In aerospace applications, an engineer must be especially cognizant of size and weight constraints which affect design decisions. Although designing at or below the material fatigue limit may be desirable in most industrial applications, in aerospace application it is almost mandatory to design certain components for a finite life at an acceptable probability of survival. Zaretsky outlined such a methodology based in part on the work of W. Weibull (1939, 1951) and G. Lundberg and A. Palmgren (1947a, 1947b, 1952). It is the objective of this work to apply the method of Zaretsky (1987) to statistically predict the life of a generic solid disk with and without bolt holes; determine the effect of disk design variables, thermal loads, and speed on relative life; and develop a generalized equation for determining disk life by incorporating only these variables

August, Richard

Smith, Todd E.

Zaretsky, Erwin V.

English

NASA Technical Reports Server

N88-22428
LIFE AND RELIABILITY OF ROTATING DISKS
Erwin V. Zaretsky, Todd E. Smith,* and Richard August*
Structures Division
NASA Lewis Research Center
INTRODUCTION
In aerospace applications, an engineer must be especially cognizant of size and
weight constraints which affect design decisions. Although designing at or
below the material fatigue limit may be desirable in most industrial applica-
tions, in aerospace applications it is almost mandatory to design certain com-
ponents for a finite life at an acceptable probability of survival. Zaretsky
(1987) outlined such a methodology based in part on the work of W. Weibull
(1939, 1951) and G. Lundberg and A. Palmgren (1947a, 1947b, 1952). Zaretsky's
method (1987) is similar in approach to that of loannides and Harris (1985).
Mahorter et al. (1985) discuss the accuracy of life prediction techniques for
military turbine engine components such as compressor and turbine disks. The
development of a 0.794 nun (0.03 in.) crack in any of the critical areas of a
disk, such as bolt holes, bore, dovetail, etc., is considered the end of its
low cycle fatigue (LCF) life. This is the life, L0.1, at which the military
requires a 99.9 percent probability of survival or a 0.I percent failure rate.
Disk retirement policy requires that the disks be removed from service or
reworked at their L0. 1 life. The statistically predicted lives for bolt-hole
cracks were shorter than the deterministically predicted values. These results
imply that a deterministic approach to life prediction is not necessarily con-
servative, and that a probabilistic approach is viable in light of all the
statistical variations in the design parameters.
In view of the aforementioned, it is the objective of this work to (a) apply
the method of Zaretsky (1987) to statistically predict the life of a generic
solid disk with and without bolt holes, (b) determine the effect of disk design
variables, thermal loads, and speed on relative life, and (c) develop a gener-
alized equation for determining disk life by incorporating only these
variables.
*Sverdrup Technology, Inc., Lewis Research Center Group. Work performed
on-site at NASA Lewis under NASA Contract NAS3-24105.
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https://ntrs.nasa.gov/search.jsp?R=19880013044 2020-03-20T06:30:16+00:00Z
OVERVIEW
FINITE ELEMENT MODEL
Parametric analytical studies were conducted to investigate the effect of
varying physical dimensions and speed on the relative lives of a generic solid
disk. The physical model of the solid disk requires that the structure be
divided into a series of components (elements). This is most easily done by
considering the disk as a collection of concentric rings. The disk was modeled
with i0 rings of equal radial increment for the parametric values of disk diam-
eter studied.
n
,-- Vn
I
I I1
I
_.__F TO
!
LT i
i=I
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GOVERNING EQUATIONS
The following equation for the 10-percent life of a uniform solid rotating
disk is obtained:
Introducing the concept of a Dynamic Speed Capacity No, which is defined by
Zaretsky (1987) as the speed that would produce a theoretical life of one
million stress cycles,
_o:_,0its) _ t_j ox,<<,_,>
and for any speed N for a given disk geometry.
LI0 = xlO 6
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SYMBOLS
A material constant, stress cycles
