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TRANSCRIPT
,v
NASA CONTRACTORREPORT
--- ._ STATI·STICAL - COKPLEX USE(NASA-CR-72835)·· E 4340 STEEL AND ITS t
. ~FATIGUE DATA FOR SA TY D. Kececiog1u. elIN DESIGN BY RELIABILI _ ) 15 Nov. 1970, . Univ •• Tucson. CSCL 11F'al (At:l-ZOna _-------\, --_. "-..
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STATISTICAL COMPLEX FATIGUE DATA FOR SAE 4340 ·STEEL ANDITS USE IN DESIGN BY RELIABILITY
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Tucson, Arizona
by Dr. Dimitri Kececioglu and John L. Smith. jI(J, L
Prepared under Grant No. 03-002-044 by The University ofArizona
for
NATIONAL AERONAUTICS AND SPACE ADMINISTRATIOND. C.
Reprod"uced by . ~!\ NATIONAL TECHNICAL. INF~~~~!.e~~~.;~~~~ICE_. __ ._ ~Sp.r.i~fi"-Id,,~22151 _
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STATISTICAL COMPLEX FATIGUE DATAFOR SAE 4340 STEEL "AND ITS USE
IN DESIGN BY RELIABILITY
by
Dr. Dimitri Kececioglu and Johti L. Smith
Prepared for
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
.November 15,1910Contract NG~ 03-002-044
Project Management
byVincent R. Lalli
NASA-Lewis Research CenterCleveland, .Ohio
College of EngineeringTHE UNIVERSITY OF ARIZONA
Engineering Experiment Station. Tucson, Arizona
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TABLE OF CONTENTS".......-.-Page
• • • •• • • •
I, I • • • •
'ix
vii
• •• • •
• • •
· .. .I I
• • •
• • •LIST OF ILLUSTRATIONS· •
LIST OF TABLES
INTRODUC'l'ION
• • • • • • • • • •
1
xi
· . -I •• •I • • I
• •• • •
I • • •• • •
• • • •• • •
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ABSTRACT
CHAF-TEll
1.1 Background on Testing Program • • • • • • 1
1.2 Data Reduction. • • I • • • • • • • • • • 7
1.3 Goodness-of-Fit Tests • • • • • • • • • • 10.i .Lt, Generation of Statistical Fatigue
Diagrams From Test Data • .. • I • • • • • 10
1·5' 110dification of One Test Nachine. • • • • 11
II DATA REDUCTION • • I • • I I I • I • • I • • • 12
2.1 Determination of Stress Levels andStress Ratios • • • • • • • • I I " I • 12
Stress-to-Failure Data (StaircaseMethod) • • • >... . . . . . . . . .. . I • 19
• • •
Endurance Tests for R = ~ andR=1.0 •••••••••••
Endurance Tests for. R = 3.5
• I25
31
• I •
2.4
Cycles-to-Failure Data at SpecificStress Ratios and Levels •••••
Normal Distribution Fit to the Data • • •
35
36
2.4.1 Determination of Mean, StandardDeviation, Skewness, and Kurtosis. 36
Preceding page blank
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TABLE OF CONTENTS (continued)
Page
CHAPTER' "
-\'-.-,
II DATA HEDUCTION (continued)
2.5 Loe-Normal Distribution Fit to the Data •. 42
2.5.1 Determination of Mean, Standard'Deviation, Skewness, and Kurtosis.
2.6.1 Chi-Square Goodness~of-FltTest •• '
2.6.2 Kolmogorov-Smirnov Goodness~ofFit Test • • • • • • • • • • • • ._
Goodness-of-Fit Tests • • • • • " • • • • " ,
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43 '
43
44
45
47• •• •• •• • • •Discussion of Results2.7
2.8 Construction of Statistical FatigueDiagrams at Various Numbers of Cycles ofLife ...'. • • • • • • • • • • • • • • 53
./
III OVERALL CONCLUS I CI~S • • • • • • • • • • • • •
IV REcom'1ENDATI01'JS • • • • • ., . • • • • • • • •
APPENDIX AI Cm·iPUTER PRCGRAr'l TO CALCUUTESTRESS LEVELS AND RATIOS.'. • • • • • •
2.8.1 Use of Statistical FatigueDiagrams in Design • • • •
Modification of One Test Machine.
• • • •
• • • •
76
84
93
95
96
APPENDIX Bl COr:IPUTER PROGRAI1 TO CALCULATEPARAHE']1ER ESTH1ATES FOR TP.:E NORr·iALAND LOG-1'~02I'~AL DISTRIBUTIONS AEDCONDUCT GOODi(ESS-OF-FIT TESTS • • • • • 115
APPENDIX C, CAlJIBRATIOl: CONSTAETS OF EACHNACHINE FO?. 'I'm:: DIFFERENT NODES • • • •
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CIL\PTER..... ''''''-'''-''-
i. ....:... ~~I .
Page
APPEl'JDIX D:
APPEHDIX EI
cmiFUTER OUTPUTS LISTING STRESSLEVELS A!'!D RATIOS OF INDIVIDUALSPECH1ENS FOR THE VARIOUS STRESSLEVELS. • • • • • • • • • ~ .'. •
COHPUTER OUTPUTS LISTING STRESSLEVELS AND RATIOS OF ItIDIVIDUALSPECINENS FOR THE ENDURANCE TESTS
• • •
• • •
APPE~~IX FI FRINTOUT OF PROGRAM TO CALCULATETHE STRENGTH DIS'I'RIBUTIOE ESTD'IATESFOR liliAN AND STANDARD DEVIATION • • • •
LIST OF REFERENCES • • • • • • ' .. . ~ . . . .
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LIST OF ILLUSTRATIONS
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page
FIGURE
1.1 Schematic Diagram of Test Machines Showingthe Loading Configuration • • ',' • • • • • • • 2
1.2 Stress Element on Surface of Test Specimen ••• 4
1.3
1.4
Test Specimen • • • • • • • • • • • •
S-N Diagram ShoHing Cycles-to-FailureDistributions for a Specific Non-ZeroStress Ratio •• '.' ••••• ~ •••
• • • •
• • • •
• '6
• 8
Illustration of a Visicorder Record Sh01'11ngthe Bending, Torque and Calibration Traces ••. 13
-2.2 Flot·, Chart of Calculations to Obtain NominalStresses in the Specimen Groove/From theStrain Gage Outputs ••••••• • • • • • • 17
2.3 Endurance StrenGth Data Obtained by theStaircase Method for Stress Ratio of ~ forSAE 43lj·0 Steel, NIL-S-5000B, Condition C4,Rockl-Tell C 35/40 • • • ~ • • ',' • -. '. • • • • 26
2.4 Endurance Strength Data Obtained by theStaircase Method for Stress Ratio of 1.0for SAE 4340 Steel, MIL-S-5000B, ConditionC4, Roc.lnlell C 35/4,0,. • • • • • • • • • • • Ii 27
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2.6
Endurance Strength Data Obtained by theStaircase Method for Stress Ratio of 3.5ror SAE 4340 Steel, MIL-S-5000B, ConditionC4, Roc~1ell C 35/40 • • • • • • • • • • •
Cycles-to-Failure Distributions andEndurance Strength Distribution for StressRatio of 00 • • • • • • ~ • • • • • • • 0 0
Cycles-to-Failure Distributions andEndurance Strength Distribution for StressRatio of 3.5 ••• o· 0 •••••••••• ' 0
• 0
o •
• •
32
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so
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LIST OF ILLUSTRATIONS (continued)
page
FIGURE
2.8 Cycles-to-Failure Distributions andEndurance Strength Distribution for StressRatio of 0.825 ••••••••.•••• ••••• 51
/
Cycles-to-Failure Distributions at StressRatio of 0.44 ••••• • ; ••••••• • • 52
2.10 Fatigue Diagram ShmTing the rlodifiedGoodman Line and the Soderberg Line. • • • • • 53
2.11 Histogram Obtained From Cycles-to-FailureDistributions •••• • • • • • • • • • • • '. ,
- 2.12
2.13
Plot of Estimated Strength Distributionsat Various Cycles of Life and EstimatedCycles-to-Failure Distributions at VariousStress Levels for Stress Ratio ,of 3.5 ••
S~N Diagram Sho1'Ting Random Data Points andCell Structure for Testing and ObtainingDistributions •••• • • • • • • • • • •
• •
· '. 59
2.14 S-N Dia.gram for Stress Ratio of 00 ShoNingthe Estimated Cycles-to-Failure Distributionsand Test Band ••••••••••• '. • • • • 62
2.15 Relationship Beh:een the Stress Ratio Lines. • 72
2.16 Statistical Fatigue Diagram for Infinite Life.' 75
2.17 Conventional Fatigue Diagram Sh01·:ing theModified Goodman Line. • • • • ·' . • • • • • • 77, "
. 2.18 Sta.tistical Fatigue Diagram. • • • • • • • • • 78
2.19 Enlargement of Stress and StrengthDistributions on H2 Axis of Figure 2.18. • • • 79
2.20 The Distribution of Z = S - s • • • • • • • • 80
2.21 Gear Coupling Used on Fatigue Te?ting Machine. 86
2.22 Modified Gear Coupling\ ,f)
• • • .'. • ~ • • • • • " -92
'-. '-r"
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. LIST OF TABLES
. ~
':'" ... ' ,page
TABLE
'~~1:
2.2
! -.""\ ;. - ':.. -' ':..................: I"~ :.' •" ,__ ~~. _ ._~___ '" .-_"" _ t,
. '-,'" , '. ~... - ~ - - ..Table' Shovl1ng- the Average Stress Le'vels andRatios at Which Specimens Were Tested. • • • • 20·
Estimates of the Mean and Standard Deviationof the Endurance Strength for Stress Ratiosof co • :3. 5 and 1. 0 • • • • • • '. • • • • •.•• 24' .
2.3 List of Specimens and the Corresponding ~
Stresses and Pan Weights for Endurance Testsat Stress Ratio = co • • • • • • • • • • • • • 28
2.4 List of Specimens and the CorrespondingStresses and Pan Weights for Endurance Testsat Stress Ratio = 1.0 • • • • • • • • • • •• 29
2.5 List of Specimens and Corresponding ActualStresses and Intended stresses for EnduranceTests at Stress Ratio = 3.5 ••• .'. • • • • 33
Normal Distribution Parameter Estimatesand the Maximum D-Values • • • • • • • • • • • 37
2.7 Log-Normal Distribution Parameter Estimatesand the I"iaximum D-Values • • • • • • • • • • •. 38
2.8 Tables of Critical Values of D to Use in theKolmogorov-Smirnov Goodness-of-Fit Test. • • • 46
Estimated Normal Parameters for StrengthDistributions at Various Cycles-or-Life. • • • 57
2.10 Cycles-to-Failure and Stress Data for StressRatio of co at Stress Levels of 73,000 psi.a~d 81,000 psi ••••••• '.' ••••••• 63
2.11 Alternatlne Strength Distribution Parametersand the Corresponding Distribution ParametersAlong the Various Stress Ratio Lines ~or
Infinite Life • • • • • • • • • • • • • • • .~ 68
-TABLE ~,
LIST OF TABLES (continued)
\ . page\'---
2.12 Data and Results From Static Tests onNotched Specimens* ••••••••• -••••• 69
2.13 Data and Results From Static Tests onUnnotched Specimens. • • -. • • • • • • ,'-. •• 70
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ABSTRACT
A brief description or the complex fatigue machines
used in the test program is presented. .The data generated
from these machines are given and discussed. T~10 methods
of obtaining strength distributions from the data are
also discussed. Then follows a discussion of the con_v
struction of statistical fatigue diagrams and their'use
in-designing by reliability. Finally, ·some of the pro-.
blems encountered in the test equipment and a corrective
. modification are presented.
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CFJiPTER I
INTRODUCTION',--
1.1 B~ckgXQund on Testipg Program
The purpose of the test program is to generate cycles
to-failure data by testing rotating specimens subjected to
a bending moment and a constant torque. The test machines
were designed and built at the Univer~ity of Arizona~
The following is a brief discussion of the test
. machines. For a more complete discussion of the design and
development of the test machines see Ref. 1. Presently
there are three machines at the Univer~ity of Arizona.
The test specimen is subjected to a bending moment by
weights hung at the end of a lever arm. Fig. 1.1 shows a
schematic diagram of the machine illustrating the loading
geometry. The torque is applied by turning the torque
coupling, which rotates shaft A with respect to shaft B
and holds the relative position of the shafts. After the
specimen is tightened in the holding collets the adjusting
nut on the torque coupling can be turned while holding
shaft A from rotating. This will rotate shaft B with
respect to shaft A. It requires a full 360 degree turn of
the adjusting nut to rotate shaft B one degree with respect
to shaft~A. Once the play' in the gear boxes 1s taken up
1
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.2
SHAFT 'A'
TORQUE COUPLING
SHAFT 'B'GEARBOX
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SPECINENTOOLHOLDER
. . SPHERICAL ROLLE~R. . /BEARIHG
TORQUE GAGES J' .t-- BENDING GAGES 1_ 'W. '
-~It::!+
GEAR COUPLING TO PROVIDE FIVOTSOTOOLROLDS~~ CAN DEFLECT D01..;1'~HARDSLIGHTLY HHEN LOAD IS AYFLIED
SHAFT FORMOTOR -.
Fig. 1.1 Schematic Diagram of Test Machines Showingthe Loading Configuration
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then torque 1s exerted on the specimen.
'I'he roagni tudes of the shear stress and the normal
stress that the specimen experiences are related to the."
stresses ";hich the toolholder experiences. Two strain
gage bridse curcu1ts are mounted on the toolholder in the
position ShO\'111 in Fig. 1.1. Acting through an amplifier
the output of one bridge is a measure of the normal stress
and the output of the other bridge 1s a meo.sure of the
shear stress that the toolholder experiences. The out
puts are in the form of a Visicorder@record. The strain
gage outputs and how they are used to determine the stresses
in the specimen will be further discussed in Sect. 2.1.
Since the machines are capable of subjecting a specl-
--men to combined stresses, tests can be'conducted for any
combination of alternating and mean stresses. The alter
nating stress is in the form of a normai stress caused by
a constant bending moment on the rotating specimen. Any
element of volu~e located at the surface of the specimen
will experience an alternating tension and c~mpresslon of
equal magnitude. If torque is applied the element will also
experience a constant shear stress perpendicular to the
normal stress. Figure 1.2 show the stress element.
Early in the research program a study was made of the
various failure theories. On the basis of ~~ls study it
was concluded that the von Mises-Hencky theory most closely
predicts fatigue failures in steel (1. p. 41).
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'tyxm
't xym
± Sxa
';:ym' ..-,.'tyxm
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Fig. 1.2 Stress Element on Surface of Test Specimen
./
Shigley (2, P. 185) makes the point that the von
Mises-Hencky theory (also called the distortion-energy
theory) "las developed to predict yield under static loads.
However, because of the good agreement between fatigue
/
de. ta and the theory it was accepted as the failul-'e governing
___criterion for the research program. ~
nitudes of the alternating and mean stresses a stress
ratio 'ias defined in terms of the von Mlses stresses
(2, P. 188)~For an ordinary element subjected to bi-axial
stresses the von Hises stresses are given by:
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[82 82 2 ]
1S = - 8 Sya + + 31: xya 2
a. xa xa ya
[S2 S + S2 + 31: 21
Sm = - 8 ] 2 \xm xm ym ym xym .,---.,
For the case of an alternating normal stress along the x
axis and a constant shear stress perpendicular to the x
axis, the above equations reduce to
S = Sxaa
S = JJt XYIDm
The stress ratio is defined as
/
where
= (3)
R = stress ratio /
S = alterne.ting normal stressa
S = mean normal stressm
t Xym= mean shear stress.''j
The specimens being used for the'test program are
made of 8AE 4J40 steel, condition C-4, heat treated to
35 - 40 Re The specimens were all manufactured from the
same heat. Fig. 1.3 ShO\'IS the specimen geometry.
The ultimate purpose of the test program is to develop
statistical fatigue diagrams which can be used to design a
specified reliability into a rotating shaft subjected to a
bending moment and a constant torque 'for a specified life.
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In order to secure data to construct the statistical
fatigue surfaces the following test plan'was proposed
(1, pp. 132-138):~
1. Tests are to be conducted at various alter~
nati~g stress levels, holding the stress
ratio constant, to determine the cycles-to-
failure distributions at the various levels.
Tests, of 18 specimens at each level, are to~
be conducted at 4 to 6 different stress
levels for each stress ratio. This data can
then be plotted on log~log paper with stress
level on the ordinate and cycles-to- failure
on the abscissa. The resulting diagram will
look like fig. 1.4. Such a diaeram can be
obtained for' each stress ratio except R = o.2. Once the cycles-to-failure diagrams are ob
tained then the strength distributions for
specified lives would be obtained and these
distributions used in constructing the fatigue
diagrams to be used to design by reliability.
..
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1.2 p~ta Reduction.J
The data as t~1cen from the testing machines is in the
form of a Visicorder record which contains two traces for
each specimen. One trace records the amplitude of the alter-
nating normal stress as seen by the strain gages on the
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CYCLES-TO-FAILURE - LOG10 SCALE
['
Fig. 1.4 S-N Diagram Showing Cycles-to-Failur~Distributions for a Specific Non-ZeroStress Ratio.
. 8
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toolholderand the other trace records the magnitude/
of the constant shear stress. From these records the
nominal normal stress and average nominal shear stress ,can".
be obtained for all the specimens which were tested at a
given level. Theretermination of these stress levels
from the Visicorder records is discussed in Section 2.1.
The terms, "nominal normal stress" and ~nominal' shear
stress" refer to the stresses the specimen i'l'ould exper-
1ence at the outermost fiber of the cross section if it
did not contain a stress riser; ie, if the test section
were of a consta.nt diameter and that diameter were equal
to the diameter across the base of the groove in the
present specimen./
Section 2.2 discusses the methods employed in deter
mining the endurance level at the various stress ratios.
