An Analysis of Stress Concentration in Compound Fillets Produced
During Dimensional Restoration Repair of Jet Engine Components
by
Steven Dallas Townes Jr.
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December 18, 2009
© Copyright 2009
by
Steven Townes
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ....................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction.................................................................................................................. 1
2. Background .................................................................................................................. 4
3. Methodology ................................................................................................................ 7
3.1
3.2
Numerical Analysis ............................................................................................ 7
3.1.1
Two Dimensional Analysis .................................................................... 7
3.1.2
Three Dimensional Analysis .................................................................. 8
3.1.3
Axi-symmetric Analysis......................................................................... 9
3.1.4
Shoulder Fillets ...................................................................................... 9
FEA Analysis ................................................................................................... 12
3.2.1
CAD Modeling ..................................................................................... 12
3.2.2
FEA Analysis ....................................................................................... 14
4. Results and Discussion .............................................................................................. 20
4.1
Numerical Solution .......................................................................................... 20
4.2
FEA Solutions .................................................................................................. 20
4.2.1
Single Fillets ........................................................................................ 21
4.2.2
Undercut Geometries ........................................................................... 24
5. Conclusion ................................................................................................................. 29
6. References.................................................................................................................. 30
iii
LIST OF TABLES
Table 1. A tabular representation of all analyzed geometries, for use in conjunction with
Figure 9. ........................................................................................................................... 14
Table 2. Numerically calculated Kt values for single shoulder fillets. ........................... 20
Table 3. Analyzed geometries with associated max stress and Kt values. ..................... 21
iv
LIST OF FIGURES
Figure 1. A cut away of a turbine engine showing hubs press fit onto the turbine shaft.
The removal of these hubs during engine overhaul can leave distress on the part
requiring dimensional restoration. [7] ............................................................................... 2
Figure 2.
An illustration showing a piece of geometry before, and then after a
dimensional restoration coating has been applied. The machined substrate now contains
two fillets, complicating stress analysis............................................................................. 3
Figure 3. A GE J85 turbine engine cut away so internals are visible. The shaft can be
clearly seen. [8] .................................................................................................................. 4
Figure 4. The hub fits onto the shaft via an interference fit. The chamfer of the hub is
designed to clear the fillet on the shaft. The blue line denotes the hub’s land area. ........ 5
Figure 5. A snap diameter pre-and post pre-machining operations. ................................ 6
Figure 6. A graphical representation of a circular bar with a shoulder fillet. Due to
geometric similarities. This geometry is used to analyze the affects of undercutting a
turbine shaft. [6]............................................................................................................... 10
Figure 7. Stress concentration factors for a stepped tension of a bar of circular cross
section with a shoulder fillet. [6] ..................................................................................... 11
Figure 8. The baseline two dimensional geometry shaft with shoulder fillet. This
geometry when rotated 360 degrees creates a complete shaft with a shoulder fillet. ...... 12
Figure 9. An illustration and chart demonstrating how undercuts were dimensioned at
the shoulder of the shaft. Use in conjunction with Table 1. ........................................... 13
Figure 10. A meshed geometry. The undercut applied to the model is visible to the right
of the mail shoulder fillet. ................................................................................................ 15
Figure 11. An illustration showing how all models were loaded in ANSYS. ................ 15
Figure 12. A .005 inch density free mesh, and subsequent contour stress plot. ............. 17
Figure 13. A .005 inch density free mesh, which has been refined on the fillet lines.
