Accounting for the Stiffness of Three-Dimensional Features in a TwoDimensional Axisymmetric Rotating Disk Analysis
by
Kurt E. Leach
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, Connecticut
April, 2012
1
© Copyright 2012
by
Kurt Leach
All Rights Reserved
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CONTENTS
Accounting for the Stiffness of Three-Dimensional Features in a Two-Dimensional
Axisymmetric Rotating Disk Analysis ........................................................................ 1
LIST OF TABLES ............................................................................................................. 5
LIST OF FIGURES ........................................................................................................... 7
TABLE OF SYMBOLS .................................................................................................... 8
ACKNOWLEDGMENT ................................................................................................... 9
ABSTRACT .................................................................................................................... 10
1. Introduction................................................................................................................ 11
2. Methodology .............................................................................................................. 12
2.1
Description of Problem .................................................................................... 12
2.2
Test Variables................................................................................................... 14
2.3
Analysis Methodology ..................................................................................... 16
2.3.1
CAD Model Creation ........................................................................... 16
2.3.2
FEA Model Creation ............................................................................ 18
2.3.3
Boundary Conditions ........................................................................... 20
3. Results and Discussion .............................................................................................. 22
3.1
3.2
3.3
Model Validation ............................................................................................. 22
3.1.1
Two-Dimensional Model ..................................................................... 22
3.1.2
Three-Dimensional Model ................................................................... 27
Two Dimensional Interface Load vs. Three Dimensional Interface Load ....... 28
3.2.1
Interface Load Results.......................................................................... 28
3.2.2
General Sensitivities............................................................................. 30
Accounting for Stiffness Difference in Two-Dimensional Axisymmetric
Analysis ............................................................................................................ 33
3.3.1
Plane Strain Elements at Tab ............................................................... 33
3.3.2
Axisymmetric Elements at Tab with Zero Out-of-Plane Modulus ...... 35
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3.3.3
Plane Stress Elements with Decrease Axial Modulus at Tab to Reduce
Bending Stiffness ................................................................................. 38
3.3.4
Plane Stress Elements at Tab with Reduced Axial Modulus in Adjacent
Full Hoop Region ................................................................................. 39
4. Conclusion ................................................................................................................. 43
5. References.................................................................................................................. 44
6. Appendix.................................................................................................................... 46
6.1
Appendix A ...................................................................................................... 46
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LIST OF TABLES
Table 2.1 – Test Matrix for Parameter Evaluation .......................................................... 16
Table 2.2 – Number of Nodes and Elements for Each Three-Dimensional Cyclic
Symmetric Model ............................................................................................................ 20
Table 3.1 – Force Summations for All Cases .................................................................. 24
Table 3.2 – Summary of Contact Results for 25% Material Removed Case .................. 24
Table 3.3 – Summary of Contact Results for 50% Material Removed Case .................. 25
Table 3.4 – Summary of Contact Results for 25% Material Removed Case .................. 26
Table 3.5 – Interface Loads for 25% Material Removed Case ........................................ 29
Table 3.6 – Interface Loads for 50% Material Removed Case ........................................ 29
Table 3.7 – Interface Loads for 75% Material Removed Case ........................................ 29
Table 3.8 – Interface Load as a Function of Mesh Size .................................................. 30
Table 3.9 – Interface Load as a Function of Planar Element Type ................................. 32
Table 3.10 – Interface Load as a Function of Contact Element Type ............................. 32
Table 3.11 -- Interface Loads for 25% Material Removed Case with Plane Strain
Elements .......................................................................................................................... 34
Table 3.12 – Interface Loads for 50% Material Removed Case with Plane Strain
Elements .......................................................................................................................... 35
Table 3.13 – Interface Loads for 75% Material Removed Case with Plane Strain
Elements .......................................................................................................................... 35
Table 3.14 – Interface Loads for 25% Material Removed Case with Axisymmetric
Elements at Tab ............................................................................................................... 37
Table 3.15 – Interface Loads for 50% Material Removed Case with Axisymmetric
Elements at Tab ............................................................................................................... 37
Table 3.16 – Interface Loads for 75% Material Removed Case with Axisymmetric
Elements at Tab ............................................................................................................... 37
Table 3.17 – Interface Loads for 25% Material Removed Case with Reduced Axial
Modulus at the Tab .......................................................................................................... 38
Table 3.18 – Interface Loads for 50% Material Removed Case with Reduced Axial
Modulus at the Tab .......................................................................................................... 39
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Table 3.19 – Interface Loads for 75% Material Removed Case with Reduced Axial
Modulus at the Tab .......................................................................................................... 39
Table 3.20 – Interface Loads for 25% Material Removed Case with Reduced Axial
