Designation: E1049 − 85 (Reapproved 2017)Standard Practices forCycle Counting in Fatigue Analysis1This standard is issued under the fixed designation E1049; 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 These practices are a compilation of acceptable proce-dures for cycle-counting methods employed in fatigue analysis.This standard does not intend to recommend a particularmethod.1.2 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.1.3 This international standard was developed in accor-dance with internationally recognized principles on standard-ization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.2. Referenced Documents2.1 ASTM Standards:2E912 Definitions of Terms Relating to Fatigue Loading;Replaced by E 1150 (Withdrawn 1988)33. Terminology3.1 Definitions:3.1.1 constant amplitude loading—in fatigue loading,aloading in which all of the peak loads are equal and all of thevalley loads are equal.3.1.2 cycle—in fatigue loading, under constant amplitudeloading, the load variation from the minimum to the maximumand then to the minimum load.NOTE 1—In spectrum loading, definition of cycle varies with thecounting method used.3.1.3 mean crossings—in fatigue loading, the number oftimes that the load-time history crosses the mean-load levelwith a positive slope (or a negative slope, or both, as specified)during a given length of the history (see Fig. 1).3.1.3.1 Discussion—For purposes related to cycle counting,a mean crossing may be defined as a crossing of the referenceload level.3.1.4 mean load, Pm—in fatigue loading, the algebraicaverage of the maximum and minimum loads in constantamplitude loading, or of individual cycles in spectrum loading,Pm5 ~Pmax1Pmin!/2 (1)or the integral average of the instantaneous load values orthe algebraic average of the peak and valley loads of a spec-trum loading history.3.1.5 peak—in fatigue loading, the point at which the firstderivative of the load-time history changes from a positive toa negative sign; the point of maximum load in constantamplitude loading (see Fig. 1).3.1.6 range—in fatigue loading, the algebraic differencebetween successive valley and peak loads (positive range orincreasing load range), or between successive peak and valleyloads (negative range or decreasing load range); see Fig. 1.NOTE 2—In spectrum loading, range may have a different definition,depending on the counting method used; for example, “overall range” isdefined by the algebraic difference between the largest peak and thesmallest valley of a given load-time history.3.1.6.1 Discussion—In cycle counting by various methods,it is common to employ ranges between valley and peak loads,or between peak and valley loads, which are not necessarilysuccessive events. In these practices, the definition of the word“range” is broadened so that events of this type are alsoincluded.3.1.7 reversal—in fatigue loading, the point at which thefirst derivative of the load-time history changes sign (see Fig.1).NOTE 3—In constant amplitude loading, a cycle is equal to tworeversals.3.1.8 spectrum loading—in fatigue loading, a loading inwhich all of the peak loads are not equal or all of the valleyloads are not equal, or both. (Also known as variable amplitudeloading or irregular loading.)1These practices are under the jurisdiction of ASTM Committee E08 on Fatigueand Fracture and are the direct responsibility of Subcommittee E08.04 on StructuralApplications.Current edition approved June 1, 2017. Published June 2017. Originallyapproved in 1985. Last previous edition approved in 2011 as E1049–85(2011)ɛ1.DOI: 10.1520/E1049-85R17.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] 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 StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.13.1.9 valley—in fatigue loading, the point at which the firstderivative of the load-time history changes from a negative toa positive sign (also known as trough); the point of minimumload in constant amplitude loading (see Fig. 1).3.2 Definitions of Terms Specific to This Standard:3.2.1 load—used in these practices to denote force, stress,strain, torque, acceleration, deflection, or other parameters ofinterest.3.2.2 reference load—for spectrum loading, used in thesepractices to denote the loading level that represents a steady-state