Note the smoother contour plots and higher calculated max stress. ................................ 18
Figure 14. The mesh from figure 13 was refined again on fillet lines and the subsequent
mesh was developed. Note little change in the contour plot, and minimal change in
calculated max stress. ...................................................................................................... 19
Figure 15. Baseline, .030 inch radius fillet. .................................................................... 22
v
Figure 16. 90 degree shoulder. ....................................................................................... 22
Figure 17. .020 inch radius fillet. .................................................................................... 23
Figure 18. .040 inch radius fillet. .................................................................................... 23
Figure 19. Undercut geometry, .010 inch offset. ............................................................ 24
Figure 20. Undercut geometry, .020 inch offset. ............................................................ 25
Figure 21. Undercut geometry, .030 inch offset. ............................................................ 25
Figure 22. Undercut geometry, .040 inch offset. ........................................................... 26
Figure 23. Undercut geometry, .050 inch offset. ............................................................ 26
Figure 24. A chart displaying the relationship between undercut offset and the change
in Kt……………………………………………………………………………………..27
vi
LIST OF SYMBOLS
Kt
Stress factor
Psi/psi
σ
Stress
lb/in2

Shear Stress
lb/in2
Directional components of the applied body force per
lb/in3
PVx, PVy
unit volume
components of force at the surface force per unit area
lb/in2


Strain
Inch/inch
Shear Strain
Inch/inch
P
Force
Pounds
D
Diameter (large shaft diameter)
Inch
d
Diameter (small shaft diameter)
Inch
t
Shoulder height
Inch
r
Fillet radius
Inch
V
Young’s Modulus
lb/in2
Px, Py
vii
ACKNOWLEDGMENT
I would like to thank my friends, family, Pratt & Whitney and the staff of Rensselaer
Polytechnic Institute.
viii
ABSTRACT
Shoulder fillets on turbine shafts are often undercut during the repair process so
that dimensional restoration coatings can be applied. This can often create compound
shoulder fillets, for which Kt can be difficult to calculate. A shaft containing a shoulder
fillet was modeled, and both FEA and numerical analysis used to characterize the effect
of both fillet size, and compound fillet geometry on Kt. As expected, results showed that
increase the radius of a shoulder fillet reduced stress, and the replacement of a fillet with
a 90 degree corner caused a sharp rise in Kt. In regards to compound fillets, it was
observed that placing an undercutting close to the existing fillet can simulate a single
large radius fillet and effectively reduce the observed Kt. As the undercut moves away
from the existing fillet, stress will begin to rise as the fillets act independently.
ix
1. Introduction
The repair of turbine engine components is a growing, competitive, high dollar,
industry. With the high cost of engines and components, there is large incentive for
operators to save money by repairing damaged hardware as opposed to replacing it.
Industry focus on engine overhaul and repair has been even further magnified in recent
times as aircraft operators have been further monetarily constricted with high energy
costs and a declining global economy.
Dimensional restoration is a common repair technique used to restore worn
surfaces on turbine engine components. These repairs are often performed on wear
surfaces where tight interference fits cause parts to wear during assembly and
disassembly. Dimensional restoration repairs are also performed on surfaces where the
parts wear in service from vibration.
The turbine engine shaft is critical component to turbine engine design. The
backbone of the engine, it is upon this shaft which all rotational components are
mounted via snap diameters. (Figure 1) Due to its high expense, the repair of this shaft
is very desirable by customers both military and commercial. The shaft’s varying
diametric geometry provides a variety of shoulder fillets along the shaft’s length. When
dimensional restoration repairs are designed, they must be done so as to not increase any
existing stresses to dangerous levels.
1
Figure 1. A cut away of a turbine engine showing hubs press fit onto the turbine
shaft. The removal of these hubs during engine overhaul can leave distress on the
part requiring dimensional restoration. [7]
To complete a dimensional restoration repair, a distressed surface is first premachined away, leaving a smooth clean surface. If necessary, the surface is prepped
using an appropriate process, often grit blasting, and then a dimensional restoration
coating is applied to restore the dimension. Popular dimensional restoration coatings
include plasma spray, high velocity oxy-fuel (HVOF) coating and plating. If required,
the coating will be finished machined to the appropriate size and surface finish.
While part repair through dimensional restoration is able to correct part dimensions,
it unfortunately has detrimental effects on a part’s robustness and thus service life. The
dimensionally inspected coatings are usually applied with a coating thickness varying
from .001 to .050 inch. These coatings also are very weak and brittle when compared to
the base metals upon which they are sprayed. For these reasons, the coatings are not
given credit for structural integrity, and the pre-machined geometries are analyzed to see
the magnitude by which part strength and life were affected by the repair. The size and
2
depth of the pre machine undercut effects the parts structural integrity, and is determined
by the parts OEM geometry, and the restrictions on the dimensional restoration coatings
thickness.
Figure 2. An illustration showing a piece of geometry before, and then after a
dimensional restoration coating has been applied. The machined substrate now
contains two fillets, complicating stress analysis.