Modulus in the Adjacent Axisymmetric Region ............................................................. 40
Table 3.21 – Interface Loads for 50% Material Removed Case with Reduced Axial
Modulus in the Adjacent Axisymmetric Region ............................................................. 41
Table 3.22 – Interface Loads for 75% Material Removed Case with Reduced Axial
Modulus in the Adjacent Axisymmetric Region ............................................................. 42
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LIST OF FIGURES
Figure 2.1 – Cross-section of a Generic High Pressure Turbine Rotor Assembly .......... 12
Figure 2.2 – Pictorial view of Two Disks in Contact at Cylindrical Appendages ........... 13
Figure 2.3 – Illustration of Tab and Slot Thickness for the 1st Disk ............................... 15
Figure 2.4 – 2D Dimensional Cross-section showing the General Dimensions of Disks
used for this Study ........................................................................................................... 17
Figure 2.5 – Schematic showing Sector Angle and Tab Thickness ................................ 18
Figure 2.6 – Finite Element Mesh used for the Two-Dimensional Analysis .................. 19
Figure 2.7 – Finite Element Mesh used for the Three-Dimensional Analysis (shown is
the 50% mat’l removed, 16 tab case)............................................................................... 20
Figure 2.8 – Boundary Conditions to be applied to the Models ...................................... 21
Figure 3.1 – Free body diagram for the Disk 1 (2D Model – 25% material removed case)
......................................................................................................................................... 23
Figure 3.2 – Free body diagram for the Disk 2 (2D Model – 25% material removed case)
......................................................................................................................................... 23
Figure 3.3 – Plot of Nodal Reaction Forces for All Three Cases .................................... 26
Figure 3.4 – 2D vs. 3D Radial Deflections for the 25% Material Removed 64 Tab Case
......................................................................................................................................... 27
Figure 3.5 – 2D vs. 3D Radial Deflections for the 25% Material Removed 4 Tab Case 28
Figure 3.6 – Radial Deflections for Different Mesh Sizes .............................................. 31
Figure 3.7 – Screenshot of Finite Element Model showing where Plane Strain Elements
are used ............................................................................................................................ 33
Figure 3.8 – Illustration Showing Elements with Reduced Axial Modulus .................... 40
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TABLE OF SYMBOLS
TPS = Thickness applied to Plane Stress with Thickness Element (in)
N = Total Number of Tabs
T = Thickness of One Tab in Circumferential Direction (in)
R = Net section radius (in)
% = Percent Material Removed to create the Tab
Fcentrif = Centrifugal Load (lbf)
m = mass of disk (lbf-s2/in)
r = radial center of mass (in)
ω = Angular Velocity (rad/s)
Fhoop = Radial Constraining Load due to Circumferential Strains (lbf)
σH = Circumferential Stress (psi)
A = Cross-sectional Area of the Fully Circumferential Section (in2)
Finterface = Force transmitted Between 1st Disk and 2nd Disk due to Contact (lbf)
Fconstraint = Reaction force at Radial Constraint (lbf)
ρnew = New Density Applied to Element (lbf-s2/in4)
ρoriginal = True Material Density (lbf-s2/in4)
%MR = Percent Material Removed
R = Radius to Center of the Element (in)
DRF = Density Reduction Factor
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ACKNOWLEDGMENT
I would like to thank Professor Gutierrez-Miravete for his guidance throughout the
project. Also, I’d like to thank my fiancé Brittagne for supporting me while I earned my
degree.
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ABSTRACT
The analysis of gas turbine engine rotating hardware is extremely important in order to
estimate the rotor’s durability as well as other design criteria. The geometric nature of
gas turbine engines lend themselves to be easily modeled in a two-dimensional
axisymmetric structural finite element model. However, most rotors are not completely
axisymmetric.
Features such as holes, broach slots, tabs and slots can provide
inaccuracies in the axisymmetric model. Here, the effect of tabs on the axisymmetric
analysis was evaluated. A typical approach to modeling tabs in a two-dimensional
axisymmetric model is to use plane stress with thickness element for the tab. The
thickness value used is equivalent to the number of tabs multiplied by the thickness of
one tab. The number of tabs and percent of material removed by the tab (metal-to-air
ratio in the tab region) were evaluated for a generic representative rotor model. The
two-dimensional axisymmetric interface loads were then compared to that of a threedimensional cyclic symmetric model, which is generally considered to be more correct.
Using the plane stress with thickness method, the difference in interface load between
the axisymmetric model and the three-dimensional cyclic symmetric model was larger
for low count tabs and small for high count tabs. The percent difference was shown to
be as high as 30%. A number of different techniques were used to try to more accurately
capture the true stiffness of the slotted interface. The techniques attempted were, using
plane strain elements in the tabbed region, using axisymmetric elements with no out-ofplane stiffness, reducing the axial modulus of the tab to match the three-dimensional
model and reducing the axial modulus of the full hoop material adjacent to the tab.
These techniques were compared for all cases.
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1. Introduction
In the gas turbine industry, both aviation and power generation, the performance and
durability of the engine’s components are of the utmost importance. In aviation gas
turbines, such as commercial turbofan engines, this attention to component durability is
magnified since public safety is in the balance. Therefore, it is essential for any aircraft
engine producer to ensure that safety and component durability are at held in high
regard.
Along with that notion, weight and cost are the biggest drivers toward
competitiveness. One of the biggest weight and cost contributors to a turbofan engine
are the compressor and turbine rotors. The rotors also pose the highest safety in the case
that they fail. With today’s modern advances in finite element analysis, a rotor system
for either a turbine or compressor can be modeled to understand the state of stress in
each rotating component.
However, with today’s computing capability, it is often
impractical to create a full three-dimensional model of a rotor system; therefore,
axisymmetry is taken advantage of for modeling these rotors. However, most rotors
have non-axisymmetric features such as holes, tabs, slots, splines, etc. These features are
necessary to rotor and engine functionality; however it can be challenging to model these
features in a two-dimensional axisymmetric model. This project will examine the ways
that some of the error caused by non-axisymmetric features, specifically tabs, can be
reduced.