condition upon which load variations are superimposed.The reference load may be identical to the mean load of thehistory, but this is not required.3.3 For other definitions of terms used in these practicesrefer to Definitions E912.4. Significance and Use4.1 Cycle counting is used to summarize (often lengthy)irregular load-versus-time histories by providing the number oftimes cycles of various sizes occur. The definition of a cyclevaries with the method of cycle counting. These practices coverthe procedures used to obtain cycle counts by various methods,including level-crossing counting, peak counting, simple-rangecounting, range-pair counting, and rainflow counting. Cyclecounts can be made for time histories of force, stress, strain,torque, acceleration, deflection, or other loading parameters ofinterest.5. Procedures for Cycle Counting5.1 Level-Crossing Counting:5.1.1 Results of a level-crossing count are shown in Fig.2(a). One count is recorded each time the positive slopedportion of the load exceeds a preset level above the referenceload, and each time the negative sloped portion of the loadexceeds a preset level below the reference load. Reference loadcrossings are counted on the positive sloped portion of theloading history. It makes no difference whether positive ornegative slope crossings are counted. The distinction is madeonly to reduce the total number of events by a factor of two.5.1.2 In practice, restrictions on the level-crossing countsare often specified to eliminate small amplitude variationswhich can give rise to a large number of counts. This may beaccomplished by filtering small load excursions prior to cyclecounting. A second method is to make no counts at thereference load and to specify that only one count be madebetween successive crossings of a secondary lower levelassociated with each level above the reference load, or asecondary higher level associated with each level below thereference load. Fig. 2(b) illustrates this second method. Avariation of the second method is to use the same secondarylevel for all counting levels above the reference load, andanother for all levels below the reference load. In this case thelevels are generally not evenly spaced.5.1.3 The most damaging cycle count for fatigue analysis isderived from the level-crossing count by first constructing thelargest possible cycle, followed by the second largest, etc.,until all level crossings are used. Reversal points are assumedto occur halfway between levels. This process is illustrated byFig. 2(c). Note that once this most damaging cycle count isobtained, the cycles could be applied in any desired order, andthis order could have a secondary effect on the amount ofdamage. Other methods of deriving a cycle count from thelevel-crossings count could be used.5.2 Peak Counting:5.2.1 Peak counting identifies the occurrence of a relativemaximum or minimum load value. Peaks above the referenceload level are counted, and valleys below the reference loadlevel are counted, as shown in Fig. 3(a). Results for peaks andvalleys are usually reported separately. A variation of thismethod is to count all peaks and valleys without regard to thereference load.5.2.2 To eliminate small amplitude loadings, mean-crossingpeak counting is often used. Instead of counting all peaks andvalleys, only the largest peak or valley between two successivemean crossings is counted as shown in Fig. 3(b).5.2.3 The most damaging cycle count for fatigue analysis isderived from the peak count by first constructing the largestpossible cycle, using the highest peak and lowest valley,followed by the second largest cycle, etc., until all peak countsFIG. 