Sections 2.3. 2.4 and 2.5 deal "11th the calculation
of the mean and standard deyiation, and the coefficients
of skewness and kurtosis of both the cycles-to-failure da.ta
and the natural logarithms of the cycles-to-feilure data.
T,-lo computer programs '\'Tere developed to aid in the'
reduction of the test data:'
1. A program to reduce the Visicorder records
to stresses and stress ratios.
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A program to calculate the mean and· stande.rd
deviation of the cycles-to~failure data and
test how well the data fits the normal dis-"tr1bution. The program also makes these
/.
calculations for the log-normal distribution.
Th~ progra.ms are d1 scussed. in Appendices A and B.
Included in the -discussion is a complete description of the
input dEl..ta card formats for ee.ch deck a.nd a flow chart for
The firs£ program mentioned above-was developed by
this au.thor whereas the second was previously written but
was mod.1f.led to be used with the data of this research
program.
1.3 £L92Q~f~~9f-FitTests
Seotion 2.6.1 discusses the Chi-square goodness-of-
fit tent and its applicability to the test data. Section
2.6.2 discusses the Kolmogorov-Smirnov goodness-of-
fit te~t and its applicability.
1.4 Q~11.s1',?t!oYL2.f Stati,stica-1.. Fatig]..~~ Di8.l!rarnsr
From
~~.~t~.. Da ta
Whe generation of tbe stati~tical fatigue diagrams
requl~es that the strength distributions at various cycles.\
of lite be knOl'l·n. The test Pl~n that 1'laS proposed at the
beg1nntng of the research effort was set up.with the
objeotlve of obtaining test data from 'Nhich estimates could
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be made of these strength distributions. Section 2.8
discusses the proposed method and its limitations. In the
same section a~other test plan and method is proposed' for
obtaining the re~uire~ distribution based onactu~l data.
Section 2.8.1, discusses how to use statisti~al fatigue
diagra.ms to design a shaft to a specified reliabi1ity. The
assumptions and 'limitations of the method are also dis-
cussed.
1.S ~ificati9n of One Test ~achine.:
-\011 th tl)e approval of Dr. Kec~ci9glu and I1r. Vincent R.
-Lalli of NASA-Lewis this author made a modification to
"
one of the three test machines in December, 1969. It is
believed that this modification '"Till eliminate some of the
problems encountered ,oJ! th the machines. None of the data
presented in this report was obtained from the modified
machine. Details of 'the modification are presented in
Section 2.9.
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<-CHAPTER II'
,'\ DATA REDUCTION.'-....:..
2.1 peterm~na~ion of Stress Levels and Stress Ratios
As stated in the introduction, the raw data is in the
form of a Visicorder record' containing two traces.
Figure 2.1 is an illustration of what such a record looks
like ~ The traces are.a m'easure of the alternating normal
stress and constant ,,:shear stress as seen by the strain
gages which are mounted on the machine toolholdef. The
values of these traces ,0 in terms of divisions, can be
converted into values of nominal normal stress and nominal/
shear stress in the groove of the specimen. However, to
be able to make this conversion several constants must be
known which relate the output of the strain gages on the~. ..
toolholder to the nominal stresses in the groove of the
specimen. The calibrations required to obtain these con
stants are qUite extensive and since the original cali
brations were not conducted by this author only a brief
discussion will be undertaken here. For a complete dis
cussion of the ca.libration 'procedure see Ref. 3.
There are five constants required to make the con~
version from divisions on the Visicorder record to nominal
12
BENDING GAGE OUTFUT TORQUE G~GE'OUTPUT
TORQUECAIIEF~ATION
TRACE
ZERO. BENDING
TRACE
ZEROTORQUE
TRACE
,.-----ROTATING BY POi.·!ER--------,
-..t-rT~~ROTATING BY EAED / ~_!
BENDInGCALIBRATIONTRACZS
L •.' \ .
Fig. 2.1 Illustration ofa Visicorder Record Showingthe Bending, Torque and Calibration Traces
"
=
1~
stresses in the test speci~en. They ares
constant to relate the output of the,strain gages on the specimen to the
, actual stress in the specimen.
'totK -bit - cr ot·
where
=
=
=
=
constant to relate the output of the
bending strain gages on the toolholderto the output of the strain gages on the
specimen.
constant to relate the actual shearstress in the specimen to the output of
th~ torque strain gages on the toolholder., .
constant to relate the output of thebending gages on the to61holder to the
./
output of the torque gages on the tool-holder ,-rhen the toolholder is subjected
to a changing torque while the bending,moment is held constant.
constant to relBtethe output of the torque
gages on the toolholder to the output ofthe bending gages on the toolholder.when
the toolholder is SUbjected to a changingbending moment While the torque 1s heldconstant.
=\
normel stress in the specimen as indicated by
the output of strain b~ees on the test speci~en.
" .
CI = actual normal stress in the specimen.as
Clot = normal stress in the toolholder a.s indicatedby the strain gages on the toolholder.
~I
las = actual shea.r stress in the specimen.
~ot = shear stress as indicated by the straingages on the toolholder.
The last, two constants listed are interaction con-
stants. Due to misalignment, a bending gnge may record an
output"'t-Jhen the shaft is sUbject~d. to pure torsion. Like
wise, a misaligned torque gage may react to a pure bending
on the shaft. These t~TO constants take this interaction
into account.
Whenever a strain gage on the toolholder ,is replaced
or when a change is made, to the michine that alters its
loading characteristics some of the calibration constants~,
may change. Therefore" "lhenever such changes are TIl:ad,e,
calibration tests must be performed to obtain corrected
values for the constants that are affected by the change.
Changes that affected the calibretion constants ....rere me.de
at three different times throughout the test program. In
order to kno;rwhat constants were in effect during given
time periods each time period was designated as a mode of
operation. The values or these constants are given in
Appendix C.
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Figure, 2.2 shows the sequence of calculations required
to convert the strain ga3e outputs, as recorded by the
Visicorder f to nomin~l stresses in the groove of the test'.
specimen. The meanings of the symbols used in Fig. 2.2
Which have not been defined earlier in this section are:
= number of Visicorder record divisions used
when adjusting the gain in the bendingchannel of the amplifier.
number of Visicorder record divisions usedwhen ad justing the gain in the torque che.nnelof the amplifier.
Reb ~- value in ohms of the calibrating resistanceused when adjusting the gain in the bendingchannel amplifier.
Rct = value in ohms of ca.librating resistance usedwhen adjusting the gain in the torque channel
~f of the amplifier.
=
=
amount of deflection, in Visicorder divisions.
caused by the toolholder bending stra.in gagebridge output when load is applied to the
specimen.
amount of deflection, in Visicorder divisions,caused by 'the toolho1der torque strain gage
bridge output when load is applied to specimen.
Output bending stressin toolholder fromVisicorder divisions,
°ot· =
Output bendtng stress into61holder corrected fortorque ~nteraction ,
~ ,Cfot = a ot - Kt,ib t ot
Actual nominal bendingstress in specimen groove
17
Output shear stressin toolholder fromVisicorder divisions
Output shear stress intoolholder corrected forbending interaction
I
tot = tot - Kb/ t aot
/
Actual nominal shearstress in specimen groove
=/a ot
K Kgr-th bgr
/Stress Ratio
R = _(Ja~.fJtas
Fig. 2.2 Flow Chart of Calculations to Obtain NominalStresses in the Specinen G~oove From theStrain Gage Outputs.
E = modulus of elasticity for steel
= 30 x 106 psi.
. \
.R b resistance of each bending gage '--='g= 190 ohms.
Rgt = resistance of each torque gage
= 120 ohms.
Gb = bending gage factor
= 3.23
J Gt = torque gage factor=. 2.05
NQ
, = number of active gages in each bridge
= 4<'
J th = polar moment of inertia of the toolholder
cross section where the gages are mounted.v
Cth = radius of the toolholder i'lhere the gages are
mounted.
= 1.000 inches.
= polar moment of inertia .of the specimencross section at the base of the groove.
= radius of the specimen at the base of thegroove.
= .• 249 inches.
In Fig. 2.2 the equation for a ot ' in the second
19
• •CJ ot - ~t(b tot1,.+ Kt / b Kb/ t
=
box on 'the left. is an a.pproximation. ·The full equation
lSI
Upon examination of the values of Kt / b and Kb/ t in
Appendix C. it is seen that they are small B.nd th~ product
ls negligllble compared to 1. Thus. Eq.(4) reduces to
the one given in Fig. 2.2 •
A program for the CDC 6400 computer at The University
.of Arizona using FORTRAN IV was developed to determine
CJas 't "Las and R. as per Fig. 2.2, from the data. . The,
program, along with user instructions and flow chart, is
presented in Appendix A.
Table 2.1 lists the average stress levels and ratios
at which test data were collected. The computer outputs
listing the stresses in the individual specimens are given
in Appendix D.
2.2 St~ss-to:.lliilureData (Sj;.ft.irca§e !'!ethod)
The method used for testing specimens to determinec
the distribution of the endurance strength was the stnir
case method (II-. p. 48 j 5) sometimes called the "up a.nd
down method". Briefly, the method consists of testing 8.
specimen subjected to a stress equal to the estimated
endurance strength. If the specimen fails, the next speci
men is subjected fo a stress one increment Imler than the
failed specl~en. If the specimen does not fail by a
'.
.~-
,J)
TAEL
E2
.1
TAEL
ESH
OWIN
GTH
EAV
ERAG
EST
RESS
'LEV
ELS,
AND
RATI
OS
ATW
HICH
SPEC
Ir1EN
SWERE~
TEST
ED
Av
erag
eA
vera
.ge
Sta
nd
ard
Num
ber
of
Av
erag
eS
tan
dar
dD
ev.
Str
ess
Nor
mal
Dev
.N
orm
alS
pec
imen
sS
hea
ro
fS
hea
rR
ati
oS
tress
*-p
s1Stress~~·::·
Test
ed
Stress~:·**
Stress*J,~
_,1
44
,00
01
50
01
20
°1
14
,00
09
00
18
°°
00
98
,00
02
70
01
80
08
1,0
00
90
0(
18
00
73
,00
01
80
01
80
0
15
1,0
00
38
50
,1
22
5,0
00
80
0
3·5
11
5,0
00
18
20
18
19
,50
07
00
83
,00
01
20
01
81
3,5
00
75
0--
--
74
,00
07
50
18
12
,50
06
00
"1
11
.00
01
25
01
27
3,5
00
20
50
92
,00
06
70
0'\
18
66
,50
03
40
00
.82
5,
76
,00
03
15
01
85
3.5
00
3L~50
65
,00
03
85
01
84
7,0
00
13
00
0.4
46
9,0
00
lL l-
OO1
89
0,0
00
'16
00
60
,00
07
00
18
78
,50
01
40
0"j'
*R
ound
edo
ffto
neare
st1
,00
0p
si.
**R
ound
edo
ffto
neare
st5
0p
si.
*~H~
Rou
nded
off
ton
eare
st5
00
psi.
( ,
(.'1
'".
t\) o
21
predetermine~ time the test is termin~ted and the next spec-." J .~ .7:" .t-.~.: 0 ':.: ,:.~. _": :': ": .~~. .,.., . ~ ,"
imen is SU?j~~t~dto~a ~tress one increment higher. This'~"':',i .... : .,. _' :-. . I •
proced~re continues until the desired sample size is,','"....
obtain~.d.· It 'should be noted that only approxlmately~. ,,; ~'. ~'-,
"/ .... ,
50~~ of the --t'ested spe'cimens are used in the calculations".. r "
of mean anq stan~ard deviation. The calculations are based
on either the successes or failures, whichever has occurred"t-. .... ~ '.,. -~ .. -- ',.,. "~. ."' .•' ("':: :._..... ". - ,I. } p.... __ ._ •• _. _ >'," • _ • _,. '__ • ,_ '\.. ~
the least number, of times. The specimen just preceding
the first change of mode is considered as the beginning~1r}::::_ :~ C-:,;:".?: ~.~" ....._: ~~ ....-~ ..- .. :,.
of the test. ,A change~of mode is a success followed byC' ~
t.·-~~l ~:. : >. .L ~~ .... ::~.- ,I :. 'L •• I .... -'
a failure or a failure followed by a success.
The equations for calculating the estimates of the').:-
mean and standard deviation of the endurance strength are:1]::>: D.. ~~~-~.··:::. 'c . - ':'. :'"~'~.:: ::
given by (5, p. 114).
wher,a.
ro, = mean,.:.: ••~. ~'." • >.-' •.• -.-
fi,,;= s.tanQard devlation
./
(5)
.(6~
_.' _,·Y. =. tjle ).O\'lest .St!BSS level at \-lhich a success or"--, ..
. .. .-.- ... ~- . --. - .
fai~ure .(whichever the analysis is based on)
" ,C>ccurred.
""", ,d. == stre~s .increment... ..-.o:~-~- ..- •..•...•._.. -.-
"":-' .. "-.
22
"-N ~ effective sample size; Ie. the number of speci-
"mens used in the calculations.
'--n
A = I: i n11=0
nB"= I:
1=0
n = number of success (or failures) ''lhich occurred at
the I th level.
The lowest level is considered the zeroth level. the nextstthe 1 level, etc. In Eq. (5) above the (+) is used if
the calculations are based on successes and the (-) if
based on failures.
The above analysis requires that the variate being
tested is assumed to be normally distributed or can be
transformed to anorroal distribution (5, p. 111). Also,
the stress increment should be in the range of .5a to 2a,
where a is the standard deviation of the distribution.
Therefore, some prior knowledge of the variance is helpful
for good results.
The staircase method is very good for the determination
of the mean of the variate being tested. However, the
estimate of the variance can be poor if the sample size" -
1s not large. Mood and Dixon state that (S, p. 112),
uNeasures of ~reliablli!l. may \-lell be very misleading if the
sample size is ;tess thanfort21. or !ifJ.y. II It is not clear,
23
i.J"}~I'1eVer,'~-l·lhether- they are referring to effective or actual
sample size, the effective sample size being approximately
50% of the act~aJ. sample size. Because of· research pro
gram limitations the effective sample sizes of the endur
anCe tests range from 16 .. to 18.
En~~r?nce tests ~ere conducted at R = ~, 3.5, 1.0 and
0.44. As sf~t~d-b~i~~e, the tests of specimens which do.' .~. -. - ... ._-~ - - . -",
not fail are terminated at a predetermined time. For1
stress ratio of ~ the tests were terminated after 90 hours----~-,;....:.._..__._.__._.._--- --~------ ._..._------
.-
W~lch--is-e~uivalent to more than 9.5 milliori cycles. In-------_._--_.~._-_."""-~._._-----.~ •.. _.'-'._-
conducting the endurance tests at stress ratio of 00, of the
eighteen specimens which failed, sixteen failed before 48• t ". .. 0-;', ,
hours of , running time. ,- - /-i" .• -. ~
<- ; -For stress ratio of 1.0 the tests were terminated at
'---- --_._-------_._-_._------_..-- ---
48 hours which is equivalent to over 5 million cycles.
Of the eighteen specimens which failed during the endurance
;:~ests at R.= 1.0 all.failed before 24 hours of running time.
~h~ endurance tests at R = 3.5 were terminated at 24 hours
which represents more than 2.5 million cycles •. .
.' The endurance .tests at R = 0.44 have not been com-c:.
pleted as of the date of this report. They are also being
terminated at 24 hours of running time.
The calculation of the means and standard deviations
from the staircase data will be discussed separately in the
next two sections. The results of those calculations are
"given in Table 2.2
"\'
--' TABLE 2.2
ESTnL~TES OF THE HEAN AND STAl\TJ)ARD DEVIATIONOF THE E1IDURANCE STRENGTH FOR STRESS
RATIOS OF «), 3.5 AND 1.0
''--- '
Stress Ratio Nean* (psi) Sta.ndard Deviation**
co 57,000 3,800~,
3.5 55,000 3.700:,
1.0 57,000 3.300
(.0
Estima.tes a.re based on the calculations discussed in
Sections 2.2.1 and 2.2.2 •
* Rounded off to nearest 1,000 psi.
** Rounded .off to' nearest 100 psi •
./
, .
L: 2.2.1 ~ndurance Test§ f£r R = ~ and R = 1.0
The tests were based on the amount of weight. '.
25
placed on the loading arm. An increment of one pound in--.-the pan was used to obtain the staircase. The calculations
for mean and standard deviation were also based on weight
and these values were then converted to psi.
Figures 2.3 and 2.lj. ShOi'1 the staircase plots
for the tests. They were taken from an earlier report
(6, pp •. 48-49). Tables 2.3 and 2.4 list the specimens
that 'Nere used in the endurance level calculations for
R = ~ and R = 1.0 respectively._I
To obtain a better estimate of the endurance
strength distribution the average of the stresses in all/
the specimens for a given pan weight was found and then the
increment between these averages for the different pan
weights was calculated. In order to obtain a uniform
Incrementan average was found for the increments.
For stress ratio of ~ the calculations are
based on successes. The average stress at each pan weight
and the number of specimens t'lhich succeeded at each level
ls:
Pan Height Stress Number ofpounds psi Successes
27 .b?~ 61,857 126 "'0 59,999 225 54,~JO 1024 51,915 223 .59,421 1
;,
.~
osu
ccess
,if
spec
imen
surv
ived
101cycl~s
ofa
ilu
re,
ifsp
ecim
enb
rok
eb
efo
re1
07
'cy
cle
s
~
:.'
Num
ber
of
Sp
ecim
ens
o5
10
1520
I 2530
35("
Fig
.2
.3E
nd
ura
nce
Str
en
gth
Dat
aO
bta
ined
by
the
Sta
ircase
Met
hod
"fo
rS
tre3
sR
at:!
.oo
feo
f.or
SAE
4-34
0S
teel,
I1IL
-S-5
000l
.),
Co
nd
itio
nC
4,R
ock
wel
lC
35
/40
.
t\)
0-.