This study will address a case where two rotating disks with cylindrical
appendages are contacting each other, where one cylindrical appendage is slotted. Here,
the difference in stiffness of the slotted cylinder in the two-dimensional model will be
evaluated against its true stiffness (determined via a three-dimensional model).
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2. Methodology
2.1 Description of Problem
Obtaining the stresses for a rotating disk is of utmost importance in the gas turbine
engine industry as it is necessary to uphold flight safety. In order to perform a structural
analysis of rotating disks, various modeling techniques are employed. Typically, the
rotor system is modeled in a two-dimensional axisymmetric model, and then local
features are evaluated with three-dimensional submodels.
To perform the two-
dimensional axisymmetric rotor modeling, some assumptions need to be made about the
non-axisymmetric features, such as tabs, holes, broach slots, etc. Figure 2.1 shows a
generic high pressure turbine rotor assembly, and Figure 2.2 shows a three-dimensional
sector of two rotating disks that are contacting at the cylindrical appendages, with only
the 1st disk contact surface being an interrupted or non-axisymmetric contact.
Figure 2.1 – Cross-section of a Generic High Pressure Turbine Rotor Assembly
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Figure 2.2 – Pictorial view of Two Disks in Contact at Cylindrical Appendages
These tabs are necessary for the rotor system’s functionality as the spaces between the
tabs allow air to pass through the cylindrical appendages and provide cooling to gas path
components such as turbine blades.
In two-dimensional rotating disk analyses, three-dimensional out-of-plane features need
to be accounted for in some manner. For this particular problem, the tab in the 1st Disk
flange shown in Figure 2.2 is not an axisymmetric feature and therefore cannot be
captured using axisymmetric elements. A popular way to account for the out-of-plane
features is to use plane stress with thickness elements and then apply the appropriate
thickness amount to account for the mass of the feature.
In the case of tabs, the
thickness that is applied is shown by Equation 2.1.
𝑻𝑷𝑺 = 𝑵 ∗ 𝑻
Equation 2.1
Assuming the same material properties are applied to the plane stress with thickness
elements as the axisymmetric elements, then the mass will be accurately captured;
however, there will be significant differences in stiffness. For the same metal-to-air ratio
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in the tab, the same thickness value will be applied regardless of the number of features.
In other words, the thickness value applied in the model will be the same whether there
are 2 large tabs of 100” thickness with or 100 small tabs of 2” thickness. In each of
these instances the stiffness will be different and therefore should be treated differently
in the two-dimensional axisymmetric model. The true stiffness of the structure will be
found using a three-dimensional cyclic symmetric model. Since the tabs are repetitive
and similar, cyclic symmetry can be taken advantage of in order to model the structure
three-dimensionally. In Mohammed Hassan’s “Vibratory Analysis of Turbomachinery
Blades,”[4] cyclic symmetry is also taken advantage of in order to perform a vibration
analysis of a bladed disk. In Hassan’s work, cyclic symmetry was validated using a full
360˚ model and a cyclic sector model. For this study, a static analysis will be performed
(as opposed to a vibratory analysis) in order to capture deflections and thus is a
simplified application of Hassan’s work. Understanding the difference between the
plane stress with thickness stiffness and the true stiffness will aid to help determine what
corrective action to take to account of this stiffness difference in the two-dimensional
axisymmetric model.
2.2 Test Variables
Gaining an understanding of the variables that contribute to the stiffness
difference between axisymmetric models and the three-dimensional models will help
define the analyses that need to be performed. As mentioned previously, the number of
tabs around the circumference can contribute to this difference as the axisymmetric
model does not have a way of accounting for the number of tabs, just the total thickness
of all the tabs. Also, the percent material removed by the tab will be evaluated. The
percent material removed (or air-to-metal ratio of the tab) is accounted for in the
axisymmetric model on a full ring basis. Figure 2.3 shows a sector of just the 1st Disk in
order to illustrate the tab thickness.
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Figure 2.3 – Illustration of Tab and Slot Thickness for the 1 st Disk
The percent material removed by the tab can be expressed as the ratio of air in the tab
region over the total circumference. Equation 2.2 expresses this ratio in terms of A
(where A is shown in Figure 2.3), the material removed by the tab.
%=
𝑁𝐴
2𝜋𝑅
Equation 2.2
It is advantageous to express the percent material removed in terms of the tab thickness
(defined as T in Figure 2.3). The result is shown in Equation 2.3.
𝑵𝑻
% =𝟏−(
)
𝟐𝝅𝑹
Equation 2.3
Using the percent material removed and the number of tabs as variables for this study, a
test matrix was developed in order evaluate these parameters and is shown by Table 2.1.
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Number of Tabs
Table 2.1 – Test Matrix for Parameter Evaluation
% Material Removed
25%
50%
75%
4
4
4
8
8
8
16
16
16
32
32
32
64
64
64
For each percent material removed, a two-dimensional axisymmetric model must be
created. For a given percent material removed, the axisymmetric model will be the same
regardless of the number of tabs, because the thickness applied to the plane stress with
thickness element only requires that the thickness of all the tabs be accounted for, as
previously discussed in section 2.1, and described by Equation 2.1. However, a threedimensional model will need to be created for all the cases shown by Table 2.1, resulting
in a total of 15 cyclic symmetric models.
