2
4
0
0
-2
-4
0
0
1.875
1.875
-1.875
-1.875
Table 5.10: Experimental Array
155
Each case was repeated three times and a constant clamp load of 2500 KN was
applied in all cases. Though the machine was programmed to apply a clamping load
of 2500 KN, the actual clamp load applied by the machine varies slightly from the
nominal. The metal injection stage was ignored in these experiments due to the
practical difficulty of moving the shot sleeve for each test. The strains on each tie bar
were obtained under clamp load only. The average of the strains measured by the four
strain gauges on each tie bar was calculated to obtain the strain on each tie bar. The
tie bar loads were then calculated from the strain values. The nominal clamp load per
tie bar was assumed to be the average of the four tie bar loads and the load on each tie
bar was estimated as the ratio between the tie bar load and the nominal load.
5.4.2 Comparison of Experimental Measurements and Model Predictions
The tie bar loads were also calculated for all the thirteen cases using the power law
model. The experimental measurements and power law predictions of tie bar loads for
the thirteen cases are shown in Table 5.11. Ideally the loads on all four tie bars should
be equal for case-1 where the dies are centered on the platen. However the
measurements show that the loads on bottom tie bars are lower than on the top tie
bars. This could be due to inaccuracy in positioning the dies on the platen causing the
measurements to be biased towards the top tie bars. Further, the squareness of the
machine was not checked before the experiments and lack of squareness and perfect
flatness of the machine platens could also contribute to this observed difference.
156
Case DPX
1
2
3
4
5
6
7
8
9
10
11
12
13
0
0
0
2
4
0
0
-2
-4
2
4
-2
-4
DPY
0
2
4
0
0
-2
-4
0
0
1.875
1.875
-1.875
-1.875
T1
1.03
1.10
1.16
0.99
0.94
0.96
0.92
1.09
1.16
1.06
1.03
1.02
1.08
Experimental
Measurements
T2
B1
1.02 0.99
1.09 0.92
1.15 0.85
1.09 0.90
1.14 0.85
0.95 1.05
0.90 1.10
0.98 1.03
0.92 1.09
1.15 0.84
1.22 0.76
0.90 1.11
0.84 1.17
B2
0.97
0.90
0.83
1.02
1.07
1.04
1.08
0.91
0.83
0.95
1.00
0.97
0.91
Polynomial model
Predictions
T1
T2
B1
B2
1.00 1.01 1.04 1.04
1.05 1.06 0.99 0.99
1.10 1.11 0.93 0.93
0.94 1.07 0.98 1.09
0.86 1.12 0.91 1.13
0.94 0.95 1.08 1.08
0.87 0.88 1.12 1.12
1.06 0.94 1.09 0.98
1.11 0.86 1.13 0.92
0.98 1.12 0.93 1.04
0.90 1.17 0.87 1.08
1.00 0.89 1.13 1.02
1.05 0.82 1.18 0.95
Table 5.11: Comparison of Tie bar Load Measurements and Tie bar Load
Predictions from the Regression Model
The comparison between model predictions and experimental measurements for the
13 cases and for all four tie bars is shown in Figure 5.5. It can be observed from these
charts that the model predictions are consistently lower than the experimental
measurements for the top tie bars and they are consistently higher than the
measurements for the bottom tie bars in all of the 13 cases. This can be attributed to
the constraint type used between the cover platen and the machine base. The edge
nodes of the cover platen and the base were tied using a multi point constraint in the
computational (FEA) experiments. Though the 250-ton machine used for the
experiments has a welded joint the multi-point constraint used in the FEA might be
157
stiffer than the actual welded joint on the machine. The lack of flatness and
squareness of the dies could also have contributed to some of these differences.
Figure 5.5: Tie bar Load Measurements vs. Predictions from the Regression
Model
The differences between the measurements and model predictions are shown in Table
5.12. The differences between the model predictions and the load measurements vary
from 0.1% to 12% depending on the die location. As expected, the worst cases are the
diagonally off center cases, case-10 and case-11, where the measurements show that
the load on the top tie bar-1 (T1) is higher than the nominal and the model predictions
show that the loads on top tie bar-1 (T1) is lower than the nominal. Similarly the load
measurements on bottom tie bar-1 (B2) are lower than the nominal in case-10 and
158
case-11, and the model predictions show that they are higher than the nominal.
Overall, the pattern of results is very good.
Case
DPX
DPY
1
2
3
4
5
6
7
8
9
10
11
12
13
0
0
0
2
4
0
0
-2
-4
2
4
-2
-4
0
2
4
0
0
-2
-4
0
0
1.875
1.875
-1.875
-1.875
Difference between Measurements and Model
Predictions (%)
T1
T2
B1
B2
2.54%
1.26%
-4.83%
-7.40%
4.50%
2.84%
-6.77%
-9.17%
6.52%
4.59%
-7.20%
-9.76%
5.40%
2.03%
-7.72%
-6.95%
7.63%
2.24%
-6.75%
-6.03%
1.98%
0.10%
-2.75%
-4.90%
4.28%
2.13%
-1.80%
-4.32%
2.80%
3.37%
-5.10%
-7.81%
5.19%
5.90%
-3.89%
-9.10%
8.15%
3.25%
-9.65%
-9.23%
12.33%
4.66%
-11.54%
-8.60%
2.49%
1.02%
-2.25%
-5.41%
3.39%
2.29%
-0.91%
-4.14%
Table 5.12: Difference between Measurements and Model Predictions
