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Nonlinear Finite Element Analysis of the Coupled Thermomechanical Behaviour of
Turbine Disc Assemblies
Davinder Singh Delhelay, B.E.
A thesis subrnitted in confonnity with the requirements for the degree of Master of Applied Science
Graduate Department of Mechanical and Indusaial Engineering University of Toronto
O Copyright by Davinder Sin* Deihelay 1999
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Abstract
Comprehensive two and three-dimensional finite element studies are made of the effect of
the critical geomevic features and interface conditions in the fr--tree region of a gas
turbine disc upon the resulting thennomechanicd contact stress distribution at the blade-
disc interface. In addition to the rotationai body forces, the thermal state of a turbine disc
made from nickel based super ailoy, was determineci using realistic boundary
temperatures and the assembly was treated as a coupled boundary value problem.
Three aspects of the work were accordingly examined- The fmt was devoted to the
vaiidation of the basic finite element model using convergence tests and photoelastic
stress analysis. The second was concemed with exarrining the effect of the flank length
and flank angle, inner radius of disc and interfacial friction upon the therrnomechanical
stress-state of an insulated disc. The third was concemed with the determination of the
influence of the skew angle upon the triaxial state of stress and load sharing between the
teeth in the turbine disc assembly. Contact between the disc and the blades was modelled
using contact elements and the anaiysis pertain to an insulated disc, with heat conduction
being the main form of heat flow.
The outcome of the study reveals: (i) the important role played by the temperature and
rotation upon the induced thermomechanical contact stresses at the blade-disc interface,
and (ii) the effect of the skew angle upon the resulting three dimensional
thermomechanical stress state, and the inability of the two-dimensional finite element
model to accurately capture the stress gradients at the frr-tree interface and through the
thickness.
1 offer my sincere gratitude to Professor S. A. Meguid for his technicd guidance and
support througfiout the course of my research and for aiiowing me tu use the extensive
computer facilities in his laboratories. 1 also wish to thank the members of the
Engineering Mechanics and Design Laboratory for their help during the undertaking of
the cument study. Especially, 1 would Iike to thank Aleksander Czekanski and Pararnjit
Singh Kanth for their help. Last, but by no means Ieast, 1 would S i e to offer my sincere
gratitude to my parents for their unconditional support.
Contents Abstract
Acknowledgments
Notation
List of Tables
List of Figures
1 Introduction and Justification 1 .............................................................. ........ 1 .1 Turbine Disc Assemblies .. 1
................................................................................ 1.2 Research Objectives 3
..................................................... ...................... 1.3 Method of Approach .. 3
..................................................................................... 1 A Layou t of Thesis 6
2 Literature Review 9
...................................................... 2.1 The Stress Field in Rotating Discs 10
...................................... 2.1.1 Solid Disc Andysis .............................. ... 10
.................................................................... 2.1.2 Hollow Disc Analysis 1 1
...................................................... 2.2 Thermal Stress of a Circuiar Disc 12
......................................................... 2.3 Earlier Finite Element Models 14
.......................................................... 2.3.1 Structural Integrity 14
............................................................ 2.3.2 Thermal Stresses 16
...................................... 2.3.3 Contact at the Blade/Disc Interface 18
......................................................... 2.3.4 Experimental Work 20
3 Thermomecbanical F i t e EIement Analysis of a Turbine Disc Assembiy 27
..................................... 3 . i Fundamentals of the Finite Element Method 27
......................... ... 3.2 Coupled Thermomechanical Contact Pro blems ... 30
............................ 3.3 Solution and Superposition of Uncoupled Problems 32
................................... 3.4 Coupled Analysis for Turbine Disc Assembly 33
...................... 3.5 Two Dimensional Modelling of Turbine Disc .... ...... 37
............................. 3.5.1 Two Dimensional Details of the Geometry 37
......................................... 3.5.2 Finite Element Modeiiing of Disc 37
.................................................. 3.6 Contact between Blade and Disc 39
Modellling of Contact for 2D Therrnomechanical Analysis ......... 44
.................................. 3.7 Three Dimensiond Modelling of Turbine Disc 45
3.7.1 Thtee Dimensional Details of Geometry ................................ 45
3.7.2 Discretization of the Disc Assembly .. .. ................. ..... .......... 46
3.7.3 Modelling of Contact for 3D Thennomechanical Analysis ........... 47
..................................... ........................... 3.8 Convergence Tests ... 48
4 Results and Discussions 55
4.1 Photoelastic Verification of Finite EIement Analysis .................................. 55
4 1 . 1 Photoelastic Materials and Geometry .................... ... .............. 55
................................. 4.1.2 Digital Imaging of Photoelastic Patterns 56
............................................... 4.1.3 Typical Photoelastic Results 57
................................................ 4.2 Two Dimensional Finite Element Analysis 59
........................................ 4.2.1 Details of Models and Applied Loads 59
................. 4.2.2 Analysis of Coupled Thermomechanical Stress Field 62
...................................... 4.2.3 Effect of Contact Angle and Friction 69
....................................... 4.2.4 Effect of Hank Angle and Friction
74
............... 4.3 Three Dimensional Finite Element Analysis ................... .... 78
............... 4.3.1 There Dimensional Mode1 with Straight Fu-Tree Slots 79
................... 4.3.2 Three Dimensional Mode1 with Skew Fir-Tree Slots 80
4.3.3 Results and Discussions of Three Dimensional Models ............... 80
5 Conclusions and Future Work 87
5.1 Description of the ProbIem ...................................................................... 87
5 -2 Thesis Contribution ............................................................................ 88
5.3 General Conclusions ................................... ... . 89
............................ Effect of Geometry and Interface Conditions 89
.................. ..................................... 5.4 Recommendations for Future Work .., 89
Appendix I
Appendix II
Notation
Inner radius of disc
Outer radius of disc
Matrix of shape functions denvatives
Elasticity matrix
Young' s modulus
Force vector
Stiffness matrix
Contact stiffness in normai direction
Contact stiffness in tangentid direction
Fiank length
Number of fir-tree teeth
Applied load
Inner radius
Outer radius
Radius
Thickness
Displacement vector
contact angle
bott~m flank angle
top flank angle
Coefficient of friction
Poisson's ratio
Density
Stress
Angular velocity
vii
T Temperature
Ti Temperature at the inner surface
T,, Temperature at the outer surface
K Thermal Conductivity
Superscript e denotes the element
Subscripts n and s represent the tangentiai and normal components, respectively.
Subscripts i and O represents the inner and outer surface components, respectively.
Subscripts r and 8 represents radial and tangentid stress components, respectively.
viii
List of tables
3.1 Details of reference geometry and applied loads ............................... 36
3.2 Details of two dimensional geometry for turbine disc assembly ............. 39
................... Photoelastic and FE reference design geometry specifications 56
Geometric specification of models used in 2 D finite element analysis ..... 61
Physical properties of a typical INCONEL 720 ailoy .......................... 61
Specification of the geometry considered for coupled analysis .............. 62
Specifications of geometry anaiysed to determine the effect of variation of
number of teeth in turbine disc assembly for fnctioniess contact ( p = 0 ) .. 64
Specifications of the geometry of models analysed to determine the
effect of contact angle for P = y = 40" and p = 0.0 ............................. 70
Specification of the geometry of different models analysed to determine the
effect of flank angle and friction on stress field ................................ 75
.. ....................... Geometric specification of 3 D non-skew geometry .. 79
......................... Geometric specification of 3D skew angle geometry 80
....................... . A 1: Materid property values used in evaluation of stresses ... 92
........................ A.2. Cornparison of analytical and finite element results 93
List of figures
.. ........... 1.1 : Different blade fastening arrangements ................... .... .,.......... 1.2. Turbine disc illustrating loads acting at critical design regions .............. 1.3. Schematic of fir-tree disc groove and blade rwt definitions ................
............................................ 1.4 Outline of the method of approach
2.1 : Stress distribution in a solid rotating disc ..................... .. ....................... 2.2. Stress distribution in a hollow rotating disc ............................................. 2.3 Thermal stress distribution in a hollow circular disc ................... .......
....................................... 2.4 Contact between two deformable bodies
................................................. 3.1 : Discretized finite element mode1 of a disc
3.2: Radiai stress distribution along the radius of disc for thermal
......................................... loading only ................... ... ... ... 3.3 Tangential stress distribution along the radius of disc for thermal
....................................................................... loading only
3.4 Radial stress distribution along the radius of disc for centrifuga1
........................................................................ loading only
3.5 Tangential s a s s distribution dong the radius of disc for centrifuga1
loading o d y ........................................................................ 35
........... 3.6 Combined radiai stress distribution dong the radius of the disc 35
3.7 Combined tangentid stress distribution dong the radius of the disc ....... 35
............ 3.8 Normalized von Mises stress dong the interface for three teeth 36
3.9 Details of two dimensional geometry for three teeth ........................ 38
........................................ 3.10 Discretized two dimensional geometry 40
..................................... 3.1 1 Contact between two deformable bodies 40
................................... 3.12 (a) Two dimensional contact elements, and 41
....................................... (b) Three dimensional contact eIements 41
3.13 Specifications and creation of three dimensional geometry .................. 46
....... 3.14 Non skew and skew angle geometry investigated in this study ... ... 46
3.1 5 Three dimensional discretized straight geornetry of the bladddisc
assembly ........................................................................... 49
3.16 Three dimensional discretized skew geometry of the bladddisc assembly 50
.............. 3.17 Norrnalized von Mises stress distribution for different meshes 5 1
........................... 3.18 Convergence test for two dimensional geometry 52-53
Images used in automated photoelasticity software for 2D analysis ......... 57
Composite FE and photoelastic maximum shear stress contour images .... 58
Photoelastic analysis of a turbine disc showing nomalized difference
between maximum and minimum principal stress dong the bottom
...................................................................... contact region 59
................ Discretized reference geometry of aeroengine turbine disc 63
Deformed (dotted) and undeformed (solid) outline of reference
........................................................... aeroengine turbine disc 63
............... Maximum principal stress contour for turbine disc assembly 65
............... Minimum principal stress contour for turbine disc assembly 66
................... Maximum shear stress contour for turbine disc assembly 66
........................... Von Mises stress contour for turbine disc assembly 67
......................... . 4 IO: Von Mises stress contours for turbine disc assembly 67
xi
4.1 1 : Von Mises stress contours for turbine disc assembly . . . . . . . . . . . . . . . . . . . . . .... 4.12: Normaiized von Mises stress contours for turbine disc assembly for thme
teeth (n = 3) ..... . . . . .. . . .-... . . . . . . . .. .. . .-. . .. .. . .. . ...... . . .. -. -. . . -. -. .. .. -. . . .. 4.13: Nomalized von Mises stress contours for turbine disc assembly for four
teeth (n = 4) . . . . . . . . . . . . . . . . .. . .., . . . . . . . . - - -. . - . . -. . . . -. . . . . . . -. -. . . . . . . . . . . . . . . . 4.14: Nonnaiized von Mises stress contours for turbine disc assembly for five
teeth (n = 5) . . . . . . . . . . . . . . . . . . . . . . . . ..S. - - -.. . . . -. - - -. -. . . . . . . . -. . . -. . . - . . . . . . . . . . . . 4.1%: Normalized von Mises stress vs. normalized distance dong interface
n = 3 , $ =40",y=40°, pm0.0 ................................................... 4.15b: Norrnalized von Mises stress vs. normaiized distance aiong interface
n=4, B =40°, y =400,p=0.0 ........................................................ 4.15~: Normalized von Mises stress vs. normalized distance dong interface
n=5, =4û",y=40°, p=0.0 ................................................... 4.16a: Normalized von Mises stress vs. normalized distance dong interface
n = 3 , ~ = 1 5 ~ , $ = 4 0 ~ , y = 4 0 ' ' .................................................. 4.16b: Nonnalized von Mises stress vs. normaiized distance along interface
n =4, a = lSO, f3 =40°,y=400 ................................................... 4.16~: Normalized von Mises stress vs. normalized distance along interface
n = 5, a= lSO, =40°, y=40° ............................ ... ..... .... ......... 4.17a: Norrnalized von Mises stress vs. nonnalized distance aiong the teeth
a = 20°, B = 40°, y = 40° . . . . . . . . . . . . ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17b: Normalized von Mises stress vs. normaiized distance dong the teeth
a = 20°, P = 30°, y = 40° ......... . . . .. .. .. . . . .. +. . . ....... . . . . . . . . .. . . . . .. .. .. . . . . . 4.17~: Normalized von Mises stress vs- normalized distance along the teeth
a = 20°, P = 40°, y= 50' ......... . . . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18a: Normalized von Mises stress vs. normdized distance along the teeth
a = 20°, = 40'. y= 30° ................... ........................................ 4.18b: Normalized von Mises stress vs. normalized distance dong the teeth
a = 20°, P = 50°, y= 40° ................................... - .......................
xii
Skew and non-skew angle geometry investigated in this study .. . ... . .-. . . 78
3D discretized geornetry of the saaight (non-skew) bladddisc assembly 79
3D discretized geometry of skew angle blade/disc assembly .. . . .. . .. . . ... 8 1
Von Mises stress contours for a turbine disc assembly with n = 3,
a=20°, =4û0.y=4@, p =0.0, skew =O0- .............................. 81
Von Mises stress contours for a turbine disc assembly with n = 3,
a=20°, $ =4@, y=40°, =O.O. skew = 10" .............................. 82
Von Mises stress contours for a turbine disc assembly with n = 3,
a = 20°, P = 4û0, y = 400, p = 0.0, skew = 20° .......................................... 82
Normalized von Mises stress contours for three dimensional geometries:
(a) dong the bottom tooth, and .. ,... .. .... ..,.... .. . .. .. . . . . . .. .. . . . . .. . . . ... 84
(b) across thickness at the lower contact line . .. .. . . .. . .. .. . ., . .. . .. .. .. . .. 84
Normalized von Mises stress for three dimensional straight slot
geometries for various flank angles
(a) dong the bottom tooth, and ...................................... .. . . . 85
(b) across thickness at the lower contact line ................................ 85
Effect of coefficient of friction on the von Mises stress through thickness85
xiii
Chapter 1
Introduction and Justification
1.1 Turbine Disc Assemblies
The safety of gas turbine engines has always been the main concern of aïrcraft
certification authorities. Economic pressure resulting from the reduced availability of
strategic materials and the high cost of engine components and the continued demand, by
dl engine suppliers/users, for longer life and higher thrust to weight ratio continue to
provide a stimulating challenge for engine designerddevelopers [ 1.1 - 1.31.