Leg
end:
()su
ccess
,if
spec
imen
surv
ived
4:7
:1'0
6cy
cle
s
ofa
.ilu
re,
ifsp
ecim
enb
rok
eb
efo
reL~
x1
06
cycl
cn
:.>,
"
o5
10
152
025
3035
40N
umbe
ro
fS
peci
men
s
Fig
.2
.4E
nd
ura
nce
Str
eng
thD
ata
Ob
tain
edby
the
Sta
ircase
Met
hod
for
Str
ess
Rat
ioo
f1
.0fo
rSA
E4
34
0S
teel,
MIL
-S-5
0CO
B,
Co
nd
itio
nc4
,R
ock
wel
lC
35/4
0.(
N.
"'-J
TABLE 2.)
LIST OF S:FECI~:.ENS AJlJl) THE CORRESPOlIDINGSTRESSES AND PAN HEIGHTS FOR
ENDURANCE TESTS AT STRESS RATIO = ~
Specimen Pan Nomina.l Norma.lNumber Height stress in Specimen
pounds psi.
87 25.0 60,51917, 25.0 37,476
--74 25.0 46,827 -202 J 25.0 51,125182 25.0 57,353124 25.0 60,lUl
216 26.0 '58,017/,
205 24.0 49;580199 25.0 54,036209 25.0 53,903177 24.0 54,249168 25.0 57,132190 26.0 61,981
-.207 ,27.0 61,857142 25.0 65,528208 23.0 . 59,422
\
----
28
29
'F~:', i:C1-~/.-: tIST' OF 'SPECIMENS AND THE CORRESPONDING
"f:(;::: DC;:::· ~< ''"/. '··STRESSES AND PAN WEIGHTS FOR '-~'- ," '" ._. '\" .' .
. E~IDURANCE TESTS AT STRESS F.ATIO = 1.0·tt;",:;; .C':,::*: c-' 'r '":.:-.:' ~.• :: _ ~ •• ~. .' . .
~ ..' . r· ~._.. .- .' ..... -
$pec1men Pan Nominal Normal Stress StressNU!l1ber Height in Specimen Ratio,
: . .p<:)1Jnds psi •.. '.. -- .. -.
248 27.0 48,979 .721..~ ::- • -"!,,,.".
288 27.0 61.723 1.072. :':"-267 '" ::.:: 26.0 58,350 1.045.. ,.
..c··"·-300 - ~ : ':. ,27 .0·. 55,537 1.005.,
~ .'.' '" 0 ~'15.. " .?b,'O: . 59,810 1.169... -.- 409···· " ...... ". . .
25.0 57,067 1.066:. ::':;64 ","",' ... -,
2~·. 0":.. .. _. : •••• f 52,863 1.040.' , ~"'372 "r:'26 0 58.885 1.042- ..-_ .... -- , . . :: '- ...... 3~2.. : ... 28.0 61,561 1.031't' .,- '.,~ ~.. - ._- -~, . " .._, , ..
429 27.0 58.918 '1.019359 27.0 58,727 1.055417 : ,r 26 0 55,581 .996· . -- -' .- .336 . .. 26.0 54,156 .909,.. _.. ' i ..435 26.0 56.298 1.114401 25.0' 57,266 1.081400 '. 29.0 60,166 1.064
.. ·
386' · 28.0 : 58,619 1.063~:::~.
402 27.0 . 58,288 1.087
"'-:,!":",:::.-",,..• __ .J...,.....:..::
.., ~ .-, ~ -...'t,:,;_", •
• _ ...... "J
.,
30
Note that the average stress for the one spcciwen at a
pan weight of 23 pounds is greater than the average stress
for both 25 and 24 pounds. To eliminate this inconsistency,'I
the one data point taken at 23 pounds ues elil:1ine. tGd from
the calculations. Then, taking the average of the incre-
ments betvTp.en the other four levels the value of 3, 31!}
psi. for the stress increment was obtained.
In order to get values for the number of spec
imens '\'lhich succeeded at each level, 1t was assumed that
each specimen tested at a given pan weight was subjected to
a stress equal to the average stress calculated for that
and
. 1:: ;.• ~
,-, ' .....
m = 57.317 psi.
s 1.620 (3.314) [15(27.2-(17),2 + 0.029J= .~.- -" (15)2
s = 3.818 psi.
Thus the estimates for the mean and standardr-"'C _: .·;deviation of the enclurance strength distribution for stress(: .--
;~~~atlo of 00 are 57.317 psi. and 3.818 psi. respectively.
Similarly. the estimates for the mean and,~r-.. '
'standard deviation. of the endurance strength distribution-! \, .. ~ -.. -----..- - .-',::for.istress ratio of"··-l~O., "Tere found to be 56.785 psi. and
The staircase method was used to conduct the..
'..
) ,_ ....---'<;3.290 psi •• respe'ctively.
~':' ~'. ----;:Endura,n.9.? Level f..Q.r R_ = ~.5.
(tests at this ratio also. but instead of basing the tests-
:;.on ,pa.n '\'Teight they l'Tere based on stress in the specimen.
The.stress increment was chosen to be 3.000 psi. Figure 2.5,1
;ShOi'lS the staircase plot for these tests. Table 2 i 5 lists
:the specimens which were used in the calculation of the
{distribution parameters. Also listed are the stress as
,recorded by the strain ga.ges. the target stress and actual~ .
~tress ratio in.each spccimen6;.'"
~.
Table 2.5 iridicates that the stresses were not.,..- . '-
held as close as targeted when the tests were conducted.
The scatter may be due: to carelessness on t~e part of the'-.- .•~. .
operators and terii ma~hine operating inconsistencies.
Leg
end
:.
6o
success
,if
spec
imen
surv
ived
2.5
x1
0cy
cle
s.
6.
Ofo
.1lu
re,
ifsp
ecim
enb
rok
eb
efo
re2
.5x
10
cy
cle
s
69 66.
'\/~6
JP
.I,
~
16
til
0ti
lC,
)
H ,~57 54 51
'-,'0
;.,
510
.15
2025
3035
40
(
.j I ;
Fig
.2.
5N
umbe
ro
fS
pec
imen
s
En
du
ran
ceS
tren
gth
Dat
aO
bta
ined
by
the
Sta
ircase
Met
hod
for
Str
ess
Rati
oo
f3
.5fo
rSA
E43
40S
teel,
MIL
-S-5
000B
,C
on
dit
ion
CL~,
Ro
ckw
ell
C3
5/4
0.
\.0
)l\
)
",. .. -.-., ... .~-" .'
TABLE 2.5
LIST OF SPECIi>1ENS AIm CORRESPONDING ACTUAL STRESSES.~.;",:. "',,"::" :"~ ~-~ ", ~
AND INTENDED STRESSES FOR ENDURANCE~ TESTS AT STRESS RATIO = 3.5
-
Specimen. Actual Normal Intended Normal Actual
Number S:tress in Stress in StressSpecimen (psi) Specimen (psi) Ratio
477 5.5,917 60,000 3 •.5446.5 .5.5,82.5 . .57,000 4.26478 .58,291 60.000 - 3.64.506 .51,2.56 57,000 . 3.22528 .57.376 ~ 60,000 3.78444 .53,410 57,000 3.26466 54,236 .57,000 ;.60.524
/
47,071 . .51.000 3.03546 .50.292 54,000 3.93499 60,903 57.000 . .5.72532 .54.533 57,000 4.43483 50,828 54,000 3.10.529 51,381 54,000 3•.56587 52,067 54.000 3 . .50581 .54,320 57,000 3.67611 .56,U26 ·60,000 3.42,
.589 .54,118 57.000 3.4-7603 53,813 - 57.000 3.41
33
These are dHrcu.s'sed·' in·:Section 2.9. Because of this the
.. average ~"actual:::stres-s ':at"each target level 1'1aS determined
'''''. '.tf· '11 --, .. :- ",-.",'- !; ": . ~ '"~C.,·, ,as 0 011s. ,. C",_, ~ -" , •. _, ,:' - .,
Average .·Ac.tu8.1..:.,~_·::.~"~,':,~'l'a~get 'Stress
Stres?.;:.~Jp~~~~:.." (psi)
Number of
.Successes
57,1'03'·': c'
·fE.s..~ t B23 " :'~.'
51,142.t,1'·47; 071 '
~~'::::'::'; '.60,000
,~-,...= ..57,000~. :"'- .. ' ~ .
.54,000
51,000
'4
941
......-.•.~. " .....
. ..:.-:/,-'-~" '-:
r'," •·•.The average stress increment bet'\<leeri these......... " ;... ,- .-- -- .' ~ '- - .". -.' ;.-' - ... -.
levels is3.37'Z"ps1.· Using Eqs: (5) and (6) l'There.;........ ..- _..' _. - .......- .-- "...- . ~ '., . ' .. - "..---'. ~-
. ,d")r'~' ; ,r,.".~...-' ~.y _=-1}7. 071 . psi.\..-..... ~-\. , .......:. ,," \..~.. "'-.. - - ... :,:.- .
,/
. -,Eo G"..:;i,.::.z,.,· +Co:. "';,.:= 1.f~ /~ ~.~ (; ', ... ::::> ...
·r~':.···,,,· .;.~ .. :~.' .. ,.... )-'" - •., ..-~,-~, -_ •.•',<. ". K'~ ~' "1 ni"=' 1(4) + 2(9) + )(4).
fE-.~:;'__:.':-: :.:: "::,:"::.:' ~~1.. c_=)4
l·}.:;·C.:).~'·<:.O-'}'c-~':·B'~'~ i 2 ni= 1 (4) + 4(9) + 9(l~)1=1
1],:',': :-..:: :::. -? :'.". , .- .- ." - - 76
.then . ., . '.. ., .' ,... , ,. .'- - ..... - . - .
::.~ =.47,071 + 3.377 ( ~'+~ )
= 55, 139 ps1 •
"
+ 0.029J
"l' 0,.-_ ~, ..... "'l>~.-."... ~_
:: - . '" -, ".-••~ .f
. .t.t,C: ,--.:6 . ,... E.:'~,· .~_
35
Thus, the estimated mean and standard devi-
. ation of the endurance strength distribution for stress
ratio of 3.5 a:r;e 55,139 psi.• and 3,738 psi., respectively.
2.3 £y.~les~-to-~·.~J1t2I.e I?C!.~gj; S~Jflc S1;;1:'~£:LB2~tJ.2§'
~nd Leve1Ei
It was desired to determine how the cyc1es-to~
failure data at each stress level and ratio vIas dis-
tributed.
Based on a study made by Broome (7) in an enrlier
report on this research program it. was decided to fit the
. normal and log-normal distributions to the data and con-
duct goodness-of-fit tests.
The parameters for these two· distributions were
estimated using the unbiased 'estimates for mean and stan
dard deviation. For the normal case the cycles-to-.
fallureis the variate and for. the log-normal case the
natural logarithms of the cycles-to-failure is the variate.
The moment coefficients of skewness and kurtosis were
also estimated from the data. For the normal distribution-the values of these two parameters are 0 and 3. respec-
tively.
The Kolmogorov-Smirnov goodness-of-fit test was used
as a. measure of hm'; 1'Jel1 the distributions fi t the data.
A computer program was used to make the calculations.
The program had been deve~oped previously but it was modifie~.
by this author to include the Kolmogorov-Smirnov test.
The original program, which was written by Patel (8), used
only the Chi-square goodness-of-fit test. The pro8ra~~and
user instructions are included in Appendix B.
Table 2.6.lists the results obtained by estimating
the normal distribution parameters from the data and con-
ducting the Kolmogorov-Smirnov goodness-of-fit test.
Table 2.7 lists the results obtained by using the natural
logarithms of the cycles-to-failure data.
2. It- Formal Dtstribution F~.t to the Data·.
The probability density function for the normal
distribution is given by
/
1 ( 2S;-~ }2f(x) - 1 e
- 2 '0 _ C<) < x <00=o.j2n' c
(7)
..
l'lhere ' A = mean
o = standard deviation
If)J... and 0 are not 1(n0i'1n from prior knowledge they
must be estimated from the data as discussed in the next,I. ~-
section.
2.4.1peter~1nationof Hean, Standard DeviRtL2DJ~
~ke\'mess and KU'l"tos1,s
By definition the moment generating function,
denoted by met), is the expected value of e tx which is
denoted by E [etXJ. For .the normal distribution
;,
f\
,.
..'
TABL
E2
.6
.NOR
MAL
DIS
TR
IBU
TIO
NPARA~!ETER
ESTI
MA
TES
AND
THE
MAX
DV
ALU
ES
Str
ess
Ave
rage
Sam
ple
Nor
mal
Dis
t.P
aram
eter
sM
ax.
Alt
ern
s.ti
ng
0<3
O<l~,
Frat:-i~
Str
ess
Siz
eS
tan
dar
dD
Lev
el(p
si)
Mea
nS
kew
-D
evia
tio
nn
ess
Kurtosl~
Val
ue*
14
4,0
00
12
2,7
73
.53
2-1
.31~4
3.5
61
..1
75
11
4,0
00
:"'-1
89
,02
9<1
,02
4-.
24
71
.99
5.0
99
00
·98
,00
0-
18'
22
,17
1:3
,815
-.0
42
1.8
55
'.082
81
,00
01
87
7,9
77
12
,54
9.8
73
3.0
60
.181
.j.
73
,00
01
81
61
,98
43
4,2
86
-.6
16
2.2
01
.17
4~~--
15
1,0
00
-12
1,4
51
:27
5-.
22
21
.72
5'~113
-3•.5
11
5,0
00
186
,20
31
,01
0-.
797
3.3
97
.111
83
,00
01
83
9,6
08
12
,99
01
.87
67
.24
1.2
39
74
,00
018
74
,21
21
9,5
31
.14
72
.05
7.1
32
11
1,0
00
126
,55
41
,00
5.8
93
2.1.
;93
.29
60.
82.5
92
,00
018
20
,46
75
,59
5.2
39
-2.2
57
.11
97
6,0
00
186
1,0
28
11
,13
2.
-.L~28
\2.1
82
.10
96
5,0
00
181
27
,57
72
1,2
27
.171
'1.9
~~2
.11}
1,6
9,0
00
185
3,5
73
15
,J~'
701
.66
74
.83
3r
O-'
•
0.44
\~??r---
60
,00
01
81
42
,55
041
,5t$
5.3
93
3.0
57
•_I.H
3
,,....
-\ i
~4-
~~~
.62.~
'C::P
::P....:
;,\:
:;:;
t:r
~.
-Jr--
.~.
~1axlmum
D-V
alue
From
K-S
Test
~
t.( ,
w ....,
TABL
E2
.7
LOG-
NORM
ALD
ISTR
IBU
TIO
NPA
RAM
ETER
ESTI~~TES
AND
MAX
D-V
ALU
ES
Str
ess
Av
erag
eSa
r.1pl
eL
og-N
orm
alD
ist.
Par
amet
ers
~H
e.x.
Alt
ern
ati
ng
Q0
(3~
0<4.
-~ "'l.......
.I...-
-....~
_~atio
Str
ess
Sta
nd
ard
1;:-
DS
ize
Mea
nSl(e·~l-
f-'•
<•
'~.'/;'"
;_
.
Lev
el(p
si)
{u
rto
sis
,.'D
evia
tio
nn
ess
Val
ue*
lL~4,
000
12
7.9
06
54
7.2
2791
4-1
.58
6,4
.25
8.2
07
11
4,0
00
189.
1019
61.1
1613
0-.
Ll'0
72
.12
3-
.09
20
09
8,0
00
18
9.9
92
12
3.1
7638
2-.
26
51
oL~5-
.094
-./
'
81
,00
01
81
1.2
52
66
7.1
53
95
4·5
75
-2
.69
7.•
159
73
,00
01
81
1.9
70
99
7.2
31 -1
-941
-.9
06
2.7
07
'-,.
19
8
15
1,0
00
12
,7
.26
26
76
.19
77
89
-.4
35
2.0
06
-.1
28
3.5
11
5,0
00
18
8.7
20
89
4.1
5724
7.L
I·03
-2
.86
9-
.08
6U
3,00
01
81
0.5
45
16
0.2
06
66
4.6
86
-4
.40
7-
.18
47
4,0
00
18
11
.18
03
38
.27
42
19
:....3
422
.50
1-
.08
4'
11
1,0
00
12
8.777701~
.ll~5966
,.7
}-I-
2-2
.29
5.2
80
0·.8
259
2,0
00
189
.89
01
29
.280
831
-.1
71
-1
.91
6.1
20
76
,00
01
81
1.Ob19L~0
.1°S
06
2-.
'70
32
.'4
57
-.1
11/.
65
,00
018
11
.7L~3320
.167
235
-.0
30
-1
.76
9.lL~7
0.4
46
9,0
00
18
10
.85
67
36
.2).}
t)421
1.2
10
-:3
.899
-.2
14
60
,00
01
81
1.8
25
35
1.3
05
28
2-.
45
73
.18
2.1
18
*M
aXim
umD
-Val
ueFr
omK
-ST
est
\~
..~
.J)
( I
\j.\
'
'v)
,
co
..~....
,..'.
" .
1
o.J2'iT'
rCOtxJ e '
..00
dx
·39
Carrying out the integration yields
met)
which is the moment generating function for the normal
distribution.
The procedure for obtaining the rth,~oment
1s to take the r th derivative of the moment generating.
function l'lith respect to t and evaluate it at t = O.
By definition the r th sample moment is given
by nE r
1=1 xi .
n •
Taking the first derivative of the moment
generating function with respect to t and evaluating at
t = 0 yields
Then equating the first distribution' moment and the
first sample moment yields the estimate for the meannE
A i=l Xi (8)/1.= -n-
It can be sh01'711 that this is an unbiased estimator.
Taking the second derivative of the moment
generating function l'lith respect to t and evaluating at
·... :.
