# ASTM E739-10 (Reapproved 2015)

Designation: E739 − 10 (Reapproved 2015)Standard Practice forStatistical Analysis of Linear or Linearized Stress-Life (S-N)and Strain-Life (ε-N) Fatigue Data1This standard is issued under the fixed designation E739; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers only S-N and ε-N relationships thatmay be reasonably approximated by a straight line (on appro-priate coordinates) for a specific interval of stress or strain. Itpresents elementary procedures that presently reflect goodpractice in modeling and analysis. However, because the actualS-N or ε-N relationship is approximated by a straight line onlywithin a specific interval of stress or strain, and because theactual fatigue life distribution is unknown, it is not recom-mended that (a) the S-N or ε-N curve be extrapolated outsidethe interval of testing, or (b) the fatigue life at a specific stressor strain amplitude be estimated below approximately the fifthpercentile (P . 0.05). As alternative fatigue models andstatistical analyses are continually being developed, laterrevisions of this practice may subsequently present analysesthat permit more complete interpretation of S-N and ε-N data.2. Referenced Documents2.1 ASTM Standards:2E206 Definitions of Terms Relating to Fatigue Testing andthe Statistical Analysis of Fatigue Data; Replaced byE 1150 (Withdrawn 1988)3E468 Practice for Presentation of Constant Amplitude Fa-tigue Test Results for Metallic MaterialsE513 Definitions of Terms Relating to Constant-Amplitude,Low-Cycle Fatigue Testing; Replaced by E 1150 (With-drawn 1988)3E606/E606M Test Method for Strain-Controlled FatigueTesting3. Terminology3.1 The terms used in this practice shall be used as definedin Definitions E206 and E513. In addition, the followingterminology is used:3.1.1 dependent variable—the fatigue life N (or the loga-rithm of the fatigue life).3.1.1.1 Discussion—Log (N) is denoted Y in this practice.3.1.2 independent variable—the selected and controlledvariable (namely, stress or strain). It is denoted X in thispractice when plotted on appropriate coordinates.3.1.3 log-normal distribution—the distribution of N whenlog (N) is normally distributed. (Accordingly, it is convenientto analyze log (N) using methods based on the normaldistribution.)3.1.4 replicate (repeat) tests—nominally identical tests ondifferent randomly selected test specimens conducted at thesame nominal value of the independent variable X. Suchreplicate or repeat tests should be conducted independently; forexample, each replicate test should involve a separate set of thetest machine and its settings.3.1.5 run out—no failure at a specified number of loadcycles (Practice E468).3.1.5.1 Discussion—The analyses illustrated in this practicedo not apply when the data include either run-outs (orsuspended tests). Moreover, the straight-line approximation ofthe S-N or ε-N relationship may not be appropriate at long liveswhen run-outs are likely.3.1.5.2 Discussion—For purposes of statistical analysis, arun-out may be viewed as a test specimen that has either beenremoved from the test or is still running at the time of the dataanalysis.4. Significance and Use4.1 Materials scientists and engineers are making increaseduse of statistical analyses in interpreting S-N and ε-N fatiguedata. Statistical analysis applies when the given data can bereasonably assumed to be a random sample of (or representa-tion of) some specific defined population or universe ofmaterial of interest (under specific test conditions), and it isdesired either to characterize the material or to predict theperformance of future random samples of the material (undersimilar test conditions), or both.1This practice is under the jurisdiction ofASTM Committee E08 on Fatigue andFracture and is the direct responsibility of Subcommittee E08.04 on StructuralApplications.Current edition approved Oct. 1, 2015. Published November 2015. Originallyapproved in 1980. Last previous edition approved in 2010 as E739 – 10. DOI:10.1520/E0739-10R15.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.3The last approved version of this historical standard is referenced onwww.astm.org.