Spur gear stress analysis results are presented for a variety of loading conditions, support conditions, fillet radii, and rim thickness. These results are obtained using the SAP IV finite-element code. The maximum stresses, occurring at the root surface, substantially increase with decreasing rim thickness for partially supported rims (that is, with loose-fitting hubs). For fully supported rims (that is, with tight-fitting hubs), the root surface stresses slightly decrease with decreasing rim thickness. The fillet radius is found to have a significant effect upon the maximum stresses at the root surface. These stresses increase with increasing fillet radius. The fillet radius has little effect upon the internal root section stresses

Chang, S. H.

Coy, J. J.

Huston, R. L.

English

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AVRADCOM
Technical Report 82-C-8
NASA
Technical Memorandum 82865
t
.4. Finite Element Stress Analysis
)f Spur Gears Including Fillet Radii
and Rim Thickness Effects
NEl-2E646(NASA-TM-828e5)	 A FINITE ELEMENT STRESS
ANALY.iIS OF SPUN GEARS IN-LULING FILLET
EADII AND 6IM THICKNESS cEt ICTS (NASA)	 1l p
HC A02/MF A01	 CSCL 131
S. H. Chang and R. L. Huston
University of Cincinnati
Cincinnati, Ohio
and
J. J. Coy
Propulsion Laboratory
AVRADCOM Research and Technology Laboratories
Lewis Research Center
Cleveland, Ohio
Uncias
G3/37 284U4
0^0o^g^
Prepared for the
Winter Annual Meeting of the
American Society of Mechanical Engineers
Phoenix, Arizona, November 14-19, 1982
embers in brackets refer to References at the end of the paper.
A FINITE ELEMENT STRESS ANALYSIS OF SPUR GEARS INCLUDING
FILLET RADII AND RIM THICKNESS EFFECTS
S. H. Chang and R. L. Huston
Department of Mechanical Engineering
University of Cincinnati
A	 Cincinnati, Ohio 45221
and
J. J. Coy
Propulsion Laboratory
AVRADCUM Research and Technology Laboratory
NASA Lewis Research Center
M	 Cleveland, Ohio 44135N
,;,	 ABSTRACT
Spur gear stress analysis results o,e presented for a variety of loading
conditions, support conditions, fillet radii, and rim thicknesses. These
results are obtained using the SAP IV finite-element code. The maximum
stresses, )ccurring at the root surface, substantially increase with decreasing
rim thickness for partially supported rims (that is, with loose-fitting hubs).
For fully supported rims (that is, with tight-fitting hubs), the root surface
stresses slightly decrease with decreasing rim thickness. The fillet radius
is found to have asig M cant effect upon the maximum stresses at the root
surface. These stresses increase with decreasing fillet radius. The fillet
radius has little effect upon the internal root section stresses.
INTRODUCTION
During the past decade there have been a number of investigations into
the nature of spur gear stresses. References [1-12]* summarize some of these
efforts. They include a variety of theoretical and experimental approaches to
determining the stresses, but the results are reasonably consistent. Of par-
ticular interest is the fact that the results obtained using finite element
techniques are consistent with results obtained using vastly different theo-
retical and experimental techniques. For example, in 1955, Jacobson [13j
studied bending stresses using photoelastic techniques. In 1913, his results
were matched by Wilcox and Coleman [11] using finite element techniques. In
19b2, Aida and Terauchi [14] studied bending stresses using stress functions
and classical elasticity theories, and more recently (1981) Cordou and Tordion
[2] studied the stresses using complex variables. Their results also confirm
results obtained using finite element techniques. Uther noteworthy theoreti-
cal studies which confirm finite element results are those of Baronet,
Tordion, and Premilhat [1,1] and of Shotter [8].
Regarding the use of the finite element method itself in gear stress
analyses, there have recently been a number of notable achievements. For
example, Chabert, Dang Tran, and Mathis [3] have used finite element tech-
niques to examine the stress distribution across the root section. Tobe, Kato
and Inoue [9,10], Winter ano Hirt [12], Cornell [5], and Wilcox and Coleman
[11] have studied root stresses using finite element methods. Finally, Oda,
Nagamura, and Aoki have examined the effect of rim thickness on the root
stresses using the finite element method.
In view of this, the objectives of the research effort of this paper are
to determine (a) the surface stress distribution for the entire tooth profile
for tip and pitch point loading; (b) the fillet stresses and the root section
stresses for a variety of loading positions and fillet radii; and (c) the
effect of rim thickness and mounting support upon the root stress. The SAP IV
finite element technique [15] was used in the analysis.
ANALYSIS
The Model
Figure 1 shows typical finite—element grids used in the analysis. The
number of elements used was varied depending upon the particular loading and
geometry being considered. Typically the grid had approximately 190 elements
and 120 nodes.
The gear tooth itself had a modulus M of 5 (pitch
number of teeth). It was a member of an 18—tooth gear
thus being 90 mm. The tooth sides are involute curves
was 20% The fillet radius and hub thickness were var
material was steel with an elastic modulus of 2.11x105
diameter divided by the
with the pitch diameter
and the pressure angle
iables. The tooth
N/mn2.
Loading and Support
The tooth was loaded with a 400 N/mm concentrated line load applied
normal to the tooth boundary at various points as shown in Figure 2. The hub
or rim was supported alternatively: a) at all points along the boundary, and
b) at only the radial points.
The Finite Element Procedure
The SAP IV technique [15] was used to assemble and solve the governing
equations
Ku=R
	