Fig. 1.1 shows the different methods adopted in fastening blades to disc. These are: pin
joint, dovetail and fu-tree. Fir-tree fasteners have been comrnonl y implemented in
turbines because they provide multiple areas of contact over which large thermai and
centrifuga1 stresses can be accommodated [ 1.3 - 1-61,
The thermomechanical integrity of turbine discs and the attached blades is crucial to the
operational safety and service life of gas turbine engines. The failure mode of a primary
member in an engine, such as turbine disc assemblies, is usually catastrophic, often
resulting in loss of Life and hardware. Aeroengine designers are constantly façed with the
challenge of estabLishing stress levels in tbese critical parts that will aiiow the use of
suitable high strength heat resistant d o y s operating in a safe thennomechanical loading
regimes 11.3, 1.5 - 1.71.
Fig. 1.1 : Different blade fastening arrangements [ 1 -21.
At this stage, it is important to identify the pertinent parameters which influence the
mechanical integrity of aeroengine turbine disc assernblies. These include: (i) the
quasistatic thermomechanical strength, toughness and rupture strength of the different
constituents of the assembly, (ii) the applied thermal and body forces associated with the
temperature of the gases and the rotational speed of the disc, and (iii) residual stress state
of the components/assembly. In this thesis, we focus our attention to the study of the
coupled thermomechanical behaviour of a turbine disc made fiom Inconel.
1.2 Research Objectives
Aeroengine turbine discs have three criticd regions for which lifetime certification is
necessary: the fir-tree rim region, the assembly holes or weld areas and the hub region.
The loads associated with these regions are the centrifuga1 forces on the disc and
associated blades, themai loads and the bending loads due to the gas pressure. Fig. 1.2
shows these areas dong with their associated load. Fig. 1.3 shows the criticai geometric
features of a frr-tree joint. These features include the number of fir-tree teeth ni, flank
length 1, contact angîe a, and fiank angles f l and ywhich define the tooth pitch.
It is therefore the objective of this study to evaluate the coupled thermomechanical
behaviour of turbine disc assemblies (Fig. 1.1) using the finite element method.
Comprehensive two and three-dimensiond finite element models of realistic disc
geornetries were developed using the commercial code ANSYS 5.4. Specifically, it was
desired to:
(i) evduate the eHect of the flank length and flank angle, inner radius of disc
and interfacial friction upon the thennomechanicd stress state of the disc.
(ii) detennine the influence of the skew angle (Fig. 1.3) upon the triaxial state
of stress and the load sharing between the teeth in a turbine disc assembly.
Contact between the disc and the blades will be modelled using contact
elements, and
(iii) validate the fundamental finite element mode1 using convergence tests and
photoelasticity.
1.3 Method of Approach
This research is divided into three main components. The f m t is concemed with the two-
dimensional therrnomechanical modelling of turbine disc assemblies assurning that plane
stress conditions dominate the behaviour of the disc. Attention is devoted to the selection
of the appropriate elements, the discretised geometry for the diffefent cases examined. In
these cases, we study the effect of Bank length, flank angle, contact angle and coefficient
of friction upon the resulting thennomechanical stress state in the assembly as
represented
FF Centrifuga1 Force Fs Blade Force FD Supporting Load T= f(r) Thermal Load
1: Rim (Fir-Tree Groove) 2: Bolt HoldWeld Region 3: Disc Hub
Fig. 1.2: Turbine disc iliustrating loads acting at critical design regions [1.3].
by von Mises and normal and tangential contact stresses dong the blade-disc interface.
The contact between the disc and the blades was modelied using special contact eiements,
which employ the combinai Penalty-Lagrange multipiiers formulations.
The second was concemed with the extension of the two-dimensional finite element
mode1 into three-dimensional in order to account for the effect of varying the skew angle
of the blades upon the resulting stress field using contact elernents.
mer Radius Ri
Top Flank Angle y
Bottom Flank Angle f3 Contact Angle a
Fig. 1.3: Schematic of fn-tree disc groove and bide root definitions.
The third was motivated by our desire to validate the basic hndamental element models.
This was carried out by: (i) conducting extensive mesh convergence test, (ii) comparing
the results of simplifïed disc models with existing closed form expressions, and (iii)
comparing the FE results with photoelastic stress measurements. Fig. 1.4 shows a
schematic of the method of approach adopted in this study.
1.4 Layout of Thesis
This thesis is divided into five chapters in total. The Fit Chapter outlines the objectives
of the work and justifies the reasons behind the undertaking of the study. Chapter Two
covers the Iiterature survey concerning the thermomechanical behaviour of the fir-tree
joints in aeroengine discs. Three aspects of the work existing in the literature are covered:
theoretical, numerical and experimental. Chapter Three describes the finite element
method and the moâels adopted for determining the coupied load sharing in the fir-tree
region. In addition, the chapter includes the details of the effect of the coupling between
the thermal and mechanical loadings upon the stress state of the different geometries and
interface conditions examined. Details of the contact between the blade and disc are also
presented in that chapter. In Chapter Four, we vaiidate the FE models, discuss the results
and outline the major findings of the work. Chapter Five concludes the work and outlines
some possible areas that may be explored for future work in this important area of
research.
Thennomechanical Analysis of a Turbine Disc As~embly
Closed Fonn Solutions
Stress F d g Technique
t t t J
Evaluation and Cornparison of Rcsuits
Conclusions and Recommendations I Fig. 1.4: Outline of the method of approach.
1.1 Papanikos P., On the Structural Integrity of Dovetail Joints in Aeroengine Dises.
M.A. Sc. Thesis, University of Toronto, 1992.
1.2 Venkatesh S., Structural Integrity Analysis of an Aeroengine Disc. M. Sc. Thesis,
Cranfield Institute of Technology, L 988.
1.3 Meguid S. A., Engineering Fracture Mechunics, Elsevier Applied Science,
London, pp. 287 - 295, 1988.
1.4 Kanth P., 2D & 3 0 Finite E h e n t Analysis of Fir-Tree Joints in Aeroengine
Disc, M.A. Sc. Thesis, University of Toronto, 1998.
1.5 Srinivasan J., Gowda R., Padmanabhan R., A Numerical Three-Dimensional
Themal Stress Analysis for Cooled Blades, The American Society of Mechanical
Engineers (ASME), 89-GT-l68,6 pp., 1989.
1 . Stjepanovic J., Three dimensional Finite Element A ~ l y s i s of Dovetail Joints in
Aeroengine Discs, M.Sc. Thesis, University of Toronto, 1 996.
1 -7 Arvanitis S. T., S ymko Y. B., and Tadros R. N., Muftiaxial Life Prediction System
for Turbine Components, Journal of Engineering for Gas Turbines and Power,
Vol. 109, pp. 107 - 114, January 1987.
Chapter 2
Literature Review
Solutions of the equations defining the stress field for elastic rotating circular discs
subjected to specific boundary conditions are well known and detailed in many solid
mechanics textbooks. These solutions are available for the anaiysis of structural members
subjected to mechanicd and thermal loadings separately. These cIosed focm expressions
cannot simply be applied to real turbine disc assemblies, because of the complex nature of
the geometry to be examined. As a result, most of the treatments of aeroengine discs have
been limited to either the use of finite element, or experimentally using photoelasticity or
a combination of the two.
This chapter which is concemed with the development of earlier theoretical and
experimental works is divided into three main sections. The fmt is concemed with closed
form expressions for the stress field in rotating discs. The second with the thermal stress
andysis in a circular disc. The third section focuses on earlier investigations undenaken
using two and three-dimensional finite element models of the contact problem. The
coupling of thennomechanical loading complicates the andysis considerably. This is due
to the nonlinear nature of contact analysis and the presence of conductive media As a
result, the literature does not address the coupling effect.
2.1 The Stress Field in Rotating Discs
An aeroengine turbine disc is generally subjected to severe stresses due to the various
forms of loads acting on it. The disc experiences large centrifuga1 forces resulting from its
rotation and from the attached blades. The blades are aiso subjected to bending, huisting
and vibratory loads that are transmitted to the rim of the disc. In addition, the blade/disc
assembly is subjected to large temperature gradients leading to high thermal stresses.
Simple analytical forrnuIae have k e n developed for the analysis of a rotating disc of
unifonn thickness. The stresses generated through the thickness are neglected, since they
are assumed to be smdl in cornparison with the in-plane radial and tangential stresses.
Using Airy's stress function, these stresses can be expressed as:
where p and v are the respective material density and Poisson's ratio, r is the disc radius,
and o is the disc rotational speed. A and B are constants to be determined from the
boundary conditions of the problem. Two cases are generally examined: solid and holIow
discs, as detailed below.
2.1.1 Solid Disc Analysis
In order to prevent the stresses at the center of the disk from becorning infinitely large the
constant B in Eqn (2.1) has to be zero,. In addition, using the free boundary condition:
(a, ) . = b = O, we obtain:
Hence, Eqs. (2.1 a) and (2.1 b) become:
The maximum stresses occur at the center of the disk ( r = O), as shown in Fig. 2.1.
Fig. 2.1 : Stress distribution in a solid rotating disc.
2.1.2 Hollow Disc Analysis
For the hollow disc, there exists a free boundary at both the inner and outer radii, such
that (a, ) , = , = (a, ) , = b = O, leading to:
The maximum radial stress can be shown to occur at r = (ab)lR, and the maximum hoop
stress occurs at the inner radius of the disc, such that:
3+v ( ~ ~ ) , = ( ~ ) ~ o ' ( b - a)', and
The distribution of the radial and centnfugal stresses within a hollow rotating disc is
shown in Fig. 2.2.
Fig. 2.2: Stress distribution in a hollow rotating disc.
The above expressions do not account for blade and themal loadings. In such cases, the
boundary conditions should be modified to reflect the fact that radial stresses at the disc
edges are no longer zero [2.1-2.21.
2.2 Thermal Stress of a Circular Discs
General expressions have been
the dis placement components
developed for the radial and tangentid stresses as well as
in circular disks under radially varying temperature
distribution 12-31. These general expressions are as follows:
where ri and ro are the inner and outer radii of the disc, a is linear coefficient of thermal
expansion, E is Young's Modulus and r is the current disc radius. The distribution of
different stress component within a hollow disc are shown in Fig. 2.3.
Fig. 2.3: Thermal stress distribution in a hollow circular disc.
The radial temperature distribution T(r) cm be expressed as follows:
where Ti and To are the temperatures at the inner and outer surface of the disc. The
maximum radial stress occurs at r = (ab)? The maximum tensile and compressive hoop
stress occurs at the inner and outer radius of the disc. These expressions are valid ody for
the case of plane stress [2.3].
2.3 Earlier Finite
Closed form solutions are
Element Models
limited to simple geometries subjected to simp!e boundary
conditions. In real gas turbine engine stress analysis, the complexity of the blade and disc
geometry and the loading limit the accuracy and hence the usefulness of closed fonn
solutions. Most aeroengine disc stress anaiysis are conducted using the finite element
method.
2.3.1 Structural Integrity
Simplified uncoupled solutions have ken developed by a number of investigators. Chan
and Tuba [2.4] analyzed the effect of the blade/disc clearance and the frictional forces on
the stress distribution in the blade root fastening. It was shown that the coefficient of
friction had a slight yet distinguishable effect on both the maximum fillet stresses and on
the load distribution at the teeth flanks. Changes in clearance revealed more significant
effects compared to those obtained by contact friction.
With the effective use of the finite element method for modeling complex geometries, the
ernphasis shifted towards developing new finite element software that can mode1 the
elastoplastic properties of the bodies in contact. Zboinski [2.5] developed a cornputer
program for the FE analysis of thee-dimensionai non-linear contact problems. The
algorithm utilized the variational principle of inmemental frictional contact mechanics. A
simplified FE analysis of a four teeth fir-tree region of turbomachinery was performed by
simple tension and bending. The results revealed that the maximum tensile stress due to
tension and bending occurs at the base of the tooth closest to the disc. Thermal analysis
were not taken into account.
