# ASTM E739-10 (Reapproved 2015)

Designation E739 10 Reapproved 2015Standard Practice forStatistical Analysis of Linear or Linearized Stress-Life S-Nand Strain-Life -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 -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 -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 -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 -N data.2. Referenced Documents2.1 ASTM Standards2E206 Definitions of Terms Relating to Fatigue Testing andthe Statistical Analysis of Fatigue Data; Replaced byE 1150 Withdrawn 19883E468 Practice for Presentation of Constant Amplitude Fa-tigue Test Results for Metallic MaterialsE513 Definitions of Terms Relating to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 19883E606/E606M Test 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 used3.1.1 dependent variablethe fatigue life N or the loga-rithm of the fatigue life.3.1.1.1 DiscussionLog N is denoted Y in this practice.3.1.2 independent variablethe selected and controlledvariable namely, stress or strain. It is denoted X in thispractice when plotted on appropriate coordinates.3.1.3 log-normal distributionthe distribution of N whenlog N is normally distributed. Accordingly, it is convenientto analyze log N using s based on the normaldistribution.3.1.4 replicate repeat testsnominally 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 outno failure at a specified number of loadcycles Practice E468.3.1.5.1 DiscussionThe 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 -N relationship may not be appropriate at long liveswhen run-outs are likely.3.1.5.2 DiscussionFor 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 -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 theperance 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. DOI10.1520/E0739-10R15.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume ination, refer to the standards 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 -N Curves Considered5.1 It is well known that the shape of S-N and -N curvescan depend markedly on the material and test conditions. Thispractice is restricted to linear or linearized S-N and -Nrelationships, for example,log N 5 A1B S or 1log N 5 A1B orlog N 5 A1B logS or 2log N 5 A1B login which S and 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 ina-tion stated in terms of some appropriate independent con-trolled variable.NOTE 1In 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 must be established for use inanalysis.5.1.1 The fatigue life N is the dependent random variablein S-N and -N tests, whereas S or is the independentcontrolled variable.NOTE 2In 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 -N tests, there is never any doubt that the fatigue life is thedependent variable.NOTE 3In plotting S-N and -N curves, the independent variables Sand are plotted along the ordinate, with life the dependent variableplotted along the abscissa. Refer, for example, to Fig. 1.5.1.2 The distribution of fatigue life in any test is unknownand 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 1The 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 alongthe abscissa to con to engineering convention.FIG. 1 Fitted Relationship Between the Fatigue Life N Y and the Plastic Strain Amplitude p/2 X for the Example Data GivenE739 10 20152N is assumed to be the same at low S and levels as at highlevels of S or . Accordingly, log N is used as the dependentrandom variable in analysis. It is denoted Y. The independentvariable is denoted X. It may be either S or ,orlogS or log ,respectively, depending on which appears to produce a straightline plot for the interval of S or of interest. Thus Eq 1 and Eq2 may be re-expressed asY 5 A1BX 3Eq 3 is used in subsequent analysis. It may be stated moreprecisely as Y X5A1BX, where Y Xis the expected value of Ygiven X.NOTE 4For testing the adequacy of the linear model, see 8.2.NOTE 5The expected value is the mean of the conceptual populationof all Ys 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 -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 ology involves use ofplanned grouping to a balance potentially spurious effects ofnuisance variables for example, laboratory humidity and ballow 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 6A 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, supplierwhen seeking a random sample of the material being tested unless thatparticular source is of specific interest.NOTE 7Procedures 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 SizeThe minimum number of specimensrequired in S-N and -N testing depends on the type of testprogram conducted. The following guidelines given in Chapter3ofRef1 appear reasonable.Type of TestMinimum Numberof SpecimensAPreliminary and exploratory exploratory research anddevelopment tests6to12Research 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 ReplicationThe 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 testedType of Test Percent ReplicationAPreliminary and exploratory research and developmenttests17 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 ExamplesGood 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 Program8.1 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 -N relationship is describedby the linear model YABX more precisely by Y X5ABX, 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 followsA5 Y2 BX4B5i51kXi2 XYi2 Yi51kXi2 X25where the symbol “caret”denotes estimate estimator,4The boldface numbers in parentheses refer to the list of references appended tothis standard.E739 10 20153the symbol “overbar” denotes average for example, Y5i51kYi/k and X5i51kXi/k, Yi log Ni, Xi Sior i,orlogSiorlog irefer 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 is25i51kYi2 Yi2k 2 26in which i BXiand the k 2 term in the denomi-nator is used instead of k to make 2an unbiased estimator ofthe normal population variance 2.NOTE 8An 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 strainlevel 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 BTheestimators and Bare normally distributed with expectedvalues A and B, respectively, regardless of total sample size kwhen conditions a through ein8.1 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 6 tp,orA6tp31k1X 2i51kXi2 X24, 7and for B is given by B 6 tpB,orB6tpFi51kXi2 X2G28in 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 9The confidence intervals for A and B are exact if conditionsa through ein8.1 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 conditiond is not met exactly, due to the robustness of the t statistic.NOTE 10Because the actual median S-N or -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,