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States15. Types of S-N and ε-N Curves Considered5.1 It is well known that the shape of S-N and ε-N curvescan depend markedly on the material and test conditions. Thispractice is restricted to linear or linearized S-N and ε-Nrelationships, for example,log N 5 A1B ~S! or (1)log N 5 A1B ~ε! orlog N 5 A1B ~logS! or (2)log N 5 A1B ~logε!in which S and ε may refer to (a) the maximum value ofconstant-amplitude cyclic stress or strain, given a specificvalue of the stress or strain ratio, or of the minimum cyclicstress or strain, (b) the amplitude or the range of theconstant-amplitude cyclic stress or strain, given a specificvalue of the mean stress or strain, or (c) analogous informa-tion stated in terms of some appropriate independent (con-trolled) variable.NOTE 1—In certain cases, the amplitude of the stress or strain is notconstant during the entire test for a given specimen. In such cases someeffective (equivalent) value of S or ε must be established for use inanalysis.5.1.1 The fatigue life N is the dependent (random) variablein S-N and ε-N tests, whereas S or ε is the independent(controlled) variable.NOTE 2—In certain cases, the independent variable used in analysis isnot literally the variable controlled during testing. For example, it iscommon practice to analyze low-cycle fatigue data treating the range ofplastic strain as the controlled variable, when in fact the range of totalstrain was actually controlled during testing. Although there may be somequestion regarding the exact nature of the controlled variable in certainS-N and ε-N tests, there is never any doubt that the fatigue life is thedependent variable.NOTE 3—In plotting S-N and ε-N curves, the independent variables Sand ε are plotted along the ordinate, with life (the dependent variable)plotted along the abscissa. Refer, for example, to Fig. 1.5.1.2 The distribution of fatigue life (in any test) is unknown(and indeed may be quite complex in certain situations). Forthe purposes of simplifying the analysis (while maintainingsound statistical procedures), it is assumed in this practice thatthe logarithms of the fatigue lives are normally distributed, thatis, the fatigue life is log-normally distributed, and that thevariance of log life is constant over the entire range of theindependent variable used in testing (that is, the scatter in logNOTE 1—The 95 % confidence band for the ε-N curve as a whole is based on Eq 10. (Note that the dependent variable, fatigue life, is plotted here alongthe abscissa to conform to engineering convention.)FIG. 1 Fitted Relationship Between the Fatigue Life N (Y) and the Plastic Strain Amplitude ∆εp/2 (X) for the Example Data GivenE739 − 10 (2015)2N is assumed to be the same at low S and ε levels as at highlevels of S or ε). Accordingly, log N is used as the dependent(random) variable in analysis. It is denoted Y. The independentvariable is denoted X. It may be either S or ε,orlogS or log ε,respectively, depending on which appears to produce a straightline plot for the interval of S or ε of interest. Thus Eq 1 and Eq2 may be re-expressed asY 5 A1BX (3)Eq 3 is used in subsequent analysis. It may be stated moreprecisely as µY ? X5A1BX, where µY ? Xis the expected value of Ygiven X.NOTE 4—For testing the adequacy of the linear model, see 8.2.NOTE 5—The expected value is the mean of the conceptual populationof all Y’s given a specific level of X. (The median and mean are identicalfor the symmetrical normal distribution assumed in this practice for Y.)6. Test Planning6.1 Test planning for S-N and ε-N test programs is discussedin Chapter 3 of Ref (1).4Planned grouping (blocking) andrandomization are essential features of a well-planned testprogram. In particular, good test methodology involves use ofplanned grouping to (a) balance potentially spurious effects ofnuisance variables (for example, laboratory humidity) and (b)allow for possible test equipment malfunction during the testprogram.7. Sampling7.1 It is vital that sampling procedures be adopted thatassure a random sample of the material being tested.Arandomsample is required to state that the test specimens are repre-sentative of the conceptual universe about which both statisti-cal and engineering inference will be made.NOTE 6—A random sampling procedure provides each specimen thatconceivably could be selected (tested) an equal (or known) opportunity ofactually being selected at each stage of the sampling process. Thus, it ispoor practice to use specimens from a single source (plate, heat, supplier)when seeking a random sample of the material being tested unless thatparticular source is of specific interest.NOTE 7—Procedures for using random numbers to obtain randomsamples and to assign stress or strain amplitudes to specimens (and toestablish the time order of testing) are given in Chapter 4 of Ref (2).7.1.1 Sample Size—The minimum number of specimensrequired in S-N (and ε-N) testing depends on the type of testprogram conducted. The following guidelines given in Chapter3ofRef(1) appear reasonable.Type of TestMinimum Numberof SpecimensAPreliminary and exploratory (exploratory research anddevelopment tests)6to12Research and development testing of components andspecimens6to12Design allowables data 12 to 24Reliability data 12 to 24AIf the variability is large, a wide confidence band will be obtained unless a largenumber of specimens are tested (See 8.1.1).7.1.2 Replication—The replication guidelines given inChapter 