The existence of pre-existing stress concentrating features further complicates
dimensional restoration repair design. The resultant compound fillets from undercuts
made close to existing stress risers can be unpredictable. The purpose of this project is
to study the effects of dimensional restoration undercut size and location when located in
close proximity to a to a pre-existing shoulder fillet. By understanding how stress
concentration changes relative to undercut location, one can better design dimensional
restoration repairs to maximize part life.
3
2. Background
A turbine engine shaft transfers torque created by the engine’s the turbine to the
engine’s compressor. In this process, the turbine shaft incurs rotational shear loads,
bending loads and a thermal gradient.
These considerations must be taken into
consideration not only when designing the shaft, but also when repairing it. The repair
of snap diameters on the shaft must be done so as to not increase stress concentrations to
levels where cracks could initiate leading to shaft failure. This project will study the
effect of undercut size and location on turbine shaft stress, particularly in the form of
tension.
Figure 3. A GE J85 turbine engine cut away so internals are visible. The shaft can
be clearly seen. [8]
It is first important to understand how a snap diameter with a on a turbine shaft is
designed. Figure 4 illustrates the design concept.
Snap diameters fit two circular
features together through the usage of an interference fit. The horizontal surface area
where one piece mates with another is known as the land area. The geometric transition
which goes from the smaller diameter section of the shaft to the larger portion of the
shaft is known as a shoulder fillet. At the end of the land area, the vertical surface which
4
transitions from the fillet is used as a “stop” to axially locate the mating component. The
hub, which mates with the shaft is designed with a chamfer which clears the fillet when
the hub is fully engaged into its axial stop. (Figure 4)
Figure 4. The hub fits onto the shaft via an interference fit. The chamfer of the
hub is designed to clear the fillet on the shaft. The blue line denotes the hub’s land
area.
Snap diameters of this design are very common on turbine shafts. Since there are
multiple compressor and turbine stages, multiple snap diameters are used axially along
the shaft to accompany them and their supporting hardware.
When the engine is
disassembled for overhaul, these hubs and other accessories are pulled off and can leave
scoring behind. If the scoring from the hub removal is beyond a prescribed depth,
known as a serviceable limit, the diameter must be repaired. Deep scoring in the turbine
shaft is detrimental to part life as scoring marks can serve as crack initiation points.
A turbine shaft with containing snap diameters is essentially a stepped shaft and this
geometry is relatively well understood in the field of structures.
It is understood that
the fillets on a stepped shaft is a geometry which leads to stress concentrations. Hand
calculations exist which can calculate this stress.
When the shaft is pre-machined for dimensional restoration repair, portion of the
part which is damaged is undercut to remove the damage. This undercut is made close
to the existing fillet so that the land area can be restored in its entirety. (Figure 5)
Sometimes the undercut may be created inside of the existing fillet. Since the coating is
5
given no credit for structural integrity, the size and shape of this undercut will determine
the resultant stress concentration.
Figure 5. A snap diameter pre-and post pre-machining operations.
While calculations have been formulated for determining the stress in fillets, what
can not be easily calculated by hand is the resultant stress from compound fillets. An
undercut created in close proximity to an existing fillet may create a compound Kt
weakening the part. In other cases, notch strengthening may occur.
While in service the shaft experiences many different types of loading and stress,
this project will focus on how the shaft responds to tensile stress both before and after
repair. A stepped shaft will be modeled, and undercuts will be applied at various axial
locations along the shaft. Not only will location be a variable, but size of the fillet as
well. A tensile load will then be applied, and the Kt calculated using FEA software.
These results will be compared to one another to gain an understanding of the true effect
of undercutting a turbine shaft on the stress experienced.
6
3. Methodology
The amount of stress that a part sees is often the limiting design factor for a
part’s life in its application. In elementary terms, stress is defined as a force divided by
an area. It is often assumed that the stress that a component sees is evenly distributed
through the object’s cross-section.
However, irregular geometry features such as
grooves and shoulders lead to stress concentrations. Stress concentrations area localized
areas of high stress within a part.
These stress concentrations must be properly
understood in order properly understand how different geometries will react under load.
