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Stress Concentration Factor Convergence Study of a Flat Plate with an
Elliptical Hole Under Elastic Loading Conditions
by
Dwight Snowberger
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, ConnecticutDecember 2008
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CONTENTS
LIST OF TABLES............................................................................................................. 3
LIST OF FIGURES ........................................................................................................... 4
ABSTRACT ...................................................................................................................... 6
1. Introduction.................................................................................................................. 7
2. Objectives .................................................................................................................... 8
3. Methodology................................................................................................................ 9
3.1 Schematics.....9
3.2 Stress Concentration Factor Equations for an Elliptical Hole .10
3.3 Boundary Conditions...11
3.4 Elements..113.5 FEA Model..12
4. Results........................................................................................................................ 14
5. Discussion.................................................................................................................. 18
6. Conclusion ................................................................................................................. 21
7. References.................................................................................................................. 22
8. Appendix A................................................................................................................ 23
9. Appendix B................................................................................................................ 26
10. Appendix C................................................................................................................ 32
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LIST OF TABLES
Table 1 Stress Concentration Factors for Various Ellipse radii
Table 2a Plane42 Element Type FEA Model Results/Data
Table 2b Plane82 Element Type FEA Model Results/Data
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LIST OF FIGURES
Figure 1. Flat Plate with an Elliptical Hole
Figure 2. FEA model Boundary Conditions
Figure 3. 4-noded Quad Element, (Reference 2)
Figure 4. 8-noded Quad Element, (Reference 2)
Figure 5. FEA Model Size Control Labels
Figure 6 Magnified view of the stress distribution near the tip of the ellipse. (a/b = 1.25,
b = 0.8, Plane 42 element type shown)
Figure 7 Ellipse Short Radius, b vs Length of Element at the Right of Ellipse needed to
obtain an accuracy of +/- 1% from the calculated stress concentration factor
Figure 8 Ellipse Short Radius, b vs. # of Elements to the Right of the Ellipse needed to
obtain an accuracy of +/- 1% from the calculated stress concentration factor
Figure 9 Plane 42 FEA model results with a short radius of b = 0.1
Figure 10 Plane 42 FEA model results with a short radius of b = 0.2
Figure 11 Plane 42 FEA model results with a short radius of b = 0.3
Figure 12 Plane 42 FEA model results with a short radius of b = 0.4
Figure 13 Plane 42 FEA model results with a short radius of b = 0.5
Figure 14 Plane 42 FEA model results with a short radius of b = 0.6
Figure 15 Plane 42 FEA model results with a short radius of b = 0.7
Figure 16 Plane 42 FEA model results with a short radius of b = 0.8
Figure 17 Plane 42 FEA model results with a short radius of b = 0.9
Figure 18 Plane 42 FEA model results with a short radius of b = 1.0
Figure 19 Plane 82 FEA model results with a short radius of b = 0.1
Figure 20 Plane 82 FEA model results with a short radius of b = 0.2
Figure 21 Plane 82 FEA model results with a short radius of b = 0.3
Figure 22 Plane 82 FEA model results with a short radius of b = 0.4
Figure 23 Plane 82 FEA model results with a short radius of b = 0.5
Figure 24 Plane 82 FEA model results with a short radius of b = 0.6
Figure 25 Plane 82 FEA model results with a short radius of b = 0.7
Figure 26 Plane 82 FEA model results with a short radius of b = 0.8
Figure 27 Plane 82 FEA model results with a short radius of b = 0.9
Figure 28 Plane 82 FEA model results with a short radius of b = 1.0
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ACKNOWLEDGMENT
I would like to thank my wife Amanda, and my son Steven for all of their support and
sacrifice throughout my graduate education experience.