3 of Ref (1) are based on the following definition:% replication = 100 [1 − (total number of different stress or strain levels usedin testing/total number of specimens tested)]Type of Test Percent ReplicationAPreliminary and exploratory (research and developmenttests)17 to 33 minResearch and development testing of components andspecimens33 to 50 minDesign allowables data 50 to 75 minReliability data 75 to 88 minANote that percent replication indicates the portion of the total number ofspecimens tested that may be used for obtaining an estimate of the variability ofreplicate tests.7.1.2.1 Replication Examples—Good replication: Supposethat ten specimens are used in research and development forthe testing of a component. If two specimens are tested at eachof five stress or strain amplitudes, the test program involves50 % replications. This percent replication is considered ad-equate for most research and development applications. Poorreplication: Suppose eight different stress or strain amplitudesare used in testing, with two replicates at each of two stress orstrain amplitudes (and no replication at the other six stress orstrain amplitudes). This test program involves only 20 %replication, which is not generally considered adequate.8. Statistical Analysis (Linear Model Y = A + BX, Log-Normal Fatigue Life Distribution with ConstantVariance Along the Entire Interval of X Used inTesting, No Runouts or Suspended Tests or Both,Completely Randomized Design Test Program)8.1 For the case where (a) the fatigue life data pertain to arandom sample (all Yiare independent), (b) there are neitherrun-outs nor suspended tests and where, for the entire intervalof X used in testing, (c) the S-N or ε-N relationship is describedby the linear model Y=A+BX (more precisely by µY ? X5A+BX), (d) the (two parameter) log-normal distributiondescribes the fatigue life N, and (e) the variance of thelog-normal distribution is constant, the maximum likelihoodestimators of A and B are as follows:Aˆ5 Y¯2 BˆX¯(4)Bˆ5(i51k~Xi2 X¯!~Yi2 Y¯!(i51k~Xi2 X¯!2(5)where the symbol “caret”(^)denotes estimate (estimator),4The boldface numbers in parentheses refer to the list of references appended tothis standard.E739 − 10 (2015)3the symbol “overbar” (–) denotes average (for example, Y¯5(i51kYi/k and X¯5(i51kXi/k), Yi= log Ni, Xi= Sior εi,orlogSiorlog εi(refer to Eq 1 and Eq 2), and k is the total number of testspecimens (the total sample size). The recommended expres-sion for estimating the variance of the normal distribution forlog N isσˆ25(i51k~Yi2 Yˆi!2k 2 2(6)in which Ŷi= Â + BˆXiand the (k − 2) term in the denomi-nator is used instead of k to make σˆ2an unbiased estimator ofthe normal population variance σˆ2.NOTE 8—An assumption of constant variance is usually reasonable fornotched and joint specimens up to about 106cycles to failure.The varianceof unnotched specimens generally increases with decreasing stress (strain)level (see Section 9). If the assumption of constant variance appears to bedubious, the reader is referred to Ref (3) for the appropriate statistical test.8.1.1 Confidence Intervals for Parameters A and B—Theestimators Â and Bˆare normally distributed with expectedvalues A and B, respectively, (regardless of total sample size k)when conditions (a) through (e)in8.1 are met. Accordingly,confidence intervals for parameters A and B can be establishedusing the t distribution, Table 1. The confidence interval for Ais given by Â 6 tpσˆÂ,orAˆ6tpσˆ31k1X¯ 2(i51k~Xi2 X¯!24½, (7)and for B is given by Bˆˆ 6 tpσˆBˆ,orBˆ6tpσˆF(i51k~Xi2 X¯!2G2½(8)in which the value of tpis read from Table 1 for the desiredvalue of P, the confidence level associated with the confi-dence interval. This table has one entry parameter (the statis-tical degrees of freedom, n, for t ). For Eq 7 and Eq 8, n =k −2.NOTE 9—The confidence intervals for A and B are exact if conditions(a) through (e)in8.1 are met exactly. However, these intervals are stillreasonably accurate when the actual life distribution differs slightly fromthe (two-parameter) log-normal distribution, that is, when only condition(d) is not met exactly, due to the robustness of the t statistic.NOTE 10—Because the actual median S-N or ε-N relationship is onlyapproximated by a straight line within a specific interval of stress or strain,confidence intervals for A and B that pertain to confidence levels greaterthan approximately 0.95 are not recommended.8.1.1.1 The meaning of the confidence interval associatedwith, say, Eq 8 is as follows (Note 11). If the values of tpgivenin Table 1 for, say, P = 95 % are used in a series of analysesinvolving the estimation of B from independent data sets, thenin the long run we may expect 95 % of the computed intervalsto include the value B. If in each instance we were to assert thatB lies within the interval computed,