ASTM E739-91(2004)e1
(Practice)Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (ε-N) Fatigue Data
Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (<bdit>S-N</bdit>) and Strain-Life (<bdit>ε-N</bdit>) Fatigue Data
SCOPE
1.1 This practice covers only S-N and ε-N relationships that may be reasonably approximated by a straight line (on appropriate coordinates) for a specific interval of stress or strain. It presents elementary procedures that presently reflect good practice in modeling and analysis. However, because the actual S-N or ε-N relationship is approximated by a straight line only within a specific interval of stress or strain, and because the actual fatigue life distribution is unknown, it is not recommended that ( a) the S-N or ε-N curve be extrapolated outside the interval of testing, or ( b) the fatigue life at a specific stress or strain amplitude be estimated below approximately the fifth percentile (P 0.05). As alternative fatigue models and statistical analyses are continually being developed, later revisions of this practice may subsequently present analyses that permit more complete interpretation of S-N and ε-N data.
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´1
Designation:E739–91 (Reapproved 2004)
Standard Practice for
Statistical Analysis of Linear or Linearized Stress-Life (S-N)
and Strain-Life (´-N) Fatigue Data
This standard is issued under the fixed designation E739; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
´ NOTE—Editorial changes were made throughout in May 2006.
1. Scope E606 Practice for Strain-Controlled Fatigue Testing
1.1 This practice covers only S-N and ´-N relationships that
3. Terminology
may be reasonably approximated by a straight line (on appro-
3.1 The terms used in this practice shall be used as defined
priate coordinates) for a specific interval of stress or strain. It
in Definitions E206 and E513. In addition, the following
presents elementary procedures that presently reflect good
terminology is used:
practiceinmodelingandanalysis.However,becausetheactual
3.1.1 dependent variable—the fatigue life N (or the loga-
S-N or ´-N relationship is approximated by a straight line only
rithm of the fatigue life).
within a specific interval of stress or strain, and because the
3.1.1.1 Discussion—Log (N) is denoted Y in this practice.
actual fatigue life distribution is unknown, it is not recom-
3.1.2 independent variable—the selected and controlled
mended that (a) the S-N or ´-N curve be extrapolated outside
variable (namely, stress or strain). It is denoted X in this
the interval of testing, or (b) the fatigue life at a specific stress
practice when plotted on appropriate coordinates.
or strain amplitude be estimated below approximately the fifth
3.1.3 log-normal distribution—the distribution of N when
percentile (P . 0.05). As alternative fatigue models and
log (N) is normally distributed. (Accordingly, it is convenient
statistical analyses are continually being developed, later
to analyze log (N) using methods based on the normal
revisions of this practice may subsequently present analyses
distribution.)
that permit more complete interpretation of S-N and ´-N data.
3.1.4 replicate (repeat) tests—nominally identical tests on
2. Referenced Documents different randomly selected test specimens conducted at the
same nominal value of the independent variable X. Such
2.1 ASTM Standards:
replicateorrepeattestsshouldbeconductedindependently;for
E206 Discontinued 1988; Definitions of Terms Relating to
example,eachreplicatetestshouldinvolveaseparatesetofthe
Fatigue Testing and the Statistical Analysis of Fatigue
3 test machine and its settings.
Data; Replaced by E1150
3.1.5 run out—no failure at a specified number of load
E467 Practice for Verification of Constant Amplitude Dy-
cycles (Practice E468).
namic Forces in an Axial Fatigue Testing System
3.1.5.1 Discussion—The analyses illustrated in this practice
E468 Practice for Presentation of Constant Amplitude Fa-
do not apply when the data include either run-outs (or
tigue Test Results for Metallic Materials
suspended tests). Moreover, the straight-line approximation of
E513 Definitions of Terms Relating to Costant-Amplitude,
the S-Nor ´-Nrelationshipmaynotbeappropriateatlonglives
Low-Cycle Fatigue Testing
when run-outs are likely.
3.1.5.2 Discussion—For purposes of statistical analysis, a
run-out may be viewed as a test specimen that has either been
ThispracticeisunderthejurisdictionofASTMCommitteeE08onFatigueand
removed from the test or is still running at the time of the data
Fracture and is the direct responsibility of Subcommittee E08.04 on Structural
Applications.
analysis.
Current edition approved May 1, 2004. Published June 2004. Originally
approved in 1980. Last previous edition approved in 1998 as E739–91(1998).
4. Significance and Use
DOI: 10.1520/E0739-91R04E01.