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ABSTRACT
A flat plate with an elliptical hole made of steel (E=30e6psi) with dimensions of 10x20
inches with an elliptical hole of 1 for its long radius and a short radius that varied in
length from 1 to 0.1 in 0.1 increments, was loaded with a 1000psi pressure load in a
quartered FEA model created in ANSYS. A comparison was made between the equa-
tion for a flat plate with a circular hole and an elliptical hole equation with the ellipse
being a circle from Reference 1. The results were found to be the same, with a stress
concentration of 2.54 using the elliptical hole equation and 2.50 for the circular hole
equation. This showed that just the elliptical hole equation could be used to calculate the
stress concentration for all cases including when the ellipse was a circle.
Two different finite elements (Plane42 and Plane82 from the ANSYS element li-brary) were used on the model. These elements were used to show that increasing the
order of the element is one way to improve an FEA model. This was shown by measur-
ing the length of the element at the tip of the ellipse. The Plane42 (4-noded quad)
element had the shorter element length with 0.0095 while the Plane82 (8-noded quad)
had a length of 0.0147. Since the Plane82 element had a longer length than the Plane42
on the same model it meant that the Plane82 element was better at determining the stress
concentration factor that is within +/- 1% of the stress concentration as calculated from
the closed form solution.
Finally, it was also possible to produce a more accurate FEA model by increasing
the number of elements in a mesh. To show this, the number of elements used to mesh
the model were recorded and compared for each ellipse size. The Plane42 model ranged
from using 7 elements at the ellipse tip for the case of the ellipse being a circle to
needing 46 elements and an element scaling factor of 32 when the ellipse had a short
radius of 0.1. The Plane 82 model needed 4 elements with no scaling factor for thecircular hole case to using 38 elements with a scaling factor of 23 when the short radius
was 0.1. All of the values for the number of elements and element scaling factors were
recorded for each model only when the model produced a stress concentration that was
within +/- 1% of the stress concentration as calculated from the closed form solution.
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1. Introduction
Changes in geometry such as a circular hole or an elliptical hole cause increases in the
amount of stress created at these discontinuities. This stress increase is more commonly
known as the stress concentration factor. This factor is a ratio between the maximum
stress produced at the discontinuity divided by the nominal stress far away from the hole.
These factors have been well-studied and documented, with closed form solutions for
more common geometries available in such texts as Reference 1. For an elliptical hole
in a flat plate, the stress concentration will be different depending on the narrowness of
the ellipse.
FEA programs such as ANSYS can be used to approximate the stress concentration
factor as calculated using a closed form equation for a given geometry. The accuracy of
the model can be increased by two ways. One is by increasing the mesh density around
the discontinuity in order to better capture the increase in stress. The other method is to
increase the order of the element, such as using an 8-noded quad element vs. a 4-noded
quad element.
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2. Objectives
The objective of this project will be to study the effect an elliptical hole has on the stress
distribution of a flat plate as the sharpness of the ellipse increases from a circle to a
narrow crack. It will be shown that as the ellipse sharpness increases, more elements
will be needed to accurately capture the stress concentration factor to be within 1% of
the calculated value from Reference (1) for the specific geometry of a flat plate that is
10x20 inches and a 2-inch long diameter elliptical hole. Two element types will be
compared, which are the 4-noded quad (ANSYS Plane42) and the 8-noded quad
(ANSYS Plane82). Finally, the equations for a flat plate with a circular hole and an
elliptical hole will be compared for the case when the elliptical hole is a circle in order to
show that the results are similar enough that the elliptical equations can be used for the
case when the ellipse becomes a circle.
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3. Methodology
3.1 Schematics
This report will focus on the specific geometry of an elliptical hole in a flat plate in
Figure 1.
b = short radiusa = long radius
D = width of the flat plate
Figure 1 Flat Plate with an Elliptical Hole.
ba
1000psi
2
D = 10
20
1000psi
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The plates material will be ordinary steel with a Youngs Modulus of 30e6 psi and the
model will use a pressure load of 1000psi (nominal stress of the model). One important
point to note, this report is a study of the stress concentration factor under elastic loading
conditions. The materials modulus does not have any effect on the outcome of the
results, as long as the stress does not exceed the materials yield point (36,000 psi for
steel), the maximum stress will always be 1000psi multiplied by the stress concentration
factor.
