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The Influence of a Taper on theThe Influence of a Taper on the
Stress Concentration Factor of aStress Concentration Factor of aShoulder Filleted ShaftShoulder Filleted Shaft
Curtis A. Schmidt
University of Tulsa
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Rolivich, Tipton, Sorem,.Multiaxial stressconcentrations in filleted shafts, Jornal ofMechanical Design, 2001
Rolivich, Tipton, Sorem,.Udated stressconcentration factors for filleted shafts in bendingand tension, Journal of Mechanical Design, 1996
TensionTension TorsionTorsionBendingBending
Stress Concentration Factor Graphs:Stress Concentration Factor Graphs:Stress Concentration Factor Equations:Stress Concentration Factor Equations:Maximum Stress Location Chart:Maximum Stress Location Chart:Maximum Stress Location Equations:Maximum Stress Location Equations:
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Stress Relief
Taking away material has been found to lower the StressConcentration factor
Shioya, 1963 shows this for semicircular notches in a semiinfinite plate as seen below (from Petersons SCFs, Pilkey)
This phenomena can be taken advantage in standard
filleted shafts by adding a taper
Notches
get closer
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DtD d
r
Geometry of a tapered-filleted shaft
Geometry range to be tested is similar to that offilleted shafts
d = 1d = 1
0.0020.002
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ANSYS Element Type:
Tension
Plane 82 Element=.3, E = 30x 106
x = 1000psi
ANSYS Element Type:
Bending
Plane 83 Element=.3, E= 30x 106
xMAX= 1000psi
ANSYS Element Type:
Torsion
Plane 83 Element=.3, E= 30x 106
= 1000psi
xxxMAX
xMAX
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The geometry was divided up
into regionsSmall element sizes could be
used when looking at the stress
case in the radius and larger
ones could be used elsewhere
The transition from small to
large was done by establishing a
gradient across the regions.
Divisions set
at Small
Element Size
(SE)
Area 1 divisions
set at GRAD times
Small Element
Size (SE
)
Area 2 divisions set
at GRAD3 times
Small Element Size
(SE
)
Divisions set at GRAD2 times
Small Element Size (SE)
Divisions set at GRAD4 times
Small Element Size (SE)
Element size transition
1
2Small
element sizes
Larger
element sizes
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Figure 12 - Percent change in tensile stress concentration factor vs. element size for different
r/d ratios
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
20.00%
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Smallest element size (in)
PercentChangef
romn
extlargerelementsi
ze
0.46
0.32
0.1
0.02
To Determine the smallest element size (SE) and transition gradient (GRAD)
simulations were run to show convergence
As the small element size approached a value of 0.003 convergence was
illustrated in the stress concentration factor
This was also shown over a range of r/d ratios
Though 0.003 was a sufficient element size for convergence of the stress
concentration factor it had to be reduced to provide enough resolution along the
arc (at least 1o increments) for all cases of r/d. Reducing the small element size
made it necessary to adjust the GRAD constant as well so the amount of nodes
would not saturate the computers being used.
Figure 10 - Tensile Stress concentration factor vs. r/d for different element sizes
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
r/d ratios
Tensilestressconcentrationfactors
0.025
0.011764706
0.005882353
0.003921569
0.002941176
r/d < .002, SE = 1/1200, GRAD = 3.85r/d < .002, SE = 1/1200, GRAD = 3.85r/d > .002, SE = 1/600, GRAD = 2.50r/d > .002, SE = 1/600, GRAD = 2.50
For all cases where SE along the radius does not provide atFor all cases where SE along the radius does not provide at
least 1 degree resolution, 1 degree resolution will beleast 1 degree resolution, 1 degree resolution will be
assignedassigned
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The resulting mesh and
model can be seen
below for a samplecase
~ 1/600~ 1/4~ 1/8~ 1/60
SCALE
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r/d D/d Eq. 1 Eq. 1 MAXAVERAGE -0.92% -1.34% -0.59% -0.37% -4.26% -0.27% -0.57% -0.04%
MAX 0.59% 6.18% 0.92% 0.72% 11.66% 0.59% 9.88% 2.33%
MIN -5.85% -7.14% -5.67% -1.59% -20.38% -2.26% -10.35% -1.80%
Tensile Loading
Bending
Loading
Torsional
Loading
Standard
Geometry
To check the accuracy of the models, data was generated for 46 standard filleted
shaft geometries under all three loading conditions.
The stress concentration factors for1, eq and MAX were found to be very similar
to published data
However the location of the maximum stress differed significantly from previous
work
The mesh size in the radius most likely explains this discrepancy.