Papanikos, Meguid and Stjepanovic [2.6] developed a FE d e l to examine the effect of
the critical geometric features and interface conditions on the stress field at the blade/disc
interface for an aeroengine compressor disc under centrifugal loading. Their results
indicated that the maximum stress occurs just below the lower contact point. The results
also showed that variations in the geometry and coefficient of friction effects the stress
distribution at the blade/àisc interface. This work concluded that three dimensionai
rnodels gave more accurate stress distri bution compared to two dimensional anal ysis-
Furthemore, it was shown that the skew angle can significantly influence the stress
distribution at the bladddisc interface. A sirnilar study by Meguid et ai. [2.7] analyzed the
structural integrity of dovetail joints by initiating and tracking the propagation of cracks
in aeroengine disc subjected to centrifugal loading. It was observed that peak stresses
occurs just below the lower contact point of the last tooth. For the disc geometry studied
by Meguid, crack propagation took place across the throat region and not through the disc
and the stress intensity factor at the crack tip was of the mixed mode type.
Singh and Ratwani [2.8] suggested an approach to analyze the fir-tree region of a turbine
disc subjected to centrifuga1 loading. Their method treated the total fir-tree region as an
assembly of several steps. Using a generalized root step, the analysis was conducted to
obtain its stiffhess and deflection using the FE method. Several of these root steps were
later assembled to fonn a fir-tree region of a disc. The loads carried by each step were
then calculated by solving a set of linear algebraic equations. This work suffers from too
many approximations.
Sarlashkar and Lam 12-91 used the FE mode1 to study the effect of machining tolerances
on the load distribution and peak stresses within a fir-tree region. A special gap element
was developed to account for friction, compressive stresses and the random gap size. The
steady state stress for an arbitrary gap combination was interpolated fiom the database
containing pre-calculated bladeldisc stress. The author proposed that the profile of the
root/groove should be improved in order to minimize the stress concentration and still be
able to keep manufacturing costs and tolerances within lirnits. A similar study was
conducted by Millwater and Wu [2.10] to accurately and efficiently de tedne the
reliability of structures with muItiple failure modes subjected to high degree of
uncertainty in loading, matenal properties, geometry and support. An advanced
probabilistic numerical structural analysis was used to analyze a turbine blade subjected
to centrifugai, pressure and temperature loading. The results reveal that by using their
software, system reliability cm be detennined in a stepby-step procedure to minimize
unnecessary computations.
2.3.2 Thermal Stresses
Thermal stresses and thermal straïns and the rupture Iifetime of a structural component
are governed by: (i) the applied thermomechanical load, (ii) the component geometry, and
(iii) the material creep properties. Simplified expressions have been developed for simple
geometries under constant loading. With the recent advances in finite element analysis
technique, however, most complex shapes can be modelled and analyzed. Srhivasan et
al. [2.11] studied the effect of thermal stresses on cooled turbine blades which was
described in terms of a series expansion. The plane blade profile was assumed to take the
form of a second-degree surface with a constant curvature. Using the equilibrium
equations for the deforrned surface, a general stress pattern for the blade surface was
obtained. Thei .ethoci provided improved estimates of radial stresses as well as
approximate estimates of other components of the thermal stress tensor.
Arvanitis et al. 12.121 developed a life prediction mode1 for turbine blades and vanes
subjected to complex thermal and mechanical loading. The concept of maximum
principal normal strain was developed to analyze the component for fatigue. It was
assumed that the components were nickel-based superalloys. A creep hinction was
developed with the help of 3D finite element creep analysis. Finally, the creeplfatigue
interaction was evaluated by the ductility exhaustion method. The developed method was
used to analyze solid uncooled blade-vane assembly and the results correlateci well with
test measurements.
Hepworth et al. [2.13] determined the life of gas turbine blades and vanes using the finite
element method. Data obtained Crom an aerotherrnai analysis was applied as boundary
conditions to obtain the stress distribution in the blades. The resulting stress and
temperature distributions were used to estimate the damage experienced by the material
due to fatigue and creep, and to estimate the remaining life of the blade. Sirnilar study
was carried out by Amagasa et al. (2.141 to investigate the effect of cooling and thermal
coating on turbine blades and vanes made of a nickel based superailoy. They showed that
their cooling method maintains the metal temperature and thermal stresses under
allowable metallurgical limits for maximum temperature. This helps to increase the creep
and fatigue Iife of the turbine blade.
Wan et al. [2.15] developed a computer program to access the thermaVmechanical
integrity of the turbine blade under high temperature and centrifugai loading. By coupling
their program with other available codes, an accurate and efficient estimate of thennal
and mechanicd stresses was obtained. Using the finite element method, the blade
temperature obtained from the previous iterations was used as boundary conditions to
calculate thermal and dynarnic stresses in order to access the thermal-mechanical fatigue
damage in turbine blades. Submodeiiing can be done for detailed analysis of the cntical
regions of turbine blades. The caIculated results were compared with published data and
experimental results and good agreement was obtained.
23.3 Contact at the Blade/Disc Interface
Mechanical problems involving contact are generally nonlinear in nature, such as the one
show in Fig. 2.4- in contact problems, the contact area is not known a priori. It is a
function of the contact stress state. Consequently, the solution cannot be obtained in a
single step. Several steps must be taken, updating the tentative solution after each step
until convergence is satisfied.
Before contact Contact with no penetration
Contact with petration
Fig. 2.4: Contact between two deformable bodies [2.16].
Original finite element formulations did not accommodate the treatrnent of contact, and
ad-hoc techniques were used. In these cases. simplifying assumptions regarding the actual
contact surface and the distribution of the contact stress were made. These simplifying
assumptions enabled the treatment of each individuai body as a separate problem. This
ad-hoc technique proved inadequate, in many cases, where neither the contact surface nor
the stresses on it could be easily estimateci. This has prompted the development of contact
elements 12-16 - 2-17].
Finite element methods treat contact problems by extending the variational formulation
upon which the finite element method is based. Contact elements are formulated and
assembled into the original finite element code in order to enforce the contact conditions.
The solution is then obtained by solving the resulting set of non-linear equations.
Masataka et al. [2.18] developed a three dimensional finite element mode1 for the stress
analysis of the fn-tree region for centrifugd loading, taking into account the slippage of
the contact surface at the bladc/disc interface. The contact surface is modelled by using a
spring coefficient. Slippage behueen the rwt and the groove was considered by varying
the coefficient of friction p from 0.0 to 0.5. It was observed that there was no large
siippage at the contact SUrfitce when the coefficient of friction is between 0.3 to 0.5. It
was concluded that accurate resuits could be obtained by linear finite element analysis of
the blade root and groove interface for this range of p.
A major study was done by Zboinski [2.19] to anaiyze the contact problem of turbine
blade attachment. Friction and sliding of the contact surface were mainly considered in
the analysis. It was shown that due to non-uniforrnity of the contact region, maximum
stresses occurred at the edge of the contact surfaces of the disc, while minimum stresses
occurred in the middle parts of the contact surface. It was concluded that friction and
contact play a major role in dictating the three dimensional stress state across the fir-tree
flanks.
Mase et al. 12.201 used finite element and linear programming methods to optirnize and
improve the reliability of a turbine blade so that the localized stresses at the blade/disc
interface can be minimized. Using the coeffkient of friction as a parameter, it was shown
that there was no slip in the contact region for p = 0.3, or higher. Most of the slip took
place when p 2 0.1 approximately. It was concluded that most of the slip was not due to
centrifuga1 loading. By optimizing the geometry, it was shown that there was a reduction
of 18 percent in the total stress in the contact region. The variation of the radius of
curvature and the angle of inclination of the contact surfaces were shown to be effective
ways for reducing the local stress in the region.
Srivastav and Redding [2.21] modelled the imprfect contact conditions usiog a standard
displacement based three-dimensional finite element procedure. The contact at the blade
and the disc of a fir-tree was modelled using two node super-elements having relatively
few degrees of freedom. Contact was assumed to occur only at pre-defined super element
boundary nodes. It was shown that uniess the contact and clearance was not accounted for
in the model, the calculated stress vaiues were either under or over estimated,
2.3.4 Experimental Work
In the design of aeroengine bladeldix assernblies, the desire for weight reduction must be
balanced against the safety and reiiability of these assemblies. in addition, highly accurate
FE analysis must be carried out since the joint design is sensitive to slight geometrical
variations. In order to validate the FE work, a number of experimental stress analysis
techniques had been used.
One of the most comprehensive and earlier experimentai works to investigate the
stmctural integrity of turbine blade attachments was performed by Durelli et al. l2.221.
This study was concerned with the improvement of the strength of the joints and divides
the problem of the blade at tachent into a number of steps:
(i) transmission of forces from the blade to the fir-tree root,
(ii) optimization of the fir-tree geometry in tenns of fillet contours, contact angle and
non-symmetric contact conditions related to stress distribution, and
(iii) high temperature fatigue testing.
The study was done using actual blades and oversize plastic models of the fir-tree joint
subjected to combined centrifuga1 and bending loads. Brittle coating, strain gages, photo-
elasticity and fatigue testing were al1 used, These experimental techniques revealed that
the distribution of loaâ between the blade and disc interface depends on the rigidity of the
interface or contact region. clearance between the mating teeth and on the slippage at the
contact region. It was also concluded that the stress concentration factor decreases as
fiange angle increases. which was not confmed for coupled loading. From the high
temperature fatigue tests, it was found that fatigue fractures were located at points of high
concentration at the bladddisc interface.
Uchino et al. [2.23] adopted smaller design margins by bringing the conventionai safety
factors close to the actual values by using three dimensional photoelastic technique for
aircraft gas turbine. They tried to modify design data using 3D photoelastic technique and
verified their results using the stress distribution obtained from 3D finite element
analysis. 3D-photoelastic anaiysis was conducted to confirm the stress concentration due
to the variation of the broach angle (dovetail corner) in disc/blade attachment for dovetail
joint. It was concluded that the highest effective stress was in the acute corners in the
groove. If high broach angles were adopted, sufficient safety margins should be
considered compared with the stresses obtained by two dimensional finite element
andysis.
Durelli and Rajaiah [2.24] developed a method to optimize the shapes of holes and fillets
in a finite plate with the objective of decreasing the stress concentration factors. The
method is based on photoelasticity in which the material is removed from the low stress
portions of the holes and fillets till the isochromatic fringes coincide at the boundary for
both the tensile and compressive segments. In this manner, the stress concentration factor
of the dovetail joint was reduced by 24 percent.
Parks and Sanford [2.25] used two experimental techniques to show that the sum of the
principal stresses throughout the field combined with directional and tangential stresses at
the bladddisc interface from photoelasticity give the complete stress field distribution in
a fir-tree joint. It was shown that 27 percent reduction in stress concentration was possible
by removal of material from the disefillet boundary.
Haake et al. 12-26] developed a procedure to calculate the components of stress in an
image using photoelastic parameters at the boundary. It was shown chat automated full-
field polariscope can be used to determine two-dimensional stresses. The verification was
done for a turbine slot, where the comparison of the results obtained by their method and
by the shear difference method showed good agreement.
In the present investigation, we employ the finite element method to study the stress state
in a turbine disc assembly subjected to themiornechanical loading. The finite element
results were verified using photoelastic stress anaiysis.
2.1 Stjepanovic J., Three dimensional Finite Element Analysis of Dovetail Joints in
Aeroengine Discs, M. Sc. Thesis, University of Toronto, pp. 13 -15, 1996.
2.2 Zhilun X., Applied Elasticity, John Wüey & Sons, New York, 3* Edition, 1992.
2.3 Boley B. and Weiner J., Theroy of Thermal Stresses, John Wiley & Sons Inc., L"*
Edition, 1 %O.
2.4 Chan S. K. and Tuba L S., A Finite Element method for Contact Problems of Solid
Bodies - Part II: Applications to Turbine Blade Fastenings, International Journal
of Mechmica1 Science, v. 13, pp. 627 - 639, 1971.
2.5 Zboinski G., Finite elernent computer program for incrernental analysis of large
three dimensio~l frictional contact problems of linear elasticiry, Cornputers and
Structures, v.46, n.4, pp 679 - 687, 1993.
2.6 Papanikos P., Meguid S. A. and Stjepanovic Z., Three dimensional nonlinear
finite elernent a ~ l y s i s of dovetail joints in aeroengine discs, Finite Element in
Analysis and Design, Vol. 29, 1998.
2.7 Meguid S. A., Refaat M. H. and Papanikos P., Theoretical and experimental
studies of structural integdy of dovetail joints in aeroengine discs, Journal of
Materials Technology, Vol. 56, pp. 1539 - 1547, 1996.
2.8 Singh G. D. and Rawtani S., Fir tree fastenings of turbomachines, bfudes - 1: Deflection analysis, International Journal of Mechanical Science, Vol. 24, no. 6,
pp. 377 - 384,1982.
Sarlashkar A. K., Lam T. and Mccloskey T. H-, Blude mot attuchment evaluation
low-cycle fatigue estimutes based on probabilisric approuch. ASME International
Joint Power Generation Conference, v. 2, pp- 51 1 - 5 16, 1996.
Millwater H. R. and Wu Y. T., Computational structural reliability unalysis of a
turbine bltz.de, The American Society of Mechanical Engineers. 93-GT-237, pp. 1
- 14, 1993.
Srinivasan J., Gowda R- M. S. and Padrnanabhan R., A numerical three-
dimensional thermal stresses analysis for cooled blades, The American Society of
Mechanical Engineers, 89-GT- 168, pp. 1 - 4, 1989.