2.3 Analysis Methodology
2.3.1
CAD Model Creation
First, three-dimensional and two-dimensional solid models must be created in a
Computer Aided Design (CAD) software. For this project, NX 6 was be used to create
the solid models. To make the model creation easier, the CAD model was built using
parametric modeling. By creating a parametric model, the tab thickness and the number
of tabs can be defined as a variable, and the rest of the model can be built incorporating
the variables. By using the Expression Manager in NX 6, the number of tabs and
percent material removed variables can be changed and the updates will then propagate
through the model. This makes the creation of the CAD model for the 15 cases much
more efficient. A cross-section sketch of the disks used for this study along with some
key dimensions is shown in Figure 2.4.
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Figure 2.4 – 2D Dimensional Cross-section showing the General Dimensions of Disks used for this
Study
For the three-dimensional CAD model, the two-dimensional cross-section served as
the basis to define the three-dimensional solid body. In the axisymmetric regions, the
two-dimensional cross-section was revolved about the engine centerline. The angle of
revolution (or sector angle) was defined as 360˚ divided by number of tabs modeled.
The tabs were created also by revolving the tab region about the engine centerline,
however, the angle of revolution for the tab region was defined as to create the
appropriate tab thickness. Equation 2.3 can be rearranged in order to come up with an
appropriate tab thickness for a given the number of tabs and percent of material removed
by the tab. Figure 2.5 shows an example of a three-dimensional CAD model that was
created, with the sector angle called out.
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Figure 2.5 – Schematic showing Sector Angle and Tab Thickness
Using this methodology, all the CAD models necessary to complete the test matrix
will be built.
2.3.2
FEA Model Creation
After the two-dimensional sheet bodies and the three-dimensional solid bodies were
created the models were imported in to the finite element analysis software. For this
project ANSYS will be used as the finite element analysis (FEA) software. The twodimensional ANSYS models will be built first.
2.3.2.1 Two-Dimensional Model
The two-dimensional models will be analyzed using PLANE42 elements. The
axisymmetric option (Key Option 3 set equal to 1) will be used for the axisymmetric
regions of the part, and the plane stress with thickness option (Key Option 3 set equal to
4) will be used to model the tab. The thickness value applied to the plane stress with
thickness region will be equal to the number of tabs multiplied by the tab thickness as
seen in Equation 2.1. To model the contact between the two disks, CONTAC12 nodeto-node contact elements will be used. An initial contact stiffness of 1E+10 will be used,
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and the penetrations and relative sliding will be checked in order to ensure the proper
contact behavior. CONTAC12 elements have a significant advantage over 2D surfaceto-surface in terms of solution time and solution convergence probability. A mesh size
study was performed in order to make sure that the loads and deflections are converged
as a function of mesh size. The two-dimensional mesh size was then carried over into
the three-dimensional model. Figure 2.6 shows the finite element mesh that will be used
for the two-dimensional analysis. There are 5621 nodes and 5328 elements in this finite
element model.
Figure 2.6 – Finite Element Mesh used for the Two-Dimensional Analysis
2.3.2.2 Three-Dimensional Model
The three-dimensional models will be analyzed using SOLID45 elements. The
axisymmetric portion of the two disks will be sweep meshed such that each of the
symmetry boundaries has a matching mesh. This facilitates the application of cyclic
symmetric boundary conditions.
The cyclic symmetric boundary condition can be
achieved by coupling the nodes on one symmetry boundary with the corresponding
nodes on the opposite symmetry boundary in all degrees-of-freedom (radial, axial and
circumferential).
The contact will be modeled with CONTA174 surface-to-surface
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contact elements with a contact stiffness of 1E+10. Again, the penetration amounts will
be checked in order to ensure that this is an appropriate value. Figure 2.7 shows the
finite element mesh that will be used for the 50% percent material removed, 16 tab case.
Figure 2.7 – Finite Element Mesh used for the Three-Dimensional Analysis (shown is the 50% mat’l
removed, 16 tab case)
Table 2.1 shows the number of nodes and elements for the resulting three-dimensional
models.
Table 2.2 – Number of Nodes and Elements for Each Three-Dimensional Cyclic Symmetric Model
2.3.3
Boundary Conditions
The boundary conditions used for this study need to generate an appropriate
amount of load at the interface. The 1st Disk will be fixed at the inner diameter in the
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axial direction and the 2nd Disk will be fixed at the inner diameter in the radial and axial
directions, as seen in Figure 2.8. A rotational speed (angular velocity) was also applied
to the model, with the axis of rotation about the engine centerline.
Figure 2.8 – Boundary Conditions to be applied to the Models
A typical high pressure turbine can operate anywhere between 10,000 RPM and 20,000
RPM, depending on the engine’s overall configuration. As a first pass, a rotational
speed of 12,000 RPM was applied to the model, and provided about 160,000 lbs of load
across the interface (for the 25% material removed case). Typical interface loads in a
high pressure turbine are on the order of magnitude of 100,000 lbs and therefore, a
rotational velocity of 12,000 RPM will be used to complete this study.