40
t = 0 yields
2 2m2 = 0 +/"\
Equating the second moment yields
nt 2
= 1=1 x tn
but
nt x;: 2
= [i:=l ...1Jn·
'.':'.
thus
n[ t x J2
i=l i-- n 2 - =
/' .,
. 't1here
nt
xi- i=lx = =n sample mean.
It can be shown that this is a biased estimator f6r 02
For unbiasedness, the expected value of the parameter
estimator must equal the parameter (9, p. 72) thus in this
case for unbiasednessnt (x _i)2
E [ i=L-i_ J =n .2 .o •
"
p-1nE [
Going through the computations yields
nt ( -)2
i=l xi-x _ =n J
,41, '.
Therefore the estimator
nI:' - 2
a 2 :: 1=..1 ~: t-X)
" n, \,','~
•
1s a biased estimate for cr 2 .' However, the unbiased
estimator for 02 is
nI: ( -)2
i=l xi-x--n-l-
The moment coefficients of skewness and kurtosis are
g1ven by
and 0(4 =
respectively, where
nI: -)2
= 1=1 (xi:.:_n
nL ( -)3
= 1=1 xi~=':n
-'
and nI: -)4i=~i-:'_
n (See Ref. 10, p.48).
Evaluating the third and fourth derivatives of
the moment generating function at t = ° and su;bstituting
into the equations for 0(3 and ..DC L~ '..;111 lead to values of
eX) = ° and 0(4 = 3 for the normal distribution •.
The estimates of the ~ean, standard deviation
/'
...42
and the moment coefficients of skewness and kurtosis have
been calculated for each group of 'cycles-to-failure data.
The estimates of the moment coefficients of skewness and~,-
kurtosis were calculated in order to give a further indi
cation of hOi'r \'lell the. data fits the normal dictrlbutlon.
However, it is important to note that even though 0(3
of a symmetrical distribution is zero, obtaining avalv.e
of zero for the estimat~ of 0<3 from the data does not
necessarily mean that the distribution is symmetrical.
Mood and Graybill make this point (9, p. 109) and state
that, "knOi'rledge of the third moment gives almost no clue
as to the shape of the distribution." Therefore, the
moment coefficient of skewness is'not a good measure of
whether or not a distribution is symmetrical.
The moment coefficient of kurtosis is a
measure of the peakedness of the distribution.
Lo~-Nor~al Distribution Fit to the Data~ - - - -
A variate is distributed log-normal if the logarithm
of the variate is distributed normal. That is, by letting
y = l08eX then
1f(y) :::: - eo~2TI'
\-There the unbiased estimates of}{ and a are
nE
Yi~ - .i~lA= Y :::: n
.'; .....
.~.. , .. :;: _..-.
Deter~ination of Mean, Standard De~iation,
§kewness and Kurtosi!
The determination of these parameters for the
log-normal distribution is the same as for the normal
distribution as covered in Section ,2.4.1 except the log
of the variate is used in the calculations.
'2.6 Goodness-of-Fit Tests---~..-...:;:;..;;;...;;.~
In o~der to determine how well the normal and log-
normal distributions fit the dat~ two goodness-of-fit
tests were proposed. They are the Chi-square and Kolmo30rov-/
Smirnov goodness-of-fit tests. However, both tests were
not used in determining whether or not to reject a
particular distribution. In all cases the Chi-square
goodness-of-fi t test "!as not valid because of the small
sample sizes. Further reasons for it not being valid are
given in the next section.
Since the COMputer program is w~itten using both
tests it should be pointed out that when analyzing data
where the Chi-square test is valid, if both tests are used
to determine rejection, the level of significance changes.
In other words, if the critical values for ~ejection are
based on a significance level of 0<.. for both tests and the
criteria for rejection of the null hypothesis is if either
..44
one 6r the other or both tests rejects the distribution
then the level of slgniflcance in rejecting is not ~.
It is not clear what the ,confidence level would be sinc~
the extent to which the two tests are correlated is not
known. Therefore, only one test should be used in deter
mining goodness-of-fit. For the data in this report only
the Kolmogorov~Smirnov test was used.
2.6.1 Phi-S~~~e Goodnes§:of-Fit Test
The Chi-square goodness~of-fit test can be .
used only '\'lith grouped data, that is, data divided intoI-~'
cells. The total Chi-square value is the sum of the Chi-
square values of each cell. This can be written as
kL
i=l
. where
'E = the expected frequency of the i th celli
°i = the observed frequency of the i th cell
lk = total number of cells
Vk _1 = total Chi-square value
·2'~k-1 can be shown to be distributed Xk _1
hence the name Chi-square test, where k-1 1s the number
of degrees of freedom. HO'lleVer, 1f the parameters of'
the hypothesized distribution must be estimated from the
data the degrees of freedom must'be decreased by the number
of parameters estima.ted. In the case '\'There the f1 t of the
46
where
Dn = Kolmogornov statistic'-,
"Fn(x}
Xl= S f(x) dx
"_ co
f,(x) = probability density function of the
hypothesized distribution
xl = any specific value of the variate X
=xN (11 )
X": = number of observations less than or
equal to xl
N "- sample size/'
If Dn 1s greater than some critical value (Dc)
the hypothesized distribution is rejected at some level of
significance (~). The probability statement is P(D > D ) =n c .oC ." Table 2.8 lists the critical values for sample sizes
of 12 and 18 at various levels of significance.
TABLE 2.8
TABLES OF CRITICAL VALUES OF D TO USE IN THEKOLNOGOROV-SiHRNOV GOODl'JESS-OF-FIT TEST
Sample Level of Significance (ex:)
Size 0.10 0.05 0.01
12 0.338 0.375 0.450
18 0.278 0.309 0.371
. .
In using the Kolmogorov-Smirnov test the\ ~
D-value is found at ea.ch data point. If at any point the
calculated D-value is greater than the critical va.lue .at~l
>the desired significance level, the hypothesis is rejected.
If not, the hypothesis is not rejected.
:·2.7 Qi.s.cussion of Resu.l~s
In 8 of the 15 stress levels tested the D-value-
.. (Kolmogol'OY Statistic) was less for the log-norme.~dis-
tribution than for the normal distribution. The moment
coefficient of kurtosis was closer to 3.0 for the log
norma.l distribution in 9 of the i5cases. The moment
coefficient of skewness was closer to zero for the log
.. 'normal distribution in 7 of the 15 cases. HOI'leve=c, asJ •
was mentioned in Section 2.4.1, obtaining a value close
to zero for the moment coefficient of skel'mess does not
necessarily mean that the distribution is sJ~metrlcal or
normal.
Based on the Kolmogorov-Smirnovt~st, in no case can
either the normal or the log-normal distribution be re
jected with 90% confidence. For stress ratio of 0.44
at stress level 69,000 psi. the normal distribution can ~•
be rejected with 85% confidence •
From the above results i~ appears that the log
normal distribution is favored.
The cycles-to-failure data for stress ratios of ~
/
48
and 0.825 were previously analyzed and reported by Broome
- (7). He also concluded that the log-normal distribution
was favored. Ee also points out that from a phenomeno~
'.logical viewpoint the log-normal distributior. 18 justified.
Figures 2.6 through 2.9 show plots on log scales at
the average stress levels of the cycles-to-failure dis
tributions for stress ratios of ~, 3.5 , 0.825 and 0.44 •
The distributions shown are log-normal. The mean line
'shown on each figure was obtained by fitting a straight
line, by the method of least squares, to the estimates of
the means of the distributions on each figure. The 3
standard deviation limits for each distribution are also
shm·m. To obtain a' smooth envelope the dashed lines ":ere/
drawn in by sight.
Specimens were tested at only two stress levels for
the stress ratio of 0.44. They were 69,000 psi. and
60.000 ps~. An attempt was made to test at a level of
75,000 psi •• but at a stress ratio of 0.44 this requires a
shear stress of 98,500 psi. The specimen began yielding
and finally broke as the torque Was. applied. Thus, for
stress ratio of 0~44 the cycles-to-failure tests were
restricted to two stress levels. The endurance tests for
stress ratio of 0.44 have not yet been completed but it
appears as thou8h the mean endurance limit ~s about
51.000 psi.
r=~
80• -' 0) p.
70~
... .s:: ~
60~ (1
) H -t->
50C
I)
-ill2
'rr
~2
J4
56
85
t40~
10
.2
34
56
~8
6.'
s::1
02
34
56t
8I
r4"
10
7I 2
.34
56
81
04
t.
+-.
II
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AE
4-34
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TE
EL
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il-
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=,'M".
,,--
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,~r-
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i\'++
++H
1'1H''''~l+!JH:r.r
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l.
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03
18
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16
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14
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0P
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. tI)
tI) ~
10
0~ ~ (/)
90to s:: oM
80
.;.> m s:: ~~ 0)
70~ r-
i<: r-
i60
~ s:: ....1 R 0 ..,. ....
50
Cy
cle
s-to
-Fail
ure
Fig
.2
.6C
ycle
s-to
-F
ail
ure
Dis
trib
uti
on
san
dE
nd
ura
nce
Str
en
gth
Dis
trib
uti
on
for
Str
ess
Rati
oo
f~
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00
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43
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40
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34
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81
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23
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Cy
cle
s-to
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Fig
.2
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ycl
es-t
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a11
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Dis
trib
uti
on
San
dE
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Str
en
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2.8
Cy
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ibu
tio
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and
En
du
ran
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tren
gth
Dis
trib
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for
Str
ess
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5.
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6
Cy
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-to
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Fig
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s-to
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Dis
trib
uti
on
sat
Str
ess
Rat
ioo
f0
.44
.
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ilure
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-,--
t""""
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..,,',,,:,,,
"....,'",
.'.
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•"I
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.,,,.,
,,w',f
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120
Po
"~
- fJ) ~31
00
.!-! ,'J
(I)
90to l::: ..-l
.80
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70.p ..~ .-:.: ,...,
60
oJ s:: .,,-1 s 0 z
50
-.-_•
.._-...
._--
-.--
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---
2.8 :ponstruction ,of StQ...t.~Gtical Fe.ti.n:u.e Diap;rams at
Various Numbers of Cycles of Life-- • _ _... .... '~_ . r.
-The ultimate objective of the research program is"
to construct statistical fatigue diagram~ from which
shafts can be designed for a specified cycle life and
reliability.
One kind of conventional fatigue diagram is one
using the modified Goodman line as shm'1n in Fig. 2.10 •
Sendura11ce
NODIFIED·· GOOD!tIAN LINE
SODERBERG LINE
NEAN srrRESS
Fig. 2.10 Fatigue Diagram Showing the ModifiedGoodman Line e.net the Soderberg Line.
The diagram is a plot of alternating stress on the
ordinate versus mean stress on the abscissa. The modified
Goodman line is a line connectil15 the endurance strength
and ultimate tensile strength. Another line, which
. ;,
! .
\---
'_•.~ .. :' ., '
-, .
,connects the endurance strength and yield strength, is
called the Soderberg line. Both the modified Goodman line
and Soderberg line are conservative (2, p. 178).
The modified Goodman line is used to deslgn for an
infinite life. Other lines can be constructed connecting
the ultimate strength to some alternating stress corre-
sponding to a specified life. This line would be used to
design for that finite life. However, these types of lines
are deterministic and do not account for variability in the
strength of a material. If the 'strength distributions can
be determined for various stress ratios and cycles of life
then fatigue diagrams can be defined in terms of a mean line
and standard deviation about the -line.
The method of determining the streneth distributions
which has been proposed for this research program has been
discussed'in earlier reports (1, pp. 134-135; 11, pp. 24-26).
Briefly, once the mean line and 3a envelopes (a = standard
deviation) have been established on the S-N diagrams as
discussed in Section 2.7 then distribution parameters can
.be interpolated at evenly spaced stress levels. Suppose,
the strength distribution is desired at N cyc1es~ A hist-
ogram can be constructed along the N cycle line such that
the mj.dpointsof each histogram cell is one' of the inter
polated stress levels, as shm'm in Fig. 2.11-. The ordin
ate of each cell is the area, to the left of N1 ' under the
55
distribution curve corresponding to that cell. For
exampl~, denoting the bottom distribution function on
Fig. 2.11 as f{NIS1 ) then the ordinate of the bottom hist-'\
ogram cell is
N1F (N1S1) = S. f (N Is1) dN.
_ co(12)
Likei'i'ise, the ordinate of the next cell '\'fill be
N1. F(NI3
2) = ) f (N IS2 j ,dN. (13)
_ co
In general the ordinate of the i th 'cell/ l'1ill be
"
.'
N1F(NIS i ) = '~ f(Nls i ) dN
_ co,/
(14)
and the total histogram will look like that shown in
Fig. 2~11. This total histogram, a~ a first approx-
1mation, is taken to be cumulative strength histogram of
specimens failing by N1 cycles. The probability density
histogram can be obtained fro~ this cl~u1ative histogram.
Denoting the strength random variable 8.long the N1 axis as
S then the value of the i th cell of the probability density
histogram ,is f(S1) = F(NIS i ) - F(NIS i _1 ). After the
probability density histogram is found the~ the normal di~-
tribution parameters can be estimated using statistical
method.s.
.:\
f(N
IS3
)
;.~
,
',-'
\
g(S
)
PRO
BA
BIL
ITY
DEN
SITY
HIS
TOG
5AI'l
f(N
IS2
)
CID
IULA
TIV
EH
ISTO
GRA
M
Ss.C
I) .~s4
idh0~~
~ U)
s3Q ....
S~
2~ ~
Sl
CLl
E-t
H <:
N1
CYCL
ESTO
FAIL
UR
E(
.Fig
.2
.11
His
tosr
amO
bta
ined
From
Cy
cles
-to
-Fa1
1u
reD
istr
ibu
t10
ns
\J\
.~
.'..
A computer program i'las developed by R. E. Smith (11)
using this method of estimating the strength distribution
parameters. A printout of the program is given in Appendix
F.The estimates of the strength distribution parameters
at cycles of 10,000, 50,000 and 100,000 for a stress ratio
of 3.5 are listed in Table 2.9. These distributions are
plotted on Fig. 2.12 • Note that the ± 30 limits are
inside the ± 30 envelope for the cycles-to-failure dis
tributions; hence, for this case the method appears to ,
yield unconservative estimates of the strength distri-
but10n parameters. The reason for this \'1ill be inves-
tigated.
TABLE ~2.9
ESTIHATED NORf':1AL PARAHETERS FOR STRENGTH
DIS'l'RIBUTIOnS AT VARIOUS CYCLES-OF-LIFE
Cycles - Parameter Estimatesof Normal Distribution
ofLife N Nean Standard Dev.
(psi) (psi)
10,000 106,639 3,256
50,000 79,021 3,253100,000 70,972 2,313
\
"
r
\J\
<:J)
T
r1
~.145
RAD
•.~
.50
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,-
roT",
SAE
4340
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.735
DIA
.-
70
.
zr-l
.+<.U
60
s:: ....-l S o~ C ....-l -lJ Q s:: H r.)
-lJ
r-l <
18
0±
!2f=
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j;ds
H]o-
Fait
ur~
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fitI
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160~
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60
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,
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04
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Lt-5
68
10
52
34
56
81
06
.2
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56
81
07
Cy
cle
s-to
-Fail
ure
Fig
.2
.12
Plo
to
fE
stim
ated
Str
en
gth
Dis
trib
uti
on
sat
Var
iou
sC
ycl
eso
f.
Lif
ean
dE
stim
ated
Cy
cle
s-to
-Fail
ure
Dis
trib
uti
on
s·at
Var
iou
sS
tress
Lev
els
for
Str
ess
Rat~o
of
3.5
.'
."
· 1
59
Another approach might be to base the strength dis-
tribution estimates on actual data so that some measure of
accuracy ca.n be obtained through the use" of an appro-\
priete goodness-of-fit test. The approximate distributions
of strength at a specified number of cycles is wanted,
then the distribution of the stresses to which the spec-I
1mens failing at N cycles rire subjected might be obtained
as follows. N 1s a random variable and the probability
"of even one specimen failing at exactly N cycles is zero.
Therefore~ the stren~th distribut~on at a discrete N isL.::,
unattainable, but a distribution of specimens failing
within a band of N is possible to obtain.
/
CONVENTIONALS-N LII\TE
Fig. 2.13 S-N Diagram Showins Random Data Pointsand Cell Structure for Testing and Obtaining Distributions.
60
Suppose the strength distribution is desired at
N1
cycles. From eXisting data an 'estimate of the con
vention~IS-N curve can be obtained and a range of stress~ . .
correspondinsto 103 cycles and the endurance limit can
be obtained; Sand S on Fig. 2.13. Then specimens',0 e
could be tested at random levels '\'1i thin this range. The
stress range ,could be divided into cells, as shown in
Flg. 2.13 and specimens tested at random levels within
each cell. This would insure a more even distribution of
data points.
Once the random data is obtained then the parameter
estimates can be calculated and goodness-of-fit tests
conducted. A cell with N1 as its midpoint can be con
structed as sh01·;n in Fig. 2.13. For a conservative
estimate the cell may be constructed so that N1 is the
lower boundary. The estimate of the mean strength at N1
cycles '\·;auld be the average of the stresses at which the
specimens were tested that failed within the N1 cell (the
darkened data points on Fig. 2.13). The width of the cell
would depend on ho'w good an estimate i-laS desired. Using
these data points the estimates of any distribution para~
meters could be obtained and goodness~of-fit tests con-
ducted to determine which distributions to reject. As an
example, Fig. 2.14 is the S-N diagram for stress ratio of ~
sh011ing the estimated distributions of the cycles-to-failure
data. Suppose it is desired to find the strength distribution., .'
1
1
, .