The stress concentration factor, K, is used to quantify stress concentrations. Kt is
the common denomination used to identify a stress factor caused by geometry and can
be formulated from theoretical calculations.
Kt 
 max
 nom
As seen, Kt is a ratio of the max stress at a feature, in relation to the nominal stress
that a part sees. Kt values can be verified through elastic theory, finite element analysis
(FEA) analysis or through strain gage tests. For this project, a turbine shaft was modeled,
undercut and analyzed using a combination of CAD software and FEA software and the
results compared and contrasted.
3.1 Numerical Analysis
3.1.1
Two Dimensional Analysis
In the case of a two dimensional analysis, a scenario of plane stress scenario is
analyzed. In plane stress, stress is only created in the x and y directions. Because of
this, it is assumed that strains in the z direction do not exist. The differential equations
of equilibrium and compatibility equation for stress in a plane strain scenario are as
follows:
7
 x  xy

 p Vx  0
x
y
 xy  y

 p Vy  0
x
y
 2

 2  2
y
 x
p
p

 x   y    f v  Vx  Vy
 x
y






1  v For plane stress

f (v )   1
1  v For plane strain
PVx, and PVx, denote the directional components of the applied body force per unit
volume in the respective directions.
At the surface the conditions are as follows:
p x  l x  m xy
p y  l xy  m y
In these equations px and py are components of force at the surface force per unit area. L
and m are directional cosines of the normal vector in relation to the boundary.
3.1.2
Three Dimensional Analysis
Three dimensional scenarios are more complex to analyze than two dimension
problems. Due to the fact that stresses are being created in all three dimensions, based
upon the Poisson's ratio that a material has, material properties may now indeed affect
the final stress distribution. When Poisson’s ratio does affect the stress distribution, it
should be provided with the equations used to estimate the said stress distribution.
Generally speaking, stress concentrations in a three dimensional model will be
affected by Poisson’s ratio when the forces applied could have the affect of changing
altering the cross sectional area. For example, a cylindrical shaft in tension’s stress
distribution could be affected by the Poisson’s ratio because tension will cause the center
of the shaft to shrink in area. The same shaft’s stress distribution in torsion however,
would not be affected by the Poisson’s ratio.
8
For a revolved solid, cylindrical coordinates (r, θ, x) can be used to analyze the
stress distribution. The equilibrium equations need to do this are shown below.
 r  rx  r   


 p Vr  0
r
x
r
 rx  rx  rx


 p Vx  0
r
x
r
 2  r  2  x  2  rx


rx
x 2
r 2
The strain elements are defined by the following equations where u is displacement in
the radial direction, and w is displacement in the axial direction.
u
r
u
 
r
w
x 
x
u w
 rx 

x r
r 
3.1.3
Axi-symmetric Analysis
The stress distribution difference between an axi-symmetric three dimensional
solid and that of a two dimensional plane element are very close. While the stresses will
not be equivalent, the shape of the distribution and the associated stress concentration
factors will be very close to one another. This is very convenient for analysis as often it
takes less effort to analyze a two dimensional geometry than it does a three dimensional
object.
3.1.4
Shoulder Fillets
This project deals analyzes a shoulder fillet on a turbine shaft. Shoulder fillets are a
very common design feature as they alleviate stress when compared to a sharp corner.
9
Stress distributions for shoulder fillets are fairly well understood. The turbine shaft
geometry can be accurately modeled using previously derived equations for the stress
distribution in a stepped shaft with a shoulder fillet.
Figure 6. A graphical representation of a circular bar with a shoulder fillet. Due to
geometric similarities. This geometry is used to analyze the affects of undercutting
a turbine shaft. [6]
For a shoulder fillet on a cylindrical bar, the following equations holds true.
Kt 
 max
 min
 nom 
4P
d 2
Through photoelastic testing, the following chart and corresponding equations
shown in Figure 7 were derived to help determine the Kt of a shoulder fillet on a
cylindrical bar in tension. The bar was analyzed in tension. Figure 7 was used to help
establish base line Kt values which were compared to those calculated using FEA
software.