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3. Results and Discussion
3.1 Model Validation
3.1.1
Two-Dimensional Model
3.1.1.1 Free-Body Diagram
In order to validate the model behavior, a free-body diagram was created of both the
components. The main loads acting on the two disk are the centrifugal load due to the
angular rotation (Fcentrif), the radial constraint due to the circumferential strains (Fhoop),
the contact load (Finterface) and the radial constraint (Fconstraint). The centrifugal load due to
the angular rotation was approximated by Equation 3.1.
𝐹𝑐𝑒𝑛𝑡𝑟𝑖𝑓 = 𝑚𝑟𝜔2
Equation 3.1
The radial constraint due to the circumferential strain is approximated by Equation 3.2.
𝐹ℎ𝑜𝑜𝑝 = 2𝜋𝜎𝐻 𝐴
Equation 3.2
The interface load and the reaction at the radial constraint were extracted directly from
the analysis results. Figure 3.1 shows the free-body diagram of the 1st disk with all the
loads for the 25% material removed case.
The first disk shows a residual radial force of 4,000 lbs. This difference can be due to
the approximation of Fcentrif and Fhoop for this system. However, this is only 0.1% of the
highest load on the disk and about 2.5% of the interface load.
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Figure 3.1 – Free body diagram for the Disk 1 (2D Model – 25% material removed case)
Figure 3.2 shows the second disk free body diagram.
Figure 3.2 – Free body diagram for the Disk 2 (2D Model – 25% material removed case)
The 2nd disk shows a residual force for 500 lbs. Again, this difference can be attributed
to the approximation of the centrifugal load and the hoop constraint. Table 3.1 shows
the force summations for all three cases.
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Table 3.1 – Force Summations for All Cases
Overall, based on these force summations, the two-dimensional axisymmetric model is
behaving as expected and all the forces are accounted for within some tolerance.
3.1.1.2 Global Deflections
3.1.1.3 Contact Checks
The contact behavior in the two-dimensional model needs to be validated in order to
ensure that the contact elements are behaving as expected.
Some of the checks
performed include: checking contact load distribution, contact penetration, relative
motion between contact node pairs, and element status. Table 3.2 shows a summary of
the aforementioned results for the 25% material removed case.
Table 3.2 – Summary of Contact Results for 25% Material Removed Case
The table shows that only the last two elements along the surface are in contact, and the
remaining elements are out of contact. This result is verified by the fact that the last two
24
elements are transmitting a load and the other contact elements are not. Also, the
elements that are not in contact show a gap between the node pairs, whereas the
elements in contact show some amount of penetration. This penetration is necessary in
order to gain model convergence, however too large a penetration can lead to incorrect
model results. A typically accepted value for contact penetration is ~0.1 mils, and here
0.01 mils of penetration is shown. Therefore, the contact stiffness chosen (1E+10) is
sufficient enough for this particular application. Also, the relative axial motion between
the contacts was evaluated. The amount of relative motion between the node pairs is
around 0.010” for all of the contact pairs. The global element size in this analysis is
0.050” and therefore, the relative axial motion of the contact node pairs only consumes
about 20% of the element length and therefore, the uses of a node-to-node contact type is
sufficient for this analysis.
For further verification, see section 3.2.2.3 where a
sensitivity to contact element type is performed. Summaries of the contact results for the
50% and 75% percent material removed case are shown in Table 3.6 and Table 3.7
respectively.
Table 3.3 – Summary of Contact Results for 50% Material Removed Case
25
Table 3.4 – Summary of Contact Results for 25% Material Removed Case
For these two cases, appropriate contact behavior is validated in the same manner that
the 25% material removed case was. The elements in contact show a radial force
transmitted and a small amount of penetration, and the elements out of contact show a
gap between node pairs and no transmission of load. The contact penetration amounts
are appropriate for this application and the relative axial motion levels are acceptable.
One difference between the cases is that as the percent material removed increases, the
radial load becomes more distributed along the contacting surface. This is validated also
by looking at nodal reaction plots of the contact surface as seen in Figure 3.3.
Figure 3.3 – Plot of Nodal Reaction Forces for All Three Cases
The nodal force plot validates the contact behavior with respect to the load distribution
along the contacting surface. Overall, the contact elements are behaving correctly and
the contact element type and contact stiffness chosen are appropriate for this application.
26
3.1.2
Three-Dimensional Model
The three-dimensional model was validated by comparison to the two-dimensional
axisymmetric model. The global deflections in the axisymmetric regions away from the
tab should be very similar between the two-dimensional axisymmetric model and the
three-dimensional cyclic symmetric models.
3.1.2.1 Comparison of Three-Dimensional Results to Two-Dimensional Results
The radial deflections were compared between the two-dimensional axisymmetric model
and the three-dimensional cyclic symmetric model for each of the cases. Figure 3.4 and
Figure 3.5 show these comparisons for the 25% material removed – 64 tab case and the
25% material removed – 4 tab case respectively.
Figure 3.4 – 2D vs. 3D Radial Deflections for the 25% Material Removed 64 Tab Case
27
Figure 3.5 – 2D vs. 3D Radial Deflections for the 25% Material Removed 4 Tab Case
The 64 tab case shown in Figure 3.4 shows a good match between radial displacements
for the whole model. However, the 4 tab case shown in Figure 3.5 shows a large
circumferential variation in the three-dimensional model. In regions away from the tab,
the global deflections show a strong match. Appendix A shows this comparison for all
of the models. Overall, the three-dimensional models are behaving as expected.