3.2 Stress Concentration Factor Equations for an Elliptical Hole
Equation (1) (Reference 1) is the stress concentration factor for an elliptical hole in a flat
plate. Equation (1) is only valid if the a/b ratio is between 0.5 and 10. For this project
the ratio of a/b varies from 1 to 10.
(1)
where:
a = the long radius of ellipse (1)
b = the short radius of ellipse (will be varied from 1 to 0.1 in 0.1 increments)D = width of the flat plate (10 inches)
K = stress concentration factor for an elliptical hole in a flat plate.
The special case of where b = a for a circular hole, the elliptical hole equation (1) yields
a stress concentration of 2.54. For the same plate, using the equation for a circular hole
from Reference (1), the stress concentration is 2.50. This is only a 1.6% difference.
Therefore, the elliptical hole stress concentration equation (1) will be used for both the
circular and elliptical hole cases.
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3.3 Boundary Conditions for FEA model
Figure 2 shows the boundary conditions for the FEA model. The flat plate from Figure
1, was modeled as a quarter plate with the left vertical edge constrained in the x direction
and the bottom edge being constrained in the y direction. The right vertical edge of the
model will be a free edge and the top edge will be where the pressure load of 1000psi is
applied.
Figure 2 FEA model Boundary Conditions
3.4 Elements
There will be two element types used in this report. The first element is a 4-noded quad
element. In ANSYS the name of the element type is Plane42 shown in Figure 3.
Figure 3. 4-noded Quad Element, (Reference 2)
1000psi
Free EdgeEdge constrained, dx = 0
Edge constrained, dy = 0x
Load Applied on Edge
Ellipse
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The second element to be used is an 8-noded quad element, called the Plane82 in
ANSYS, shown in Figure 4.
Figure 4. 8-noded Quad Element, (Reference 2)
3.5 FEA Model
The 2-D FEA model in ANSYS (see Appendix A for log file code) was created to
accurately determine the stress concentration within 1% of the closed form solution of
equation (1) for an elliptical hole in a flat plate. In order to do this it was necessary to
set up manual controls for ANSYS to mesh the model and easily capture how many
elements were used at the point of highest stress, which is at the ellipse tip, to calculate
the stress concentration factor to be within 1% of the closed form solution calculated by
equation (1). Figure 5 is a schematic of the FEA models size control limits, which can
be changed by the user to either increase or decrease the mesh density at the hole and
around the whole model. These labels are referred to in the FEA model code of Appen-
dix A.
Figure 5. FEA Model Size Control Labels
Each highlighted edge
of the model will have
its own size controls.
Ellipse
Right Side of Flat PlateLeft Side of Ellipse
To of Plate
Right Side of
Ellipse Tip
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Additional manual mesh control of the FEA model is achieved by use of the LESIZE
command in ANSYS. Below is a sample line from the FEA model text file.
LESIZE,_Y1, , ,10, 3, , , ,1
This command will generate 10 elements on the line that it is assigned to, and the 3 will
size those elements based on a scaling factor where the first element will be 3 times
smaller than the last node in the line. This means that the elements will become gradu-
ally larger the farther away they are from the hole. The scaling factor command allows
the user to optimize the number of elements used in a model, since the farther away an
element is from the elliptical hole, the less its stress is going to change. Because the
greatest stresses will be produced on the right side of the ellipse tip, smaller elements
will be needed than at the top of the plate where larger elements can be used.