5.5 Response Surface Plots for the Effect of Die Location and Cavity
Location on Tie Bar Loads
The relative effects of the die location and cavity location on the tie bar load
imbalance can be explained using the power law models shown in equations (5-7) and
(5-8). These equations can be used to generate the response surface plots showing the
tie bar load imbalance as a function of two variables. Figure 5.6 shows the effect of
the location of cavity center of pressure on the load on top tie bar-2, when the
pressure load is 20% of the clamp load. Based on the sign conventions used in the
power law models, a positive value for CPX and CPY means the center of pressure is
159
oriented towards the top tie bar-2 and a negative value indicates a movement away
from top tie bar-2.
The dies are centered on the platen for the case shown in Figure 5.6. Since the cavity
load is very negligible the tie bar load is almost equal to the nominal load irrespective
of the location of the center of pressure.
Figure 5.7 shows the same case but with a cavity load that is 80% of the clamp load.
It can be seen from this figure that there is a maximum of 9% imbalance in the
positive and negative directions, when the center of pressure is 4 inches off-center
from the platen center both horizontally and vertically.
Top Tie Bar Load / Nominal Load
1.1
1.08
1.06
1.04
1.02
1
0.98
0.96
4
2
4
2
0
0
-2
CPY (inches)
-2
-4
-4
CPX (inches)
Figure 5.6: Effect of Cavity Location on Tie Bar Load
(Cavity Load = 20% Clamp Load)
160
Load on Top Tie Bar-2 / Nominal Load
1.07
1.06
1.1
1.05
1.04
1.05
1.03
1.02
1
1.01
1
4
2
4
0.99
2
0
0
-2
CPY (inches)
0.98
-2
-4
-4
CPX (inches)
Figure 5.7: Effect of Cavity Location on Tie Bar Load
(Cavity Load = 80% Clamp Load)
Figure 5.8 and Figure 5.9 show the effect of die location on the tie bar load imbalance
when the cavity load was 20% of clamp load and 80% of clamp load respectively. In
both of these cases the center of pressure is located on the geometric center of the
platen. It can be seen from these figures that a die that is off-center by 4 inches from
the platen center causes an imbalance of approximately 10% irrespective of the
magnitude of the cavity load.
161
1.2
1.05
1.1
1
1
0.9
0.95
4
2
4
2
0
-2
-4
DPY (inches)
0.9
0
-2
-4
DPX (inches)
Figure 5.8: Effect of Die Location on Tie Bar Load
(Cavity Load = 20% Clamp Load)
Load on Top Tie Bar-2 / Nominal Load
Load on Top Tie Bar-2 / Nominal Load
1.1
1.1
1.2
1.1
1.05
1
1
0.9
0.95
4
2
4
2
0
0
-2
DPY (inches)
0.9
-2
-4
-4
DPX (inches)
Figure 5.9: Effect of Die Location on Tie Bar Load
(Cavity Load = 80% Clamp Load)
162
It can be concluded from these plots that the effect of the location of the dies on the
platen has the primary effect on the tie bar load imbalance. When the clamp load is
applied, it is distributed among the four tie bars depending upon the location of the
dies on the platen. In the next stage when the pressure load is applied, it relieves and
redistributes the contact load on the parting surface. The redistribution of the contact
load on the parting surface will depend upon the location of the cavity center of
pressure and the magnitude of the cavity pressure. Due to the redistribution of the
contact load, the moments on the tie bars are altered and the tie bar load is also
redistributed. Therefore ignoring the location of the die could lead to poor predictions
of tie bar load.
5.6 Summary and Conclusions
Power law model to predict tie bar loads as a fraction of the nominal load has been
developed using dimensional analysis and finite element modeling experiments. The
functional form obtained from the dimensional analysis ensures that the model
predictions would be independent of the machine size and tonnage. The adequacy of
the model has also been verified against experimental tie bar load measurements from
a 250-ton two toggle machine. The power law models were constructed using the tie
bar load data obtained by finite element modeling of a 1000 ton four-toggle machine.
Considering this fact and the magnitude of the loads involved, 0.1%-12% difference
between the model predictions and the load measurements from a 250 ton two-toggle
163
machine is very reasonable. The model seems to predict well independent of the
machine size.
The model predictions were also compared against predictions from finite element
model with boundary conditions and constraints that are different from the ones used
in the computational experiments. The comparison showed that the power law model
predictions for the bottom tie bar are about 2% higher than the nominal load due to
the tied constraint between the nodes on the bottom of the cover platen and the
machine base. The power law also shows that ignoring the die location, as done in the