(1)
where K is the global stiffness matrix, u is the array of nodal displacements
and R is the force array. The solution is obtained using Gauss elimination
through the linear equation solver SESOL [15]. After the nodal displacements
are found, element stress displacement relations are used to obtain the
element stresses.
RESULT
Surface Stress Distribution
First, the stress distribution along the tooth surface was calculated for
a tip loading and for a load applied near the pitch point. Figure 3 contains
2
a representation of the results for the maximum principal surface stress.
They show that, except for a local concentration, the maximum stress occurs at
the root of the tooth.
Root Surface Stresses as a Function of the Root Radius
.	 The above analysis led to a closer examination of the maximum fillet
4
	
	
stresses as a function of the fillet radius. Specifically, the tooth was
loaded at the points shown in Figure 2. The fillet stresses were then calcu-
lated for radii of 0.2M, 0.3M, and 0.4M or 1.0 p an, 1.5 own, and 2.0 inn, respec-
tively. The results for the 1.0 mm and 2.0 own f illet radii are shown in
Figure 2. The results for the 1.5 mm fillet radii are intermediate to these.
An examination of the numerical values associated with these results shows the
stresses increase linearly with the inverse of the fillet radii, for the range
of radii considered. Also, the stresses are seen to increase linearly with
the distance from the fillet to the point of application of the load. These
results are consistent with those obtained and recorded by Chabert, et al.
[3], and with those of short beam theory as recorded by Roark and Young [16].
Internal Root Section Stresses as a Function of Root Radius and Loading
Position --	 -----
Figures 4 and 5 show the principal stress distribution across the root
section for tip loading and for pitch point loading for the various fillet
radii. As expected, the stresses are smallest at the center of the section
ana the largest stresses occur at the surface. Interestingly, the fillet
radius has little effect upon the internal stress distribution.
Effect of Rim Thickness and Support Conditions
Figures 6 to 10 show the effect of the rim thickness and the hub or rim
support upon the fillet stresses and the stresses across the root section.
Specifically, Figure 6 shows the surface stresses for a gear tooth with a
fully supported rim (that is, supported at the rim base and along the radial
sides, simulating tight fitting hubs). The fillet radius was 2.0 nni ana the
rim radii were 35.0 mm and 37.1 nni. The loading was the same as that shown in
Figure 3.
Similarly, Figure 7 shows the fillet stresses for a gear tooth with a
partially supported rim (that is, supported only along the radial sides,
simulating loose fitting hubs). The fillet radii, rim radii and loading were
the same as with the fully supported rim.
These results show that when the rim is fully supported the fillet
stresses decrease slightlj as the rim thickness decreases. However, wher the
rim is on Ty partfally supported the fillet stresses increase substantiallz as
the rim thickness decreases. Moreover, for partially supported rims the
compressive stresses, at the fillet opposite the loaded side increase at a
greater rate than the tensile stresses at the fillet of the loaded side.
These results are sununarized graphically in Figure ft.
Finally, figures ya and 10a show the internal root section principal
stresses for a fully supported rim with the same root radii and rim radii as
above. The loading was at the tip as shown with a magnitude of 4UU N/nrn as
before. Similarly, Figures 9b and lUb show the internal root section princi-
pal stresses for a partially supported rim with the same fillet radii, rim
radii, and loading. These results also show that the stresses decrease
slightly with decreasing rim thickness for fully supported rims, but they
increase with rim thickness for partially supported rims.
These results are consistent with recent experimental findings recorded
by Drago and Lutthans [17]. 	 6
SUMMARY
1. The rim thickness and rim support have a significant effect upon the
stresses - particularly for partially supported thin rims, with the compres-
sive root stresses, opposite the loading side, being most affected. The
stresses increase with decreasing rim thickness for partially supported rims
(such as with loose-fitting hubs). However, for fully supported rims (such as
with tight-fitting hubs) the stresses decrease slightly with decreasing rim
thickness. For large rim thickness the rim support has little effect upon the
stresses.
2. The maximum stresses occur at the root surfaces, except for local
stress concentrations immediately beneath the load. These root stresses
increase with decreasing fillet radii.