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4. Results
Table 1 shows the stress concentration factor for each of the 10 ellipses used. The first
column is the a/b ratio, which is the ratio between the long and short radius of the ellipse
(as seen in Figure 1), the 4 constants C1-C4 needed in equation (1), and the stress
concentration factor using equation (1). The last two columns of Table 1 are the stress
concentration factor tolerance, which will be used to determine if the FEA model from
ANSYS (Appendix A) has an adequate mesh density. This tolerance was chosen as +/-
1% from the calculated stress concentration (K) in equation (1).
a/b a b C1 C2 C3 C4 K K+1% K-1%
10.00 1.00 0.10 21.00 -25.25 32.17 -25.92 17.03 17.20 16.86
5.00 1.00 0.20 11.00 -12.81 14.50 -10.69 8.93 9.02 8.84
3.33 1.00 0.30 7.67 -8.67 9.14 -6.14 6.25 6.31 6.19
2.50 1.00 0.40 6.00 -6.59 6.66 -4.07 4.92 4.96 4.87
2.00 1.00 0.50 5.00 -5.35 5.28 -2.93 4.12 4.16 4.08
1.67 1.00 0.60 4.33 -4.52 4.42 -2.24 3.59 3.62 3.55
1.43 1.00 0.70 3.86 -3.92 3.85 -1.78 3.21 3.24 3.18
1.25 1.00 0.80 3.50 -3.48 3.44 -1.47 2.93 2.96 2.90
1.11 1.00 0.90 3.22 -3.13 3.15 -1.24 2.71 2.74 2.68
1.00 1.00 1.00 3.00 -2.86 2.93 -1.08 2.54 2.56 2.51
Table 1 Stress Concentration Factors for Various Ellipse radii
Table 2a and 2b show the results of the FEA models for the Plane 42 and the Plane 82
elements. This data was recorded for each model only when the maximum stress
produced at the ellipse tip divided by the nominal stress (1000psi) resulted in a stress
concentration factor that was within the +/- 1% tolerance from the stress concentration
calculated by equation (1).
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This is an explanation of Table 2a and 2bs columns, and Figure 5 gives a graphical
representation for the naming/location of the edges of the model.
b = short radius of the ellipse (inch),
Kroarks = Stress concentration factor calculated using equation (1).
K+1% and K-1% is the stress concentration factor, Kroarks +/- 1% in order to establish
a tolerance which will determine if the model has been meshed properlyGnom = Nominal stress in the plate far away from the hole. This will be the 1000psi
pressure load. (psi),
Gmax = Maximum stress created at the ellipse tip which is where the highest stresses
are produced. (psi),
Kmodel = Stress concentration factor obtained from the FEA model in ANSYS, whichis calculated by dividing Gmax/Gnom.
Right = Number of elements that were on the bottom edge of the model, which was to
the right of the ellipse tip.
Scale = Scaling factor used in the LESIZE command, which scaled the elements,
Left = Number of elements that were used to generate the mesh on the left vertical edge
of the model. This is also the side that was constrained to not move in the x-direction.Right Side = Number of elements that were applied to the right vertical edge of the
model. This was the unconstrained, free edge of the model.
Top = Number of elements that were applied to the top of model, which was the edge
that the 1000psi pressure load was applied.
Element Length on Right of Ellipse = Length of the element that is on the Right Side
edge of the model at the tip of the ellipse. Figure 6 shows the length dimension of the
element that was recorded for this column.
Figure 5. FEA Model Size Controls Labels, re-shown from page 12
Each edge of themodel will have its
own size controls.
Ellipse
Right SideLeft
To
Right
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Figure 6 Magnified view of the stress distribution at the tip of the ellipse. (a/b = 1.25, b =
0.8, Plane 42 element type shown)
Length of element
y
x
Ellipse
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5. Discussion
The first objective of this report is to show that the classical solution for a circular hole
in a flat plate is the same as using the closed form solution for an elliptical hole in a flat
plate with a short radius of b = a = 1 which is a circular hole. The results of the closed
form equation (Reference 1) for a flat plate with a circular hole produced a stress con-
centration of 2.50 and the elliptical hole equation (Reference 1) had a stress
concentration of 2.54. These results are within 1.6% of each other, which meant just the
elliptical equations were used to calculate the stress concentration factor for all cases
including when the ellipse become a circle.
The second objective was to show that it is possible to improve the results of an
FEA model by increasing the order of the elements used in the model. Based on the
results gathered from the FEA models from Tables 2a and 2b, the following figure was
created.