10
Figure 7. Stress concentration factors for a stepped tension of a bar of circular
cross section with a shoulder fillet. [6]
11
3.2 FEA Analysis
Analytical results for Kt were compared to those calculated using FEA software. In
order to do this, geometries had to be modeled, using a CAD program, and then
imported into a FEA program. Appropriate loading could then be applied to the model
and the results recorded and analyzed.
3.2.1
CAD Modeling
A stepped shaft containing a shoulder fillet, representing a turbine shaft snap
diameter was drawn using Unigraphics CAD software. While a three dimensional model
could be used to analyze stress for this project, a two dimensional model cross section of
a shaft was used instead as it could provide equivalent results with a lower computing
power requirement.
This would allow for a quicker, less computer intense Finite
Element Analysis (FEA), while still yielding accurate results. All two dimensional
models were exported from Unigraphics in .STL format.
Figure 8. The baseline two dimensional geometry shaft with shoulder fillet. This
geometry when rotated 360 degrees creates a complete shaft with a shoulder fillet.
The baseline geometry shaft is shown in Figure 8. This is geometry when revolved
creates a solid shaft, with a radius of one inch on the larger diameter, stepping down to a
half inch radius with a .030 inch shoulder fillet. Eight other geometries were created to
compare to the baseline. Of these eight geometries, three were variations of the singular
fillet, exploring how the fillet size and shape affected the Kt. These geometries included
12
a right angle, a larger than baseline radius (.040 inch), and a smaller than baseline (.020
inch) radius.
Five of these geometries explored undercut location. Undercuts were all .010 inch
in depth, and extended completely from the undercut’s radius to the end of the shaft. All
undercuts were made with the same size radius onto the baseline geometry, but the
distance from the shoulder was varied in .010 inch increments.
An outline of all
analyzed geometries can be found in Table 1.
Figure 9.
An illustration and chart demonstrating how undercuts were
dimensioned at the shoulder of the shaft. Use in conjunction with Table 1.
13
Analyzed Geometries
Shoulder Radius
Undercut?
A
(Yes or No)
(Distance from shoulder)
.030 inch (baseline)
N
n/a
.020 inch
N
n/a
.040 inch
N
n/a
0 inch (90°)
N
n/a
.030 inch
Y
.010 inch
.030 inch
Y
.020 inch
.030 inch
Y
.030 inch
.030 inch
Y
.040 inch
.030 inch
Y
.050 inch
Table 1. A tabular representation of all analyzed geometries, for use in conjunction
with Figure 9.
3.2.2
FEA Analysis
3.2.2.1 FEA Technique
All of the analyzed geometries, after being modeled in Unigraphics were exported
in the .STL format and brought into ANSYS FEA software, version 11. The models
were all free-meshed using “8node82” quadratic elements, with a mesh density of .005
inch. An example of this mesh can be seen in Figure 10. Though material type is not
important for determining the stress distribution in this analysis, ANSYS requires that a
material type be inputted in order to run the analysis, so it was assumed that the material
was Nickel with a .310 Poisson’s ratio. Nickel was chosen because it is a common
material for the construction of turbine shafts.
14
Figure 10. A meshed geometry. The undercut applied to the model is visible to the
right of the mail shoulder fillet.
All models were loaded in the exact same matter. A tension load of 100 pounds was
applied to both ends of the shaft for a combined load of 200 pounds of tension. (Figure
11) After the model was run, the fillet was selected and a contour plot of the object’s
first principal stress plotted over top of the geometry.
This plot gave a visual
representation of the stress distribution within the geometry. Specific nodes around the
fillet were then selected and magnified so that the stress was easily visible. The location
and value of the of the max stress value were recorded and compared and contrasted
amongst analyzed geometries.
Figure 11. An illustration showing how all models were loaded in ANSYS.
ANSYS supplies a maximum stress value and with each contour plot. It was this
maximum stress value which was used to calculate the change in Kt. To determine the
difference in Kt between any undercut geometry and the base shoulder fillet, a ratio was
taken of both geometry’s max stresses values.
15
3.2.2.2 FEA Mesh Analysis
To ensure the mesh used was correctly sized to accurately calculate the Kt and
display accurate contour lines, a mesh study was performed. The proper mesh density
was determined by first meshing a selected geometry using a .005 inch density, free
mesh. This mesh size was considered more than ample at calculating stress through the
simple areas of the geometry. It was not accepted though that this mesh size was
capable of accurately determining stress in the small fillets modeled in this study.
