1、Designation: E 739 91 (Reapproved 2004)e1Standard Practice forStatistical Analysis of Linear or Linearized Stress-Life (S-N)and Strain-Life (e-N) Fatigue Data1This standard is issued under the fixed designation E 739; the number immediately following the designation indicates the year oforiginal ado
2、ption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.e1NOTEEditorial changes were made throughout in May 2006.1. Scope1.1 This practice
3、 covers only S-N and e-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 e-N
4、 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 e-N curve be extrapolated outsidethe interval of testing, or (b) the fatigue life at a specific
5、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 e-N d
6、ata.2. Referenced Documents2.1 ASTM Standards:2E 206 Definitions of Terms Relating to Fatigue Testing andthe Statistical Analysis of Fatigue Data3E 467 Practice for Verification of Constant Amplitude Dy-namic Forces in an Axial Fatigue Testing SystemE 468 Practice for Presentation of Constant Amplit
7、ude Fa-tigue Test Results for Metallic MaterialsE 513 Definitions of Terms Relating to Constant-Amplitude, Low-Cycle Fatigue Testing3E 606 Practice for Strain-Controlled Fatigue Testing3. Terminology3.1 The terms used in this practice shall be used as definedin Definitions E 206 and E 513. In additi
8、on, the followingterminology is used:3.1.1 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 wh
9、en 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 methods based on the normaldistribution.)3.1.4 replicate (repeat) testsnominally identical tests ondifferent randomly s
10、elected 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
11、 loadcycles (Practice E 468).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 e-N relationship may not be appropriate at long liveswhen run-outs are likely.3.1.5.2
12、 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 in
13、terpreting S-N and e-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 of1This practice is under the jurisdiction ofASTM Committee E08 on Fatigue andFracture and is th
14、e direct responsibility of Subcommittee E08.04 on StructuralApplications.Current edition approved May 1, 2004. Published June 2004. Originallyapproved in 1980. Last previous edition approved in 1998 as E 739 91 (1998).2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact AS
15、TM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3Withdrawn.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.material of inter
16、est (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.5. Types of S-N and e-N Curves Considered5.1 It is well known that the shape of S-N and e-N curvesc
17、an depend markedly on the material and test conditions. Thispractice is restricted to linear or linearized S-N and e-Nrelationships, for example,log N 5 A 1 B S! or (1)log N 5 A 1 B e! orlog N 5 A 1 B log S! or (2)log N 5 A 1 B log e!in which S and e may refer to (a) the maximum value ofconstant-amp
18、litude 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 the constant-amplitude cyclic stress or strain, given a specific value of themean stress or strain, or (c) analogous information stated int
19、erms of some appropriate independent (controlled) 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 e must be established for use inanalysis.5.1.1 The fatigue life
20、N is the dependent (random) variablein S-N and e-N tests, whereas S or e is the independent(controlled) 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 da
21、ta 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 e-N tests, there is never any doubt that the fati
22、gue life is thedependent variable.NOTE 3In plotting S-N and e-N curves, the independent variables Sand e 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 ind
23、eed may be quite complex in certain situations). ForNOTE 1The 95 % confidence band for the e-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 L
24、ife N (Y) and the Plastic Strain Amplitude Dep/2 (X) for the Example Data GivenE 739 91 (2004)e12the 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 fati
25、gue 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 logN is assumed to be the same at low S and e levels as at highlevels of S or e). Accordingly, log N is used as the dependent(r
26、andom) variable in analysis. It is denoted Y. The independentvariable is denoted X. It may be either S or e,orlogS or loge, respectively, depending on which appears to produce astraight line plot for the interval of S or e of interest. Thus Eq1 and Eq 2 may be re-expressed asY 5 A 1 BX (3)Eq 3 is us
27、ed in subsequent analysis. It may be stated moreprecisely as Y ? X= A + BX, where Y ? Xis the expected valueof Y given 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
28、 mean are identicalfor the symmetrical normal distribution assumed in this practice for Y.)6. Test Planning6.1 Test planning for S-N and e-N test programs is discussedin Chapter 3 of Ref (1).4Planned grouping (blocking) andrandomization are essential features of a well-planned testprogram. In partic
29、ular, 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 t
30、hatassure 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 thatconceivab
31、ly 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
32、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 e-N) tes
33、ting 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 andspecimens6to12D
34、esign 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:% repli
35、cation = 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 allo
36、wables 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
37、 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: Suppo
38、se 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. Statistica
39、l 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 (a
40、ll 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 e-N relationship is describedby the linear model Y=A+BX (more precisely by Y ? X= A+BX), (d) the (two parameter) log-normal distributiondescribes the fatigu
41、e life N, and (e) the variance of thelog-normal distribution is constant, the maximum likelihoodestimators of A and B are as follows: 5 Y2 BX(4)B5(i 5 1kXi2 X! Yi2 Y!(i 5 1kXi2 X!2(5)where the symbol “caret”()denotes estimate (estimator),the symbol “overbar”( ) denotes average (for example, Y=( i 5
42、1kYi/k and X=(i 5 1kXi/k), Yi= log Ni, Xi= Sior ei,orlog Sior log ei(refer to Eq 1 and Eq 2), and k is the totalnumber of test specimens (the total sample size). The recom-mended expression for estimating the variance of the normaldistribution for log N is4The boldface numbers in parentheses refer t
43、o the list of references appended tothis standard.E 739 91 (2004)e13s25(i 5 1kYi2 Yi!2k 2 2(6)in which Yi= + BXiand the (k 2) term in the denomi-nator is used instead of k to make s2an unbiased estimator ofthe normal population variance s2.NOTE 8An assumption of constant variance is usually reasonab
44、le 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 statistic
45、al 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 k)when conditions (a) through (e)in8.1 are met. Accordingly,confidence intervals for parameters A and B can be establishedu
46、sing the t distribution, Table 1. The confidence interval for Ais given by 6 tps,or 6 tpsF1k1X2(i 5 1kXi2 X!2G, (7)and for B is given by B6 tpsB,orB6 tps (i 5 1kXi2 X!2#2(8)in which the value of tpis read from Table 1 for the desiredvalue of P, the confidence level associated with the confidenceinte
47、rval. This table has one entry parameter (the statisticaldegrees 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 conditions(a) through (e)in8.1 are met exactly. However, these intervals are stillreasonably accurate when the actual life distr
48、ibution 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 10Because the actual median S-N or e-N relationship is onlyapproximated by a straight line within a specific interval of stress
49、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