Arvantis S. T., Symko Y. B. and Tadros R. N., Multiaxïal lrlfe prediction system
for turbine components, Journal of Engineering for Gas Turbines and Power, Vol.
109, pp. 107 - 1 14, lanuary 1987.
Hepworth J. K., Wilson J. D., Ailen J. M., Quentin G. H. and Touchton G., Life
assesment of gus turbine blades and vanes, The Arnerican Society of Mechanical
Engineers, 97-GT-446, pp. 1 - 6, 1997.
Amagasa S., Shimomura K., Kadowaki M., Takeishi K., Kawai H., Aoki S. and
Aoyama K., Snuiy on the turbine vane and blade for a 1500" clars industrial gas
turbine, The American Society of Mechanical Engineers, 93-GT414, pp. 1 - 7,
1993.
Wan S. M., Lam T- C. T., Allen J. M. and McCloskey T. H., A gas turbine blade
thennaUstructura1 program with linked flow-sulid modeiing capabiiity, The
American Societv of Mechanical Engineers, 94-GT-270. OD. 1 - 7, 1994.
2.16 ANSYS Version 5.4 Engineering Analysis System, User's Munual ( sofrware
online), Ansys Engineering S ystems, Houston, 1 998.
2-17 Kenny B., Patterson E., Said M. and Aradhya K., Contact Stress Distribution in a
Turbine Disc Dovetail Type Joint - A Cornparison of Photoelastic and Finite
Element Results, Strain, pp. 2 1 - 24, 199 1
2.18 Masataka M., Yasutomo K., Thom K, Katsuhisa F., Yoshiki K. and Seigo L, Root
and Groove Contact AnaIySIS for Steurn Turbine Bïades. Japan Society of
Mechanical Engineers International Journal, Series I: Solid Mechanics and
Strength of Materîals, v. 35, No. 4, pp. 508 - 514, 1992.
2.19 Zboinski G., Physicul and geomem'c non-lineurities in contact problems of elustic
turbine blade attachments, Proceedings of the Institute of Mechanical Engineers:
Part C: Journal of Mechanicd Engineering science, v. 209, N. C4, pp. 273 - 286, 1995.
2.20 Mase M., Kaneko Y. and Watanabe E., Study on the contact technoiogy and
optimal figure design of the roof and groove for steam turbine long blades,
Intenational Conference on Cornputer Aided Optimum Design of Structures,
OPTI, Proceedings, Computational Mechanics Publications, Southampton, United
Kingdom, pp. 489 - 498,1997.
2.21 Srivastav S. and Redding M., 3D Modelling of irnperJect contact conditions
between turbine blades and disk, Advances in Steam Turbine Technology for the
Power Generation Industry: ASME International Joint Power Genetarion
Conference, v. 26, pp. 197 - 204, 1994.
2.22 Durelli A. J., Dally J. W. and Riley W . F., Stress and Strength studies on turbine
blade attachments, Proceedings of the Society for Experimentai Stress Anaiysis,
v. XVI, n. 1, pp. 171 - 186, 1958.
2.23 Uchino K., Kamiyarna T., Inarnura T., Simokohge K., Aono H. and Kawashima
T., Three-dimensional photoelastic analysis of aeroengine rotary parts,
Proceeding of the International Symposium on Photoelasticity, Tokyo, pp. 209-
214, 1986.
2.24 DureIli A. J. and Rajaiah K., Lighter and stronger. experimental Mechanics, pp.
369 - 379, November 1980.
2.25 Parks V. I. and Sandford R. I., Photoelastic und holographie analysis of a
turbine-engine component, Experimentai Mechanics, pp. 328 - 333, September
1978.
2.26 Haake S. J., Patterson E. A. and Wang 2. F., 2 0 and 3 0 seperation stresses using
automnted photoelasticity, Experimental Mechanics, Vol. 36, no. 3, pp. 269 -
276, September 1996.
Chapter 3
Thermomechanical Finite Element Analysis
of a Turbine Disc Assembly
3.1 Fundamentals of the Finite Element Method
The fundamental concept of the finite element method is that a physical domain is
discretised into a small number of subdomains, known as elements, over which a
continuous field variable such as velocity, stress, pressure, or temperature can be
approximated. These elements are connected at specific points known as nodes or nodal
points. Since the actual variation of the field variable is not known inside the domain,
approximating functions are needed to describe this variation. These approximating
functions interpolate the values of the field variable at the nodal points of each element.
Since the geometric and the required material properties of each element are known,
suitable field equations such as equilibrium or heat balance c m be written for each
element. Using the principal of minimum potential energy of each unit, the elemental
stiffness rnatrix can be obtained for each element [3.1 - 3.21.
Each element is connected to the other element at nodal points to f o m a continuum of the
entire model. The global stiffness matrk is obtained by assembling the element
stiffhess matrices for each element for the entire domain. The new unknowns obtained by
the assemblage of elements are the nodal values of the field variable. Boundary
conditions are taken into account, which heip to modify the overail equilibrium
equations. This then yields the global equilibrium [equation 3-11 in terrns of banded
matrices [3.3 - 3.43:
Wmg) = Wl (3- 1 )
where
{ug} represents the global displacement vector, and
{F) represents the global applied load or force vector.
The general solution of an engineering problem can be descnbed in a step-by-step
procedure. This sequence of steps describes the actual solution process that is followed in
setting up and solving equilibrium or heat balance equations. A step-by-step approach
adopted in finite element problems is summarized below:
Ideaiization of the structure: the geometrical feamres of the structure are
simplif ied in order to accommodate sensible discretisation.
Discretisation of the structure: in this case, the body is subdivided into an
equivalent system of finite elernents. The type, size and number of elements is
dictated by the geomeuical features of the component, applied loads and
restraints, accwacy needed and CPU floating point power.
Choice of interpolation or displacement function: the assumed displacement
function approximates the achial or exact distribution of the displacement field
within the continuum. in general, the interpolation function is taken in the fonn of
a polynomial; the number of terms that c m be retained in the polynomial is
limited by practical considerations.
Derivation of the element stiffness mtriK: the stiffness matrix is composed of
the coefficients of the equilibrium equations derived from the material and the
geometric properties of an element and obtained by the use of the principle of
minimum potential energy (equilibrium condition). The stifhess [K('>] relates the
displacements at the nodal points (ufe)} to the applied forces at the nodal points
{E'C'} where (e) denotes the element number, namely Cgr.9 {de)} = (pl),
AssemMy of element eqmtions for the overali àiscretised body: this process
includes the assembly of the global stifhess matrix p] for the entire body from
the individual element stifniess matrices @J and the global vector FI from the
element nodal force vectors (F'@)}, such that [ICg ]= 2 [K"'] with n k i n g the total C
number of elements.
Solution for the unknown nodal displacements: the overall equilibrium
equations have to be modifed to account for the boundary conditions of the
problem. After the incorporation of the boundary conditions, the global
equilibrium equations can be expressed as mg](ug} = F). For linear elastic
problems, the displacement vector can be easily obtained. But for non-ïinear
problems, the solution is obtained in a sequence of steps, each step involving the
updating of the global stiffhess matrix Ig.] andor load vector el. Computation of element strains and stresses from nodal displacements:
having determineci the primary unknowns (nodal displacements), it is often
necessary to use these nodal displacements to detemiine the element strains and
stresses by using the appropriate solid mechanics equations.
With the availability of many powerful linear and nonlinear finite element packages, it
was felt unnecessary to develop the solution programs for the current study. The work
concentrates on the mechanics and design aspects of the fir-tree joint in an aeroengine
turbine disc rather than the programrning aspect of the work.
Throughout this work, use has k e n made of A N S Y S 5.4; a finite element package that
contains a pre-processor, a number of solvers and a post-processor. The pre-professor
allows the user to rapidly create two and three-dimensionai finite element m d e l s and
prepares the model for anal ysis through automatic model checking routines. The frontal
solver (two-dimensional d e l ) and the preconditional conjugate gradient solver (three-
dimensionai model) were used in the present problem to accommodate the geometrical
nonlinearities occurring in the fir-tree joint. The general pst-processor allows the user to
review results over the entire mode1 at specific load steps and substeps [3.3,3-5 - 3-71.
3.2 Coupled Thermomechanical Contact Problems
There are a large number of mechanicd and electrical problerns for which the interaction
of more than one field of analysis needs to be considered in order to soive the problem
accurately. This is known as coupled-field andysis, where the coupled interaction
between the two fields is significant and influence the solution. Examples of these
include: an electric field interacting with a magnetic field, magnetic field producing
structural forces, thermomechanical problems, piezoelectric materials and thenno-
electro-mechanical coupling. In coupted-field andysis one finite elernent type is used for
both fields, which helps to simplify the modeliïng of this class of problems.
Coupled-field analysis is conducted using a coupled-field finite element that has ail the
necessary degrees of freedom. Coupling is ensured by including ail the necessary tenns in
the element stiffness matrix. This method is suited for situations in which the solution of
one field affects the solution of the other field, and vice-versa. 13.61.
Two methods. as defined by ANSYS 5.4, are used to solve coupled problems using the
finite element method:
(i) Matrix Coupling - The stifiess matrix can be written as:
[K:: ::]{::} = {F,} KI*, Klz, Kzi and I& are the stiffness terms, u~ and u;? are displacement terms and
FI and F2 are load terms. The coupling effect is accounted by the diagonal
stiffness terms K12 and K2,. The solution may provide coupled response in the
solution after one iteration.
(ii) Load Vector Coupling - The stiffhess matrix equation can be written as:
The coupled effect is accounted for in the loaà terms F and Q. At least two
iterations are required to achieve a coupled response. Included in this method is
the case where on-diagonal matrices are functions of the other unknowns from
previous iterations. This load vector coupling method was used for the coupled
thennomechanicd analysis of the fir-tree region in a turbine disc assembly
13 .a w here:
IKtl = ~ t b ] + lK*l (FI = Pd} + {Pl + {pl + (P"J (Q) = (~''9 + (Pl + (QCl
w here:
K = structural stiffness matrix
ICt = thermal conductivity matrix (may consist of 1 or 2 of the following matrix)
K&= thermal conductivity matrix of the materid
K== thermal conductivity matrix of convection surface
F = load matrix
pd= applied nodal force matrix
l?'= thermal strain force vector
Pr= pressure load vector
force vector due to acceleration effects (e-g. gravity or body forces)
Q = heat (temperature) mauix
Q"~= applied nodal heat flow rate vector
= heat generation rate vector for causes other than Joule heating
QC = convection surface vector
u = displacement vector
T = thermal potentiai (temperature) vector
3.3 Solution and Superposition of Uncoupled Problems
Two cases were examined. The fmt was concemed with thermal stresses resulting from a
radial temperature distribution, while the second was concemed with the body forces
resulting from the rotation of the disc. In both cases, single circular hollow disc (Fig. 3.1)
was considered and no contact analysis was necessary. For the thermal analysis, a radial
temperature distribution was assumeci with O°C king at the inner surface and 100°C at
the outer surface nodes (AT= 10O0C), and, the model was run using ANSYS 5.4. In the
case of centrifugai loading, the angular vtlocity was taken to be 365 rpm. The following
values were selected for the disc in both of these cases: p = 85 10 kg/m3, v = 0.29, inner
radius a = 0.03 17Sm, outer radius b = 0.1905m, K = 1 13 W/m°C and a = 1 1 .7ebfc.
0°C Fig. 3.1: Discretized finite elernent model of a disc.
Radial (a,) and tangentid (a,) stress distributions were obtained as a function of the
radius of the disc. Comparisons between the radial and tangentid stresses obtained using
the theoretical expressions (Eq. 2.5) and the finite element method are shown in Figs. 3.2
and 3.3 for the case where only thermal stresses were considered. These figure show that
the values obtained for a, and a, using finite element method match well with the
theoretically obtained results for simple geometries with no contact. Let us devote Our
attention to the effect of body forces. Figs. 3.4 and 3.5 show the cornparison between a,
and a, obtained using the theoretical expressions (Eqs. 2.1 and 2.3) and the finite
element method for the cases involving centrifugal loading. Again, the results obtained
match well with the finite element predictions. These results are only applicable for
simple bodies, where coupling and contact does not exist. In this case, the combined
thermal and mechanical loading can be evaluated by simply superimposing the results for
the thermal and the centrifugai loading obtained separately. Figs. 3.6 and 3.7 show the
combined radiai and tangential stress distribution dong the radius of the disc analytically
and numericaily using the finite element method. It can be seen that gocxi agreement
exists between the two.
3.4 Coupled Analysis for Turbine Disc Assembly
To model the effect of thennomechanical coupling, a turbine assembly was modelled
using realistic data with a= 6 6 0 rpm, as given in Table 3.1. This same data has been
used throughout the course of this research. Contact was modelled at the bladddisc
interface using contact elements. Tests were carried out for centrifugal and thermal
ioading separately and the von Mises stresses were plotted for each indîvidual loading as
shown in Fig. 3.8. In order to identify if the problem is coupled or uncoupleci, the results
obtained from centrifuga1 and thermal loading were added together as shown in Fig. 3.8
(Rotation + Thermal). Furthemore, the analysis was extended to treat this coupled
problem properly using the appropriate element under thermomechanical loading.
