ASTM E739-91(2004)
(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 (<i>S-N</i>) and Strain-Life (ε-<i>N</i>) 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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Designation:E739–91 (Reapproved 2004)
Standard Practice for
Statistical Analysis of Linear or Linearized Stress-Life (S-N)
and Strain-Life (e-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 (e) indicates an editorial change since the last revision or reapproval.
1. Scope 3.1.1 dependent variable—the fatigue life N (or the loga-
rithm of the fatigue life).
1.1 This practice covers only S-N and e-N relationships that
3.1.1.1 Discussion—Log (N) is denoted Y in this practice.
may be reasonably approximated by a straight line (on appro-
3.1.2 independent variable—the selected and controlled
priate coordinates) for a specific interval of stress or strain. It
variable (namely, stress or strain). It is denoted X in this
presents elementary procedures that presently reflect good
practice when plotted on appropriate coordinates.
practiceinmodelingandanalysis.However,becausetheactual
3.1.3 log-normal distribution—the distribution of N when
S-N or e-N relationship is approximated by a straight line only
log (N) is normally distributed. (Accordingly, it is convenient
within a specific interval of stress or strain, and because the
to analyze log (N) using methods based on the normal
actual fatigue life distribution is unknown, it is not recom-
distribution.)
mended that (a) the S-N or e-N curve be extrapolated outside
3.1.4 replicate (repeat) tests—nominally identical tests on
the interval of testing, or (b) the fatigue life at a specific stress
different randomly selected test specimens conducted at the
or strain amplitude be estimated below approximately the fifth
same nominal value of the independent variable X. Such
percentile (P . 0.05). As alternative fatigue models and
replicateorrepeattestsshouldbeconductedindependently;for
statistical analyses are continually being developed, later
example,eachreplicatetestshouldinvolveaseparatesetofthe
revisions of this practice may subsequently present analyses
test machine and its settings.
that permit more complete interpretation of S-N and e-N data.
3.1.5 run out—no failure at a specified number of load
2. Referenced Documents cycles (Practice E468).
3.1.5.1 Discussion—The analyses illustrated in this practice
2.1 ASTM Standards:
do not apply when the data include either run-outs (or
E206 Definitions of Terms Relating to Fatigue Testing and
suspended tests). Moreover, the straight-line approximation of
the Statistical Analysis of Fatigue Data
the S-Nor e-Nrelationshipmaynotbeappropriateatlonglives
E467 Practice for Verification of Constant Amplitude Dy-
when run-outs are likely.
namic Forces in an Axial Fatigue Testing System
3.1.5.2 Discussion—For purposes of statistical analysis, a
E468 Practice for Presentation of Constant Amplitude Fa-
run-out may be viewed as a test specimen that has either been
tigue Test Results for Metallic Materials
removed from the test or is still running at the time of the data
E 513 Definitions of Terms Relating to Constant-
analysis.
Amplitude, Low-Cycle Fatigue Testing
E606 Practice for Strain-Controlled Fatigue Testing
4. Significance and Use
3. Terminology 4.1 Materials scientists and engineers are making increased
use of statistical analyses in interpreting S-N and e-N fatigue
3.1 The terms used in this practice shall be used as defined
data. Statistical analysis applies when the given data can be
in Definitions E206 and E513. In addition, the following
reasonably assumed to be a random sample of (or representa-
terminology is used:
tion of) some specific defined population or universe of
material of interest (under specific test conditions), and it is
ThispracticeisunderthejurisdictionofASTMCommitteeE08onFatigueand
desired either to characterize the material or to predict the
Fracture and is the direct responsibility of Subcommittee E08.04 on Structural
performance of future random samples of the material (under
Applications.
similar test conditions), or both.
Current edition approved May 1, 2004. Published June 2004. Originally
approved in 1980. Last previous edition approved in 1998 as E739–91(1998).
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
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website.
Withdrawn.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
E739–91 (2004)
5. Types of S-N and e-N Curves Considered 5.1.1 The fatigue life N is the dependent (random) variable
in S-N and e-N tests, whereas S or e is the independent
5.1 It is well known that the shape of S-N and e-N curves
(controlled) variable.
can depend markedly on the material and test conditions. This
practice is restricted to linear or linearized S-N and e-N
NOTE 2—In certain cases, the independent variable used in analysis is
relationships, for example,
not literally the variable controlled during testing. For example, it is
common practice to analyze low-cycle fatigue data treating the range of
log N 5 A 1 B ~S! or (1)
plastic strain as the controlled variable, when in fact the range of total
log N 5 A 1 B ~e!or
strain was actually controlled during testing.Although there may be some
question regarding the exact nature of the controlled variable in certain
log N 5 A 1 B ~log S! or (2)
S-N and e-N tests, there is never any doubt that the fatigue life is the
log N 5 A 1 B ~log e!
dependent variable.
NOTE 3—In plotting S-N and e-N curves, the independent variables S
in which S and e may refer to (a) the maximum value of
and e are plotted along the ordinate, with life (the dependent variable)
constant-amplitude cyclic stress or strain, given a specific
plotted along the abscissa. Refer, for example, to Fig. 1.
value of the stress or strain ratio, or of the minimum cyclic
stress or strain, (b) the amplitude or the range of the constant-
5.1.2 Thedistributionoffatiguelife(inanytest)isunknown
amplitude cyclic stress or strain, given a specific value of the (and indeed may be quite complex in certain situations). For
mean stress or strain, or (c) analogous information stated in
the purposes of simplifying the analysis (while maintaining
terms of some appropriate independent (controlled) variable. sound statistical procedures), it is assumed in this practice that
thelogarithmsofthefatiguelivesarenormallydistributed,that
NOTE 1—In certain cases, the amplitude of the stress or strain is not
is, the fatigue life is log-normally distributed, and that the
constant during the entire test for a given specimen. In such cases some
variance of log life is constant over the entire range of the
effective (equivalent) value of S or e must be established for use in
analysis. independent variable used in testing (that is, the scatter in log
NOTE 1—The95%confidencebandforthe e-NcurveasawholeisbasedonEq9.(Notethatthedependentvariable,fatiguelife,isplottedherealong
the abscissa to conform to engineering convention.)