The geometry was then loaded as shown in section 3.2.2.1, and first principal stress
solution plotted using a contour plot overtop the geometry. The stress contours and max
stress values were observed. After the solution was recorded, the geometry was remeshed with a higher fidelity mesh, re-loaded and the solution once again processed, and
results compared until a solution was repeated.
To produce a higher fidelity mesh, the area was re-meshed by lines using the
ANSYS “refine mesh” feature. This feature automatically refined the mesh in the fillet,
allowing areas of geometry distant from, the area of interest to remain coarse. The mesh
was refined a total of 3 times before a consistent solution was plotted. This method
gives higher fidelity results where needed, while helping to reduce computer processing
time. The results of this study can be seen in figures 12 through 14.
As a result of this study, it was shown that only one iteration of mesh refinement
was needed to accurately calculate the Kt in the shoulder fillet. This was partly due to
the fact that the baseline .005 inch mesh density which was already relatively fine
compared to the dimensions of the fillets investigated, which ranged from .020 to .040
inch.
16
Figure 12. A .005 inch density free mesh, and subsequent contour stress plot.
17
Figure 13. A .005 inch density free mesh, which has been refined on the fillet lines.
Note the smoother contour plots and higher calculated max stress.
18
Figure 14. The mesh from figure 13 was refined again on fillet lines and the
subsequent mesh was developed. Note little change in the contour plot, and
minimal change in calculated max stress.
19
4. Results and Discussion
Kt solutions from both numeric and FEA software were determined and compared.
The objective of analyzing results was to gain a complete understanding of the effects of
undercut location on Kt.
4.1 Numerical Solution
Numerical solutions are only capable of accurately predicting the solution for
singular shoulder fillets. Thus, numerical solutions were used to understand the effect
that fillet size had on Kt without an undercut existing. Kt was solved directly using the
dimensions of the shaft. It is noted that Kt values derived from this methodology are
predictions, and shouldn’t be taken as concrete answers.
The analyzed baseline
geometry is shown in Figure 9. Calculations were performed as illustrated in Figure 7
and the result can be seen in Table 2.
Fillet Radius
Kt Equation
.030 inch
1.6196
Table 2. Numerically calculated Kt values for single shoulder fillets.
4.2
FEA Solutions
After each geometry was loaded into ANSYS and forces applied, contour plots of
the first principal stress were plotted on top of each geometry. This gave a visual
representation of the stress distribution within the cross section. ANSYS also identifies
the max stress that a geometry experiences and the location where this maximum stress
value takes place.
ANSYS was not used to solve for Kt, but to analyze the change in Kt. The
baseline Kt was taken from the numerical solution determined in the previous section.
The baseline geometry was created and modified with various undercuts, and the change
in max stress values at the fillet used to determine the change in Kt. Since maximum
stress and Kt are directly proportional, the percentages of change in maximum stress
directly translated determine the change in Kt over the baseline geometry. Table 2
tabulates the change in Kt and the max stress for each geometry analyzed.
20
Shoulder
.030 inch
A
Radius
Undercut?
(Distance
from shoulder)
(Yes or No)
Figure 15
.030 inch
Kt
Change
in Kt
N
n/a
1.62
n/a
(baseline)
Figure 16
0 inch (90°)
N
n/a
2.2
+36.1%
Figure 17
.020 inch
N
n/a
1.88
+16.5%
Figure 18
.040 inch
N
n/a
1.45
-10.7%
Figure 19
.030 inch
Y
.010 inch
1.62
0%
Figure 20
.030 inch
Y
.020 inch
1.57
-3%
Figure 21
.030 inch
Y
.030 inch
1.458
-10%
Figure 22
.030 inch
Y
.040 inch
1.411
-12.7%
Figure 23
.030 inch
Y
.050 inch
1.48
-8%
Table 3. Analyzed geometries with associated max stress and Kt values.
4.2.1
Single Fillets
The base geometry when loaded in tension showed a maximum stress
concentration located at the fillet, as was expected.