Fig. 3.8 shows the clearly the importance of treating the analysis as a coupled problem. It
can be seen that the stresses in the contact region are much higher than those obtained by
simply adding the stresses. This shows clearly that the problem cannot be simply
simplified into two subproblems: one thermal and one mechanical, since interaction
between the two can influence the stress state of the turbine disc assembly considerably.
O 0.04 0.08 0.12 0.1 6 0 2
Radius (m)
Fig. 3.2: Radial stress distribution dong the racüus of disc for thermal loading oniy.
2,00E+08 - -c- Analytiil -O- FEM
Ê 1.çOE+08 - u 2 '-"" - 2 5.00E+07 - 2 G O.ooE+oo
4 1 -5.00€+07 *
Radius (m)
Fig. 3.3: Tangentid stress distribution dong the radius of disc for thermal loading only.
O 0.04 0.08 0-12 0-16 0.2
Radius (m)
Fig. 3.4: Radial stress distribution dong the radius of disc for centrifuga1 loading only.
O 0.04 0.08 0.12 0.16 0.2
Radius (m)
l.ooE46 9.00E45
8.OOE+OS h
E 7.00E45
S.WE+OS - S.ûûE+ûS
U: 4.00E45 Q $$ 3.00E45
Fig. 3.5: Tangentid stress distribution dong radius of disc for centrifuga1 loading only.
- - -t- Analyücal - + FEM - - - -
Radius (m)
6.00E+07
5.WE+07
A
E A.00€+07 s 2 3.WE+07 U> cn
2.00E+07 cn
1.00€+07
O.WE+OO
Fig. 3.6: Combined radial stress distribution along the radius of the disc.
- -Com bined Analytical -Combined FEM
-
-
-
-
-Corn bined Analytical +Combined FEM
L
O 0.04 0.08 0.12 0.16 0.2
1 Radius (m)
Fig. 3.7: Combined tangentid stress distribution along the radius of the disc.
Contact angle, a
Upper flank angle, y
20"
Lower flank angle, f3
Inner radius, ri
40°
1 Outer radius, r, 1 0.25 m 1
1 Temperature at the outer surface, T, I
1 956°C 1 Temperature at the inner surface, Ti
1
Speed of rotation 1 6600 rpm
0°C
Table 3.1 : Details of reference geometry and applied loads.
Nomialized distance dong the interface
Fig. 3.8: Nonnalized von Mises stress dong the interface for three teeth.
In Fig. 3.8, the von Mises stresses were normalized by pa2a2, where p is the density of
the material, o is the speed of rotation and a is the disc bore radius.
3.5 Two Dimensional Modelling of Turbine Disc
Let us begin the coupled thennomechanical analysis by limiting our attention to two
dimensional finite element mode1 of the structure. Effectively, a plane stress slice was
modelled using the contact geometric features of the disc. The mode1 was analyzed for
three, four and five fu-tree teeth.
3.5.1 Two Dimensional Details of theGeometr y
Fig. 3.9 shows the details of the two dimensional geometry of the reference design for the
fir-tree region. In this case, the number of teeth n = 3, contact angle a = 20°, bottom Bank
angle f3 =40°, top flank angle y=40° and flank length 1 = 3.62 mm are defmed and
show in the figure. Table 3.2 shows the details of the different geometries undertaken in
this study for the two dimensional finite element model. The table clearly indicates that
by changing the contact angle cx and flank angles P and y , the flank length changes
which in turn changes the contact length between the two mating teeth of a turbine blade
and a disc.
3.5.2 Finite Element Modelling of Disc
Strictly speaking, the distribution of stresses in a fir-tree joint is a three dimensional
problem. However, it is assurned that the variation of load in the thickness direction is not
significant and that there is no dramatic thickness variation near the fir-tree region.
Furtherrnore, it is believed that one can obtain significant and vaiuable results from the
two dimensional model.
The original mode1 of the turbine disc contains sixty fir-tree slots. Because of the radial
symmetry, only one sector of the turbine assembly, consisiing of half of the fir-tree disc
and half of the fir-tree blade, was rnodelled, Fig. 3.10. The plane stress two dimensional
results will then characterise the stress distribution of the blade and disc assembly. This
can result in a significant saving of the effort and cost and stiil can provide a significant
insight into the problem.
Outer Radius % = 0.55 mm
Inner Radius R, = 0.55 mm
Top Flank Angle y = 40"
Bottom FIank Angle P = 40" d 2 - Contact Angle a
Fig. 3.9: Details of two dimensional geometry for three teeth.
) Number of 1 Contact Angle, 1 Bottom Fia& 1 Top Flank 1 Fi& Length, 1
Table 3.2: Details of two dimensional geometry for turbine disc assembly.
3.6 Contact between Blade and Disc
Contact problems involve kinemafic contact conditions as well as unknown boundary
conditions. The unknowns are the actud contacting surface, the stresses and the
displacements on the contact surfaces. Solutions of problems involving contact cannot be
detennined accurately within a single step analysis. Severai steps must be taken, updating
the tentative solution after each step, until convergence is reached. Fig. 3.11 shows the
contact between two deformable bodies.
Early finite element formulations did not accommodate for the treatment of contact, and
ad-hoc techniques were used. In these cases, simpliwng assumptions regarding the
Fig. 3.10: Discretized two dimensional geometry.
Before contact Contact with no penetration
Contact with penetration
Fig. 3.1 1 : Contact between two deformable bodies [3.6].
actuai contact surf" and the distribution of the contact stress were d e . These
sirnplifying assumptions enabled the treatment of each individuai body as a separate
problem. This ad-hoc technique pmved inadequate in many cases where neither the
contact surface nor the stresses on it could be easily estimated. This has prompted the
development of contact elements c3.8 - 3.101. Finite element methods treat contact
problems by extending the variational formulation upon which the finite element method
is based. Contact elements are formulated and assembled into the original finite element
code in order to enforce the contact conditions. The solution is then obtained by solving
the resulting set of non-linear equations.
The two and t h e dimensionai contact elements used in this study adopt a node-to-
segment interface model, as shown in Fig. 3.12. The amount of the open gap or the gap
penetration of the contact node on the target plane is calcuiated dong with the point of
projection of the contact node. Contact is indicated when the contact node penetrates the
target surface defmed by the target nodes. The penetration represented by the magnitude
of the gap is a violation of compatibitity. In order to satis@ contact cornpatibility, forces
are developed in a direction normal to the target that will tend to reduce the penetration to
an acceptable numerical level. In addition to compatibility forces, friction forces are
developed in a direction that is tangent to the target plane (refer to Ref. [3.3], and Section
2.2).
Contact surface and node Contact d a c e and node
Target d a c e and nodes
Target mrface and nodes X
Fig. 3.12: (a) Two dimensionai contact elements. and
(b) Three dimensional contact elements [3.6 1.
Regarding the normal forces, three methods of satis€ying contact compatibility exist: a
penalty method, Lagrange multiplier method and a combined method. The penalty
method approximately enforces compatibility by means of a contact stiffness (Le., the
penalty parameter). The combined approach satisfies compatibility to a user-defined
precision by the generation of additional contact forces that are referred to as Lagrange
forces. The Lagrange multiplier method can mostly be used for frictionless problerns.
For the penalty method,
where K, is the contact stiffness and g defines the magnitude of the gap.
For the combined method, the Lagrange multiplier component of force is computed
locdly (for each element) and iteratively. It is expressed as
where:
hi+, =Lagrange multiplier force at iteration i + 1
E = user defined compatibility toleranc
a= an intemally computed factor (a c 1)
Tangentid forces are due to fnction that arises as the contact node meets and moves
dong the target. Three friction models are available: frictionless, elastic Coulomb
friction, and rigid Coulomb fnction. The Coulomb friction representations require the
specification of the coeffkient of sliding friction.
For the frictionless case, the tangentid force is f,= O .
For the elastic Coulomb friction the tangentid displacement u, of the contact node
relative to the target is calculated and decomposed into elastic (or sticking) and sliding
(inelastic) components:
where: u ," = elastic tangential deformation
u : = sliding (inelastic) tangential deformation
The tangentid force is then
t, ={K, u:<F< if sticking
C if sliding
K, = sticking stiffhess
where: fs = sliding force limit of the Coulomb friction mode1 (Fs = - pf, ) F = static/dynamic friction factor
Elastic contact tangential deformations are ignored in the rigid Coulomb fnction mode[.
The contact node is always assumed to be sliding on the target, where the tangentid force
f, is calculated as:
The choice of K,, (contact stiffness) and K, (sticking stiffhess) is very critical for the
convergence of the analysis. Unfortunately, no data for the estimation of these quantities
c m be found in the literature. Hence, convergence tests were carried out by the author
and the choice of K,, and Ks was based on the critenon that the element should converge
rapidly for al1 reaiistic values of coefficient of fnction C( (O. 1 to 0.5).
Modeliing of Contact for 2D Thermomechanical Anaiysis
When contact is established in case of thermomechanical analysis, heat is transfemd
across the interface in a direction normal to the target surface. The total heat flow from an
insulated target surface to the contact node is given as:
KC(Tœ-T,) ifincontact if open
where:
q, = total heat flow at contact node.
K, = contact conductance.
T* = temperature of the target plane at the contact point.
Ti, Tj = current nodal temperahire.
Ti, = temperature at contact node K.
S* = contact position of the contact node.
The thermal conductivity mauix is:
K,{N,XN,)' ifincontact if open
The element thermal load vector is comprised of the Newton-Raphson restoring heat
flows, and cm
where:
be expressed as:
{FP 1= q. 0% 1
{qm)= element t h e d load vector.
The thermal load vector and conductivity rnatrix are assembled with the structurai load
vector and stifkess ma&, respectively in a rnanner consistent with the defined degree of
freedom.
3.7 Three Dimensional Modelling of Turbine Disc
Three dimensionai modeiiing of turbine disc assembly was also carried out to criticaily
compare the results with those obtained from the two dimensional anaiysis. Furthemore,
three dimensional modelling helps us to analyse the effect of the skew angle upon the
stress distribution in the thickness direction, which is not possible with two dimensional
modelling. The analysis assumes that the turbine disc assembly is insulated and that heat
flow is mainly by conduction.
3.7.1 Three Dimensional Details of Geometr y
Fig. 3.13 shows the three dimensional geometry of the reference design for the fir-tree
region. In this case, the number of teeth n = 3, contact angle cc = 20°, bottom flank angle
p = 40°, top flank angle y= W and flank length 1 = 3.62 mm are defined and shown in
the figure. In this study, two types of three dimensional models have been developed.
One is the straight or non-skew fir-tree bladddisc assembly, while the other takes into
account the skew angle. Fig. 3.14 shows the effect of variation of a straight geometry and
a skew geometry. In this study, IO0 and 20" skew angles have been considered for three
fir-tree teeth. The three dimensional model is created by extniding the two dimensional
rnodel in the thickness (Le. Z direction). The values of contact angle, flank angle and
flank length given in Table 3.1 are equally applicable in the case of the three dimensional
model. However, in this study, only the cases with three teeth have been considered for
the reference design ( a = 20°, $ = 40". y = 40° ). This is because of the increased cost and
time associated with the increase in the nurnber of teeth.
Top Fl& Angle y
~ttom Flank Angle $
Contact Angie
Fig. 3.1 3 : Specifications and creation of three dimensional geometry .
l S traight Geometry Skew Angle Geometry
Fig. 3.14: Non skew and skew angle geometry investigated in this study.
3.7.2 Discretization of the Disc Assembly
In the case of the three dimensional analysis, both the skew and non skew geometries
were anaiysed. For the case of non skew or straight geometry, only one sector of the fir-
tree consisting of haif of the fir-tree disc and half of the fu-tree blade were modelled. Fig.
3.15 shows the non skew discretized geometry used for modeUing the blade/disc
assembly in three dimension.
The finite element analysis of the skew fir-tree geometry is more complex than a saaight
fir-tree geometry. The geomeuic symrnetry of the skew design cannot be extruded and
converted into mode1 symmetry, and therefore, the entire bladddix assembly has to be
modelled. Fig. 3.16 shows the discretized geometry of bladddisc assembly having a skew
angle.
3.7.3 Modehg of Contact for 3D Thermomechanical Analysis
Modelling of contact for three dimensional finite element analysis is done by using five
noded element. This element is capable of modelling structural and thermal contact.
When contact is established between the mating surface, heat is m s f e m d across the
interface in a direction normal to the target surface. The total heat flow from an insuIated
target surface to the contact node is given as:
- {K. (:O - Tm ) if in Contact 9, - if open
w here:
q, = heat flow at the contact node.
K, = contact conductance.
T* = temperature of the target plane at the contact point.
= q i T I ~ z T ~ - ~ ~ T ~ ~ o T ~
Ti, Tj, Tk, Tl = current temperature of the target nodes.
Tm = temperature of the contact node m.
The themal conductivity matrix is:
K, {N; IN; if in contact if open
The element thermal load vector is cornpnsed of the Newton-Raphson restoring heat
flows, and can be expressed as:
where:
The thermal load vector and conductivity ma& are assembled with the stnicîural load
vector and stifiess matrix, respectively, in a manner consistent with the defined degree
of freedom 13.61.