current approach in industry, will lead to inaccurate tie bar load predictions. The
significance of the effect of the die location on the tie bar load imbalance is also
illustrated by the response surface plots.
164
CHAPTER 6
CONCLUSIONS AND FUTURE WORK
Empirical correlations to predict parting plane separation and tie bar loads were
developed in this research. In addition, a methodology to characterize the platen
stiffness and determine the equivalent stiffness of dies and machine was also
developed. These findings can be used together as tools for structural die design and
machine selection. The closed form models also provide an understanding of the
contribution of the structural design variables of the die and the machine on their
mechanical performance. The major conclusions from this study are summarized in
this chapter.
6.1 Conclusions from the Power Laws to Predict Parting Plane
Separation
The model forms for the power laws to predict parting plane separation were obtained
using dimensional analysis and semi empirical plate deflection equations. These
models can be used during the initial die design stages to compare the mechanical
performance of dies and machines and they can be used as a decision tool for
structural die design and machine selection.
165
The following conclusions can be made about the relative contribution of the
structural design variable to the parting plane separation:
• The ejector side parting plane separation has the highest sensitivity to the ratio
of the weighted average of unsupported span between the pillar supports and
the die thickness.
• The platen thickness has only a first order effect on the ejector side separation.
The effect of platen thickness decreases as the number of support pillars is
increased.
• A die that is smaller relative to the platen area between the tie bars performs
better than a large die
• The largest contributor to the cover side separation is the length and width of
the die. A smaller die results in less separation. As the cover platen wraps
around the cover die, a smaller die will have less unsupported span behind it as
compared to a large die and hence it will result in lower separation
• The second major factors affecting the separation on the cover side are the
cover platen thickness and the distance between the tie bar centers. The
sensitivity of the cover side separation to these two factors is of the same order
of magnitude. A thicker platen provides better support to the dies and results in
less separation. The distance between the tie bar centers has an interaction with
die footprint. A larger platen area relative to the die foot print provides better
support and results in less cover side separation.
166
• The die thickness is the next significant variable affecting the cover side
separation. Thin compliant dies are squeezed on to the platen surface and they
follow the deflection path of the platen surface. Therefore thin dies receive
better support from the platens on the cover side.
• Small thin dies on a large thick platen performs better on the cover side,
whereas on the ejector side, the span between the pillars and the die thickness
alone play the major role on the parting surface separation.
• The die thickness ratio was found to have no effect on the ejector and cover
side parting surface separation. This could be due to the tied contact
formulation used between the rear of the insert and the front face of the die
shoe pocket. Due to the tied contact constraint the dies and inserts behave as a
single monolithic block and hence the effect of die thickness ratio or the effect
of the insert thickness could not be confirmed from these computational
experiments.
The model adequacy study showed that the predictions of the ejector side model have
a standard deviation of ±0.0022 inches and the cover side models have a standard
deviation of ±0.0005 inches even for the test cases that are outside the model domain.
6.2 Conclusions from the Machine characterization Study
A methodology to characterize the stiffness of the machines was developed in this
study. This method can be used to determine the stiffness of the platens with complex
geometric features and also to obtain an equivalent thickness parameter of a solid
167
platen with same stiffness characteristic. The equivalent stiffness parameter obtained
by this method can also be used in the power law models to predict the performance
of the dies on the chosen machine. This study has also shown that the location of the
toggle mechanism has a significant effect on the stiffness of the ejector platen. A four
toggle platen has a higher stiffness than a two toggle platen of same thickness.
However this does not imply that two toggle platens are less stiff than a four toggle
platen of same tonnage on the actual machines. In the actual machines two toggle