3. The fillet radius has very little effect upon the internal root
section stresses.
4. The SAP IV finite-element method is a very effective procedure for
investigating gear tooth stresses with a variety of loading, support, and
geometrical shapes. This method can provide a benchmark analysis of year
tooth stresses, replacing many of the currently used handbook formulae.
REFERENCES
1. Baronet, C. N., and Tordion, G. V., "Exact Stress Uistribution in
Standard Gear Teeth and Geometry FActors," ASME Journal of Engineering
for Industry, Vol. 95, No. 4, Nov. 1973, pp. 5^9- 163.
2. Cardou, A., and Tordion, G. V., "Numerical Implementation of Complex
Potentials for Gear Tooth Stress Analysis," ASME Journal of mechanical
Uesign, Vol. 103, No. 2. Apr. 1981, pp. 461-465.
3. Chabert, G., Dang Iran, T., and Mathis, R., "An Evaluation of Stresses
and Deflection of Spur Gear Teeth Under Strain," ASML Journal of
Engineering for Industry, Vol. 96, No. 1, Feb. l ye. 85-
4. Chan, S. K., and Tuba, 1. S., "A Finite Element Method for Contact
Problems of Solid Bodies - Part 1. Theory and Validation," International
Journal of Mechanical Sciences, Vol. 13, No. 7, Apr. 1971, NIi. b1^-b25.
5. Cornell, k. W., "Compliance and Stress Sensitivity of Spur bear leech,"
ASML Journal of Mechanical Design, Vol. 1U3, No. 2, Apr. 1981, pp.
4
6. Oda, S., Nagamura, K., and Aoki, K., "Stress Analysis of Thin Rim Spur
bears by Finite Element Method," Bulletin of the Japanese Society of
MechanicalEngineers, Vol. 24, No. T1r3Ty 1981, pp. 123-f28U.
7. Premilhat, A., Tordion, G. V., and Baronet, C. N., "An Improved
Determination of the Elastic Compliance of a Spur Gear Tooth Acted on by
a Concentrated Load," ASME Journal of _Engineering for Industry, Vol. 9b,
No. 2, May 1974, pp. 382-384.
8. Shotter, B. A., "A New Approach to Gear Tooth Root Stresses," ASME
Journal of Engineering for Industry, Vol. 9b, No. 1, teb. 1974, pp. 11-18.
9. Tobe, T., Kato, M., and Inoue, K., "Bending of Stub Cantilever Plate and
Some Applications to Strength of Gear Teeth," ASME Journal of Mechanical
Design, Vol, IOU, No. 2, Apr. 1978, pp. 374-38 r.
10. Tobe, T., Kato, M., and Inoue, K., "True Stress and Stiffness of Spur
Gear Teeth," Proceedings of the Fifth World Conference on Theory of
Machines and Mechanisms, A ME, 979979pp. 1U5-1108.
11. Wilcox, L., and Coleman, W., "Application of Finite Elements to the
Analysis of Gear Tooth Stresses," ASME Journal of Engineering -for
Industry, Vol. 95, No. 4, Nov. 1973, pp. 1139-1148.
12. Winter, H., and Hirt, M., "The Measurement of Actual Strains at Gear
Teeth, Influence of Fillet Radius on Stresses and Tooth Strength," ASNE
Journal of Engineering for Indust, Vol. 96, No. 1, Feb. 1974, pp. 3-3-5U.
13. Jacobson, M. A., "bending Stresses in Spur Gear Teeth: Proposed New
Design Factors Based on a Photo-Elastic Investigation," Proceedings,
Institution of Mechanical Engineers, Vol. 169, 1955, pp. 587-bU9.
14. Aida, T., and Terauchi, Y., "On the Bending Stress of a Spur bear,
Bulletin of the Japanese Society of Mechanical Engineers, Vol. 5, No. 17,
Feb. 1962, pp. 161-170.
15. Bathe, K. J., Wilson, E. L., and Peterson, F. E., "SNP-IV - A Structural
Analysis Program for Static and Dynamic Response of Linear Systems,"
kehor-^ No. EEkC 73-11, Earthquake Engineering Research Center, University
of California, Berkeley, CA, 1974.
If). koark, R. J., and Young, W. C., Formulas for Stress and Strain, 5th ed.
h,cGraw-Hill, New York, 1975, p. 187, 	 —
17. Dra('ao, R. J., and Lutthans, k. V., "An Experimental Investigation of the
Combined Lffects of Rim Thickness and Pitch Diameter on Spur Gear Tooth
moot and Fillet Stresses," ALMA Paper No. P229.22, American Gear
hanufacturers Association Fall Technical Meeting, Toronto, 1961.
(a) Small fillet radius. lb) Small rim thickness.
ORIGINAL PAGE 19
OF POOR QUALITY
(a) 1.0 mn fillet radius. Ibl 2.0 mm radius.
400 P 400 Nimm
Figure 1. - Typical finite element grids.
Figure 2. - Surface stress distribution at the fillet
LOAD: 400 Nlmm
COMPRESSION	 TENS ION
u
200 N1mm2
Ilmm
600
400
elE 200
Z
V!r	 0NW
^- -200N
-400
-600
K) NJmm
OF POOR QUALITY
(a) Tip loading.	 (b) Pitch point loading.
Figure 3. - Surface maximum stress distribution.
a - NORMAL STRESS
t - SHEAR STRESS
LOAD: 400 Nimm
(a) Tip loading.	 (b) Pitch point loading.
Figure 4 - internal root section principal stresses for 1.0 mm fillet radius.
600
400
E 200Z
Nvi 0
W
-2w
-400
-600
): 400 Nimm
o - NORMAL STRESS	 ORIGINAL PAGE IS
- SHEAR STRESS 	 OF POOR QUALITY
LOAD: 400 N/mm
(a) Tip loading	 (b) Pitch point loading.
Figure 5. - Internal mot section principal stresses for Z Omm fillet radius.
(a) Rim radius: 35mm.
	 (b) Rim radius: 37.1 mm.
Figure 6. - Surface stress distribution at the fillet for a fully supported rim.
o f PWR QUALITY
A
A
J200 N/mm'	 `Z00 NImm2
	