3.8 Convergence Tests
The basis of the finite element method, as described earlier, is to represent the continuum
by a finite number of elements interconnected through their common nodes. This is
known as discretization. Two rnethods of discretization are available for the generation of
finite element meshes in non-linear contact analysis; namely: (i) mapped mesh generation
and (ii) free mesh generation. In mapped mesh generation, the user specifies that the
program must use ail quadrilateral area elements or ail hexahedral (brick) elements to
generate a mapped rnesh. Mapped mesh requires that an area or volume be "regular", that
is, it must meet certain special cnteria determined by the program. In free meshing
operations, no specific requirements restrict the solid model. Free meshing can use either
mixed area elements shapes or else ail triangutar area or al1 tetrahedral volume elements.
The choice of meshing technique used for the fir-tree region of the model was due to the
necessity to model a complex geometry where large transitions in stress and strain fields
Fig. 3. I 5: Three dimensional discrettized sîdght geometry of the blade/disc assembly.
Fig. 3.1 6: Three dimensional discretized skew geometry of the bladddisc assembly .
are expected. To obtain a maximum amount of information from the region, the use of a
high density mcsh was required. The E'ree meshing routine is a very flexible and usefil
tool for automatic mesh generation, such that transitional meshes can be coastnicted with
relative ease on highiy imgular geometric shapes, such as the fir-tree joint. The higher
number of smailer elements will also better modelüng of the interface conditions. Away
from the interface region, where steep gradients in stress and strain are not expected, only
a few elements are used, just to maintain continuity within the model.
The size and number of elements used influences the solution accuracy required which in
itself affects the computing time required for the analysis. To obtain convergence in finite
element solutions, it is the usual practice to alter the size and number of elements within
the mesh until a compromise between solution accuracy and computing time (cost) is
obtained, The two dimensional models were discretized using four-noded quadrilaterai
and three noded triangular plane stress elements. The three dimensional models were
discretized using eight noded brick element and five noded triangular element. The
contact region was modelled with three node triangular elements for two dimensional and
five node element for three dimensional modelling. Convergence tests were c h e d out in
the present work on difierent numbers of elements together with evaluation of the
computing time. Fig 3.17 shows the nomalized von Mises stress distribution obtained for
the last tooth for of the meshes. Five cases were analysed of which mesh 3 was chosen as
the best possible mesh that could provide accurate results and required less computing
time. Figs. 3.18 shows the different meshes analysed to obtain the best possible mesh.
0.2 0.4 0.6 0.8
Normalized distance almg bottom tooth
Fig. 3-17: Normdized von Mises stress distribution for different meshes.
REFERENCES
3.1 Meguid S. A,, Integrated Computer-Aided Design of Mechanical Systems,
Elsevier Appiied Science, London, 1987.
3.2 Meguid S. A., The Finite Element Method in Mechical Engineering, MIE 1804
Course notes, Toronto, pp. 1-6, 1996.
3 -3 S tjepanovic 2, ntree-dimensional Finite Element Adysis of Dovetail Joints in
Aeroengine Discs, Diplom-Ingenieur Thesis, University of Toronto, 1996.
3.4 VenkateshS.,StnrcturalIntegrityA~lysisofanAeroenglileDisc,M.Sc.Thesis,
Cranfield hstitute of Technology, 1988.
3 -5 Papanikos P., On the Structural Integrity of Dovetail Joints in Aeroengine Discs,
M.A. Sc. Thesis, University of Toronto, 1992.
3.6 ANSYS Version 5.4 Engineering Analysis System, User's Manual (sofrware on-
line), Ansys Engineering Systems, Houston, 1998.
3.7 Bathe K. J., Finite Element Procedures, Prentice-Hall, London, 1996.
Chapter 4
Results and Discussions
This chapter is divided into thtee main sections. The fmt is concemed with the
verification of the finite element model using photoelastic analysis. The second is
devoted to the two dimensional finite element results, while the third addresses the three
dimensional modeiiing of the current coupied thermomechanical problem
4.1 Photoelastic Verification of Finite Element Analysis
In addition to the use of analytical expressions to validate the fmite element model,
photoelastic experiments were also conducted to validate that rnodel. In this section, we
address these important aspects of the photoelastic stress analysis: (i) photoelastic
materials and geometry, (ii) digital imaging of photoelastic patterns, and (ui) photoelastic
results. These are sumrnarised below.
4.1.1 Photoelastic Materials and Geometry
One of the important factors for a successhil photoelastic expriment is the proper
selection of the photoelastic materiai. A photoelastic material should exhibit several
unique properties such as birefringence, transparent y, high sensitivit y, good linearit y,
isotropy and homogeneity, resistance to thne edge effects and machinability. There are
several materials commercially available which possess most of these properties.
However, Araldite Cï200, a heat setting epoxy resin is one of the most widely used
material for making photoelastic models. It has good optical properties, good stress
linearity and exhibits high sensitivity [4.1, 4-21. Table 4.1 shows the contact angle a,
Iower flank angle f3, upper flank angle y and flank length 1 of a photoelastic model
compared to the reference design having three teeth. The photoelastic model was tested to
the softening temperature of the Araldite material (13S°C), soaked at this temperature for
two hours, and then cooled to the room temperature at a uniform rate of 2.S°C per hour. A
sin test speed of 360 rev/min was selected during the stress fieezing process (4.2,4.3].
Specimen Type
Photoetastic
Table 4.1 : Photoelastic and FE reference design geormtry specifications.
Contact angle,
Finite element
4.1.2 Digital Imaging of Photoelastic Patterns
1 9 O
In order to validate the stress distribution at the blade/disc interface in the finite element
model, preliminary experimental results were O btained from an existing two dimensional
photoelastic spin test of a turbine blade/disc assembly. The three teeth photoelastic
model, which was spun at 360 rpm during the stress freezing process, was provided by
Prof. S.A. Meguid. Four isochromatic and four isoclinic h g e patterns were captured
digitally and were analysed using an autornated software developed by the EMDL, the
details of which are given in Refs 14.21 and [4.4]. The purpose of this prelimïnary
photoelastic work was to examine the accuracy of the fnite ekrnent model.
Lower tlank
20°
In this context, four images of isochromatic pattern were obtained by rotating the
analyser to O, ir/4,~/2,3n/4 while four isoclinic images were obtained by angles of
39"
Upper h k
I 1
Flank length, I
42'
40"
3 .56
40° 3.62
O, n/8, ~/4,3x/8, as shown in Fig. 4.1. The isochromatic h g e pattern obtained from a
two dimensional model gives contours dong which the principal stress difference
(a, - a,) can be obtained if the fringe order N, material ûinge f, and the model
thickness are known. The isociinic Wnge pattem obtained in the pl- polariscope is used
primarily to provide the direction of the principal stresses at any point in the model [4.4].
A polariscope with circular polarised light beam and CCD camera was used to capture
the resulting isochromat ic fringe pattern images. An automated photoelastic technique
was used to calculate the stresses along the interface of the specimen.
Fig. 4.1 : Images used in autornated photoelasticity software for 2D analysis.
4.1.3 T ypical Photoelastic Results
Fig. 4.2 illustrates the fmite element predictions and photoelastic results in a composite
manner, while Fig. 4.3 shows the maximum stress dflerence along bottom tooth. The
stress contours in Fig. 4.2 have been obtained using ANSYS 5.4. The following can be
deduced:
(i) nie bottom tooth of the blade experiences the highest 1 d This can be seen by
refeiriog to Fig. 4.2, where the stress disinbution obtained by photoelasticity and
f i t e element method show similar trends. This results also reveal that peak
stresses and stress concentration occm within the contact region at the bottom and
top of each flank.
(ii) The isochcornatic h g e pattenu, which are constant principal stress difference
contours, compare weli with the maximum shear stress contours plots obtained
using the nnite element method. In particular, the h g e patterns predict similar
contact stresses and stress concentration around the different fillets.
Fig. 4.2: Composite FE and photoelastic maximum shesr stress contour images.
These resuits indicate that the nnite element mode1 is capable of capturing the main
features of the resulting stress field due to the cenrifiiggal loading.
Fig. 4.3: Photoelastic analysis of a turbine disc showing normalized difference between
maximum and minimum principal stress dong the bottom contact region.
4.2 Two Dimensional Finite Element Analysis
4.2.1 Details of Models and Applied Loads
S trictly speaking, the distribution of stresses in a fir-tree joint is three-dimensional.
However, if it is assumd that the variation of load in the thickness direction is not
significant and that there is no dramatic thickness variation near the fu-tree region, then it
is possible to mode1 only a slice of the blade and disc assembly as a two-dimensional
plane stress problem. The two-dimensional results wiU then characterise the stress
distribution of the blade and disc assembly to a certain extent. This can result in a
substantial saving of effort and cost and still can provide a significant insight into the
problem In this section, only two-dimensional stress analysis results are reported. The
three-dimensio na1 results are reported in the next section.
The fmite e l emnt analysis was carried out on the turbine biade and the disc for different
values of the contact angle a, flank angles B and y, and the number of teeth in contact.
The interface conditions were studied by varying the coefficient of friction. Although
eight node quadrilateral elements could have been employed for the bladeldisc assembly,
ANSYS 5.4 does not accommodate such elements in the contact region. Accordingly,
four noded quadrilateral plane stress elements were used to mode1 the assembly.
Three reference models were considered, involving three, four and five teeth,
respectively. The reference design has a contact angle a =20°, lower contact angle
B = 40" and an upper flank angle y =4û0 in each of these modeIs. The contact angle a
was varied from IO0 to 2S0 in step of SOC, while the flank angle has been kept constant at
40°.
It was found that for a equal to 10°, the size of the contact zone was too small to allow
for adequate contact elernents at the interface of the blade/disc assembly. A lower Limit of
three elements in contact dong the flanges of each tooth was maintained throughout the
fmite element simulation. In another an, the flank angles $ and y were varied in steps
of 10" to study their effect on the resulting stress field. Table 4.2 shows the number of
teeth n, lower flank angle $ , upper flank a g i e y and the flank length 1 for the different
models used in the research. In total, thirteen models were generated for two dimensional
fmite element analysis. It was shown in Ref. 14.21 that the coefficient of friction is not
constant but varies fiom 0.1 to 0.4 between the blade and disc interface at room
temperature. To investigate the effect of the difTerent interface conditions. the coefficient
of friction p was varied such that p = 0.0,O. 1,0.25 and O.S.
Al1 the models were subjected to thermal and centrifugai loading. The contacting surfaces
were modelled by using the contact interface elements, described in chapter three. No
attempt was made to accuraîely mode1 the blade. The main aim was to determine the
effect of the coupled thermal and centrifuga1 loading of the blade at the interface.
The material properties used for modeiiing the insulated blade and the disc were that of a
typical nickel-based superalloy used in the design of aeroengine turbine components,
INCONEL 720. This material is creep and fatigue resistant. The same properties were
used for the blade and the disc. Table 4.3 iists the material properties o f the turbine disc
assembly [4.5].
1 Number of 1 Contact Angle, 1 Bottom FtPnk 1 Top h k 1 Flank Length, 1
Table 4.2: Geometric s ?ecification of models used in 2D finite element analysis.
Pm~ertY
1 1
I 0.1 % Proof stress 1 1300 I MN/^'
Modulus o f Elasticity, E
Poisson's ratio, v
Value Unit
220 G N / ~ '
Density, p
expansion, a
Coefftcient of linear
Thermal Conductivity, K
85 10
Table 4.3: Physical properties of a typical INCONEL 720 aUo y.
K m 3
1 1 .7x104 PC
4.2.2 Analysis of Coupled Thermomechanical Stress Field
Fig. 4.4 shows the basic fui te element mode1 for the reference blawdisc design used in
this study. In this case, the foliowing values of the fu-tree joint were used: a = 20'. $ =
40°, y = 40° and n = 3 teeth. The rotational speed was taken as 6600 rpm and the
temperature range was assumed to k fkom O°C t o 956°C. The 0°C was maintained at the
inner surface of the disc bore and 956°C was assurned at the outer surface of the blade. In
all the analysis considered, coupled thermal and rotational loadings were considered in
both the two and three dimensional studies.
In the foregoing discussions, only an enlarged view of the frr-tree region of the bladddisc
assembly wiii be shown, since our intention is to study and examine the stress
distribution in the frr-tree region only, where the stress gradients and are very high due to
complex geometry. It should also be noted that the analysis of the results would mainly
be concentrated on the disc rather than on the blade.
The stress distribution at the blade/disc interface is obtained for the reference design. The
geometry of the reference design is given in Table 4.4. It should be noted that aii the
graphs have been normaiized by the factor po2a2, where p = 8510 kg&, o = 6600 rpm
and a = disc bore radius. in addition, the distance dong the interface has been nonnalized
by the contact flank length plus haif the fillet radii in both sides of the flank-
1 Teeth,n 1 h g k u 1 Aagle, f3 ( Angle, y 1 Length, I 1 Friction, p I Number of
Table 4.4: Specification of the geometry considered for coupled analysis.
Contact Flank Bottom Flank Coefficient of Top FIank
Fig. 4
Undeformed
\ Deformed
geomebcy of aeroenghe turbine disc.
Fig. 4.5: Deformed (dotted) and undefocmed (solid) outiine of reference aeroengine
turbine disc.
Fig. 4.5 shows the deformed and undeformed shapes of the disc assembly as a result of
the coupled themmechanical loadings. Fig- 4.6 shows the maximum principal stress
contours for the same reference design. Many interesthg points can be observed. The
fdlet radii in the disc just below the contact fianks experience high stress concentration.