61
for 70,000 cycles. If a cell of width 30,000 cycles is
constructed on the conservative side of 70,000 cycles it
passes through two distributions. The actual data pO~Dts
which fall within the cell can be obtained fro~ Table 2.10
which lists the cycles to fe.ilure data at these tl'iO stress
levels for stress ratio of ~. For stress level 73,000
psi. one point falls in the cell and for stress level
81.000 psi. 13 points fall in the cell for a 'total of 14
points. Calculating the mean and standard deviation of
the 'strength for these points yields
Then. using the Kolmogorov-Smirnov goodness-of-fit test./
it can be determined whether or hot the normal distribution
1s a good estimate for the strength distribution at 70,000
cycles.
The D statistic, as given by Eq. (10). is
where
and
e
2~2
dz
0\
N
T
11
SAE
43
40
STE
EL
CO
ND
.C
-43
5-4
0R
•c
~--+-
+---4-
.-aL
\:.~
45H
AD
.
.50
0D
IA.
-r
~'~11lJ
III:I
III
,"'1I
II,
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rt-+(h1ajJ~s-to-faiiur~
IDffi
1f' f'.
-.",-
W,
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t:::A
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(":f
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r~.
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I}80
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=I
17
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60
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c·'"
IT'I'.
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40
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68
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56
81
05
2,3
L~5
68
10
62
345~'8
10
7
Cy
cle
s-to
-Feil
ure
Fig
.2
.14
S-N
Dia
gram
for
Str
ess
Rati
oo
f~
Sho
win
gth
eE
stim
ated
Cy
cle
s-to
-Fail
ure
Dis
trib
uti
on
san
dT
est
Ban
d•
18
0;,
16
0
14
0.
• -e ~1
20
·~
.. t/) ~
10
0~ .;..>
U)
90
.
...
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.'-
'-'.
-..
e~
..--
,
TABLE 2.10
CYCLES-TO-FAILURE AND STRESS DATA FORSTRESS RATIO OF ~ AT STRESS LEVELS
~OF 73,000 PSI. A1ID 81.000 PSI.,'--
63
Average Stress Averoge StressLevel of 73,000 psi. Level of 81,000 psi.Actual Cycles-to- Actual Cycles-to-
Stress-psi. Failure Stress-psi. Failure73,796 125,950 81,820 64,99773,886 103,413 81.282 67,75970.969 . 145,783 81,321 83,812*72,029 195.824 80,421 78. 852':-~
71.973 93,303* '82.641 60,00274,3 l}2 178.192 81,027 6l },85773.404 196.894 80.667 71,968*72,572 183,306 79,201 86,349o
;l-
74,931 177,508 81,832 82,2lj·8*72,374 203,019 80,032 74,690*72,925 155.178 /80,871 75,662*74,,819 205,041 ' 83,112 71. 166'~75,982 172,661 82,001 71,556*'73,647 124,13'7 81.224 '73,089'K72,961 127,794 81,423 96,004*74,629 172,572 81,363 107,41568,029 172,275 81.363 98, 468~}72.164' 182.860 81,784 74,698~;'
-
* Indicates Specimens Whose Cycles-to-Failure Fall-
Within the Range of 70,000 to 100,000 Cycles.
b~~_. S is the value of the stress of the data point being
examined and
S (x)n .•
§'"== N
._~here1 .:: •
S == the number of points "Thich have a. stress less
than 01' equal to s.
and":'_. :.. ~ ~
·N == total nu..111ber of points.
For example, using'the last data p~int uri4er the
.~l,OOO psi. stress level column in Table 2.10;
S = 81, 78l.j· psi.
N == 14
Since 81,78l.j· psi. is the sec:ond highest stress va.lue
of the data points there are 13 points less than or equa.l to
it, so
The value of z is
81. ,78h-80 t 51.6 4zl = -Z,b32"- - == o. 82
- From standard normal tables this vo.lue of zl corresponds to
Fn(X) = 0.620
Therefore-
. D = 0.309
From a table of critical D-vo.lues (\.;hich can be found in
most statistics books which discuss the Kolmogorov~Smirnov
_~_"" - P(D> 0.292) = 0.15
test) it 1s seen that for a sample size of 14
ITpus, since th/~ D obtained here is greater than 0.292---
this normal distribution can be rejected with 85% 06n
fidence that the distribution does not fit the data, 'but
oan not be rejected with 90% confidence.
Past studies by others indicate that for a given '?cycle life the fatigue strength distribution is normal
(12, p. 351).TV,J.<) ,;w'.'.4
One reason the normal 'distribution can be ~.
rejected here with 85% confidence may be that the sample
used to estimate the mean and standard deviation is not.
random and therefore the estimates may be biased. Definite
stress levels were aimed for and although there is some./
scatter about those levels, the sample cannot be considered
a random one. The properties of the estimators for the mean.-
and standard deviation are knOl'into be good if the sample
1s random. .If the sample is not random the estimates may
be biased or insufficient. It is for this reason, if this
method is used, that the data can give ~ittle more than
. gross approximations of the strength distribution parameters.
The present data might be useful if more tests were con-
ducted randomly throughout the stress range to give an
even distribution of points.
It must be pointed out that a very larg~ number of
specimens may.have to be tested to obtain enough specimens,
,:..
66
so tha t preferably more than 35 fail \"1i thin the narrO\-T
cycle life range desired for sufficient accuracy, par
:~1cularly for strength distributions at lower cycles of. life •
.. The staircase. method should also be tried for finite
life to see how the results from the three methods compare.'
Once the random data is obtained and the strength
distribution parameters have been calculated at the
- desired cycles of life, construction of the fatigue dia
"_.grams, is ~e.latively simple.
AssQ~ing the stress ratio is held constant for the data
points on each S-N diagram then the alternating strength
distribution can be transformed to the mean strength
distribution through the constant R. When either dis-
tribution and the stress ratio, R, is known then the. other
distribution is completely defined, so only the alter
nating strength distributions will be \-lorked 1'1i th.
The means and variances are related by
S = R Sandn m C1 S 2 = R2C1 S .2,
. a m
1'1here
§ =a
S =m
Os =a
mean of the alternati~g strength distribution'1~
mean of the mean strength distribution
-standard deviation of the alternatingstrength di~tribution
. I
\."..standard deviation of the.mean strength
dis tr i bu t i 0:1
As an example, a fatigue diagram using the endurance,'- -
~,
strength distributions calculated in Se~tions 2.2.1 and
2.2.2 will be constructed. The endurance strength dis-
tribution parameters are listed in Table 2.11 •
The distribution for stress ratio of 0 is the
ultimate strength distribution for unnotched specimens as
obtained from tensile tests. The details and results of
these tests were reported in an earlier report (6) and
therefore will not be discussed here. The results are
presented in Tables 2.12 and 2.13 " Twenty specimens
• were tested. Ten were identical to those used in the
fa tigue tests (Fig. 1.3) and ten ,-tere unnotched specimens.
The unnotched specimens eXhibited a yield point, which was
recorded, and the notched specimens did not have a notice-
able yield point.
The reasons for using the ultimate strength· dis
tribution obtained from the unnotched specimens rather than
that for the notched specimens is as .follm'ls. The mean or
constant stress that the specimens are subjected to in the
test program is a shear stress. However, a mean normal
,stress is desired for the mean stress axis of the fatigue
diagram. If the specimens would have. been tested to fracture
in static torsion the shear stress distribution ~rould have
been converted to normal stress b;>r the relationship Sm= .J)' t .
~
TABL
E2
.11
ALT
ERN
ATI
NG
STRE
NG
THD
ISTR
IBU
TIO
NPA
RAM
ETER
SA~~
THE
CORRESPO~roING
DIS
TRIB
UTI
ON
PARA
MET
ERS
ALON
GTH
EV
ARI
OU
SST
RES
SRA
TIO
LI1~S
FOR
INFI
NIT
EL
IFE
,,'
Str
ess
En
du
ran
ceS
tren
gth
Co
rres
po
nd
ing
Dis
t..
Rati
oD
istr
ibu
tio
nP
ara
mete
rs'
Par
amet
ers
Alo
ng
RA
Xis
!ole
anp
si*
..~S
tan
dar
dD
ev.
!'1ea
np
si**
Sta
nd
ard
Dev
.R
Sp
si*
**
Sps
lol'<
**a
CiS
r'
CiS
ar
00
57
,00
03
,80
05
7,0
00
3,8
00
3.5
55
,00
03
,70
05
7,0
00
3,8
00
."1
.05
7,0
00
3,3
00
81
,00
01.
!-,7
00,
~'"0
17
8,0
00
2,5
00
17
8,0
00
2,5
00
*T
hes
ev
alu
es
are
the
dis
trib
uti
on
par
amet
ers
of
the
ult
imate
stre
ng
tho
fu
nn
otc
hed
spec
imen
so
bta
ined
fro
mte
nsil
ete
sts
.,
,**
Rou
nded
off
ton
eare
st1
,00
0p
si.
***
Rou
nded
off
ton
eare
st1
00
psi
.,(
"l.f.
.,",:'
0'\
Q:)
? .
TABLE 2.12
DATA AND RESULTS FROH STATIC TESTSON NOTCHED SP3CH1EIJS*
(stress Ratio = 0)
'\......
TrueTest Ultimate Breaking Ulti83.te Bree.king
Load Load Stl:'ength StrenGthNo. 1. OOO.lbs. 1,000 Ibs. psi. *~. psi. .:<*
1 49.3 47.0 253,500 305.0002 49.6 . 47.0 255,000 305.0003 49.4 46.3 254,000 299,5004 50.3 47.4 259,000 299,5005 48.8 46.0 251,000 306,5006 49.2 46.0 253,000 302,5007 49.6 46.8 255,000 304,5008 49.8 47.1 256,000 305.5009 50.5 47.7 260,000 309,500
10 lj·9.9 47.5 256,500 302,000
Hee.n 255,500 30!~" 000
Sta.ndard 2,500 3,000Devia.tion
* Average specimen diameter at the base of thenotch 1s 0.4975 inches.
** All strengths rounded to nearest 500 psi.
TABL
E2
.13
,.
DATA
AND
RESU
LTS
FROM
STA
TIC
TEST
SON
UNNO
TCHE
DSP
ECIM
ENS
(Str
ess
Rati
o=
0)
Test
Yie
ldU
ltim
ate
Bre
akin
gD
iam
eter
Are
aY
ield
~ltimate
i{e~\ing
No.
Loa
dL
oad
Loa
dA
vera
geAve~.
Str
en
eth
fStr
eng
thS
tren
gth
1000
lb10
00Ib
1000
lbIn
.In
'Os
i-l~
psi·:~
"psi
~.
131
8.5
32•.5
24
.30.
4753
p.1
77
41
77
,50
01
83
,00
02G
L t-,0
002
30
.63
1.5
23
.50.
4764
0.17
841
71
,50
017
6,.5
0025
1 .t ,,0
00
33
0.6
31
.62
3.3
o.475'~
0.17
751
72
,50
01
78
,00
024
9,00
04
20
.83
1.0
22
.80.
4755
0.17
76,
16
8,0
00
174
,50
025
1,50
0
52
9.9
'3
1.1
22
.90.
4758
0.:1
.778
16
8,0
00
17
5,0
00
254,
50Q
63
0.1
31
.32
3.2
0.47
220.
1752
17
2,0
00
17
8,5
00
256
,500
.'7
30
.43
2·5
'25
.00.
4787
0.18
001
69
,00
01
81
,00
025
6,0
008
30
.23
1.4
24
.20.
4757
0.17
771
70
,00
01
78
,00
025
3,00
0'.
9-
30
.63
1.6
23
.60.
l.l-7
63kJ
.178
21
72
,00
017
7,.5
0025
9,00
0.
24
.20.
4755
0.17
661
76
,50
010
30
.33
1.2
17
1,0
00
250,
500
11ea
n1
71
,00
01
78
,00
025
5,00
0
Sta
.nd
ard
.3
,00
02
,50
04
,50
0D
evia
:tio
n
*A
llS
tren
gth
sR
ound
edto
the
Nea
rest
500
psi
.
""o
. }
It is generally accepted that the static strength
of ductile steel is not affected by a stress concentration.
ThercfoTe, theoretically, the same shear strength distri-~ .
bution should be obtained whether the specimens are notched
or not.
However, the mean normal strength distribution was
obtained directly from the tensile tests. In the case of
tensile tests a notch does have an affect on the strength.
The notched specimen has a higher ultimate strength due
to a radial stress being introduc~d into the specimen
at the root of the groove (6,. p."19). A grooved specimen
subjected to a static torque load would not experience
this radial stress. Therefore, the strength distribution
used at stress ratio of 0 is the ultimate strength dis-
tribution for the unnotched specimens.
In order to construct the fatigue diagram the dis
tributions along the various stress ratio lines must be
calculated. The reqtiired relationships can be derived
from Fig. 2.15. They are: .
.. .
. but
gTan e = _a = R = stress ratio
Sm
S~=Si~
. "
·.
./
Sa
'
.S.....---
a
Sm
..'
/.~
R-
AX
IS
\.
S.
m
-'
Fig
.2
.15
Rel
ati
on
ship
Bet
wee
nth
eS
tres
sR
ati
oL
ines
J~
.,.::...~
4·
:'ji
-"~'
(
"l\)
- -
Sin 0 Tan e R- 1 ' - 1[l+(Tan e )2J2 [l+R 2J 2 \
'--"\
thus
- 1~ (l+R 2) 2
S -a (16)- .R R
,
Using the same derivation the expressions for the
standard deviation along the R axis are
'18
aSaS
a== SinO-
R
'and
1Os (1+R 2 ) 2 /
Os_ 9.
== HR(18)
SR,3.5
Referring to Fig. 2.16 the distribution along the
alternating stress axis (R = (0) can be plotted directly.- 0
For this axis 0 = 90 and Eq. (15) yields S = S 00H, 00" a,
and Eq. (17) ytelds aS 00 == aS 00 .~_ H' a'
For the distribution along the stress ratio line
3.5 Eqs. (16) and (18) give
2 1= ~..0(l+R )"2
H
1.52~=- 3.5 57,201 psi.
for the mean and
=
\'--
for the standard deviation.
The same procedure would be used along any other stress
ratio axis except the appropriate values of S and a· ~.a sa
would be u~ed. The results of the calculations for the
parameters for endurance are listed in Table 2.11 and the ~.
resulting fatigue diagram is Fig. 2.16. The endurance
tests for R= 0.44 have not yet been co~pleted but from the/
tests that have been run it appears as though the endurance
level will be in the range of 49,000 to 53,000 psi. and the
scatter will be wide. The actual distribution will probably
be very close to \-That the distribution sho1'm on Fig. 2.16,
in dashed lines, looks like.
It p~obably will not be possible to conduct tests at
ratios nuch 1m'Ter than 0.4-4, especially at high stress
levels. The stress ratio is R = S / S and the 1mver limita" m
on S is the endurance 1ev~1. The upper limit on Sis the. a myield strength 'l'1hichis 171,000 psi. Thus the lOi-rer limit
on stress ratio, assuming the endurance does not fall much
below 51,000 psi. 1s
=:0
Ult
imat
eS
tren
gth
Dis
t.u
nn
otc
hed
speci~en
iR
:=1
.0
20
Yie
ldS
tren
gthD1stribution~'
~'"'\
.u
nn
otc
hed
sp
ecim
.en
."'-
._~'Jj\l
,\
,\f~\
"\
l""
.".
J,I
I..
,\I
¥~
IY
II
r!
II
I,
II
I!
II!
'r-;
-';!
-~~i
~:-'
\~~
'1I)
40
6080
100
120
14
0:6
0le
Oo
NEAN
STH
ESS
-kp
si•.
{ \
Fig
.2
.16
Sta
tisti
cal
Fati
gu
eD
iagr
amF
or
Infi
nit
eL
ife.
'""vc.;-
.
·.' '19
This may not be a very good assumption since the
fatigue diagram shows that it does falloff rapidly at'\
101'1 stress ratios. At these 10\'T ratios yielding "lill be
the predominant failure mode. Unless the capability exists
to monitor the torque on the specimen continuously. it will
not be knO"Nl1 ,'rhether or not the initial torque load w'as
constant.throughout the test.~.
2.8.1 Qse of ~~~t;stiQal Fatigue ~la~rams in De~
The conventional use of fatigue diagrams is
to determine what combinations of mean and alternating
stress are safe for a given number of cycles of operation.
For example. in Fig. 2.17 the stress combination of Sa/.1.
and S would be considered safe for a design life ofm1.105 cycles but not for infinite life. The combination of
Sand S is not safe for 105 cycles of life. The8 1 m2 s
question arises, "Hith "That confidence can one say that a
combination is safe?" To anm'rer this question the dis
tributions of the alternating and mean stresses to which
the component in question is.subjected and the distri
bution of the. limiting fatigue boundary must be considered.
A method of obtaining t~e distribution of the fatigue
boundary at various cycles of life has been previously
discussed.
MODIFIED GOODNAN LINE
-sy
MEAN STHESS
Fig. 2.17 Conventional Fatigue Diagram Sho~1ing theModified Goodman Line
/
If the distributions of the alternating and
mean stresses to which a shaft is subjected are known or
can be estimated then they can be plotted on the fatigue
diagram as shown in Fig. 2.18. If the shaft stress
distributions are normal and the stress ratio can be
assumed to be constant then the distribution along the
stress ratio axis (R2 axis in Fig. 2.18) can be obtained
from the relationships
77·
.. and
"lhere
78
R = 00 , .:
(I)0(1)ZWH~E-rE-t~(I)p::
§(S~las)
a
R = 0
MEAN STRESS.', « .- ~
-
Fig. 2.18 Statistical Fatigue Diagram-
- of the stress distribu.tion a.long~-. ." sR = mean
.;t. 4''-
2__l.v -- the R2 axis..
r-
a = standard deviation of ·the stress. sR'
2 distribution along the R2
axis.
- of the alternating stress of thes = meanashaft.
a = standa.rd deviation of the alternatingsa
stress of the shaft.