(a) Rim radius: 35mm.
	 1bi Rim radius: 37.1mm.
Figure 7. - Surface stress distribution at the fillet for a partially supported rim.
2.4
2.2
— PARTIALLY-SUPPORTED
2.0	 i	 --	 RIM
E	 i	
FULLY-SUPPORTED
RIM
z LS	 1	 TIP LOAD • 1 N/mm
o	 I cc • COMPRESSIVE STRESS
W	
^i	 Ot - TENSILE STRESS
L L5	 ,I
LA
a
1.4  
cc
L2
Ot
LO
0t
0	 2.5
	
5.0
	 7.5	 10.0	 12.5	 15.0
RIM THICKNESS, mm
Figure B. - Variation of maximum fillet stresses with rim
thickness.
600
400
"E 200Ez
V 0
to
°C -200N
-400
-600
LOAD: 40ON/mm LOAD: 4D0 N/mm
600
400
E
200
z	 0
NyLA
N -200
-400
-600
121 Fully supported rim. 	 _
Figure 10. - Internal root section principal stresses with tip loading with a rim radius of 37. lmm.
t ^^
0001Q%); ,
o - NORMAL STRESS	
OF pO R
T - SHEAR STRESS
LOAD: 400 N/mm
	 LOAD: 400 N/mm
lai Fully supported rim.	 ibi Partially supported rim.
Figure 9. - Internal root section principal stresses with tip loading with a rim radius of 35mm.
o - NORMAL STRESS
t - SHEAR STRESS


A finite element stress analysis of spur gears including fillet radii and rim thickness effects

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