platens are usually made thicker or ribs are added to the platens to achieve the
required stiffness.
A static lumped element model that uses springs with appropriate stiffness values to
represent the dies and machine was also presented and this model can be used to
determine the equivalent stiffness of the die/machine system.
6.3 Conclusions from the Power Laws to Predict Tie Bar Loads
Power law models to predict the loads on the tie bars as a fraction of the nominal load
were presented in the previous chapter. These models show that the die location on
the platen is the primary factor affecting the distribution of the loads on the tie bars.
The cavity load redistributes the contact load on the parting surfaces and changes the
moments on the tie bars. The redistribution of clamp load after the pressure stage will
depend on the location of the cavity and the magnitude of cavity pressure.
The die length and die width were found to have no effect on the tie bar load
predictions. The power law model interprets the clamp load from the dies and the
168
cavity load as point loads acting on the geometric center of the die and the center of
pressure of the cavity respectively.
The predictions from the power law models were compared against tie bar load
measurements from a 2.2 MN (250 ton) two toggle die casting machine by varying
the die locations. The comparisons showed a 0.1% to 12% difference between the
predictions and the measurements depending upon the location of the dies on the
platen. The power law models were obtained by curve fitting to tie bar load data from
finite element models of an 8.9 MN (1000 ton) four toggle machine. The 12%
difference was observed for one of the extreme off-center cases. A part of these
differences could have been caused due to the lack of perfect flatness on the machine
platen and the inaccuracies in positioning the die on the platen during experiments.
Given the differences in the machine design, clamping capacity and the magnitude of
loads involved, 0.1%-12% difference between the predictions and measurements are
reasonable and it can be said that the model predicts well independent of the machine
size.
The power law predictions were also compared against FEA predictions for machines
of different sizes. Though the boundary conditions were different in two of these test
FEA models from the ones used in the computational experiments, the power law
predictions and FEA predictions showed a good correlation. This comparison also
provided an insight that the power law predictions for the bottom tie bars might be
169
slightly higher than the nominal value (approximately 2% higher than nominal for the
a case with die and center of pressure centered on machine platen)
6.4 Future Work
The power law model to predict parting plane separation are specific to single cavity
open close dies. It is possible to extend this model to multiple cavity dies by
characterizing the span between the supports with reference to the centers of pressure
of the cavities. However the adequacy of the model for multiple cavity dies has to be
verified.
The models to predict tie bar load are also specific to open-close dies. The slide lock
mechanism in non-open close dies could cause asymmetric loads on the parting
surface depending on the locations of the slide carriers. Therefore the power law
models can be reconstructed by including tie bar load data from non-open close dies
with different locations of the slide carrier.
Currently the pressure load in the die cavity is modeled as a pressure boundary
condition on the cavity surfaces with a magnitude equal to that of the intensification
pressure and the dynamic impact load caused by the deceleration of the plunger
mechanism is ignored. There are instances where the pressure hike in the cavity due
to the impact load might exceed the intensification load. Therefore including the
pressure distribution in the cavity caused by the impact load might give better
predictions of parting plane separation from the finite element models.
170
Finally the constraint between the cover platen and the machine base is an issue that
needs further investigation. Though the explicit modeling of bolted joint showed tie
bar load predictions that are closer to the nominal, the constraint is still stiffer than the
joint in the actual machine. Including the clearances of the bolted joint and the
keyways might improve the tie bar load predictions of the finite element model.
171
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177
APPENDIX A
DESCRIPTION OF METHODS TO REMOVE PSEUDO
RIGID BODY MOVEMENT
The pseudo rigid body translation and rotation of the dies and inserts caused by the
stretching of the tie bars was described in Chapter 3. The equation used to define this
affine transformation is given by
R 0
 + [E 0]
1
T