3.2 Two Dimensional Interface Load vs. Three Dimensional Interface
Load
After running the model at the aforementioned boundary conditions, the results were
post-processed in order to understand the differences in interface load between the twodimensional axisymmetric model and the three-dimensional cyclic symmetry model.
3.2.1
Interface Load Results
Table 3.5, Table 3.6 and Table 3.7 show the differences between the two-dimensional
axisymmetric model and the three-dimensional cyclic symmetric model for the 25%,
50% and 75% material removed for the tab cases respectively.
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Table 3.5 – Interface Loads for 25% Material Removed Case
Table 3.6 – Interface Loads for 50% Material Removed Case
Table 3.7 – Interface Loads for 75% Material Removed Case
Here it is evident that the interface load difference between the two-dimensional
axisymmetric model and the three-dimensional cyclic symmetric model increases as the
number of tabs decrease. This difference was shown to be as high 34.3% for the 75%
material removed – 4 tab case. For the cases with 16 tabs or less, the two-dimensional
interface load at least 8% different than the three-dimensional interface load and
therefore it is clear that the method of using plane stress with thickness elements to
model the tab in the two-dimensional axisymmetric model is not sufficient in cases
29
where there is a small quantity of tabs with a large thickness. As the number of tabs
increase and therefore the tab thickness decreases, the interface load for the threedimensional cyclic symmetric model begins to converge on the two-dimensional
axisymmetric model interface load. This is because the tabs decrease in width and the
tabbed region becomes a plane stress application.
For plane stress theory to be
completely applicable, the thickness must be one-tenth of the smallest in-plane
dimension. For this case, that would mean that the tab thickness would need to be
0.040”. However, for the purposes of only capturing the overall stiffness of the tab (and
not the stresses in the two-dimensional axisymmetric model), the limits may need to
expanded.
3.2.2
General Sensitivities
In order to ensure that the solid element type, mesh size and contact element type used
were appropriate for this particular problem, a number of sensitivity studies were
performed, including mesh size, axisymmetric element type, and contact element type.
3.2.2.1 Mesh Size
The mesh in the two-dimensional axisymmetric model was iterated from a global
element size of 0.050” down to 0.010”. The following table shows the interface load for
a given case as a function of mesh size.
Table 3.8 – Interface Load as a Function of Mesh Size
30
The data shows that the maximum difference between a mesh size of 0.010” and
0.050” is about 0.2%. Therefore a mesh size of 0.050” is sufficient for this analysis.
Figure 3.6 shows the radial deflections for different mesh sizes. The differences in the
global radial deflections between the different mesh size cases are negligible.
Figure 3.6 – Radial Deflections for Different Mesh Sizes
The 0.050” element mesh size was used for this analysis. This allows for accurate
results, meanwhile providing sufficient computational efficiency. This mesh size will
also be used for the three-dimensional analysis, using this study as a basis.
3.2.2.2 Solid Element Type
There are two different four node planar element types available for use in ANSYS, the
PLANE42 and the PLANE182 (both have axisymmetric and plane stress with thickness
options). Here a general sensitivity was performed in order to be sure that these element
types yield consistent results. Table 3.9 shows that there is essentially no difference in
interface load between the two different element types.
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Table 3.9 – Interface Load as a Function of Planar Element Type
3.2.2.3 Contact Element Type
In the two-dimensional model, there are a number of different options for contact
elements. Here, node-to-node contact (CONTAC12) and two-dimensional surface-tosurface contact (CONTA172) were evaluated. Table 3.10 shows the interface load
results for the two contact element types.
Table 3.10 – Interface Load as a Function of Contact Element Type
The node-to-node contact element is not suited well for analysis where large deflections
and large rotations come in to play, however, they are far more robust and
computationally efficient than the surface-to-surface contact elements. Here it is shown
that the load does not change significantly between the contact element types, and
therefore the node-to-node contact elements were used for this study.
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3.3 Accounting for Stiffness Difference in Two-Dimensional
Axisymmetric Analysis
Taking into account everything that was learned with respect to the percent differences
due to the number of tabs and the percent material removed by the tab, the twodimensional axisymmetric model will need to be adjusted in order to account for the true
stiffness of the structure. The following sections show the results of studies that were
performed to more accurately account for the true stiffness in the two-dimensional
axisymmetric model.
3.3.1
Plane Strain Elements at Tab
To better capture the stiffness at lower tab counts, plane strain elements were used for
the tabbed region. Figure 3.7 shows the tabbed region and where the plane strain
elements are used.
Figure 3.7 – Screenshot of Finite Element Model showing where Plane Strain Elements are used
One of the challenges with using these elements is that it does not have a thickness input
for the out-of-plane thickness. Instead, these elements (PLANE42, with Key Option 3
set equal to 2) assume a unit thickness in the out-of-plane direction. Therefore, in order
to accurately account for the mass of the tabbed region, the density of the elements had
33
to be scaled up. For this case, a separate material property ID was applied to each
element and the density was scaled up by
𝜌𝑛𝑒𝑤 = 𝜌𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 ∗ (1 − %𝑀𝑅) ∗ 2𝜋𝑅
Equation 3.3
Using the same boundary conditions as previously discussed, the model was solved and
the results were compared to the three-dimensional interface loads.