Figure 2.19 shm'1s the stress a.nd streneth
distributions along the R2 axis. Once the parameters of
the distributions are knmm the probability of failure can
be calculated. Defining the random variable Z as
Z ::; S - s
79
"Fig. 2.19 Enlargement of Stress and Strength Distributions on R2 Axis of Fig. 2.18 •
where
s = the random variable, strength~
s = the random variable, stress.
then the mean and standard deviation of Z can be shmm to
be normally distributed "Nith parameters (13, p. 113)
z = s - s
(20)
Failure occurs i':hen s> S or "'Then S - s < o.In Fig. 2.20 the shaded area represents the probability of
,failure. Stated in mathematical terms
o
P(Z < 0) ~ S... _ 00
1 e, dz
feZ= s - s
z
80
Fig. 2.20 The Distribution of Z = S - s.
,.,Putting into the standard normal from by letting
then
S - sJo 2 + 0 2"1
S S
./
(21)
P(X< Xl) = e
For a knovm xl the value of P(X < Xl)' '\'lhich is the .prob
ability of failure, can be obtained from standard normal
distribution area tables. The reli?-bility Re is defined as
Re = 1 - P(failure) = 1 - P(X < Xl)
,
which.can be expressed as
dx (22)
81
Thus the' reliability of the shaft can be determined.
In most cases the designer must start with a''--
level of reliability and determine the shaft diameter such
that for the imposed loads the specifiedre1ic.bility
\'1i11 bernet. The procedure in this case is to 1-lorlc back-)
wards through the preceding derivation. The alternating
stress in the shaft can be expressed in terms of the dia-
meter by
. (23)
\'1here
M = the bending moment on the section in
question.
- ./
D = diameter of the shaft.
4 S'
I 1TD moment of inertia of the shaft= M = cross
section.
The standard deviation of the stress can be
expressed in terms of the mean and the standard deviation
of the diameter of the shaft. Using the approximate
partial derivative method (13. p. 90) for the standard
deviation of the alternating stress yields
(24)
Theexpression.obtained using the approximate
partial derivative method may not be valid in certain cases.
82
For example, in the case of a product Z = XY the standard
~deviation 6f Z is given by (13, p. 123) .
. whereas the expression given by the partial derivative
method is
0z ~ . J(X 0 )2 + (y ox)2' . (26)Y.
When the term (oXOy)2 is small compared to
other terms on the right side of.Eq. (25) then the approx-
imation \1ill yield good answers. If the values of ~X
and 0y differ from each other by 3 or 4 orders of mag-/
nltude then the term is no longer. negligible. Then exact
expressions for the standard deviations must beJ.used.
They can be found in Chapter 3 of ne~. 13.
The standard deviation of the shaft diameter
is a function of the tolerance on the diameter. Assuming
the distribution of the shaft diameter is normal then
99.73% of the shaft diameters will fall within + 3 0D •, ~t
Denoting the' shaft diameter as D - , where in most cases
the tolerance2t is known, then 3an ~ t.
The mean and standard deviation of the moment
can be determined from the loading distribution. The
only unknOl",rn in equations. 21, 23, and 24 is D the diameter
of the shaft and it can be solved for.
This, then, is a method of designing a
rotating shaft, subjected.to a bending moment and a torque,,
. for a specified number of cycles of life i'1ith a predeter-.
mined reliability.
HOi'1eVer it is important to recall the limi ta.-
tions and assumptions.
i . . ·The amount of data used in estimating the
distribution parameters can have a sub-
stantial effect on the shape of the dis-
tribution. The most pronounced effect is
1n the tails. Since the calculation of
reliability is based on the overlap of the
tails of tl'TO distributions it j.s neces-
ary to have good estimates of the distri
butions in order to obtain accurate values
of reliability.
2. The tests were conducted on specimens of
given geometry; hardness, and material at
room temperature. The fatigue surfaces
generated from these tests are valid only
for this type of specimen, environment and
loading. The loading was a bending moment
and a torque causing an alternating normal
stress acting parallel to the centerline of
the shaft and a constant shear stress acting
perpendicular to the centerline.
84
3. It was assumed that along any given stress
ratio axis the strength distribution is
normal. '\.:..-" . "._,
4. It "Tas also assumed that the ratio of the
alternating and mean stresses, to 'Nhich the
shaft being designed is subjected, is con-
stant. . ,
..
Obviously, it '\'Tould be an almost impossible task~.
typeto generate statistical fatigue surfaces to cover every
of material and donfigUration. Ho\,rever, it is feasible that
enough tests could be conducted to allow empirical equations
to be developed relating different stress concentration
'factors and conventional physical properties for the more/ J
common steels. This '\tould make ft possible to design from
a diagram which was not generated for that particular con-
figuration and material.
2.9 Mpdification of. One Test Macpip2IJ.
This section will discuss pome of the problems
encountered in working with the test machines and the
measures taken to overcome some of the problems. Two of
the most troublesome areas are the instrumentation and the
gear couplings. The gear couplings are the means by which
the toolholder arms are allmled to s\'ring dm'm'(?ard slightly
when the bending load is applied. Referring to Fig. 1.1
it is seen that the bending load configuration also applies
a shear load to the couplings. However,the couplings
were not designed to withstand radial shear loading,
their purpose being to transmit torsionalJloads only.
The couplings consist of three major parts. Two of the",
parts are pressed onto the shafts that are being coupled
and have an external gear. The-third part is an internal
gear sleeve "rhich slides over with the other t"t'lO ps.rts
with the gears meshing as shoi'm in Fig. 2.21 •
A radial shear load causes the sleeve to cock allowing
the centerline of one shaft to fall belrn~ the centerline-
of the other. This misalignment. along with the angular
deflection downward caused by the bending load, causes
varying degrees of vibration in the toolholders and speci-
men. vlhen the, gear couplings are nei'l the mis8,lignment is
small and there is very little vibration, but as the teeth,
in the couplings wear the misalignment and vibration
become greater.
Note, also, that there is no definite pivot point in
the couplings, thus the length of the moment arm betw'een
the loading bearing and the pivot point can change while
the machine is running causing a change in stress applied
to the specimen. H01'1eVer, from observing tests in progress
it appears as though the magnitude of the alternating stress
remains quite constant for runs of up to about two hours.
This investigator has never monitored a run,lonser- than about
two hours. The reason for this 1s that it is necessary
.,
:...::
-'-'
TOO
LHO
LDE
RS
P'.A
FT
SNA
PRI
NG
I!liII
II;:
1Ii
'!l!
'1_
__
.1p\p
[til-
__,...)1
lill
li-~
I.
[IiI
,I>
I,f•
,/>
'fI
f>
/.,),
1I
"n
.rIJ 77-;?7~7j/j~'7-~'~---
//
/.
\:".
-'
o
.G
E!i..'
qB
OX
SF'
.AFT
Fig
.2
.21
Gea
rC
ou
pli
ng
Use
don
Fat
igu
eT
est
ing
Mac
hine
.(
co.'
0\
to have the contact points off the slip rings when the"." ....
specimen breaks. Just prior to the specimen breaking, it"' ~
and the toolholder arms vibrate quite a bit. If the
points are dOi'ln on the sliprings1'Then this happens they
can become damaged or broken. Usually the endurance tests
are monitored for about 20 minutes.
Another problem is that there is a difference in the
number of divisiO!ls, as recorded by the Visicorder, bet''ieen
the static and d;~lnamic outputs. In other words, after the
~pecimen is loaded, the machine is. rotated by hand and the-
Visicorder trace is observed. Then the loads are adjusted'
until the desired output is attained. However, when thes
machino" is rotated under pOHer tht1 output is a different
amount, sometimes as much'asfour divisions. This can
represent as high as 8,000 psi. depending upon the gain
setting of the amplifier. The change can be easily cor-
rected for the bending stress simply by changing the
weight'on the loading arm. This can be done while the
machine is running. To make. the correction for torque
load the machine must be shut dO"m and the torque coupling
adjusted. HO'·rever, the amount of change behleen static
and dynamic outputs is not constant. After adjusting the
torque to allOiq for the change, "1hen the machine is
restarted the change may be a different amount so the
machine must again be "shut dO".vn and read just-ed.
' ..:....
86
There is another complication entering the problem.
After about 30 seconds of running time the torque trace\
begins to drift quite a bit. The amount of drift depends
upon '-ihich machine is being used and the torque load
applied to the specimen. The highe~ the torque load the
less the drift. Tests were made to determine if the torque
load was changing or if the drift was due to amplifier
drift. Several factors led to the conclusion that the
drift is caused by drift in the amplifier; one being the
fact tha. t the torque output drifts up"lard in t\'10 machines.
It is hard to believe that if the torque were changing,
it would become greater. The tests seemed ~o indicate
that the zero datum point was drifting. However, it
drifted only when the machine was running. The drift,
when the machine was not running would amount to one
division, at the most, in about 2 hours whereas the drift
amounted to as high as 3 divisions in 2 minutes ~lith the
machine runnine;. One result of the drift is. tha t if, ''Then
adjusting the torque load, the proper number of divisions
1s not obtained in one or tHo tries the torque \'Till start
to drift and no accurate measure of the amount of torque on
the specimen can be obtained •.. When this happens the loads
must be removedfro!ll the specimen, one holding collet
must be loosened and the amplifier recalibrated.· This is
very time consuming, especiallyif.this procedure must
A
89
be gone through bro or three time s.
The drift problem could proba.bly be eliminn.'ted by
obtainin5 sol~d state amplification equipment. The pre-"\
sent equipment has been in use since the start of the
research program and is not solid state. OV~~lning solid
state equipment is plaru1ed for the ncar future.
It is not qUite clear what is causing the difference
betNeen the static and dynamic outputs. For the bending
stress it seems feasible that such a difference may be
caused by the difference in the rate of elastic deformation
of the toolholder. ~hen the machine is rotated by hand the
rate of deformation is S10H but under pONer th8 machine
rotates at about 1,750 rpm and the deformation of an element
on the surface of the toolholder may lag behind the applied,
load. Using this line of reasoning it would seem that as
the peak load is reached, if the correspondlngdeformation
1s lagging, then as the load begins to decrease the deform-
stion will continue to increase only to that point where
the deformation corresponds to the decreasing load. In
other "rords, the toolholder never reaches the a,llount of
deformation corresponding to the peak load. This renson-
1ng would mean that the divisions of Visicorder output
should be greater for the static case than for the dynamic
case. HOT::cver, the opposl te is true. In all cnses, Hhere a
difference in the,bending output 'has been experienced, the
90
bendlnggagc output has been greater with the machine
runnir.s tmder pO\>Ter than ''lhen turned by hanel.
The above,reasoning cannot explain the differen~e in
outputs in the torque gage bridge either, because the
torque load is not alternating but is constant.
'The torque gage bridge output for the stntic case hr.ts
been greater than for the dynamic case in most tests.
This can be easily explained. Torque 1s exerted on the
specimen through the torque coupling on the backshaft.
(Fig. 1.1). The torque" is transmitted from the backshaft.
to the toolholder shafts through the gear boxes. When
torque 1s applied through the torque coupling the rate at
which it is applied is very slow. When torque is applied
the play in the gearboxes 1s taken up along with any other
play ~"n the system. Then '\':hen the machine is started the
1nitial impact causes more play to be taken up and the
torque decreases. At very high torque levels all the play
1~ taken up in the initial application of the torque.
The initial starting impact causes very little decrease in
torque gase bridge output, but the lower the torque level
the greater is the amount of decrease. For the lower levels
it 1s more difficult to take the decrease into account bv" .
setting the torque higher because at any given torque level
the amount of decrease fro~ specimen to specimen is not the
same. Several trials enable one to determine the change
91
and set the divisions accordingly.
Late in November, 1969 one machine was partially
. dismantled for~the purposc·of modifying the gear couplings.
The modification consisted of placing a spherical bearing
inside of each of the two couplings. The bearings do
several things to eliminate some of the problems. They
absorb the radial shear load and create a definite point
about which the toolholder arms can pivot. The bearings
have not been in use long enough to determine 'Nhether or
not the rate of gear tooth wear will decrease in the
couplings, but it is believed that it will. Since there
ls nO"1 a definite pivot point, even if there :ts excessive
tooth l'lear the outer l.'ing '\'Till not be able to misalign and
cause vibration. In Fig. 2.22 note that the inner diae.,
meter of the spherical bearing has a sliding fit \'li th the
shaft through it. This is true of both bearings. This
eliminates the possibility of subjecting the specimen to an
axial load when tightening the specimen in the collets.
Since the spherical bearings have been installed, about
34 endurance tests at a,stress ratio of 0.44 have been run
on the modified machine. No vibrational problems have been. ,
encountered and it appears as though the difference between
the static and'dynamic outputs is minimal.
;,
,,:,;,
:.',
..,.
TOOL
HOLD
ERSH
AFT
BEA
RIN
GRETAI~mR
RIN
G
BEA
HIN
GSE
AI'T
K'-
-;r-
--
--t~.
I/«
«C
dt
/II J
I!r
I.I~-
\I
\l._
!L_
_
SPH
ERIC
AL
BEA
RIN
G
GEAR
BOX
SHA
FT
Fig
.2
.22
Mo
dif
ied
Gea
rC
ou
pli
ng
.(
'0.
N
'..
CHAPTER III
OVERALL CONCLUSIONS
1. Estimates. for the normal distribution parameters
were obtained for the endurance level at stress ratios of
~; 3.5 and 1.0. The tests at stress ratio of 0.44 are
in progress.
2. Estimates for the normal and log-norma~ distri
bution parameters ~Tere obtained from the cycles-to-failure
de,ta. By comparing the D-values and the coefficients of
skel'mess and kurtosis it "Tas concluded that the data tends
tOHards the log-normal distribution•. It l'laS also conoluded
that in order to make a'more conclusive decision a larger
sample size at each stress level would be reqUired.
3. .The method previously proposed to obtain strength
distribution parameter estimates yields optimistic'values
and should be studied further.
4. If random data is obtained by ~he method proposed
in this report estimates which might have favorable
properties may be obtained.
5. Fatigue surface Has generated using the endur
ance data which is among the first few attempts to obtain
such design data. Efforts to generate such data based on
larger samples, with solid state instrmnentation, and with
9~
greater operator care, should be continued.
6. On th~ basis of the 34 specimens tested on the
"machine 11i th the modified couplings it is concluded that
the spheric~l bearings improved the running ch~r~cter-
lstics of the machine. It 1s hoped that the long range
performance would also be satisfactory.t"''-
. .-.. ~ ..'" ~
/
5
CHAPTER IV~. ,
RECOH!>iEUDATIONS
,J.
1. Ne'\'l' 'solid-state instrumentation should be pur-
chased. ' The ne\'1 instrumentation should be drift free a.nd
should have the capabili t~T of constantly monitoring the
strain gage outputs.
2. The gear couplings should be modified on the
remaining~1o machines.
3., The endurance tests for stress ratios of ~ and
1~O should be checked.
4. Before the next phase of the test program is,/
embarked upon the machines should be recalibrated to see
if the calibration constants are the same.
, S. Larger sample sizes should be used in future
tests,preferably 30 or more at each stress level tos' ,
obtain the much needed cycles-to-fa1lure design data.
6. The feasibility ofincol"pora.ting random stress
levels to produce de.ta from \'1hich better strength distri
bution parameters might be estimated should be investi-
gated~
".::
. ~'" .
APPENDIX As
COMPUTER PROGRAM TO CALCULATESTRESS LEVELS AND RATIOS
The purpose of this program is to convert the visi-
corder records into normal stresses, shenr stresses and
stress ratios. The program also calculates the means and ',.'
standard deviations of the stresses and ratio of each group, ,
of data. It also calculates the,cyc1es-to-fai1ure from
times to failure data, but the program will also accept
cyc1es-to- failure data. The pro~ram distinguishes be
tween the two through the use of a code number. The code/
also tells the program whether or not a group of data are
endurance test data. This discrimination is necessary
because of the calculation of mean and standard deviation
of the stresses. The program will not calculate the endur-f ' "
ance level distribution parameters. The discrimination
code is fed in as dal"E:l. alid is as follm'ls:
0 if the failure data is in times to failure
1 if the data is an endurance test2 if the failure data is in cycles-to-failure.
The input format for the code will be discussed shortly.
The program will accept as many sets of data as
desired and the groups may be mixed; ie, endurance test,
group with cycles-to-failure data and group with times
96
97
to failure data. 'A group is all the data for one stress
level. A group of data consists of the follm'iing. The
first card contains in this order, the number of speci-~,-
~,
mens in the level, the mode of the run and the code. TheJ'
mode is dependent upon the date the run was r::::::(~e. For a'
further discussion on mode see Section 2.1.
The fields,on the data card are as follows:
spac,es 1 to 5 number of specimensspaces 6 to 10 modespaces 11 to 15 - code
The number of specimens. mode and code are fixed point
numbe'rs and have no dec1mals but the numbers must be
placed to the right 1n each field.
The next sequence of cards reads in the cycles-to~
fa~lure or times tofa1lure 1n hours. minutes and seconds.
If the data 1s1n times to failure there are ten groups on
a card,. so the number of cards required ,'rill depend upon~
how many specimens are in the level. The format across
.. '
the card is:
spaces 1 and 2 blankspaces :3 and 4 - hoursspaces 5 and 6 - minutesspaces 7 and 8 - secondsspaces 9 and 10 blank
-and the sequence continues 1n this manner. If the failure
data is in cycles-to-failure the format 1s 8 fields of 10
spaces each and the decimals apperir in the last space of
"
'each fiel&, ie, spaces 10; 20, 30, etc. If the group of
,data is for an endurance test there is no failure data
and thet:e cards are left out. The program \'1il1 auto": "
maticsl11 handle it if the proper code number is put on
the firr;t card.