4.1 Materials scientists and engineers are making increased
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
use of statistical analyses in interpreting S-N and ´-N fatigue
Standards volume information, refer to the standard’s Document Summary page on
data. Statistical analysis applies when the given data can be
the ASTM website.
reasonably assumed to be a random sample of (or representa-
Withdrawn. The last approved version of this historical standard is referenced
on www.astm.org. tion of) some specific defined population or universe of
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
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E739–91 (2004)
material of interest (under specific test conditions), and it is terms of some appropriate independent (controlled) variable.
desired either to characterize the material or to predict the
NOTE 1—In certain cases, the amplitude of the stress or strain is not
performance of future random samples of the material (under
constant during the entire test for a given specimen. In such cases some
similar test conditions), or both.
effective (equivalent) value of S or ´ must be established for use in
analysis.
5. Types of S-N and ´-N Curves Considered
5.1.1 The fatigue life N is the dependent (random) variable
5.1 It is well known that the shape of S-N and ´-N curves
in S-N and ´-N tests, whereas S or ´ is the independent
can depend markedly on the material and test conditions. This
(controlled) variable.
practice is restricted to linear or linearized S-N and ´-N
relationships, for example, NOTE 2—In certain cases, the independent variable used in analysis is
not literally the variable controlled during testing. For example, it is
log N 5 A 1 B S or (1)
~ !
common practice to analyze low-cycle fatigue data treating the range of
log N 5 A 1 B ~´!or
plastic strain as the controlled variable, when in fact the range of total
strain was actually controlled during testing.Although there may be some
log N 5 A 1 B ~log S! or (2)
question regarding the exact nature of the controlled variable in certain
log N 5 A 1 B log ´!
~
S-N and ´-N tests, there is never any doubt that the fatigue life is the
dependent variable.
in which S and ´ may refer to (a) the maximum value of
NOTE 3—In plotting S-N and ´-N curves, the independent variables S
constant-amplitude cyclic stress or strain, given a specific
and ´ are plotted along the ordinate, with life (the dependent variable)
value of the stress or strain ratio, or of the minimum cyclic
plotted along the abscissa. Refer, for example, to Fig. 1.
stress or strain, (b) the amplitude or the range of the constant-
amplitude cyclic stress or strain, given a specific value of the 5.1.2 Thedistributionoffatiguelife(inanytest)isunknown
mean stress or strain, or (c) analogous information stated in (and indeed may be quite complex in certain situations). For
NOTE 1—The 95% confidence band for the ´-N curve as a whole is based on Eq 10. (Note that the dependent variable, fatigue life, is plotted here
along the abscissa to conform to engineering convention.)
FIG. 1 Fitted Relationship Between the Fatigue Life N (Y) and the Plastic Strain Amplitude D´ /2 (X) for the Example Data Given
p
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E739–91 (2004)
the purposes of simplifying the analysis (while maintaining
Research and development testing of components and 6to12
specimens
sound statistical procedures), it is assumed in this practice that
Design allowables data 12 to 24
thelogarithmsofthefatiguelivesarenormallydistributed,that
Reliability data 12 to 24
is, the fatigue life is log-normally distributed, and that the
A
Ifthevariabilityislarge,awideconfidencebandwillbeobtainedunlessalarge
variance of log life is constant over the entire range of the
number of specimens are tested (See 8.1.1).
independent variable used in testing (that is, the scatter in log
7.1.2 Replication—The replication guidelines given in
N is assumed to be the same at low S and ´ levels as at high
Chapter 3 of Ref (1) are based on the following definition:
levels of S or ´). Accordingly, log N is used as the dependent
% replication = 100 [1 − (total number of different stress or strain levels used
(random)variableinanalysis.Itisdenoted Y.Theindependent
in testing/total number of specimens tested)]
variable is denoted X. It may be either S or ´,orlog S or log
A
´, respectively, depending on which appears to produce a Type of Test Percent Replication
straight line plot for the interval of S or ´ of interest. Thus Eq
Preliminary and exploratory (research and development 17 to 33 min
1 and Eq 2 may be re-expressed as
tests)
Research and development testing of components and 33 to 50 min
Y 5 A 1 BX (3)
specimens
Design allowables data 50 to 75 min
Eq 3 is used in subsequent analysis. It may be stated more
Reliability data 75 to 88 min
precisely as µ = A+ BX, where µ is the expected value
Y ? X Y ? X
A
of Y given X.
Note that percent replication indicates the portion of the total number of
specimens tested that may be used for obtaining an estimate of the variability of
NOTE 4—For testing the adequacy of the linear model, see 8.2.
replicate tests.