[X s 1] = [X f 1] 
(A-1)
Where Xs and Xf are the starting and final coordinates of the sample nodes
respectively, R is the rotation matrix, T is the translation vector and E is the distortion
component. In this method Xs and Xf are obtained for a few sample nodes on the
parting surface of the dies. The rotation matrix R and translation vector T are given
by
cosΦ.cosγ
- cosΦ.sinγ
sinΦ 


R = sinθ.sinΦ.cosγ + cos θ. sin γ - sinθ.sinΦ.sinγ + cos θ. cos γ - sinθ.cosΦ  (A-2)


- cosθ.sinΦ.cosγ + sin θ. sin γ cosθ.sinΦ.sinγ + sin θ. cos γ cos θ. cos Φ 
T = [x
y z ]′
(A-3)
178
The variables θ, Φ and γ in equation (A-2) are the angles of rotation about the X, Y
and Z axis respectively. The variables x, y and z in equation (A-5) are the translations
along the X, Y and Z axis respectively.Least squares method that minimizes the sum
of squares 'trace(ET E)' was employed to estimate the transformation matrix in
equation (A-1). By differentiating the 'trace(ET E)' and equating it to zero and doing a
few matrix manipulations it can be shown that, the best estimate for the translation T
is the difference between the average starting and finishing position after rotation and
it is given by
T* = X s − X f * R
(A-4)
Where, X s and X f are vectors with the average starting and finishing coordinates
as their components. It follows from (A-1) and (A-4) that
E = [X s − 1 * X f ] − [( X f − 1 * X f ) * R ]
(A-5)
The ‘1’ in equation (A-5) indicates the homogenous coordinates used to represent the
affine transformation which is a translation followed by a rotation. Equation (A-5) is
used to minimize the sum of squares 'trace(ET E)'.
179
Therefore the necessary conditions for minimum error based on the angles of rotation
are given by
∂f (ϕ, T )
 ∂E T
∂E 
=0
E + ET
= tr