Follo'Yd.ng the cycles-to-failure cards are' the cards
bontnin1ne the information for each specimen in the stress
level. The information must be placed on each card as
follO\H.l ,
.! .
pan weight./
Bp~COB 1 to 5 test numbersp~CCG 6 to 10 - specimen numberspaces 11 to 15 - machine numberspacos 16 to 20 with a
deo1mal in space 20
. ,•. t.
spaoes 21 to 30 with a
decimal in space 28 bending calibration resistance
spacos 31 to 40 with adQo1mal in space 3~ number of bending calibration
divisions
apnoea 41 to 50 with a
decimal in space 48 number of divisions 6f bending
spnoea 51 to 60 with adocimal 1n space 58 torque calibration resista.nce
opnocs 61 to 70 with a(l(lolmal, in space 68 number of torque calibration
divisions
sl\.l\ces 71 to 80 '\'1i th a
daolmal in space 78 nurnberof divisions of torque
99
.~'he test number, specimen number and machine number
are fixed point numbers and mus~be placed to the right
in each field. There is one card for each test specimen'---
1"\
and the cards must be placed in the same order as the
failure data is placed.· on the cards precedinc ~~:~dse c~rds.
For data at stress' ratio of 00 there 1'lill be no torqu.e
stress data. In this case these fields can be left blank.
The computer reads blanks on data cards as zeTOs •
. This makes up one group of data at a given stress
level and ratio. As many groups.may be run as desired by
simply placing the groups one behind the other.
FolloHing is a listing of important variables in the
program and Fig. A - 1 is a flow/chart of the program.
Figure A - 2 is a program listing. The outputs from this
program are given in Appendixes D and E.
\
\
•
. .List of Definitions for Program to Find
Stress Levels and Ratios (PROGRAM STRESS)I.
100
= 0 if failure data is in times to failure.- ·1 if data is from an endurance level.
. J
= 2 if failure data is in cycles-to-failure.
times to failure in hours, minutes and seconds.
NCARDS
!10DE
NCODE
XHOURS(I)XMIN(I) SECS(I)
TOTCY(I)
NOTEST
NOSPEC
=
=
=
n~~ber of specimens tested at given level.. ."
number of mode depending on date·of test.
cycles-to-failure.
test number.
speciman number,'
MACHNO = machine number. /
PANWT
RCALB
ENCALB
ENVISB
BCALT
ENCALT
= amount of ''leight on loading arm.
= calibration resistance used in bending channel.
= number of visicorder divislons used ''lhencalibrating bending channel.
= number of divisions during actual test.
= calibrationresistarice used in torque channel.
= number of visicorder divisions used whencalibrating torque channel.
ENVIST =ENA =RGAGEB =RGAGET =GB =
-..
n~'llber of divisions during actual test.
number of active arms in strain gage bridge.
resistance of:bending strain gages.
resistance of torque strain gages.
bending gage factor.
.. ...
= torque gage factor.GT
CBGR
CGRTH
CT
CTB
CBT
RPl1
=
=
=
=
calibration constantKBGR •
c,~libratiol1 constant KGR•.THcalibration constant KT •
cali~rationconstant KT/ Bcalibration constant KB/T •
revolutions per minute of machine.
101
SOUTH = output normal stress corrected for interaction.
TAUTH = output shear stress corrected for interaction.
STRGR(I) = normal stress in specimen groove.
TAUGH(I) = shear stress in specimen groove.
SOUTHP = output stress not corrected for interaction.
TAUTHP = output 'stress not corrected for interaction •
. \
G
102
Program to Calculate Stress Levels and Rat10s
"~r.·'\IN PROGHA?'l (STRESS)
READ NCARDS. MODE. NCODE ~--~--~ 9
," YES
PRINT HEADINGSENDURANCL!: TESTSAND r~ODE
NO
PRINT HEADINGSNON ENDURANCEAND i'10DE
spnIATOD ==TID ==
SPJC ==
THJC ==
COi'JST
F1g. A - 1
== .49752001.3125
12tl~2f~~PDIA~
lh2-1.J.15l)(TOD4 _TID1.J.L
(1 ) (TaD)
THJC== SP·JC
".. :,. ...'. " .
"
103
.'
READ NOTEST, NOSPEC, MACHNO, ~ f:\PANY1T, RCALB, E}·~CALB, EilrvrsB,·2 ~
RCALT, EtJCALT, E!\TVrST
E = 30000000.ENA = 4.RGAGEB = 190.RGAGET = 120.GB = 3.23GT = 2.05.
/;/./y------------'-------
Logical if statements to choose correctcalibration constants for HACHNO and i';ODE
' ..'-,. .!~.~..
NO YES
SOUTH (ENVISB] (R) (RGAGEB)= -r1E--:t-'ic-'-"i~LB1TEl~AT( GB11RcALB J
STRGR(l)
TAUGR(I)
Fig. A - 1 (continued)
SOU1~H .= rCGRTHT~CBGH)
== 0.0
..
104
s
.. ' ~
PRINT }!O~~ST, l!CSPSC I ;TACENO,PAl\Jl'IT, TCY, RCALE, E~rC~'\LB,
EINI SB, RCALT, E~JCALrr, En'll ST ,STRGR(I), TADGE(I) ,:;i""
NO
CYBH = (XHOURS (I» (60) (RPM)CYMIN = (XMIN(I»(RPM)
CYSEC = .lSEP§tb·~) (~'pNl
TCY = CYHR+CYMIN+CYSEC
/
TOTCY(I)
PRINT No'rEST, NOSPEC, r'!AC!'Il'~O,
PAWNT, Tey, RCALB, ENCALB,ENVISB, RCALT, ENCALT, E~~IST,
STRGR(I), TAUGR(I)
6
Fig. A - 1 (continued)
105
.' -!-ENVISB 2..l~ (RGAGEe.LSOUTHP = (ENCA~BrrEN~(GBrnT~~LB)
'l'AUTHP . (ENV1sTt(1tJ~'yt\GET) )= TENCALT I(El..JA GT rRCALT-
SOUTH = SOUTHP-(CTB)(TAUTHP)
THUTH ='TAUTHP-(CBT)(SOUTH)
SOUTHSTRGH( I) == TCGR'l'H reCBGR'}
TAUGR(I) = (CT)(TAUTH)(CONST).:"
.--' -~TI31tflL.--.,,:.... -.-....-.-----... R(I) - TTAUGRrr ) 1,732)'--------r------//---'
. ( /
,.:1'
YES
YES
PRDJT NOTEST, NOSPEC, .J1ACHNO, PA~J1!!T, RCALB,ENCALB, EIJVISB, RCALT,ENCAVf, ElJVIST, STRGR ( I ),TAUGR(I), R(I)
Fig. A-l (continued)
TOTCY(I)
YES
4
NO
5
° '.
CYHR = (XHOURS(I»(60.)(RPM)CYMIN = (XMIN(I»(RPM)
CYSEC = lSECS{g~~iRPl~)
TCY = CYHR+CYMIN+CYSEC
PRINT NOTEST, NOSPEC, MACHNO, ,PA~dT, TCY, RCALB, ENCALB,ENVISB, RCALT, ENCALT. ENVIST,STRGR(I), TAUGR(I), R(I)
6
Fig. A - 1 (continued)
0'---_
106
.,:~° ,.
YES9 kT----< NO
107
Calcul ['~ -ce mean 8.nd ~-=-- --=-}l':.::'0,---<-standard deviationof norms.l stressSUBROU'I'INE MEAN
PRINrr XI'I~:AN, DEV
9
NO'
Calculate mean andstandard deviationof shear stressSUBROUTINE ME1\3
PRINT XMEAN, DEV
r r--~-:---L_----:--l
CRlculate mean andstandard deviation'01' stress ra.tioSUBROUTINE HEAN
PRIN~~ XHEAH, DEV
Fig. A-l (continued)
Program to Find I'lea.n a.nd Standard Devla tion
SUBROUTINE BEAN
108
SIG~rA
TOP2= O.
= o.
SIGMA = SIGMA+X(I)TOP2 = TOP2+(X(I)-MEAN)2
YES
I...--~XMEANSIGHA
= DATA
DEV = TOP2DNrA--l.O
FEINT XHEAN, DEV
"
Fig. A-l (continued)
APPE1~IX A (continued)
1,' ,
. /I
Fig. A-2 Computer Printout ofPROGRAt-l (STHESS)
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APPENDIXB "'----
1-
Progra~r: to Calcu'late the Parameters of the !'Jormal
and Log-Normal Distributions and Conducts
Goodness-or-Fit Tests
This program calculates the mean and standard devi-
ation of the cycles-to~failure data for both the normal and
log~·normf:\l distributions and calculates the moment<'
coefficients of skewness and kurtosis. It also performs
the Chi-square and Kolmorgorov-Smirnov goodness-of~flt
tests. The input consists ofa/
1. a card containing the number of data points at
. that stress level, the accuracy of the data, the
stress level and the stress ratio.
2. a card or series of cards containing the cyclesto-failure data listet in descending order.
3. a card or series of cards containing the cumulative frequency of the cycles-to-failure'up tothat point. These must be listed in the sameorder as the cycles-to-failure data.
The input format for the first card iSI
spaces 1 to 3 - number of data points in fixed pointformat.
spaces 4 to 8 with a
decimal in space 7 ~ number of data points in floatingpoint format
115
116
/
J.
spaces 9 to 17 with adecimal in space 13
spaces 18 to 27 with a
decimal in space 26
accuracy of the cycles-tofailure data. ~-
stress level at which the datawas taken.
spaces 28 to 35 with a~
decimal in. space 30 stress ratio at i'1hich the data.
was taken.
The cards containing the cycles-to-failure data have
eight fields of ten spaces each with the decimal at the
right of each field. In other ~ords the first data point,0
1s in spaces 1-10 1'11 th the decira.~l in space lOs the second
data point in spaces 11-20 with the decimal in space 20, et6.
The· format of the cards containing the cumulative
frequency ~ata is 26 fields of three spaces each with the
decimal to the right of each field. The first value is 1n
spaces 1-3 with the decimal 1n space 3, the secon~ value
1s 1n spaces4~6 with the decimal in space 6, etc.·
The program 1s set u.p t.o accommodate as many sets
of data as desires.
Follow~ng is a list of important v&D.d)les used in the
program. Figure B-1 is a flm.; diagram of the program and
Fig. B-2 is a printout of the program. Figure B-3 is
an example of the output for one set of data. The output
shoin1 is for data tested at a stress level 114,,000 psl. and
stress ratio of (.0.
117
List of Definitions for Pro~ram to Fit Normaland Log-Normal Distributions to Cycles
to-Failure ·Data (PROGRAM CYTOFR)
l>1ai11 Progr?m:
l'IDATA -- DATA = number of observations.
STRLV = stress level in psi.
AKURCY = accuracy to "lhich cycles-to-failure data.are knm'ln.
RATIO
X(I)
= stress ratio
= cycles-to-failure data
CUNFRQ(I) = cumulative frequency of each XCI), ie,number of X's less than or equal to XCI) •
./
PCAREA(I) = CUHFRQ(I)/NDATA
. Subroutine to calculate the mee,n and standa.rd deviatton of
the cycles-to-failure data (SUBROUTI~ffi MEAN)
SIGf1A = sum of the X(I)' s
XMEAIil = average of the X(I) 's
n{X{I)-Xr1EAN)2TOP2 = L
i=l
DEV = standard deviation of the X(I)' s
Function subroutine to find the. area under the normal
curve (FUNCTION PROB(X».
Subroutine for Chi-square goodness-of-fit test (SUB
ROUTI1\'15 CRISQA). \"--'-
118
largest value of cycles-to-failure.
smallest value of cycles-to-failure.
cell upper bound.
number of observations in 3th cell~
expected value of 3 th cell.'.~
K =XHAX :::
XMIN =
CSV =CEV -CLB :::
CUB :::
FREQ(3) :::
REQAREA ( J ) ==
CHISQR :::
U(I) :::
number cells.
cell starting value.
cell end value.
cell lower boundi
total Chi-square value.
Chi-square value of I th ·cel1.
.)
Subroutine for l~olmogorov-Smirnov test· (SUBROUTINE DTEST).
ABUNCN
::: abscissa value on standard normal curve fora given XCI).
::: area under standard normal curve fro!l1 to Z (I) •
DSTAT(I) = absolute difference between the data cumulative frequency Bnd the hypothesized cumulative frequency.
XMEAN ::: average of the X(I)'s.
DEV ::: . standard deviation of the X(I)'s
PHOB('r) ::: area under the standard normal curve from
-T to +1'.
119
...-- - :.-
Subroutine to calculate the tllOment coefficients of Ske\'1-
ness and klITtosis (SUBROUTINE ALPHA). ~_
.- .J .... -
ALPHA)
ALPHA4 =
moment coefficient of skewness.
moment coefficient of· skewness.
VAH
TOP)
SKEH
STDEV
Top4
TKURT
. n ... -.. 2 .= L (X(I)-X)
"-1=1 . -
= third moment of the data.
::: biased estimator for standa.rd deviation.
n L~:::. L (XCI )-X)
1:::1,/
<"
= fourth moment of the data.
Probr:t"'nm to Calculate Parameter Estimates for theNormal ancl Log-Normal Distributions and ConductGoodness-of-Fit Tests
}~IN PROGRAM (CYTOFR)
READJJDATA, DATA, AKURCY, STRLEV, RATIO
NO
READ Cur·1FHQ (I) , s
'--
120
PCAREA(I)
YES
:: £1!.2EFi9J1l....NDA'l'A ~'-------r
NO ~">--:;.;..."-----::>'j I :: 12:!-1
~ PRINT STRLV, RATIO, X(I)'s
Calculate mean and standard deviation of X(I)'sSUBROU1'IHE HEA!'T
Fig. B-1
Do Chi-square goodness-of-fit testSUJ3HOUTnm CHISQA .I-----
Cb
121
\:.....
.J AKURCY ~ •ooo~
..~
Do Kolmogorov-Smirnov Goodness-of-Fit TestSUBROUTINE DTEST
!Ca.lculate Homent Coefficients· of
SkevTness and KurtosisSUBhOUTIiE ALPHA
,
\t
. LOG X(I)NX (I) = [( .2"6- -I- LOGe (20) ) (10000. )J + .5
/
X(I) == NX(I)
XCI) - JJIl. - 10000 •._t-~__-=-.:....:....:....:...=---- -,- ..-J
YES T
">-~':I<:--_-~~ I = 1+1
PRINT STRLEV,·RATIO, X(I)'s
Calculate mean and standard deviation of X(I)'sSUBROUTD7E r·1EAH
Fig. B-1 (continued)
· .
Do Chi-square
goodness-of-fit testSUBROUTIlJE CRISQA'
Do Kolmogorov-Smirnovgoodness-of-fit test
SUBROUTINE DTEST
Calculate moment coefficientsof skewness and kurtosis
/
SUBROUTINE ALPIL\
Fig. B-1 (continued)
\ ....-
122 .
Subroutine to Find Hean a.nd Sts.ndard Deviation
SUBj10UT'INE NEAN
0~l'J:.@
.I~SIGHA = O.
'.!'OP2 == o. .
ty~-= 1[2; __[
r----~-'L_----:-~-......SIGHA - SIGI·IA+X (I)
TOP2 == TOF2+(X(I)-MEA~)2
123
YES NO:>-----{.. I -- 1-:-1
'l'OP2DEV:= DA'l'A--=T:Q
I~P-R-I-NT Xl-JEAN I !2EV7
Fig. B-1 (corttinued)
124
Subroutine to Find Arei Under-Standard Normal Curve
FUNCTION PROB(X)
\..-
(
<0
XSQ ::: X2
FROB::: 1.0PTERH:= 1.0FACTOR::: 1.0ODDINT ::: 3.0
6
/' ...
> 0
1RECXSQ ::: ~- X2
FROB::: 1.0-(.79788453) EXP[_X 2/2 ,-' -(--x--)(l.0-nECXSQ(1.~RECXSQ
(3.-RECXSQ(15.-RECXSQ(10S»»)]
Fig. B-1 (COi1til1ued)
125
·.1._
\
'--
PTERM _ _ PTERM~~L- U)'FAC'fOR
P'J'ERJvlTERM:: OD5fwf
PROB == PROB+TERt1
>0
J
..
FROB ::: {. 797G81l55)(X) (PROB)
-..
Fig.· B-1 (conti~ued}
/'
\
FACTOR == FACTOR+l.0ODDIHT = ODDINT+2.0
6
'. 126
Subroutine to Conduct Chl-SquB.!'e Goodness-of~Fit Test
SUBROUTnm CHISQA
.. '\...-
CHISQH :: 0.0
K = 1.5 + (3.322) LOG 10 (DATA)
REALK = KXNAX ::: X(1)XJ1D\' == X (1)
YES
NO
NO
XHIN =: XlI
.YES
. I ::: 1+1
1
Fig. B-1 (continued)
. 127
RANGE··= XHAX-XIUN
DIVIDE = j~~D~CY .
KJ.1 - [RAf1GE+AKURCY + ( 5)AKUnCV] (T'!T"IDF)\. - .HEALK •. .b .-. - - v -
RIel = KH
RKlH = DIvIDE
PRINT xr'1AX, XIHN, vI, l....-- 113' 11
\.......-
I = 1+1
NO
A == IB = (. 5) AKURCY
CSV(I) = XMIN+(A-l.0)(W)CEV(I) :;: CSV(I)+H-AKURCYCLB(I) - CSV(I)-BCUB(I) = CEV(I)+B
YES
CUB(I) = CEV(I)+BCEV(K) ::: XHAXCUB(K) = C~V(K)+E
. Fig. B-1 (continued)
IJ '= ·3+11·....,...·---!>Il '---__-.--__.....J
. (
FREQ(J) = FREQ(J)+1.0
128
NQ
~../.' Fig •. B-1 (contim.wd)
. 129
·.i..- .
T = Z(1)
AREA(I) ~ Eli~~~Tl
\ ---
NO YFS
REQAREA(l) = .5-ABEA(1)l·lAHU ::: K··l
0-;: I = 1+1 !------------v;
.S
REQAREA(I)IAHEA (I) - AREA (r·n I
.j Fig. B-1 (continued)
..-
D
REQAREA(I) =AREA{I) + AREA(M)
T .~R-E-'Q-.P-iR-E-A--= .5-AREA(K-l) I
''---
130.,.. ,~.
