ASTM E739-91(1998)
(Practice)Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (e-N) Fatigue Data
Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (e-N) Fatigue Data
SCOPE
1.1 This practice pertains only to S-N and e-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 e-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 e-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 [similar] 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 e-N data.
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Designation: E 739 – 91 (Reapproved 1998)
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 E 739; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number 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 pertains only to S-N and e-N relationships
3.1.1.1 Discussion—Log ( N) is denoted Y herein.
that may be reasonably approximated by a straight line (on
3.1.2 independent variable—the selected and controlled
appropriate coordinates) for a specific interval of stress or
variable (namely, stress or strain). It is denoted X herein when
strain. It presents elementary procedures that presently reflect
plotted on appropriate coordinates.
good practice in modeling and analysis. However, because the
3.1.3 log-normal distribution—the distribution of N when
actual S-N or e-N relationship is approximated by a straight line
log ( N) is normally distributed. (Accordingly, it is convenient
only within a specific interval of stress or strain, and because
to analyze log (N) using methods based on the normal
the actual fatigue life distribution is unknown, it is not
distribution.)
recommended that ( a) the S-N or e-N curve be extrapolated
3.1.4 replicate (repeat) tests—nominally identical tests on
outside the interval of testing, or ( b) the fatigue life at a
different randomly selected test specimens conducted at the
specific stress or strain amplitude be estimated below approxi-
same nominal value of the independent variable X. Such
mately the fifth percentile ( P . 0.05). As alternative fatigue
replicate or repeat tests should be conducted independently; for
models and statistical analyses are continually being devel-
example, each replicate test should involve a separate set of the
oped, later revisions of this practice may subsequently present
test machine and its settings.
analyses that permit more complete interpretation of S-N and
3.1.5 run out—no failure at a specified number of load
e-N data.
cycles (Practice E 468).
2. Referenced Documents
3.1.5.1 Discussion—The analyses illustrated herein do not
apply when the data include either run-outs (or suspended
2.1 ASTM Standards:
tests). Moreover, the straight-line approximation of the S-N or
E 206 Definitions of Terms Relating to Fatigue Testing and
e-N relationship may not be appropriate at long lives when
the Statistical Analysis of Fatigue Data
run-outs are likely.
E 467 Practice for Verification of Constant Amplitude Dy-
3.1.5.2 Discussion—For purposes of statistical analysis, a
namic Loads on Displacements in an Axial Load Fatigue
run-out may be viewed as a test specimen that has either been
Testing System
removed from the test or is still running at the time of the data
E 468 Practice for Presentation of Constant Amplitude Fa-
analysis.
tigue Test Results for Metallic Materials
E 513 Definitions of Terms Relating to Constant-
4. Significance and Use
Amplitude, Low-Cycle Fatigue Testing
3 4.1 Materials scientists and engineers are making increased
E 606 Practice for Strain-Controlled Fatigue Testing
use of statistical analyses in interpreting S-N and e-N fatigue
3. Terminology data. Statistical analysis applies when the given data can be
reasonably assumed to be a random sample of (or representa-
3.1 The terms used in this practice shall be used as defined
tion of) some specific defined population or universe of
in Definitions E 206 and E 513. In addition, the following
material of interest (under specific test conditions), and it is
terminology is used:
desired either to characterize the material or to predict the
performance of future random samples of the material (under
This practice is under the jurisdiction of ASTM Committee E-8 on Fatigue and
similar test conditions), or both.
Fracture and is the direct responsibility of Subcommittee E08.04 on Structural
Applications.
Current edition approved April 15, 1991. Published June 1991. Originally
e1
published as E 739 – 80. Last previous edition E 739 – 80 (1986) .
Discontinued, see 1986 Annual Book of ASTM Standards, Vol 03.01.