Length of Element at the Right of Ellipse vs Ellipse Short Radius
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b, Ellipse Short Radius (inch)
LengthofEllementtoRightofEllipse(inch)
Plane82
Plane42
Figure 7 Ellipse Short Radius, b vs Length of Element at the Right of Ellipse needed toobtain an accuracy of +/- 1% from the calculated stress concentration factor
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Figure 7 shows that as the elliptical hole became narrower a more refined mesh den-
sity was required. More importantly it shows that the Plane42 element type required the
most amount of refinement, because the element to the right of the ellipse tip (as shown
in Figure 6) needed to be smaller in order to capture the stress concentration that was
within +/- 1% of the actual stress concentration as calculated from closed form solutions.
In Figure 7, it can be seen that the Plane42 element type always used a smaller length of
element compared to the Plane82 element. The Plane42 element had a length of 0.0095
vs. the Plane82s length of 0.0147 when the ellipse had a short radius of 0.1.
The third objective of this report was to show that an FEA model could increase in
accuracy by increasing the number of elements used in a model. Figure 8 shows that as
the ellipse became narrower, more elements were needed at the ellipse tip in order to
obtain a stress concentration factor that was within +/- 1% of the actual stress concentra-
tion as calculated from closed form solutions. The figure also shows the Plane82
element was more efficient at meshing the model since it required less elements in order
to capture the stress concentration factor within the specified tolerance of +/-
1%.
Ellipse Short Radius, b vs. # of Elements to the Right of the Ellipse
0
5
10
15
20
25
30
35
40
45
50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b, Ellipse Short Radius (inch)
#ofElementstotheRightoftheEllipse Plane82
Plane42
Figure 8 Ellipse Short Radius, b vs. # of Elements to the Right of the Ellipse needed toobtain an accuracy of +/- 1% from the calculated stress concentration factor
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Finally, this report shows the mesh density needed to calculate the stress concentration
factor within 1% of the closed form solution. The values needed to generate a mesh
density that falls within 1% of the closed form solution for the stress concentration factor
are shown in Tables 2a and 2b. These values are for the specific geometry of an elliptical
hole in a flat plate with an a/b ratio varying from 1 to 10, a width of 10 and subjected to
a pressure load of 1000psi, as can be seen in Figure 1, shown below for convenience.
Figure 1 Flat Plate with an Elliptical Hole.
ba
1000psi
2
D = 10
20
1000psi
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6. Conclusion
Discontinuities in a geometry such as an elliptical hole create an increase in the stress
distribution known as the stress concentration factor. It was shown that the results for
the stress concentration calculated using the elliptical hole equations when the ellipse
became a circle were the same. Because of this, the elliptical hole equation was used for
all cases including the circular hole case.
It was also shown that the accuracy of an FEA model can be increased by two
methods. The first method was to increase the order of the element used in the model. It
was shown that the 4-noded quad element (Plane42) needed smaller elements and more
of them in order to capture the stress concentration factor to be within +/- 1% of the
closed form solution for the stress concentration factor for this specific geometry, as
compared to the 8-noded quad element (Plane82). The second method was to increase
the number of elements in the model. This was done by showing that as the ellipse
became narrower, more elements were needed in order to obtain the stress concentration
factor within 1% of the closed form solution to equation (1) for this specific geometry.
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7. References
1 Young, Warren; Budynas, Richard,Roarks Formulas for Stress and Strain
2 ANSYS Help Menu, ANSYS INC., Release 10.0A1 UP20060105
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8. Appendix A
This appendix contains the FEA code used to create all of the ANSYS models for this
project.
/CLEAR,START
/PREP7!
ET,1,PLANE82
!
/REPLOT,RESIZE
!
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,30e6
MPDATA,PRXY,1,,.3
/REPLOT,RESIZE
CYL4,0,0,1
FLST,2,1,5,ORDE,1
FITEM,2,1
!
! ------------------ Ellipse Size ---------------
!
ARSCALE,P51X, , ,1,.1,1, ,0,1
!