From: From: Updated Stress ConcentrationUpdated Stress Concentration
Factors from Filleted shaftsFactors from Filleted shafts, Journal, Journalof Mechanical Design, Sep 96. Oneof Mechanical Design, Sep 96. One
can see that there are approximatelycan see that there are approximately
26 division along the radius26 division along the radius
compared to the >90 criteria of thecompared to the >90 criteria of the
current modelcurrent model
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0.00
0.50
1.00
1.50
2.00
2.50
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Dt/d
Kq
Reduction in Kq = 0.03%Reduction in Kq = 0.32%Reduction in Kq = 1.92%Reduction in Kq = 3.86%Reduction in Kq = 6.83%Reduction in Kq = 11.45%Reduction in Kq = 14.66%Reduction in Kq = 24.60%Reduction in Kq = 31.44%
D/d = 1.5D/d = 1.5
r/d = .05r/d = .05
=75=75oo
0.00
0.50
1.00
1.50
2.00
2.50
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Dt/d
Kq
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0
1
2
3
4
5
0 10 20 30 40 50 60 70 80
Theta (degrees)
Kq
0
1
2
3
4
5
0 10 20 30 40 50 60 70 80
Theta (degrees)
Kq
D/d = 3.0D/d = 3.0
r/d = .01r/d = .01
Dt/d = 1.2Dt/d = 1.2
Reduction in Kq = 0%INCREASE in Kq = 2.4%INCREASE in Kq = 2.6%Reduction in Kq = 0%Reduction in Kq = 6.1%Reduction in Kq = 15.3%
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1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Dt/d
Kq
Theta = 75Theta = 60
Theta = 45
Theta = 30
Theta = 15
Nominal
radius intersection
D/d = 1.5
r/d = .05
Kq,nom=2.43
Illustration of the effects of a changing transition diameter for
different values of theta for a single standard fillet geometry
DtD d
r
DtD d
r
DtD d
r
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0
5
10
15
20
25
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Dt/d
Theta
Theta = 75
Theta = 60
Theta = 45
Theta = 30
Theta = 15
D/d = 1.5
r/d = .05
Thetanom=20
Illustration of the effects on the maximum stress location for achanging transition diameter for different values of theta for asingle standard fillet geometry
DtD d
r
DtD d
r
DtD d
r
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2
2.5
3
3.5
4
4.5
5
5.5
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
Dt/d
Kq
Theta = 60
Theta = 75
Theta = 45
Theta = 30Theta = 15
Nominal
radius intersection
D/d = 3
r/d = .01
Kq,nom=5.04
Illustration of the effects of a changing transition diameter for
different values of theta for a single standard fillet geometry
DtD d
r
DtD d
r
DtD d
r
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2
2.5
3
3.5
4
4.5
5
5.5
0 10 20 30 40 50 60 70 80
Theta (degrees)
Kq
Dt/d = 1.02
Dt/d = 1.025
Dt/d = 1.1
Dt/d = 1.2
Dt/d = 1.3
Dt/d = 1.5
Dt/d = 2
Dt/d = 2.8
Nominal
D/d = 3
r/d = .01
Kq,nom=5.04
Illustration of the effects of a changing taper angle for
different values of Dt for a single standard fillet geometry
DtD d
r
DtD d
r
DtD d
r
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0
5
10
15
20
25
30
35
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
Dt/d
Theta
Theta = 60
Theta = 75
Theta = 45
Theta = 30
Theta = 15
D/d = 3
r/d = .01
Thetanom=31
Illustration of the effects on the maximum stress location of achanging taper angle for different values ofDt for a singlestandard fillet geometry
DtD d
r
DtD d
r
DtD d
r
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0
2
4
6
8
10
12
0 0.05 0.1 0.15 0.2 0.25 0.3 0.Dt/d
Kq
Theta = 0
Theta = 15Theta = 30
Theta = 45
Theta = 60
Theta = 75
Special case of tangency between the fillet and the taper
yields the highest reduction in stress concentration factor
D/d = 2
DtD d
r
DtD d
r
DtD d
r
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!!Develop a set of equations similar to ones for filleted shaftsDevelop a set of equations similar to ones for filleted shafts
!!Predict the location of the maximum stress over entirePredict the location of the maximum stress over entire
range of geometryrange of geometry!!Predict the change in maximum stress compared to thePredict the change in maximum stress compared to the
standard filleted geometrystandard filleted geometry
!!Develop a program that can interpolate current resultsDevelop a program that can interpolate current results
!!Program will linearly interpolated between the data pointsProgram will linearly interpolated between the data points(4,000 per loading scenario 12,000 total) to minimize the(4,000 per loading scenario 12,000 total) to minimize the
inherent error that may develop by fitting a curve throughinherent error that may develop by fitting a curve through
datadata
!!Will show the corresponding geometry on the screen toWill show the corresponding geometry on the screen toaid designer in visualizing the effects of the differentaid designer in visualizing the effects of the different
variablesvariables
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