The contour plots for these
geometries are shown in figures 15 through 18. Through observation of the contour
plots, it can be see how stress is concentrated around the base of the fillet and then tapers
off in intensity axially as distance from the fillet increases. It is also notable that stress
quickly falls to a minimum moving when moving in the radial direction, along vertical
section portion the shoulder fillet. The minimum stress value was always found in this
portion of the fillet.
By manipulating the size of the fillet, the Kt was able to be manipulated as expected.
Decreasing the baseline .030 inch radius fillet to a .020 inch radius fillet increased the K t
by 16.5% while increasing it by .010 inch decreased the Kt by 10.7%. The purpose of
shoulder fillets in machine design is to reduce stress risers and this was evident as a 73%
increase in Kt value was observed when the fillet was replaced with a 90 degree corner.
21
Figure 15. Baseline, .030 inch radius fillet.
Figure 16. 90 degree shoulder.
22
Figure 17. .020 inch radius fillet.
Figure 18. .040 inch radius fillet.
23
4.2.2
Undercut Geometries
The base geometry was undercut using a .030 inch radius fillet. When loaded in
tension, these geometries showed a maximum stress concentration located at the fillet
region. The contour plots for these geometries are shown in figures 19 through 23.
Through observation of the contour plots, it can be see how stress is concentrated around
the base of the fillet and then tapers off in intensity axially as distance from the fillet
increases. It can be noted how when the undercut fillet moves further away from the
original fillet in offset distance, two distinct areas of stress concentration form. The
stress contours created by these fillets are similar in size and shape, which is to be
expected as the fillets are the same size.
Figure 19. Undercut geometry, .010 inch offset.
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Figure 20. Undercut geometry, .020 inch offset.
Figure 21. Undercut geometry, .030 inch offset.
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Figure 22. Undercut geometry, .040 inch offset.
Figure 23. Undercut geometry, .050 inch offset.
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The undercut geometries all contained a .030 inch radius undercut performed on the
baseline geometry, resulting in a compound fillet. FEA software was used to analyze the
resultant Kt as there is not an accurate way to calculate this value numerically. The
original single fillet was also varied to explore how its dimension would affect Kt. A
summary of this behavior can be seen in Figure 24.
Change in Kt vs. Undercut Offset
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
-2
Change in Kt (percentage)
-4
-6
Series1
-8
-10
-12
-14
Undercut Offset
Figure 24. A chart displaying the relationship between undercut offset and the
change in Kt.
When compound fillets were analyzed, the results were not as easily predictable. In
all of the tested offset geometries, the Kt never increased over the OEM geometry. This
is because, when the fillets are close to one another the resultant geometry essentially
simulates one large fillet. This was very evident in the results varying from .010-.020
inch offset. This effectively larger large shoulder fillet reduced the Kt by up to 10%.
Initially, maximum Kt’s were all physically on the undercut fillet’s radius as
opposed to the baseline geometry’s radius. As the undercut fillet was moved further
away from the original fillet, it was observed that two distinct areas of high stress
concentration were formed. This phenomenon first becomes apparent at the .020 inch
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offset undercut. (Figure 20) The .030 inch offset has the largest reduction in K t of all
undercut geometries. It was on this geometry as well that the stresses in both fillets begin
to approach one another in both size and amplitude. After the offset moves past .030
inch, the Kt begins to increase once again as the geometry transitioned from acting like
one large diameter into two distinct fillets. At offsets greater than .030 inch, the location
of the Kt also shifts from the undercut fillet, back to the original fillet.
While testing did not go out beyond a .050 inch offset, there is no reason to believe
that the Kt would have ever increased to an amount greater than the OEM. As the fillets
increase in distance from one another, the Kt from each fillet begins to act
independently. If the original and undercut fillet were to be analyzed independently, it
would be expected to see a very slight increase in Kt on the undercut fillet, because the
undercut shaft has a slightly smaller cross sectional area. With a constant load being
applied through the shaft, the section with the smallest cross sectional area would be
expected to have the greater stress, and thus register a higher local Kt.