Annual Book of ASTM Standards, Vol 03.01.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
E 739 – 91 (1998)
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 constan-
5.1.2 The distribution of fatigue life (in any test) is unknown
tamplitude 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 herein that the
logarithms of the fatigue lives are normally distributed, that is,
NOTE 1—In certain cases the amplitude of the stress or strain is not
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—The 95 % confidence band for the e-N curve as a whole is based on Eq 9. (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 De /2 (X) for the Example Data Given
p
E 739 – 91 (1998)
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) variable in analysis. It is denoted Y. The independent
% 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 10 specimens are used in research and development for the
for the symmetrical normal distribution assumed herein for Y.)
testing of a component. If two specimens are tested at each of
five stress or strain amplitudes, the test program involves 50 %
6. Test Planning
replications. This percent replication is considered adequate for
6.1 Test planning for S-N and e-N test programs is discussed
most research and development applications. Poor replication:
in Chapter 3 of Ref (1). Planned grouping (blocking) and
Suppose eight different stress or strain amplitudes are used in
randomization are essential features of a well-planned test
testing, with two replicates at each of two stress or strain
program. In particular, good test methodology involves use of
amplitudes (and no replication at the other six stress or strain
planned grouping to (a) balance potentially spurious effects of
amplitudes). This test program involves only 20 % replication,
nuisance variables (for example, laboratory humidity) and (b)
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 which Testing, No Runouts or Suspended Tests or Both,
assure a random sample of the material being tested. A random 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 X used in testing, ( c) the S-N or e- N relationship is
NOTE 6—A random sampling procedure provides each specimen that
conceivably could be selected (tested) an equal (or known) opportunity of described by the linear model Y=A+BX (more precisely by
actually being selected at each stage of the sampling process. Thus, it is
μ =A+BX), (d) the (two parameter) log-normal distribu-
Y ? X
poor practice to use specimens from a single source (plate, heat, supplier)
tion 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 5 1
ˆ
B 5 (5)
required in S-N (and e- N) testing depends on the type of test
k
¯
program conducted. The following guidelines given in Chapter ~X 2 X!
(
i
i 5 1
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 5 1 i i 5 1 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
If the variability is large, a wide confidence band will be obtained unless a large ˆ
~Y 2 Y !
(
i i
i 5 1
number of specimens are tested (See 8.1.1).
ˆs 5 (6)
k 2 2
ˆ ˆ
in which Y = Â + B X and the (k − 2) term in the
i i
denominator is used instead of k to make sˆ an unbiased
The boldface numbers in parentheses refer to the list of references appended to
this standard. estimator of the normal population variance s .
E 739 – 91 (1998)
NOTE 8—An assumption of constant variance is usually reasonable for
the value B. If in each instance we were to assert that B lies
notched and joint specimens up to about 10 cycles to failure. The variance
within the interval computed, we should expect to be correct 95
of unnotched specimens generally increases with decreasing stress (strain)
times in 100 and in error 5 times in 100: that is, the statement
level (see Section 9). If the assumption of constant variance appears to be
“ B lies within the computed interval” has a 95 % probability
dubious the reader is referred to Ref (3) for the appropriate statistical test.
of being correct. But there would be no operational meaning in
8.1.1 Confidence Intervals for Parameters A and B—The
the following statement made in any one instance: “The
ˆ
estimators  and B are normally distributed with expected
probability is 95 % that B falls within the computed interval in
values A and B, respectively, (regardless of total sample size k)
this case” since B either does or does not fall within the
when conditions ( a) through (e) in 8.1 are met. Accordingly,
interval. It should also be emphasized that even in independent
confidence intervals for parameters A and B can be established
samples from the same universe, the intervals given by Eq 8
using the t distribution, Table 1. The confidence interval for A
will vary both in width and position from sample to sample.
is given by  6 t sˆ ,or
p Â
(This variation will be particularly noticeable for small
2 ½
samples.) It is this series of (random) intervals “fluctuating” in
1 X
 6 t sˆ 1 , (7)
p k
k size and position that wi
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