Table 3.11 shows the interface load results for the 25% material removed case.
Table 3.11 -- Interface Loads for 25% Material Removed Case with Plane Strain Elements
By using plane strain elements in the tabbed region, the two-dimension approximation
for the true stiffness at low count tab improves. However, for extremely low count tabs
(like the 4 tab case), there is still as high as 21.3% difference between the twodimensional axisymmetric model and the three-dimensional cyclic symmetric model. At
high count features, such as 64 tabs, the out-of-plane tab thickness become smaller; the
plane stress method accounts for high count tabs more accurately. This general trend
carries through to the other percent material removed cases, as seen in Table 3.12 and
Table 3.13.
34
Table 3.12 – Interface Loads for 50% Material Removed Case with Plane Strain Elements
Table 3.13 – Interface Loads for 75% Material Removed Case with Plane Strain Elements
The plane strain method may be more accurate in moderate tab count sizes (between 8
and 16 tabs), but as the number of tabs increases beyond 16, the two-dimensional
axisymmetric model using plane strain method becomes too soft. Conversely, as the
number of tabs decreases below 8, the two-dimensional axisymmetric model using the
plain strain method remains too stiff.
3.3.2
Axisymmetric Elements at Tab with Zero Out-of-Plane Modulus
Another method to account for the tabs in the two-dimensional axisymmetric model is to
use axisymmetric elements for the tab and then turn the out-of-plane modulus to zero.
This method provides a similar challenge as the plane strain case in the fact that the
densities of the elements in the tab region need to be adjusted in order to accurately
account for the mass in the two-dimensional axisymmetric model.
By using
axisymmetric elements (PLANE42, with Key Option 3 set equal to 1), it is assumed that
the mass of each element is given by Equation 3.4.
35
𝑚𝑒𝑙𝑒𝑚𝑒𝑛𝑡 = 𝜌(2𝜋𝑅)𝐴
Equation 3.4
In order to account for the mass of the tab, the density for the elements in the tabbed
region has to be adjusted according to Equation 3.5.
𝜌𝑛𝑒𝑤 = 𝜌𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 ∗ (1 − %𝑀𝑅)
Equation 3.5
Using the same boundary conditions as previously discussed, the model was solved and
the results were compared to the three-dimensional interface loads. Table 3.14 shows
the interface load results for the 25% material removed case.
36
Table 3.14 – Interface Loads for 25% Material Removed Case with Axisymmetric Elements at Tab
Here it is visible that the axisymmetric elements at the tab with zero out-of-plane
stiffness perform about as good as the plane stress with thickness elements at the tab.
Here the same trend is visible, high count tabs show low percent differences between
two-dimensional axisymmetric and three-dimensional cyclic symmetric and low count
tabs give larger percent errors with the axisymmetric model being too stiff. Table 3.15
and Table 3.16 show the results for the 50% and 75% material removed cases
respectively.
Table 3.15 – Interface Loads for 50% Material Removed Case with Axisymmetric Elements at Tab
Table 3.16 – Interface Loads for 75% Material Removed Case with Axisymmetric Elements at Tab
37
Again, the same trend exists here as the 25% material removed case. Low count tabs
produce large percent errors (as high as ~40% in the 75% material removed – 4 tabs
case). Based on this information, it is suggested that this method not be used to account
for the tabs in an axisymmetric model.
3.3.3
Plane Stress Elements with Decrease Axial Modulus at Tab to Reduce
Bending Stiffness
Another method to account for the tabs in an axisymmetric model is to reduce the
modulus of elasticity in the tabbed region by the percent difference between the twodimensional axisymmetric model and the three-dimensional cyclic symmetric model.
By reducing the axial modulus, the bending stiffness is reduced proportionally, therefore
obtaining the correct bending stiffness of the tab. One caveat for using this method is
that the percent difference between the models must be known ahead of time. Under
practical circumstances, the axisymmetric model is created first and used as the global
model, and then the three-dimensional cyclic symmetric model is used as a submodel to
more accurately capture the stresses in the tab region. Therefore, this process alone
would create a feedback loop back to the axisymmetric model. Table 3.17 shows the
results of this test case.
Table 3.17 – Interface Loads for 25% Material Removed Case with Reduced Axial Modulus at the
Tab
This method proved not to provide the results that were expected. The interface load
was largely unaffected by changes to the axial stiffness in the tab region. Table 3.18 and
Table 3.19 so similar results for the 50% and 75% material removed cases, respectively.
38
Table 3.18 – Interface Loads for 50% Material Removed Case with Reduced Axial Modulus at the
Tab
Table 3.19 – Interface Loads for 75% Material Removed Case with Reduced Axial Modulus at the
Tab
The fact that reduction in the axial modulus of elasticity of the tab is does not
significantly affect the stiffness of the overall structure is very telling to the actual
mechanisms that are driving this difference. It is evident that tab itself is the reason
behind the stiffness difference, rather some other influence such as the full hoop material
adjacent to the tab.
3.3.4
Plane Stress Elements at Tab with Reduced Axial Modulus in Adjacent Full
Hoop Region
Based on the learning of the previous study, it was determined that the axisymmetric
elements adjacent to the tab may need to have their material properties altered to
accurately capture the true stiffness of the structure.