NOTE 5—The expected value is the mean of the conceptual population
7.1.2.1 Replication Examples—Good replication: Suppose
of all Y’s given a specific level of X. (The median and mean are identical
that ten specimens are used in research and development for
for the symmetrical normal distribution assumed in this practice for Y.)
the testing of a component. If two specimens are tested at each
of five stress or strain amplitudes, the test program involves
6. Test Planning
50% replications. This percent replication is considered ad-
6.1 Testplanningfor S-Nand ´-Ntestprogramsisdiscussed
equate for most research and development applications. Poor
in Chapter 3 of Ref (1). Planned grouping (blocking) and
replication: Suppose eight different stress or strain amplitudes
randomization are essential features of a well-planned test
are used in testing, with two replicates at each of two stress or
program. In particular, good test methodology involves use of
strain amplitudes (and no replication at the other six stress or
planned grouping to (a) balance potentially spurious effects of
strain amplitudes). This test program involves only 20%
nuisance variables (for example, laboratory humidity) and (b)
replication, which is not generally considered adequate.
allow for possible test equipment malfunction during the test
program.
8. Statistical Analysis (Linear Model Y =A + BX, Log-
Normal Fatigue Life Distribution with Constant
7. Sampling
Variance Along the Entire Interval of X Used in
7.1 It is vital that sampling procedures be adopted that
Testing, No Runouts or Suspended Tests or Both,
assurearandomsampleofthematerialbeingtested.Arandom
Completely Randomized Design Test Program)
sample is required to state that the test specimens are repre-
sentative of the conceptual universe about which both statisti-
8.1 For the case where (a) the fatigue life data pertain to a
cal and engineering inference will be made.
random sample (all Y are independent), (b) there are neither
i
run-outs nor suspended tests and where, for the entire interval
NOTE 6—A random sampling procedure provides each specimen that
of Xusedintesting,(c)the S-Nor ´-Nrelationshipisdescribed
conceivablycouldbeselected(tested)anequal(orknown)opportunityof
by the linear model Y=A+BX (more precisely by µ
actually being selected at each stage of the sampling process. Thus, it is
Y ? X
poor practice to use specimens from a single source (plate, heat, supplier) =A+BX), (d) the (two parameter) log-normal distribution
when seeking a random sample of the material being tested unless that
describes the fatigue life N, and (e) the variance of the
particular source is of specific interest.
log-normal distribution is constant, the maximum likelihood
NOTE 7—Procedures for using random numbers to obtain random
estimators of A and B are as follows:
samples and to assign stress or strain amplitudes to specimens (and to
¯ ˆ ¯
establish the time order of testing) are given in Chapter 4 of Ref (2).
 5 Y 2 B X (4)
k
7.1.1 Sample Size—The minimum number of specimens
¯ ¯
~X 2 X! ~Y 2 Y!
(
i i
required in S-N (and ´-N) testing depends on the type of test
i 51
ˆ
B 5 (5)
k
programconducted.ThefollowingguidelinesgiveninChapter
¯
~X 2 X!
(
i
3ofRef (1) appear reasonable.
i 51
Minimum Number
Type of Test
A where the symbol “caret”(^) denotes estimate (estimator),
of Specimens
¯
the symbol “overbar”( ) denotes average (for example, Y =
Preliminary and exploratory (exploratory research and 6to12 k k
¯
Y /k and X = X/k), Y =log N, X = S or ´,or
( i 51 i (i 51 i i i i i i
development tests)
log S or log ´ (refer to Eq 1 and Eq 2), and k is the total
i i
number of test specimens (the total sample size). The recom-
mended expression for estimating the variance of the normal
The boldface numbers in parentheses refer to the list of references appended to
this standard. distribution for log N is
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E739–91 (2004)
k
confidence intervals for A and B that pertain to confidence levels greater
ˆ
~Y 2 Y !
(
i i
than approximately 0.95 are not recommended.
i 51
ˆs 5 (6)
k 22
8.1.1.1 The meaning of the confidence interval associated
ˆ
ˆ with,say,Eq8isasfollows(Note11).Ifthevaluesof t given
in which Y = Â + BX and the (k − 2) term in the denomi-
p
i i
in Table 1 for, say, P=95% are used in a series of analyses
nator is used instead of k to make sˆ an unbiased estimator of
involving the estimation of B from independent data sets, then
the normal population variance s .
in the long run we may expect 95% of the computed intervals
NOTE 8—An assumption of constant variance is usually reasonable for
toincludethevalue B.Ifineachinstanceweweretoassertthat
notchedandjointspecimensuptoabout10 cyclestofailure.Thevariance
B lies within the interval computed, we should expect to be
ofunnotchedspecimensgenerallyincreaseswithdecreasingstress(strain)
correct 95 times in 100 and in error 5 times in 100: that is, the
level (see Section 9). If the assumption of constant variance appears to be
dubious,thereaderisreferredtoRef (3)fortheappropriatestatisticaltest.
statement “B lies within the computed interval” has a 95%
probability of being correct. But there would be no operational
8.1.1 Confidence Intervals for Parameters A and B—The
ˆ meaning in the following statement made in any one instance:
estimators  and B are normally distributed with expected
“The probability is 95% that B falls within the computed
values Aand B,respectively,(regardlessoftotalsamplesize k)
interval in this case” since B either does or does not fall within
...
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