By observing the size and shape of the contour plots, it can be observed that the
stress concentrations simulated in this project were are small enough to not combine
with one another and cause any synergistic increase in Kt. This however is all relative to
the size of the undercut fillet. As shown with the analysis of the singular fillets,
reduction in fillet size can easily increase the Kt. If the undercut fillet is reduced in
radius, stresses from the undercut may combine with those from the original fillet and
have detrimental effects. To be conservative with undercut design, fillets for undercuts
should be kept as large as possible.
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5. Conclusion
Insufficient numerical methods exist for accurately estimating a compound shoulder
fillet Kt. This project explored the effect of the size and location of compound shoulder
fillets which are created when turbine shafts are repaired by dimensional restoration.
Understanding of the effect of compound shoulder fillets on Kt is useful to engineers
who design these undercuts for dimensional restoration repairs.
As expected, it was shown that manipulating the size of a fillet can affect the value
of Kt. Smaller fillets resulted in higher Kt than larger fillets. Absence of a fillet,
demonstrated with a 90 degree corner was shown to have the highest Kt observed. It
was then shown that undercutting an existing shoulder fillet close to the OEM fillet can
effectively lower the fillet Kt based upon the fact that the undercut geometry effectively
mimics a large single fillet radius. As the undercut moves further away from the original
fillet, the Kt will begin increase back to its original value as the stress values near the
fillets begin to act independently.
In order to keep Kt to a minimum, thus increasing part life, a few procedures could
be followed when designing dimensional restoration undercuts. The first of which
would be to use as large of an undercut radius as possible. This reduces the Kt of the
undercut, as well as helps to prevent synergistic effects from the stress contours of both
fillets interacting. To keep the Kt of the undercut geometry predictable, undercut fillets
should be no smaller than the OEM fillet size. The second precaution would deal with
the placement of the undercut. If possible, the undercut should be cut very close to the
original fillet, essentially increasing its effective radius, and reducing Kt.
In practice undercut location will have both geometric and practical limitations. In
some circumstances, it may be impossible to undercut a land area with a cutter large
enough to not effectively reduce Kt. While this project cannot be used as an estimator to
predict the final stresses which will be seen in a real world application with different
dimensions, Kt behavior on any undercut shoulder fillet should be similar. Larger fillets
will effectively reduced Kt over their smaller counterparts, and undercuts performed
close to existing fillets will simulate these larger fillets, effectively reducing stress.
Independent Kt calculations should always be performed however to ensure part strength
and life will be met.
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6. References
[1] Pellicano, Francesco, 2009. “Dynamic stability and sensitivity to geometric
imperfections of strongly compressed circular cylindrical shells under dynamic axial
loads.” Common Nonlinear Sci Numer Simulat 14. 3449-3462.
[2] Catelliani, G., Pellicano, F., Dall’Asta, D. Amabili, M. 2004. “Parametric Instability
of A Circular Cylindrical Shell With Geometric Imperfections.” Computers and
Structures 82. 2635-2645.
[3] Kardomateas, G.A., 1995. “Benchmark Three Dimensional Elasticity Solutions For
the Buckling of Thick Orthotropic Cylindrical Shells.” Composites Part B 27B. 569-580.
Guedes, R.M. 2009. “Stress-Strain Analysis of a Cylindrical Pipe Subjected to a
Transverse Load and Large Deflections.” Composite Structures 88. 188-194.
[4] Kumar, J., Srivathsa, B., Kumar, Vikas. 2009. “Stress triaxially Effect on Fracture
Behavior of IMI-834 Titanium Alloy: A Micromechanics Approach.” Materials and
Design 30. 1118-1123.
[5] Whalley, R., Abdul-Ameer, A. 2009 “Countoured Shaft and Rotor Dynamics”
Mechanism and Machine Theory 44. 772-783.
[6] Pilkey, Walter D. Peterson’s Stress Concentration Factors. Hoboken: John Wiley
& Son’s Inc, 2008.
[7]
"Pratt
&
Whitney Media
Center".
Pratt
&
Whitney.
Dec.
9,
2009
<http://www.pw.utc.com/Media+Center/Image+Library/Products>.
[8] "Machine Engineering, Electro Engineering, and Informatics Engineering".
Politeknik Cilacap. Dec. 9, 2009 <http://poltecilacap.blogspot.com/2009/11/earlydesigns-of-turbojet-engines.html>.
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