c stress-life exponent
D disk diameter, m (in.)
E material Young's modulus, N/m 2 (psi)
e Weibull slope or modules
KD proportionality constant, m (in.)
K L proportionality constant, m2/N (psi -1 )
KN proportionality constant, rpm
KT proportionality constant, K (°F)
Kt proportionality constant, m (in.)
L life, stress cycles
L s system life, stress cycles
LI0 ten-percent life, life at which 90 percent of a population survive,
stress cycles
N speed, rpm
N o dynamic speed capacity, rpm
T temperature, K (°F)
t disk thickness, m (in.)
V stress volume, m3 (in. 3)
AT temperature difference, K (°F)
shear stress, N/m 2 (psi)
_L fatigue endurance limit, N/m 2 (psi)
Subscripts:
i ith component, and denotes inner disk radius
o reference point, and denotes outer disk radius
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POSTER PRESENTATION"
EFFECT OF DISK DIAMETER
The effect of disk diameter on life for a speed of 9000 rpm and a stress-life
exponent, c, of 9 is shown. As the disk diameter is increased, both the stress
and the stress volume increase and life will decrease. The effect of Weibull
slope is negligible. D is the disk diameter, KD is a constant equal to
0.61 m (24.0 in.), and LI0 is the life at a 90-percent probability of sur-
vival (or the life where i0 percent of the disks have failed).
NORMALIZED
RELATIVE
DISK LIFE,
LIO
1000
100
10
1
.1
.01
/KD_20
40 50 60 70
DISK OUTEI_DIAMETER, CM
I I I I
16 20 2q 28
DISK OUTER DIAMETER, IN.
80
I
32
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EFFECTOFDISK THICKNESS
The effect of disk thickness on the LIO
volume is increased, life will decrease.
gible, t is the disk thickness, and Kt(i.0 in. ).
life is shown. Becausethe stressed
The effect of Weibull slope is negli-
is a constant equal to 0.0254 m
NORMALIZED 3
RELATIVE
DISK LIFE,
LIO
2
L10-
_ WEIBULLeSLOPE,
_.,,,_//-- 2, 3.57
1-/
I I I I
1 2 3 q
DISK THICKNESS, CM
I I I
• 50 1.00 1.50
DISK THICKNESS, IN.
V
I
2.00
CD-88-32849
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EFFECTOFSPEED
As disk speed is increased, stresses within the disk will increase. These
stress increases will cause a decrease in life. The effect of disk rotational
speed on LI0 life is shown. N is the disk speed and KN is a speed
constant equal to 9000 rpm.
NORMALIZED
RELATIVE
DISKLIFE,
LIO
I000 _--
100
10
I
.I
01
6
o-_)
WEIBULIeSLOPE"
I I ] I I
7 118 9 10
DISK ROTATIONAL SPEED, RPM
I
12x103
CD-88-32850
. C._/ 3-277
EFFECTOFSTRESS-LIFEEXPONENT
For each stressed elemental volume within the body of the disk, the life was
determined by using an inverse stress-life relation. Not all materials will
exhibit the samestress-life relation. The stress-life exponent is generally
determined experimentally. For the previous calculations, a stress-life expo-
nent of 9 was assumed. Using varying values of c with a reference disk, the
effect of the stress-life exponent on the LI0 life of the disk was deter-
mined. These results are shown.
NORMALIZED
RELATIVE
DISK LIFE.
LIO
1.50 --
1.25 --
1.00
.75
6
I,EIBIJLL SLOPE,
e L /9_0"606
2, 3.57-_
_U1.0
I I I
8 10 12
MATERIAL STRESS-LIFE EXPONENT
I
14
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EFFECT OF ENDURANCE LIMIT
A concept first espoused by loannides and Harris (1985) was the application of
a fatigue limit in the determination of the lives of the elemental stressed
volumes. Basically, the concept is that where the shearing stress is equal to
or less than the value determined or assumed for the fatigue limit, the proba-
bility of survival for that elemental stressed volume is i00 percent. The
results of assuming varying values of a fatigue limit for the shearing stress
is shown where x L is the fatigue endurance limit at or below which stress no
failure is expected to occur and KL is equal to 4.02xi0 -8 m2/N
(2.77xi0 -4 psi-l).
NORMALIZED
RELATIVE
DISK LIFE,
LIO
I00 000
10 000
1000
100
10
.!
_=-
I I I I
20 40 60 80
FATIGUE ENDURANCE LIMIT, N/M2
I I 1 I
4 6 8 10
FATIGUE ENDURANCE LIMIT, PSl
I
I00xi06
I I
12 14xi03
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EFFECTOFTEMPERATUREGRADIENTS
The effect of a radial temperature difference upon the overall disk relative
life was determined by considering the thermal stresses superimposed upon the
centrifugal disk stresses due to disk rotation. A steady-state radial tempera-