This is tnie for ail the three teeth of the disc, which are subjected to coupled thermai and
centrifiigal loading. These stresses are rnaîniy due to the complex nature of the geometry.
The highest stress intensity occurs in the bottom tooth closest to the disc, as compared to
the upper two teeth. The same findings were observed in the p hotoelastic stress analysis.
This information can be used to redesign the fillet radii so that lower s u e s concentration
can be obtained. Similar stress distribution results were obtained by Venkatesh [4-21 and
Durelii et al. r4.q in their study of the fx-tree attachrnents accounting only for centrifuga1
loading. The contour plots of the minimum principal stress, the maximum shear stress
and the von Mises stress are shown in Fig. 4.7, Fig. 4.8 and Fig. 4.9, respectively. These
stresses helps us to analyze how stresses can be reduced by varying a specific radius
andior contact angles in a given geometry.
In order to study the effect of the number of teeth n on the stress distribution at the
bladddisc interface, von Mises stress contours were plotted for rhree, four and five teeth,
respectively. Table 4.5 gives the detaiis of the geometry of the model. Figs. 4.9, 4.10 and
4.11 show the von Mises stress contours for the three cases.
Table 4.5: Specifications of geometry analysed to determine the effect of variation of
number of teeth in turbine d i x assembly for frictionless contact (p = 0) .
Number of
Teeth, n
3
4
5
Contact
Angle, a
20°
20°
20°
Bottom Flank
Angk l3 40°
40°
40" -
Top Fiank
An&, Y
40°
40"
40"
Fiank
Length,
3.62 mm
2.58 mm
2.01 mm
The figures reveai that the magnitude of the equivalent stress inmeases as the number of
teeth increases. Whüor contour plots give a qualitative qresentation of the stress fieià,
the x-y plots provide a quantitative description of the stms field Figs. 4.12. 4.13 and
4.14 shows a series of x-y plots of von Mises stress distributions dong tbe fir-tree
boundary of the cüsc for three diffant designs. Both the wupled thennomechanical
stresses as weii as StressCs resultmg h m centrifuga1 loading done have been iacluded
for cornparison. These figures show the variation of the normalized von Mises stress
dong the fi-tree bouncia~~ of the disc. It is seen that high stresses occur at the lower
contact point far PU the teeth. Howwer, the highest stress occurs for the fast tcah at the
point just below the last blade contact point. These points aie areas of high stress
concentration and thmfore are possibk sites for crack initiation and propagation. These
figures a h show that increasing the number of teeth in the fir-tree region resulîs in
increashg the maximum equivaknt stress.
Fig. 4.6: Maximum p ~ c i p a i stress contour for turbine disc assembly.
Fig. 4.7: Minimum principal sîress contour for turbine disc assembly.
Fig. 4.8: Maximum shear stress contour for turbk disc assembly.
Maximum equivalent stress
Fig. 4.9: Von Mises stress contour for turbine disc asembly.
Maximum equivak~ stress
Fig. 4.10: Von Mises stress contour for turbine d i s assembly.
Maximum equivaknt stress
Fig. 4.1 1 : Von Mises stress contour for turbine disc assembly.
Fig. 4.12: Nonnalitod von Mises stress w n t o n for turbine disc assembly for thm teeth
(n = 3).
0.2 0.4 0.6 0.8
Nomiaiized distance along the interface
Fig. 4.13: Normalized von Mises stress contours for turbine disc assembly for four teeth
(n = 4).
O 0.2 0.4 0.6 0.8
Numaiiaed dstanae along the inter fa^^
Fig. 4.14: Norrnalized von Mises stress contours for turbine disc assembly for five teeth
(n = 5).
4.2.3 Effect of Contact Angle and Friction
The effect of the variation in the contact angle a on the stress distribution at the
blade/disc interface was investigated. The effect of friction on the resulting stress field
was also investigated in this section. Table 4.6 shows the number of models analysed by
varyïng the contact angle a, while keeping the flank angles and y constant.
Fnctioniess conditions were also assumed in our analysis. Fig. 4.15 shows the result of
the stress distribution dong the blade/disc interface for the bottom tooth.
Table 4.6: Specification of the geometry of models analysed to determine the effect of
contact angle for f3 = y = 40" and p = 0.0.
In order to investigate the effect of friction, a similar analysis was carrïed for three, four
and five teeth fr-tree joint by keeping the contact angle a equal to 1 SO, lower flank angle
p equal to 40° and upper flank angle y equal to 40°, whüe changing the value of
coefficient of friction p to 0.0, 0.1, 0.25 and 0.5. Fig. 4.16 shows the normalized von
Mises stress distribution along the bottom tooth.
As the contact angle a is varied from 15' to 2S0 in steps of 5 degree, it is seen that
the intensity of the stress distribution increases as contact angle a is increased.
Higher stresses are obtained at the lower contact point for the bottom tooth. This is
because changing a results in reducing the number of contact elements, which in
tum causes the load to be distributed in a very small contact area at the bladddisc
interface, The variation of a increases the blade width and hence increases the blade
loading on the disc stress distribution. This results in high stress concentration at the
contact surface. Since the bottom tooth takes the highest load, it experriences the
highest stress concentration.
In most cases, the stress distribution at the bladddisc interface increases as the
coefficient of fiction p is Ulcreased fiom 0.0 to 0.5 for al1 t h e , four and five teeth.
The equivalent stresses are observed to be minimum for frictionless or near
frictionless contact ( p is close to 0.0). Higher equivalent stress were observed for p
dose to 0.5.
Srnaller vaiues of a and p tend to produce stress distri'butions that are more uniform
and more Hertizan in nature-
In most cases, the stresses at the blade/disc interface increase as the contact angle a,
coefficient of ûiction p and number of teeth n are increased.
(a) n = 3, $=40°, Y=4û", p 0 . 0
Nomlaized distance dong bottom tooth
(b)n=4, g=40°, p=0.0
O 0.2 0.4 0.6 0.8 1 Nomialued distance dong bottom todh
Fig. 4.15: Normalized von Mises stress vs. normalized distance dong interface.
0.2 0.4 0.6 0.8 Noniralied distance dong battom todh
(a) n = 3, a = ISO, p = 40°, y = 40"
0.2 0.4 0.6 0.8 Normalized distance along bottom tooth
(b) n = 4, a = Fi0, P = 40°, y = 40°
0.2 0.4 0.6 0.8
Normalized distance along bottom tooth
( c ) n = 5 , a = l5", B =a0, y =4û"
Fig. 4.16: NormalUed von Mises stress vs. normaiized distance dong interface.
4.2.4 Effect of Flank Angle and Friction
Further investigations were carried out to determine the effect of the variation of the flank
angles $ and y , whiie maintainhg the contact angle a = 20" constant upon the coupled
stress field. Each fiank angle was varied in steps of 10'. In this section, only three teeth
fi-tree joints were examined and the coefficient of fnction p was varied fiom 0.0 to 0.5.
Table 4.6 shows the number of cases analysed for the different flank angles. The resulting
von Mises stress distribution dong the entire interface for the three teeth geornetry is
shown in Figs. 4.17 - 4.18. The results reveal the following:
In some cases, there is not much difference in the stress level between the second and
the third teeth fiom the top. Most of the stress concentration was observed at the
lower contact point for each tooth. However, the bottom tooth closest to the disc
experienced the highest stress.
In cases where y>p, there is a significant increase of the stress level for al1 the teeth,
and in particular the bottom tooth. This is because when y>B, the length of the
contact zone decreases, and thereby the number of elements in contact decreases. The
coupled thermomechanicd loading is distributed over a small contact area, hence the
stress level increases. A two fold increase in stress level is observed compared to the
reference design. This suggests that designs having y<p should be sought. The
results also indicate that when p = 4@ and y= 30°, significantly low stress
distribution at bladddisc interface was observed compared to the reference design.
The stress distribution at the interface becomes highly non-uniform in nature as the
coefficient of friction p is increased from 0.0 to 0.5. It was observed that the center
portion of each tooth had lower stresses compared to the top and bottom portions of
each tooth. This is due to the fact that the distributions are afcected by the shear stress
component. The highest stress field distribution was obtained for p = 0.5. As the
value of p increases, the bottom portion of each tooth takes the greatest load.
Table 4.7: Specification of the geometry of different mdels analysed to determine the
effect of f l d angle and friction on stress field.
Number of
Teeth, n
3
3
3
3
Contact angle,
a
20°
20"
20°
20"
Bottom Flank
Angle, l3 40°
30°
40°
40°
i I -
I --
3 20" 50" 1 40" 0.0,0.1,0.25,0.5
Top Fiank
An* Y
40°
40°
50"
30"
Coefncient of
Friction, p
0.0,O. 1,0.25,0.5
0.0,O. 1,0.25,0.5
0.0,O. 1,0.25,0.5
0.0,O. 1,0.25,0.5
(a) a = 20°, = 40°, y = 40°
Nonnalized distance along the interface
(b) a =20°, $ =30°, y =W
0.2 0.4 0.6 0.8 Nonnalized distance along the interface
( c) a = 20°, $ = 4û0, y = 50' Fig. 4.17: Normalized von Mises stress vs. norrnalized distance along the teeth.
Normalaed distance abng the interface
(a) a = 20°, f3 = 4û", y = 30°
O 0.2 0.4 0.6 0.8
Nomalized distance along the interface
@) a = 20°, = 50°, y = 400
Fig. 4.18: NormaIized von Mises stress vs. normalized distance alona the teeth.
4.3 Three Dimensional Finite Element Analysis
The two dimensional analysis reported in the previous section provides valuable insight
into the problem However, it suffers from two maior drawbacks:
(i) the stress variation dong the disc thickness is ignored because a plane stress
condition was assumed for the geometry. As a r d t , the triaxial state of stress
affecting stress concentration regions was ignored, and
(ii) the effect of the skew angle could not be examined.
In order to examine the three dimensionai effects on the stress distribution, two types of
three dimensional models were developed. One assuming a straight fir-tree slot through
the disc thickness, while the other takes into account a skew angle. In this study, 10' and
20' skew angles were considered for a blade with three teeth. Fig. 4.19 shows the effect
of the variation of a straight geornetry and a skew angle geometry. Skew angle models
are desirable for the purpose of mechanical integrity, since they allow for an increased
number of blades on the disc and increase the contact area in the thickness direction. It
should be noted that the straight ffu-tree geometry is a special case in which the skew
angle is zero.
Straight Geometry Skew Angle Geometry
Fig. 4.19: Skew and non-skew geometry investigated in this study.
4.3.1 Three Dimensional Mode1 with Stnight Fir-Tree Slots
In the case of a straight bladed fir-tree three dimensional mode& the same sector of the
disc used for two dimensional analysis was modelied. The three dimensional structure
was created by extruding the two dimensional mode1 m the direction normal to the plane.
Fig. 4.20 shows a close up view of the discretized geometry for the straight fi-tree slots.
O d y half of the d i i thichess was modeiled due to the geometric and 1-g symmetry
with respect to the centrai plane in the thickness direction. Table 4.8 shows the details of
the geometry.
Fig. 4.20: 3D discretized geometry of the straight (non-skew) bladddisc assembly.
Table 4.8: Geometric specification of 3D non-skew geometry.
Number of
Teeth, n
Contact Aigk,
a
Skew Angk Bottom Flrik
Angk, B
Top Flrnk
A i g k Y
4.3.2 Three Dimensional Mode1 with Skew Fir-Tree Slots
The fmite element analysis of the skew ru-tree slot is more complex than that of the
straight fr-tree slot. Due to the lack of symrnetry, the entire blade disc assembly has to be
modelled. The three dimensional model with a skew angle was developed using a
cylindrical polar coordinate system. ANSYS 5.4 has the option that allows the user to
define its own coordinate systern. A cylindrical CO-ordinate system with its otigin at the
disc centre of rotation was defined- The back surface of the model was rotated witb
respect to this coordinate system to create the skew angle three dimensional model. Table
4.9 shows the details of the geometry analyseci for skew Fi-tree slots. Fig. 4.2 1 shows
three dimensionai discretized geometry of a skew angle bladddisc assembly. Only the
left hand side of the bladddisc interface was examined-
1 Number of 1 Contact Angle, 1 Bottom Flnnk ( Top F h k 1 Skew Angie 1
Table 4.9: Geornetric specification of 3D skew angle geometry.
4.3.3 Results and Discussions of Three Dimensionai Models
Figs. 4.22, 4.23 and 4.24 show the von Mises stress distribution dong the fir-tree
boundary of the disc for coupled thermal and centrifuga1 loading. The figures also shows
the case where centrifugai loading was applied. These graphs are based on the geometric
specifications defmed in Tables 4.8 and 4.9. The von Mises stresses were normalized by
po2a2, where p is the density of the material, o is the rotational speed and a is the disc
bore radius. These figures confi i the deductions made fiom the two dimensional
analysis, that high stresses occur at the lowest contact point for al1 the teeth. The highest
stress occurs in the last tooth at a point just below the last blade contact point. in Fig.
4.22, the results obtained h m the two dimeasional and three dimensional h i t e element
analysis are compared. The resuits show that the two dimensional results foliow the trend
of the nonkew three dimensional model,
Fig. 4.21: 3D dismitizcd geometry of skew angle blade/dïsc assembly.
Fig. 4.22: Von Mises stress contours for a turbine disc asembly with n = 3, a = 20°, B =
40°, y = Mo, c = 0.0, skew = 0'.