"
EXFl1EQ(n) :: DATA~<-REQAREAOn
. EXFREQ(~2-FB~QJH}2U(~1) =: ~- EXF"'REQ (;:fr~ ..1-
CHISQR :: CHISQR+U(Io1)
PRINT I, CLB(l), CUB(l),EXFREQ(I), FREQ(I), D(I)
NO
\
Fig. B-1 (continued')
=_.~.Subroutine to Conduot Kol~ogorov-Sm1rnov:v~. . : . GoodnessO"of-Fi t Test
. SUBROUTINE ·DTESTi
4 Z ( I ) ~j I L-XHEAnl---~ = DEV--
\-.~
131
T .- -2 (I ) 1--"'_00-__<-:;°_<
_. Ds'rAT (I)
ARUNCN == 1--P~B<Tl
DSTAT = ARUHC"'7PCABEA(I)
.>-.....;>.--=O_--.:{2 == Z (I) I=0
= • 5-PCARSA ]
/
ARUNCP =PRO~LTl + .5~-------1
DSTAT(I) =ARUNCF-PCAREA(I)
NO YRS
I == 1+1
4
PRINT DSTAT(I)'s
Fig. B-1" (continued)
Subroutine to Find the l'ioment Coefficients ofSkewness and Kurtosis
SUDI.{OUTI1JE· ALPHA
TOP) = 0TOp4 = 0VAH :::: 0
VAH :::: VAR+(X(I)-XEEAN)2TOP) = TOP)+(X(I)~XMEAN»)rropl.j· := TOP4-:-(X(I)-Xi'1EAN)4
,..... -
132
~
YES
STDEV =: YAJLDA'l'A
SKEHALPHA) - ------ (srrDEV)3
. TOP!f·TKURT :::: DA'r1
ALPHA4:: TKURT(~ri'D'~V'> [j:
I--.._-!~>A, PRINT ALPHA] I ALFHA4
Fig. B-1 (continued)
I .I'i
APPENDIX B (continued.)
..
/
Figo' B-2 Computer Printout ofPHOGHAH (CYTOFR)'
\
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APPENDIX C
CALIBRATION COlTSTAH'1'S OF EACH NACHINE
}i'OR r'BE DIFFERENT f'JODES
Mode 11 26 July, 1966 to 10 August, 1967
l10de 2: 11 August, 1967 to 1 June1 1969
Machine Kb~r K Kt Kt / b Kb/ tgr-th
1 1.0123 0.0208 0.8752 -0.0459 0.0290
2 1.0123 0.0188 0.8201 0.0344 0.0422.'
:
3 1.09L:-6 0.02U. 0.9330 0.0 -0.0:tLI-9:
Mode 3: 2 June, 1969 to 1 February, 1970
Nachine K N' Kgr-th Kt Kt / b Kb/ tbor
1 1.0123 0.0208 0.8752 -0.0459 0.0290
2 1.0123 0.0188 0.8201 0.0344 0.0422
:3 1.0946 0.0211 0.7721 0.0 -0.0127
· :.'. ..;". ~ :.
APPENDIXC (continued)
CALIBRATION CONSTANTS OF EACH I'-1ACHINE
FOR THE DIFFERENT MODES
110de 4: 2 February, 1970 to next change
146
MachinE K Kgr-th Kt Kt / b Kb/ t··be;r
1 1.0123 0.0208 0.8752 -0.01.:·59 -0.0290/
2 1.0123 0.0188 0.8201 0.0344· 0.042·2I
0.7721 -0.0123 1.01.23 0.0197 0.0 --
.'
APPENDIX D
COI1PUTEH OUTPUTS LISTING STRESS
LEVELS AND RATIOS OF nmIVIDUA.L
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40
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30
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MOD
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50
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49
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50
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49
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49
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44
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51
39
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3
50
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19
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19
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55
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31
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6
.43
5
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0
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8
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4
.43
1
STR
ESS
RA
TIO
71'to~
76
'i3
4
18
(01
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19
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80
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3
71
14
5
78,+
99
19
c:3
4
80';4
4
79u4
9
17
(21
)
78
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4
78~01
78~93
71
10
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19
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59
73
d
58
45
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59
85
0
59
82
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59
45
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59
70
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60
91
5
59
77
9
59
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5
600'
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59
16
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59
25
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59
21
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60
12
8
61
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25
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59
51
b
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ESS
35
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35.1
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35
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35
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35
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35
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35
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35
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35
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35
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0
33
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35
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35
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45
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45
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45
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45
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45
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44
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44
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44
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44
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00
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2000
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011
20
00
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20
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20
00
00
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20
00
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2000
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39
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39
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39
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39
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40
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39
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39
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39
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/j.O
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39
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40
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40
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TEST
MOU
E=
NV
ISI:I
EN()I
NG
50
.00
50
.22
49
.95
50
.00
50
.10
49
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50
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49
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78
43
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8
49
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49
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49
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50
.00
49.B
O
49
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49
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49
.95
49
.10
49
.85
NeA
LBE
ND
ING
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
3.0
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
RC
Al
REN
DiN
G
13
60
34
14
03
50
16
53
24
15
36
26
11
66
85
16
13
95
16
16
93
14
86
84
11
51
37
i88
21
5
11
69
23
12
16
56
10
32
90
20
63
12
84
65
6
14
45
17
i62
67
5
67
39
2
CYCL
ESTO
FAIL
UR
E
7.0
7.0
7.0
7.0
7.0
6.0
7.0
6.0
7.0
6.0
6.0
6.0
6.0 7.0
7.0
6.0
6.0
6.0
PAN
IlIT
.
1 1 11 1 1 1 1 1 1
MAC
H.N
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50
0
53
8
45
1
56
9
55
0
48
1
51
5
65
i
44
5
54
8
47
0
46
2
54
5
48
9
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48
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48
4
496
SPE
C.
NO
.
?
50
1
502
50
3
504
50
5
50
6
50
7
508
49
7
49
9
TEST NO
.
50
9
51
0
'5i2
51
3
51
4
51
5
51
6
633
~ j
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ING
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ESS
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DEV
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r.BE
NO
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ESS
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=
NEAN
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ESS
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D
STD
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ESS
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=M
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7
.43
2
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8
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9
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9
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8
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0
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6
STR
ESS
RA
TIO
89
29
2
89
05
4
89
22
6
90
;)4
7
90\1
23
89
08
1
90
12
8
89~38
90~40
90~65
89
25
3
89~83
89~12
89
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902~1l
89
41
1
88
27
5
9U
49
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t.SS
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4
71
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68
50
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50
2
69
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3
68
70
7
68
04
8
67
07
3
69
78
2
67
55
0
68
21
1
67
70
0
69
34
4
67
62
2
68
77
4
68
52
b
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67
70
2
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STR
ESS
20
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21
.94
20
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20
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20
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21
.94
21.4
!:!
21.9
!:!
22
.20
22
.20
22
.70
21
.69
21
.94
21
.93
22
.30
21
.72
22
.55
cl.
5'1
NV
ISTO
RQUE
22
.52
22
.3'"
22
.50
22
.53
22
.47
22
.46
22
.50
22
.57
22
.53
22
.48
U.3
3
22
.56
22
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22
.&0
22
.62
22
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22
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22
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NeA
l.T
OH
I.lU
E
20
00
00
2000
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2000
fJO
20
00
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20
00
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2000
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20
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20
00
00
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20
00
00
20
00
00
2000
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20
00
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20
00
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2000
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20
00
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23
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22
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23
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23
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22
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22
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23
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23
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22
.87
23
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23
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23
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24.5
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24
.08
23
.12
23
.07
23
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22.!
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NV
ISbE
NO
ING
TEST
MOD
E=
4
/
25
.03
24
.93
25
.02
25
.04
25
.02
24
.96
24
.91
24
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24
.96
14
24
.63
PS
I.
24
.91
24
.88
24
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24
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24
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24
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25
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24
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24
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89
77
0.6
PS
I.7
61
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PS
I.
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29
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OQ
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68
86
8.3
PS
I.
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BEN
DIN
G
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
30
00
00
RCAL
BEN
DIN
G
90
04
4
53
87
8
40
67
5
44
51
1
55
95
1
36
63
1
40
97
3
67
43
2
49
42
5
46
51
7
50
40
4
48
01
9
50
52
2
93
25
9
43
75
7
50
96
3
54
76
9
46
53
3
CYCI
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TOFA
IL.U
RE
9.0
9.0
10
.0
il.0
10
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12
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12
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12
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12
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11
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11
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11
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11
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10
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12
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11
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PAN
WT.
2 2 2 Z3 21 2 2 2 3 3 3 3 31
MAC
H.N
O.
67
5
6'3
60
1
57
1
61
9
55
8
63
8
66
6
64
36
41
64
45
51
64
55
94
63
4
63
5
63
7
63
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64
i
64
2
64
66
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76
46
64
86
23
65
06
85
65
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51
66
26
59
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6&4
66
0
TEST
SPE
C.
NO
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I0.
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-:.
--.-'-~
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/
APPEi'IDIX E
\
/
COI1PUTER OUTPUTS LISTING STRESS LEVELS AND RATIOS
OF INDIVIDUAL SPECHlENS FOR THE ENDURANCE TESTS
\ .
. 163
SPEc
iMEN
NO
.
.
ENDU
RANC
ETE
ST
.q,,
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INFI
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IN.
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N.
11~F
IN.
INFI
N.
INF
IN.
iNF
IN.
INFI
N..
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lN.
STH
ESS
RA
TIO
\
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IN.
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IN.
o oo ooo o oo ooooo o
SI'tE~R
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54
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b
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9
53
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61
98
1:
61
85
7
65
52
8
59
42
2
31
47
6
4&82
1
51
12
5
51
35
3
60
41
1
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STR
ESS
60
51
9-0
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0
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0
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0
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0
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00
NV
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l:.
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0
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0
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LTO
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TEST
1M
ouE
=1
RC6l.
TORo
UE
24
.50
26
.20
24
.40
25
.40
26
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31
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24
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25
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24
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22
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46
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0
25
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25
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21
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47
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48
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NV
ISI:l
ENlll
NG
/49
.60
49
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49
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50
.00
50
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49
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49
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49
.90
49
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50
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49
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49
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50
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49
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50
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53
.40
NeA
LB
END
iNG
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
19
00
00
ReA
LBE
ND
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29
00
00
49
00
00
49
00
00
39
00
00
~~
24
.0
25
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25
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25
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27
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24
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25
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25
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25
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25
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25
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25
.0
26
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25
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23
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PAN
WEi
GH
T
,I
2 2 2 212 2
I14A
CHIN
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;;
74
87
17
216
205
19
9
209
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16
8
190
20
7
142
208
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18
2
124
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14
9
15
8
16
i
169
175
185
186
190
195
64
10
i
102
110
11
7
122
14
1
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TEST NO
.
()
t~\
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"
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3.7
80
3.2
64
3.5
59
3.5
90
3.0
2&
3.4
98
3.9
33
5.7
17
4.2
63
3.&
42
3.2
21
4.4
32
.
3.1
01
3.6
1)-
3~419
3.4
71
STR
ESS
RA
TIO
3.5
38
Blb
3
9..
48
B(2
3
B~81
7~B4
6~50
71
05
9..
64
9100
\3
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4-..
8~3S
85
94
8~39
9~97
.
90
02
7~&O
944
0
91
89
SMEA
HStR~SS
91
25
51
37
0
41
07
1
52
06
7
54
32
0
56
82
6
5411
11
53
81
3
51
25
b
54
53
3
50
29
2
60
90
3
50
82
8
513l
H
53
41
0
54
23
6
BENOll~G
STR
ESS
55
91
1
55
82
5
S8
2'n
3.8
5
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Ob
3.8
2
3.9
4
3.4
2
3.2
3
3.9
4
3.9
0
4.0
5
3.2
tl
2.9
7
3.8
1
3.7
5
3.6
4
4.1
0
4.0
3
3.7
5
NV
ISrO~Qut:
3.9
1
22
.42
22
.50
22
.50
22
.58
22
.53
22
.52
22
.53
22
.50
22
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2C
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22
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22
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22
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22
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22
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22
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22
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NeA
LTO
HQ
UE
22
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TE
StM
OUE
=4
3040
00
30
40
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30
40
00
3040
00
3040
00
RCnL
TO~:)U£
3040
00
30
40
00
30
40
00
304
000
3040
00
30
40
00
30
40
00
304
000
304
000
3040
DO
304
000
30
40
00
304
000
14
.00
16
.48
13
.70
14
.73
14
.66
12
.81
13
.83
15
.20
15
.95
1i.
92
i5.5
9
15
.20
14
.62
14
.55
14
.78
15
.42
-
14
.20
14
.78
NV
ISlJE
NO
ING
24
.95
25
.00
25
.04
24
.96
24
.98
25
.00
24
.96
25
.07
25
.07
25
.08
25
.00
25
.06
24
.91
25
.17
25
.03
24
.95
24
.98
NCAL
BEN
OiN
G
25
.00
ENDU
RANC
ETE
ST
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
25
00
00
ReA
LRE
ND
ING
25
00
00
22
.0
24
.0
21
.0
25
.0
26
.0
26
.0
23
.0
22
.0
22
.0
25
.0
21
.0
23
.0
25
.0
26
.0
25
.0
_2
7.0
PAN
WEI
GHT
22
.0
2ie
O
22 22 ? 2 2 22 2 2 2 2 2 2 2 2 2
MA
CHIN
EN
O.
50
6
52
8
44
4
46
6
52
4
54
6
49
9
53
2
48
3
52
9
58
7
581
6it
58
9
60
3
SPEC
IMEN
NO
.
47
7
46
5
47
8
44
5
45
5
45
7
46
5
TEST NO
.
48
4
52
0
52
2
52
7
52
8
52
9
53
3
53
6
53
9
58
7
59
)
6'00
60
3
--60
6
C
o~.t~I~
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o~
•I
..
0,0
)·~
•(.
MEA
NST
RES
SRA
TIO
•ST
D.
DEV
.OF
STR
ESS
RATI
O=
3.6
96
75
.61
98
7
.......
'
1I
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<•.
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MEA
NST
RES
SR
ATI
O=
STD
.D
EV.
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. APPENDIX F
/
PRIN'l'OUT OF FROGRAI'l TO CALCULATE
THE STHEl~GTH DIS'fRIBUTIOH ESTHIATES
FOR BEAN AND STANDARD DEVIATION
167
<'
,..
00
00
24
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0000
2100
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0063
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0001
1500
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0001
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0001
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0131
0001
320
00
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0001
40oo
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6001
4300
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0001
5100
0154
0001
5560
11i6
000
0163
0001
6300
0164
PRO
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MST
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LIST OF REFERENCES
Kececicglu, Dr. Dimitri, McKinley, J. N., and Saroni,Mauric~ J. "A Probabilistic Method of DesigningSp~clfied Reliabilities Into Mechanical Componentsl':i th Time Dependent Stress and Strength Dish'ibutions."Report to NASA on Grant NGR 03-002-044·, Janu.8.ry 25, 19G7.331 pp.
Shigle~T, J. E. r.echaniqa~ Engineerinl2: Delli:.D' 1st edtNeN York: McGraw-Hill Book Company, Inc., 1963, 631 pp.
Kececioglu, Dr. Dimitri, and McCom1ell, J~ffrey B."Ca) i O}'H tion of Combined Bending-Torsion FatigueR~liab1lity Research Machines and Reliability Data
. Redu'c t1 on. " Report to NASA on Grant NGR 03-002-04·!~·,
July 31, 1969. 151 pp.
A Guide r6r Fati~ue Testin~ and the Statisticalxr;o.lxiL~_2f·Fatig.ll8 Da'ta. 2nd' ed. As rn1 Special Te-chnicalI)ublic~::.tion No. 91-A,1963. 83 PP. ;
Dixon, H. J., and Hood, A. r·l."A ~lethod for Obtainingand Analyzing Sensitivitv Data." Jou~np' or th~ ~~~~·c~n
g!lti~tical Assoc,iatioD, v Vol. J}3, 1948. -'.~ pp:--i·09--12·g-:--~-..:::..:..
Kececioglu, Dr. Dimitri, Saroni, M. J., Broome, H. Wilsonand HcConnell, Jeffery. "Design and Development of and.
. Results Fro~ Combined Bending-Torsion.Fatigue ReliabilityRese::trch l':achincs." Report to NASA on Grant NGR 03-002··044, July 15, 1969. 57 pp.
Ke(jec~.o[SlD., Dr •. Dimttri ,. 8.nd Broome t H •.\-1 ilson •.·'P:cob::::b.i~~st~~-Gr3.phical.. Hnd Phenoraenolcgical_ ..~nalys isof Combined. Bending-Torsion Fatigue Reliability Data."Report to NASA on Grant NGR 03-002-044, July 30, 1969.79 pp.
a, Fatel, Anilkumar V. "A Statistical Study of FatigueStrength of AISI h3 l l-0 Steel Under Combined Stresses."Masters Report. Tucson: The University of Arizona, 1969.;
175
10. Hahn, G. J. and Shapiro. S. S. Statistical Models------_.__.._--.,........tD.-1~~n.stu~er~D.B.. NeN York: John i-I iley & Sons, 19b7.355 pp. ,
'--
11. Sm!th, Bicihard E. "Time Dependent Reliability ofCo:npOn(~!lts Subjected to Simple Fatigue." MastersThosis. Tucson: The University of Arizona, 1965150 pp.
12. - Ju"{inall,R •. c. Stres~ Strain and Stre~gth. Nel'l York:1'lcG:::-8.H-Hill Bool~ Comp;-ny, 196"7:---5"80pp:----
13. !1~1u6e11 f !!:.Ncu YOl'\{:
B. PrQ..babilistic A12..DJ:'08.ches to DesLe;.u.John Hiley & Sons. 19i58:" ,32,3 pp.