ture distribution was applied to a disk with a very small central hole. A disk
with a small central hole was considered because this configuration relieved a
mathematical singularity which occurs for isothermal boundary conditions upon a
solid disk. The effect of various uniform radial linear temperature gradients
on disk relative life is shown. AT is the total radial temperature difference
and KT is a constant equal to 0.56 K when AT is in Kelvin or 1F when
AT is in Fahrenheit, and where AT is equal to or greater than KT.
2
I _ WEIBUL_ SLOPE. LIO- \AT/
- ,I I I I
0 40 80 120 160
RADIAL TEMPERATURE DIFFERENCE, AT, K
1 I 1 i I
0 80 160 240 320
RADIAL TEMPERATUREDIFFERENCE, AT, OF
NORMALIZED .5
RELATIVE
DISK LIFE,
.2
klo
.1
.05
52
200
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EFFECTOFHOLESIZE
As discussed by Mahorter et al. (1985), tie-bolt holes will be the critical
location for failure in a disk. Using the finite element model, the effect on
disk life of bolt-hole size, location, and numberwas determined. The effect
of bolt diameter on disk fatigue life is shown. It would appear that at bolt
holes having a diameter of less than 10.2 nun(0.4 in.), the effect of hole size
is nominal. At bolt holes larger in diameter, the effect is most significant.
This analysis would suggest that the bolt holes in a disk should be smaller
but morenumerous.
NORMALIZED
RELATIVE
DISKLIFE,
LIO
10 O00xlO-6
1000
100
10
Z-
m
m
- I
0 0.5
I i
1.0 1.5
BOLT HOLE DIAMETER, CM
I I I
.2 .q .6
BOLT HOLE DIAMETER, TN.
I
2.0
I
.8
CD-88-32854
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EFFECTOFHOLE NUMBER
The effect of increasing the number of bolt holes is shown. Increasing the
number of holes and keeping the hole diameter less that 10.2 mm (0.4 in.)
appear to have less of an effect on life than having fewer holes with a larger
diameter.
NORMALIZED
RELATIVE
DISK LIFE,
LIO
10 000xl0 -6
5000
2000
1000
500
200
100
B
n
m
m
m
WEIBULL
SLOePE•
I I I I I
q 6 8 10 12
NUMBER OF CIRCUMFERENTIAL HOLES
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3-282
EFFECTOFHOLELOCATION
The effect of hole location in the disk is shownas a percent of the disk
radius measuredfrom the axis of rotation. The results indicate a general
trend of increasing life as the holes are movedradially outward. This would
not be unexpected because stresses decrease with increasing distance from the
center. The trend is not totally convincing, however, since there is an inver-
sion of the trend at a location between 40 and 50 percent of the disk radius;
that is, life decreases and then begins to increase again. It cannot be deter-
mined with reasonable certainty that this is correct or whether the finite ele-
ment meshsize was properly selected for these calculations. Further analysis
is required. However, it can be reasonably concluded that the bolt holes
should be placed as far from the center of the disk as is practical.
NORMALIZED
RELATIVE
DISKLIFE,
LIO
100O00xlO-6
10 000 ---
1000 _--
100 --
10
20
WEIBULL SLOPE,
e
1 ..-x\ _
_ \'- 3.57
I I
40 60
HOLE LOCATION, PERCENT DISK RADIUS
I
80
CD-88-32856
3-283
REFERENCES
loannides, E., and Harris, T.A., 1985, "A New Fatigue Life Model for Rolling
Bearings," Journal of Tribology, Vol. 107, No. 3, pp. 367-378.
Lundberg, G., and Palmgren, A., 1947a, "Dynamic Capacity of Rolling Bearings,"
Ingeniors Vetanskaps Akadmie_-Handlinger, No. 196.
Lundberg, G., and Palmgren, A., 1947b, "Dynamic Capacity of Rolling Bearings,"
Acta Polytechnica_ Mechanical Engineering Series, Vol. i, No. 3, pp. 1-50.
Lundberg, G., and Palmgren, A., 1952, "Dynamic Capacity of Rolling Bearings,"
Acta Polytechnica_ Mechanical Engineering Series, Vol. 2, No. 4.
Mahorter, R., London, G., Fowler, S., and Salvino, J., 1985, "Life Prediction
Methodology for Aircraft Gas Turbine Engine Disks," AIAA Paper 85-1141.
Weibull, W., 1939, "The Phenomenon of Rupture in Solids," Ingeniors Vetanskaps
Akademien-Handlinger, No. 153.
Weibull, W., 1951, "A Statistical Distribution Function of Wide Applicability,"
Journal of Applied Mechanics, Vol. 18, No. 3, pp. 293-297.
Zaretsky, E.V., 1987, "Fatigue Criterion to System Design, Life and Reliabil-
ity," Journal of Propulsion and Power, Vol. 3, No. i, pp. 76-83.
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Life and reliability of rotating disks

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