V) V)
E 100 C e n t rif ugal Stress 5 V)
C o u p l e d Stress 80
5 C 60 s O
3 - 40 - Q E 20 s
O 0.2 0-4 0.6 0.8
Norrnalized distance along the interface
Fig. 4.23: Von Mises stress contours for a turbine disc assembly with n = 3, a = 20°, $ =
40°, y = 40°, p = 0.0, skew = 10°.
Mmaiized dstance dong the i r t e r f a ~ ~
Fig. 4.24: Von Mises stress contours for a turbine dix assembly with n=3, a=20°,
p =40°, y=4û0, p=û.0,skew=20°.
Since it was observed that the highest stresses occur at the last tooth closest to the disc,
norrnalized von Mises stress trajectories are plotted along the front and back surfaces for
O0 skew, 10' skew and 20' skew angles. The results shown in Fig. 4.25(a) show that the
highest stress occurs for the middle surface fiom which the mode1 was extruded. Also the
stresses increase as the skew angle increases. The finite element prediftions of the
stresses at the lower contact point for the bottom tooth in the thickness direction also
support this trend, as show in Fig. 4.25(b). Both figures show that the two dimensional
results foilow the three dimensional trend It is also seen that very good agreement is
reached between the two dimensional results and the three dimensionai zero skew angk
results in the thickness direct ion.
In order to analyse the effect of different fIank angles, von Mises stress distributions were
obtained dong the bottom t w t h for seiected values of Bank angles f3 and y for straight
groove geometry. As shown in the previous section (Fig. 4.17(b) and Fig. 4.17(c)), in
cases w here y > f3 , significant ly high stresses were obtained. This can be attributed to the
decrease in the contact area for that particular geometry. Fig. 4.26 shows the variation of
the normalized von Mises stress distrïïution dong the bottom tooth for selected values of
flank angles. The values selected ($ = 40' and y= 30") lead to the lowest values of von
Mises stress. The same trend was observed in the two dimensional analysis. In addition,
von Mises stresses at the outer edges are slightly lower than at the centre for the
investigated geornetries, as show in Fig. 4.26(b). The straight or nonskew model was
also examined to determine the influence of the coefficient of friction p upon the
coupled stress field. Fig. 4.27 shows the effect of variation of coefficient of friction on
the von Mises stress across the thickness of the model. It is observed that fiction
increases peak stresses. It is also worth noting that the von Mises stress does not vary
significantly for values of p between O. 1 and 0.5 for the case of three dimension.
Miaaddsmœdorigbattomtoath
(a) dong the bottom tooth, and
(b) across thickness at the lower contact line
Fie. 4.25: Normalized von Mises stress for three dimensional eeometries:
Nonnalized distance along bottom tmth
(a) along the bottom twth, and 120 1 I
-4 -3 -2 - 1 O 1 2 3 4 5
Distance through thickness, mm
(b) across thickness at the lower contact line.
Fig. 4.26: Norrnalized von Mises stress for t h e dimensional straight slot geometries for various flank angles.
- 5 - 3 -1 1 3 5
Distance through thickness, mm
Fig. 4.27: Effect of coeff~cient of fiction on the von Mises stress through thickness-
4.1 Papanikos P., On the Structural IntegnSy of Dovetail Joints in Aeroengine Dix,
M.A. Sc. Thesis, University of Tmnto, 1992.
4.2 Venkatesh S., Structural Inreg* Analysis of an Aeroengine Disc, M. Sc. Thesis,
Cranfield Institute of Technology, 1988.
4.3 D d y J. W. and Riley W. F., Experimental Mechanics, McGraw Hili Book
Company, 2n* Edition, 1978.
4.4 Kanth P., 2 0 & 3 0 Finite Elernent Analysis of Fir-Tree Joints in Aeroengine
Disc, M-A. Sc., University of Toronto, 1998.
4.5 Ross R., Metallic Muterïuls specification Handbook E. & F. N. Spon Ltd.,
London, pp. 303 - 309, 1972.
4.6 Dureili A. J., Dally J. W. and Riley W. F., Stress and Strength Studies on turbine
blade aitachments, Proceedings of the Society for Experimental Stress Anaiysis,
Vol. 16, pp. 171 - 186, 1958.
4.7 Masataka M., Yasutorno K, Thom K., Katsuhisa F., Yoshiki K. and Seigo I.,
Roof and Groove Analysis for Steam Turbine Blades, Japan Society of
Mechanical Engineers International Journal, Series 1: Solid Mechanics and
Strength of Materials, Vol. 35, No. 4, pp. 508 - 5 14, 1992.
Chapter 5
Conclusions and Future Work
5.1 Description of the problem
Turbomachines such as turbines, cornpressors and pumps involve the use of fast rotating
disc assemblies. In order to optirnize their geometric features for maximum efficiency
and to ensure their efficiency and reliability, a precise knowledge of their performance
under coupled thermal and mechanical loading is necessary. Conventional methods of
strength determination are no Ionger considered suffxient , when taking into account the
dernand for increasing safety, reliability and high strength requirements of suc h rotating
dix.
Turbine discs subjected to high complex loading contains inherently highly stresseci
components. While many of these components can be contained within the engine casing
during failure, such as blade Ioss, the catastrophic failure of the turbine disc on the other
hand causes the larger fragments of the disc to puncture the engine casing. The
consequences of such a failure are particularly costly resulting in the destruction of the
engine and ultimately in the loss of Me.
The distribution of stresses around the regions of high stress concentration such as the fi-
tree regions of the disc are sources of high concem to the designer. The high stresses
generated due to complex geornetry and under coupled thermomechanical loading
becornes particularly severe when fiettmg at the contact surface is experienced by the
assembly. It has been found that fatigue cracks tend to initiate in regions of high stress
and cumulative fretting damage.
In this thesis, attention was devoted to examining the effect of the criticd geometric
features and interface conditions in the fir-tree region, such as the contact angle, the
upper and lower fiank angle, the number of teeth and the effect of the coefficient of
fkict ion at the bladddisc interface under the coupled thermomec hanical loading . The effect of the skew angle was dso examined for the case of three-dimensionai finite
eIement rnodeiiing. Three aspects of the work were accordingly examined. The first was
concerned with the two dimensional fmite element analysis of the stress field in the
turbine d i x - The second was concerned with the three dimensional finite element
analysis of the turbine disc assembly so as to examine the effect of the skew and nonskew
angle upon the triaxial state of stress field present in the disc. The third was concerned
with the validation of the findamental fmite element mode1 using convergence tests and
photoelasticity.
5.2 Thesis Contribution
The contribution of the current study can be summed up as follows:
(i) Use of non-linear two and three-dimensionai fmite element analysis to evaluate
the coupled thennomechanical stress field in the fïr-tree region of a turbine disc
assembly using contact elements.
(ii) Examination of the eEect of the critical geometrical features and interfixe
conditions in fir-tree joints on the stress field for coupled thermal and centrifuga1
loaduigs using appropriate operating conditions.
(iii) Assessrnent of the two dimensional results in comparison with the three
dimensional fmdings for cases invo lving thermomec hanical loading.
5.3 General Conclusions
A number of conclusions can be deduced fiom the current study. They can be
summarized as fo llo w s:
Effect of Geometry and Interface Conditions
The results reveai that:
(i) the maximum stress concentration occurs at just below the lower contact point
between the blade and the disc for separate centrfigal loading as well as and for
coupled thermomechanical loading,
(ii) an increase in contact angle increases the stresses at the lower contact point,
(üi) a change in the upper flank angle and lower flank angle away from the reference
design case ( a = 20°, f3 = 40°, y = 4û0 ) severely affixts the magnitude of the
stresses. For the case where the upper flank angle is greater than the lower flank
angle by approximately ten degree, the stress magnitude increases signi ficant ly
compared to the reference design stress distribution,
(iv) an increase in the coefficient of fiction increases the stress distriiution at the
blade/disc interface.
5.4 Recommendations for Future Work
This investigation has successfully determined some of the geometric features that could
influence the structurai integrity of an aeroengine fir-tree turbine disc assembly using the
finite element rnethod. This work can be extended to account for:
(i) Stresses due to gas bending.
(ii) Creep and plast kit y under coupled thermomechanical loading.
(iü) Possible blade loss and its influence on the integrity of the assembly.
(iv) Effect of residud stresses introduced by shot peerïing and coatings on the fretting
fatigue behavior of the disc.
(v) Effect of other form of heat flowAoss such as convection and c o o h g of turbine
bIades on the stress distribution for turbine disc assemb lies.
(vi) Three dimensional crack growth predictions and the detennination of design
curves for the fatigue crack growth under mixed mode loading.
Appendix 1
Calcolation of Bore Stresses in a Rotating Disc Using Closed-
Form Solutions
This Appendix gives a detailed account of the procedure for the calculation of tangential
stresses in the bore of a rotaîing disc usuig closed-form solutions for centrifbgal loading.
The purpose of such calculations is to establish the validity of fmite element solutions as
the closed-form theory c m provide an estimate of the expected stresses. The physical
properties and geometric details for the blade and the disc are given in Table (Al) (see
Ref. [4.5]).
Specifically, R is the distance of the blade center of gravity fiom the disc center and A is
the total area of the blade and disc geometry. Both of this was obtained fiom the CAD
drawing developed in patametric f o m in ANSYS 5.4. The thickness for both the blade
and the disc is defmed as unity in ANSYS 5.4 for a two dimensional mode1 as plane
stress condition was assumed. In such case, the thickness is assurned as 1 m
The total centrifugai force (CF) due to the blade at the disc rïm r, is given by:
CF = M m 2 n b (A- 1)
The radiai stress at the rim of the disc due to this force is given by:
For a rotating disc without t h e d loading, the generai expressions for radial and
tangential stresses are given by the equations (2. la) and (2.1 b) given in chapter 2 of this
thesis. The constants A and B are calculated by applying the suitable boundary
conditions:
om = O at r = q (fiee boundary condition at the bore)
O, = CF at r = ro (Centrifuga1 Ioad at disc rim)
Ro~ertY Value
Area, A
Blade center of gravity fiom disc center, R 983.66 mm2 214.61 mm
Disc outer radius, r, 190.50 mm I
Disc inner radius, ri
Ultirnate Tende Strength, 6 ms
31.75 mm
1210 ~/mrn' at 2I0C
Young's Modulus, E 220 GPa
B Iade mass, a
Rotational speed, o
8.37ee3 kg
Density (blade and disc), p
6600 RPM
85 10 kglrn3
Thickness of blade (2D), t Taken as unity
Poisson's ratio, v 0.29
Angular pitch between blades 6°C
Number of bIades, n b 60 L
Coefficient of thermal expansion, a 1 1 .7e4Pc
Thermal Conduct ivity, K
Table (Al): Materiai property values used in evaluation of stresses
In rnatrix form, the above conditions can be written as:
1 13 W/m°C
Gas inlet temperature
The values of A and B becorne 6.2399e6 and -6.12e4. Upon substitution into equation
2. lb, the tangential stress at the bore due to centrifuga1 loading is:
956 O C
For two dimensionai case, the summary of results is given in Table(A2):
O, at disc bore (inner radius ri) Vaiue of stress in&ïiV/rnZ- I Analyt ical method
- - - -- - - - --
Table (A2): Cornparison o f andyticd and fmite element results
122.17
FE method (nodes dong bore) 1 18.7642
Appendix II
Stress Distribution obtained using Photoelasticity
This appendix discusses how the difference between the maximum and minimum
principal stresses (a, -a,) were obtained fiom photoelastic specimen using the
automated photoelasticity method. In ?tu's method eight images: four isochromatic and
four isoclinic images of the same specimen are caphued under various polariscope
arrangements. M e r capturing the images, these are then brought to the photoelasticity
software to determine the whole field isoclinics and fringe order. if the stress dong a tÏee
boundary of the specimen is known, then stress separation technique can be used to
resolve the individual stresses within the specimen. However, as these values are prone to
error, only the tïinge orders were used to calculate the stresses.
Calculation of stresses
The stresses computed from the photoelastic analysis could be counted in tenns of the
number of fringes on the blade tooth flanks, as these represent the isochromatics where
the maximum principle stresses are constant. The benefit of using automated
photoelasticity is simply to aüow automated fiinge determination. Using the stress optic
Iaw, we have:
w here
a,. a, are the principle stresses,
N is the f i g e order,
fa is the material h g e value, and
t is the specimen thickness.
The material h g e value, f, is determîned from the material obtained fiom the supplier
to be 7 KN/fkbgp/m. Assuming that at a distance of 5mm dong the bladeldisc interface,
we have
Number of automated fringe orders counted = N = 2.5
BIade model thickness = t = 3.2 mm
Thus, it was cakulated that:
(a, - O,), = 5.46875 ~ lmm' , and
(O, - a,), = (a, -a2), * (Scale factor)/po2r2
where:
p = actual FE blade model density = 85 10 kg/m3
o = blade speed of rotation = 66Oû rpm = 69 1.15 radis
r = disc bore radius = 3 1.75 mm
Scale façtor = 20.84
In the above calculations, subscripts FE and PE refers to the fmite element and
p ho toelastic models respectively . The scale factor is deterrnined using the similarity
equation:
[y], =[FI PE
w here:
6 is the stress at a given point
P is the applied load = po2a2
T is the thickness
L is the typical length dimension