Here the axial modulus was
reduced in the region adjacent to the tab up to a length of half the thickness of the tab.
Figure 3.8 shows the elements that had reduced axial modulus applied to them.
39
Figure 3.8 – Illustration Showing Elements with Reduced Axial Modulus
In reality, the axisymmetric material adjacent to the tab will not truly act as
axisymmetric. There will be a patch effect from the tabs until some distance away from
the tabs where the effect dies out as a function of distance away from the tab. Therefore,
the reduction in axial modulus should not be applied as a constant, but rather a function
that dies out as the distance away from the tabs increases. Here, the reduction factor
applied to the axial modulus will be interpolated from no reduction at a distance of half
the thickness away from the tab to a reduction defined by Equation 3.6 at zero distance
away from the tab.
𝐷𝑅𝐹 =
𝑁𝑇
2𝜋𝑅
Equation 3.6
The reduction factor is a geometric relationship that is simply a ratio of the thickness of
all the material occupied by tabs to the material thickness of the axisymmetric section.
Table 3.20 shows the results for the 25% material removed case.
Table 3.20 – Interface Loads for 25% Material Removed Case with Reduced Axial Modulus in the
Adjacent Axisymmetric Region
40
Again, a similar trend surfaces, the higher tabs count show a smaller departure from
the true stiffness, whereas the lower tabs counts have a large difference from the true
stiffness. In this case, the difference between the two-dimensional axisymmetric and the
three-dimensional cyclic symmetric models is still greater than 20% for the 4 tab case.
However, this method proves to accurately account for the true stiffness (within 5%) for
up to 16 tabs, which shows a slightly better correlation than the plane stress method.
Table 3.21 and Table 3.22 show the interface load results for the 50% and 75% material
removed cases, respectively.
Table 3.21 – Interface Loads for 50% Material Removed Case with Reduced Axial Modulus in the
Adjacent Axisymmetric Region
41
Table 3.22 – Interface Loads for 75% Material Removed Case with Reduced Axial Modulus in the
Adjacent Axisymmetric Region
As the percent material removed increases, the percent difference from the true stiffness
begins to increase as well. This is evident by looking at the percent difference for the 16
tabs cases at 25%, 50% and 75% material removed by the tab. The percent differences
are 0%, 4.8%, and 12.9%, respectively. This shows a clear trend that the accuracy of
this method is dependent of not just the number of tabs but the percent material removed
as well.
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4. Conclusion
On the whole, the results of this study show that using plane stress with thickness
elements for the tab is not accurate for all cases. Using the three-dimensional cyclic
symmetric model as the true solution, it was shown that for low count tabs (16 tabs or
fewer), that the interface load in the two-dimensional axisymmetric model could be
incorrect by as anywhere between 8% and 30%. This drove a need to come up with a
method for more accurately accounting for the true stiffness in the axisymmetric model.
Four different studies were performed, and it was found that none of them were the allencapsulating solution. The plane strain method proved to be more accurate at low
count tabs, between 8 and 16 tabs. However, at very low count tabs (4 tabs), the
axisymmetric model was shown to be too stiff by about 20% still. Although a resolution
was not found that can apply to all situations, the concept is shown that the plane stress
with thickness method is not accurate at low count tabs. Therefore analysts must be
conscious of this when creating two-dimensional axisymmetric models and evaluate
whether this difference will have an effect on the end results.
43
5. References
[1]
ANSYS 11.0 Theory Reference, ANSYS Corporation, 2007
[2]
Budynas, Richard Gordon, Joseph Edward Shigley, and J. Keith Nisbett.
Shigley’s Mechanical Engineering Design. 8th ed. New York, NY: McGraw-Hill
Science/Engineering/Math, 2008.
[3]
Davis, Erica. “A Study of a Combined 2D Axisymmetric and 3D Cyclically
Symmetric Finite Element Model of a Turbine Disk” RPI Master's Project
http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Davis-FinalReport.pdf
[4]
Hassan, Mohammed. “Vibratory Analysis of Turbomachinery Blades” RPI
Master's Project http://www.ewp.rpi.edu/hartford/~ernesto/SPR/Hassan-FinalReport.pdf
[5]
Cook, Robert D. Concepts and Applications of Finite Element Analysis. 4th Ed.
John Wiley and Sons, 2002
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Meguid, S.A., P.S. Kanth, and A. Czekanski. "Finite element analysis of fir-tree
region in turbine discs." Finite elements in analysis and design. 35.4 (2000): 305-317.
Web. 1 Feb. 2012.
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Sun, W. “Bolted joint analysis using ANSYS superelements and gap elements”.
4th Int ANSYS Conf Exhib 1989 Part 2, p 6.66-6.75, 1989. Publ by ANSYS
[8]
Jettappa, Richard Rodrigues. “Pseudo-density approach to the solution of a thin
rotating turbine engine disk”. Proceedings of the ASME Turbo Expo, v 6, PARTS A
AND B, p 1087-1095, 2010, ASME Turbo Expo 2010: Power for Land, Sea, and Air, GT
2010. Publ by ASME
44
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attachments with and without crowning”. Journal of Turbomachinery, v 130, n 2, April
2008. Publ by ASME
45
6. Appendix
6.1 Appendix A
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