RECTNG,0,5,0,10,
FLST,2,1,5,ORDE,1
FITEM,2,2
FLST,3,1,5,ORDE,1
FITEM,3,1
ASBA,P51X,P51X,SEPO,DELETE,DELETE
!
FLST,5,1,4,ORDE,1
FITEM,5,10CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!
!--------- Right Side Ellipse (Horizontal) -----------
!
LESIZE,_Y1, , ,38, 23, , , ,1
!
FLST,5,1,4,ORDE,1
FITEM,5,9
CM,_Y,LINE
LSEL, , , ,P51XCM,_Y1,LINE
CMSEL,,_Y
!
!------------ Ellipse ------------
!
LESIZE,_Y1, , ,32,12 , , , ,1
!
FLST,5,1,4,ORDE,1
FITEM,5,13
CM,_Y,LINE
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LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!
!---------------- Left Side of Ellipse (Vertical) -----------
!
LESIZE,_Y1, , ,38, 2, , , ,1
!FLST,5,1,4,ORDE,1
FITEM,5,11
CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!
!------------- Right Side of Flat Plate -------------
!
LESIZE,_Y1, , ,22, 1, , , ,1
!
FLST,5,1,4,ORDE,1
FITEM,5,12
CM,_Y,LINE
LSEL, , , ,P51X
CM,_Y1,LINE
CMSEL,,_Y
!
!--------------- Top Side of Flat Plate -------------------
!
LESIZE,_Y1, , ,11,1 , , , ,1
!
MSHKEY,0
CM,_Y,AREA
ASEL, , , , 3
CM,_Y1,AREA
CHKMSH,'AREA'CMSEL,S,_Y
!
AMESH,_Y1
!
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!
FINISH
/SOL
FLST,2,1,4,ORDE,1
FITEM,2,13
!/GO
DL,P51X, ,UX,0
FLST,2,1,4,ORDE,1
FITEM,2,10
!
/GO
DL,P51X, ,UY,0
FLST,2,1,4,ORDE,1
FITEM,2,12
/GO
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!
! ---------- Pressure Load (-1000psi) -----------
!
SFL,P51X,PRES,-1000,
/STATUS,SOLU
SOLVE
FINISH
/POST1!
/DSCALE,ALL,OFF
/EFACET,1
PLNSOL, S,EQV, 1,1.0
!
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9. Appendix B
This appendix shows plots of the final FEA models for the Plane42 element type.
Figure 9 Plane 42 FEA model results with a short radius of b = 0.1
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Figure 10 Plane 42 FEA model results with a short radius of b = 0.2
Figure 11 Plane 42 FEA model results with a short radius of b = 0.3
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Figure 12 Plane 42 FEA model results with a short radius of b = 0.4
Figure 13 Plane 42 FEA model results with a short radius of b = 0.5
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Figure 14 Plane 42 FEA model results with a short radius of b = 0.6
Figure 15 Plane 42 FEA model results with a short radius of b = 0.7
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Figure 16 Plane 42 FEA model results with a short radius of b = 0.8
Figure 17 Plane 42 FEA model results with a short radius of b = 0.9
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Figure 18 Plane 42 FEA model results with a short radius of b = 1.0
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10. Appendix C
This appendix shows plots of the final FEA models for the Plane82 element type.
Figure 19 Plane 82 FEA model results with a short radius of b = 0.1
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Figure 20 Plane 82 FEA model results with a short radius of b = 0.2
Figure 21 Plane 82 FEA model results with a short radius of b = 0.3
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Figure 22 Plane 82 FEA model results with a short radius of b = 0.4
Figure 23 Plane 82 FEA model results with a short radius of b = 0.5
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Figure 24 Plane 82 FEA model results with a short radius of b = 0.6
Figure 25 Plane 82 FEA model results with a short radius of b = 0.7
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Figure 26 Plane 82 FEA model results with a short radius of b = 0.8
Figure 27 Plane 82 FEA model results with a short radius of b = 0.9
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Figure 28 Plane 82 FEA model results with a short radius of b = 1.0
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