∂θ
∂
θ
∂
θ


T
 ∂E
∂f (ϕ, T )
∂E 
E + ET
= tr 
=0
∂Φ
∂
Φ
∂
Φ


∂f (ϕ, T )
∂γ
(A-6)
(A-7)
 ∂E T
∂E 
E + ET
= tr 
=0
∂
γ
∂
γ


(A-8)
Using (A-5) in (A-6), (A-7) and (A-8)we obtain,
∂f (ϕ, T )
∂θ
∂f (ϕ, T )
∂Φ
∂f (ϕ, T )
∂γ


∂R (θ)T
= tr R Tz (γ )R Ty (Φ ) x
cov (X f , X s ) − cov(X f , X f )R x (θ )R y (Φ )R z (γ )  = 0 (A-9)
∂θ


[
]


∂R y (Φ )T T
(A-10)
T
R x (θ ) cov(X f , X s ) − cov(X f , X f )R x (θ )R y (Φ )R z (γ )  = 0
= tr R z (γ )
∂Φ


[
]

 ∂R (γ )T T
R y (Φ )R Tx (θ ) cov(X f , X s ) − cov(X f , X f )R x (θ)R y (Φ )R z (γ )  = 0 (A-11)
= tr  z

 ∂γ
[
]
Equations (A-9), (A-10) and (A-11) are only a function of the coordinate data of the
sample nodes and the rotation component R. The term ‘cov’ in these equations
denotes the covariance between the respective variables. These equations are solved
for the angles of rotation and the rotation matrix is formed. Then equation (A-4) can
be used to estimate the translation component. The estimates for the affine
transformation will be sensitive to the sample nodes that are used. As a result the
location of the nominal parting plane will also be sensitive to the sample nodes.
180
A.1. Procedure to Select Sample Nodes and Predict Pure Distortion
of the Parting Plane from Finite Element Models
The nominal location of the parting plane is defined as the plane that passes through
all of the contact nodes. Figure A.1 shows the contact pressure plot of an ejector
parting surface superimposed on the undistorted nominal parting plane. The blue
region in the plot is the region that is not in contact. The sample nodes from this
region should be avoided and the sample should be selected from the corner regions
that are in contact. These sample nodes are then input to a Mathcad program that
estimates the transformation matrix. Once the transformation matrix is obtained from
the Mathcad program, the pure distortion component is obtained using the relation

∗
[E 0] = [X s 1] − [X f 1] R∗
T
0

1
(A-12)
Where, Xs and Xf are the initial and final coordinates of the entire node set of the dies
and inserts and R* and T* are the best estimates for the rotation and translation
components respectively. The pure distortion values were then input as nodal
boundary conditions to a static finite element model with dies and inserts only. A
dummy step with no other loads was run in this finite element model and the
displacement plots were obtained from this finite element model. This pure distortion
of the parting surface is equivalent to the separation of the respective parting surfaces
from the nominal location. The ejector side separation plot obtained by this method is
181
shown in the Figure A.2. It can be observed from Figure A.2 that the nominal plane
after removing the transformation passes through all of the contact nodes.
Figure A.1: Contact Pressure Plot
182
Figure A.2: Ejector Side Separation Obtained by Sampling Nodes in
Contact Regions Only
A.2. Alternate method to Remove Pseudo Rigid Body Motion Using a
Local Coordinate System in ABAQUS
To test and compare the affine transformation procedure, local coordinate systems
were created internally in ABAQUS using three nodes for each coordinate system
definition. The three nodes should be in contact and they were chosen on three
different corners of the parting surface, consistent with a right handed coordinate
system. Then the Z-displacements of the parting surface nodes with respect to each of
these local coordinate systems were directly obtained from ABAQUS and the
displacements were averaged. The average Z-displacement is now the measure of the
183
ejector side separation and the contour plot for the displacement is shown in
Figure A.3. Comparing Figure A.2 and Figure A.3, it can be seen that the affine
transformation method and the local coordinate system methods give similar
predictions of parting plane separation. The difference in maximum separation
predictions among the two methods varied from 0.1% to 5% for the cases in which
the dies and the cavity center of pressure were centred on the platen. For cases with
off-center dies and/or center of pressure, the parting surface might be completely out
of contact on one or more corners. In such cases the affine transformation procedure
will yield better predictions of the nominal location of the parting plane.
Figure A.3: Separation With Respect to a Local Coordinate System on the
Parting Surface
184
APPENDIX B
LINEAR MODELS FOR TIE BAR LOAD PREDICTION
B.1 Linear Model for Top Tie Bar






 DPX 
 + 60.4 DPY  − 11.9 ± CPX  − 8.7 CPY 
T = 99.1 + 70 ±







 Ltb 
 Ltb 
 Ltb 
 Ltb 


 ± CPX × CPR 
 + 2.8 CPY × CPR 
+ 0.23(CPR) + 3.7



Ltb
Ltb




(B-1)
B.2 Linear Model for Bottom Tie Bar






 DPX 
 + 58.1 DPY  − 12.2 ± CPX  − 17.5 CPY 
B = 101.6 + 59.4 ±







 Ltb 
 Ltb 
 Ltb 
 Ltb 


 ± CPX × CPR 
 + 4.4 CPY × CPR 
+ 0.4(CPR) + 4.1



Ltb
Ltb




185
(B-2)