FIG. 1 Fitted Relationship Between the Fatigue Life N (Y) and the Plastic Strain Amplitude De /2 (X) for the Example Data Given
p
E739–91 (2004)
N is assumed to be the same at low S and e levels as at high 7.1.2 Replication—The replication guidelines given in
levels of S or e). Accordingly, log N is used as the dependent Chapter 3 of Ref (1) are based on the following definition:
(random)variableinanalysis.Itisdenoted Y.Theindependent
% replication = 100 [1 − (total number of different stress or strain levels used
in testing/total number of specimens tested)]
variable is denoted X. It may be either S or e,orlog S or log
e, respectively, depending on which appears to produce a
A
Type of Test Percent Replication
straight line plot for the interval of S or e of interest. Thus Eq
Preliminary and exploratory (research and development 17 to 33 min
1 and Eq 2 may be re-expressed as
tests)
Y 5 A 1 BX (3) Research and development testing of components and 33 to 50 min
specimens
Eq 3 is used in subsequent analysis. It may be stated more
Design allowables data 50 to 75 min
precisely as µ = A+ BX, where µ is the expected value Reliability data 75 to 88 min
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 e-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-
8.1 For the case where (a) the fatigue life data pertain to a
sentative of the conceptual universe about which both statisti-
random sample (all Y are independent), (b) there are neither
i
cal and engineering inference will be made.
run-outs nor suspended tests and where, for the entire interval
of Xusedintesting,(c)the S-Nor e-Nrelationshipisdescribed
NOTE 6—A random sampling procedure provides each specimen that
conceivablycouldbeselected(tested)anequal(orknown)opportunityof by the linear model Y=A+BX (more precisely by µ
Y? X
actually being selected at each stage of the sampling process. Thus, it is
=A+BX), (d) the (two parameter) log-normal distribution
poor practice to use specimens from a single source (plate, heat, supplier)
describes the fatigue life N, and (e) the variance of the
when seeking a random sample of the material being tested unless that
log-normal distribution is constant, the maximum likelihood
particular source is of specific interest.
estimators of A and B are as follows:
NOTE 7—Procedures for using random numbers to obtain random
samples and to assign stress or strain amplitudes to specimens (and to
¯ ˆ ¯
 5 Y 2 B X (4)
establish the time order of testing) are given in Chapter 4 of Ref (2).
k
¯ ¯
X 2 X Y 2 Y
7.1.1 Sample Size—The minimum number of specimens ~ ! ~ !
( i i
i 51
ˆ
B 5 (5)
required in S-N (and e-N) testing depends on the type of test
k
¯
programconducted.ThefollowingguidelinesgiveninChapter ~X 2 X!
(
i
i 51
3ofRef (1) appear reasonable.
where the symbol “caret”(^) denotes estimate (estimator),
Minimum Number
Type of Test
¯
A
the symbol “overbar”( ) denotes average (for example, Y =
of Specimens
k k
¯
Y /k and X = X/k), Y =log N, X = S or e,or
( (
i 51 i i 51 i i i i i i
Preliminary and exploratory (exploratory research and 6to12
log S or log e (refer to Eq 1 and Eq 2), and k is the total
i i
development tests)
number of test specimens (the total sample size). The recom-
Research and development testing of components and 6to12
specimens
mended expression for estimating the variance of the normal
Design allowables data 12 to 24
distribution for log N is
Reliability data 12 to 24
k
A
Ifthevariabilityislarge,awideconfidencebandwillbeobtainedunlessalarge ˆ
~Y 2 Y !
(
i i
i 51
number of specimens are tested (See 8.1.1).
ˆs 5 (6)
k 22
ˆ ˆ
in which Y = Â + BX and the (k − 2) term in the denomi-
i i
nator is used instead of k to make sˆ an unbiased estimator of
The boldface numbers in parentheses refer to the list of references appended to
this standard. the normal population variance s .
E739–91 (2004)
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
statement “B lies within the computed interval” has a 95%
dubious,thereaderisreferredtoRef (3)fortheappropriatestatisticaltest.
probability of being correct. But there would be no operational
8.1.1 Confidence Intervals for Parameters A and B—The
meaning in the following statement made in any one instance:
ˆ
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
when conditions (a) through (e) in 8.1 are met. Accordingly,
the interval. It should also be emphasized that even in
confidence intervals for parameters Aand B can be established
independent samples from the same universe, the intervals
using the t distribution, Table 1. The confidence interval for A
givenbyEq8willvarybothinwidthandpositionfromsample
is given by  6 t sˆ ,or
p Â
to sample. (This variation will be particularly noticeable for
2 ½
small samples.) It is this series of (random) intervals “fluctu-
1 X
 6 t sˆ 1 , (7)
p k
k ating” in size and position that will include, ideally, the value
F G
~X 2 X!
(
i
B 95 times out of 100 for P=95%. Similar
...
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