1 Basic Fatigue Loading ParametersE1049 − 85 (2017)2are used. This process is illustrated by Fig. 3(c). Note that oncethis most damaging cycle count is obtained, the cycles could beapplied in any desired order, and this order could have asecondary effect on the amount of damage. Alternate methodsof deriving a cycle count, such as randomly selecting pairs ofpeaks and valleys, are sometimes used.5.3 Simple-Range Counting:5.3.1 For this method, a range is defined as the differencebetween two successive reversals, the range being positivewhen a valley is followed by a peak and negative when a peakis followed by a valley. The method is illustrated in Fig. 4.Positive ranges, negative ranges, or both, may be counted withthis method. If only positive or only negative ranges arecounted, then each is counted as one cycle. If both positive andnegative ranges are counted, then each is counted as one-halfcycle. Ranges smaller than a chosen value are usually elimi-nated before counting.5.3.2 When the mean value of each range is also counted,the method is called simple range-mean counting. For theexample of Fig. 4, the result of a simple range-mean count isgiven in X1.1 in the form of a range-mean matrix.5.4 Rainflow Counting and Related Methods:5.4.1 A number of different terms have been employed inthe literature to designate cycle-counting methods which aresimilar to the rainflow method. These include range-pair(a)—Level Crossing Counting(b)—Restricted Level Crossing CountingFIG. 2 Level-Crossing Counting ExampleE1049 − 85 (2017)3counting (1, 2),4the Hayes method (3), the original rainflowmethod (4-6), range-pair-range counting (7), ordered overallrange counting (8), racetrack counting (9), and hysteresis loopcounting (10). If the load history begins and ends with itsmaximum peak, or with its minimum valley, all of these giveidentical counts. In other cases, the counts are similar, but notgenerally identical. Three methods in this class are definedhere: range-pair counting, rainflow counting, and a simplifiedmethod for repeating histories.5.4.2 The various methods similar to the rainflow methodmay be used to obtain cycles and the mean value of each cycle;they are referred to as two-parameter methods. When the meanvalue is ignored, they are one-parameter methods, as aresimple-range counting, peak counting, etc.5.4.3 Range-Pair Counting—The range-paired methodcounts a range as a cycle if it can be paired with a subsequentloading in the opposite direction. Rules for this method are asfollows:5.4.3.1 Let X denote range under consideration; and Y,previous range adjacent to X.(1) Read next peak or valley. If out of data, go to Step 5.4The boldface numbers in parentheses refer to the list of references appended tothese practices.(a)—Peak Counting(b)—Mean Crossing Peak Counting(c)—Cycles Derived from Peak Count of (a)FIG. 3 Peak Counting ExampleE1049 − 85 (2017)4(2) If there are less than three points, go to Step 1. Formranges X and Y using the three most recent peaks and valleysthat have not been discarded.(3) Compare the absolute values of ranges X and Y.(a)IfX Y.Count |A-B|asonecycle and discard points A and B. (See Fig. 5(b). Note that acycle is formed by pairing range A-B and a portion of rangeB-C.)(2) Y = |C-D|; X = |D-E|; and XY.Count |E-F|asonecycle and discard points E and F. (See Fig. 5(c).)(5) Y = |C-D|; X = |D-G|; and XY.Count |C-D| as onecycle and discard points C and D. (See Fig. 5(d).)(6) Y = |G-H|; X = |H-I|; and XY.Count |H-I|asonecycle and discard points H and I. (See Fig. 5(e).)(8) End of counting. See the table in Fig. 5 for a summaryof the cycles counted in this example, and see Appendix X1.2for this cycle count in the form of a range-mean matrix.5.4.4 Rainflow Counting:5.4.4.1 Rules for this method are as follows: let X denoterange under consideration; Y, previous range adjacent to X; andS, starting point in the history.(1) Read next peak or valley. If out of data, go to Step 6.(2) If there are less than three points, go to Step 1. Formranges X and Y using the three most recent peaks and valleysthat have not been discarded.(3) Compare the absolute values of ranges X and Y.(a)IfXY.Ycontains S, that is,point A. Count |A-B| as one-half cycle and discard point A;S=B.(See Fig. 6(b).)FIG. 4 Simple Range Counting Example—Both Positive and Negative Ranges CountedE1049 − 85 (2017)5(2) Y = |B-C|; X = |C-D|; XY.Ycontains S, that is, pointB. Count| B-C| as one-half cycle and discard point B; S=C.(See Fig. 6(c).)(3) Y = |C-D|; X = |D-E|; XY.Count |E-F| as one cycleand discard points E and F. (See Fig. 6(d). Note that a cycle isformed by pairing range E-F and a portion of range F-G.)(6) Y = |C-D|; X = |D-G|; XY; Y contains S, that is, pointC. Count |C-D| as one-half cycle and discard point C. S=D.(See Fig. 6(e).)(7) Y = |D-G|; X = |G-H|; XY. Count | E-F| as one cycleand discard points E and F. (See Fig. 7(c).) Note that a cycleis formed by pairing range E-F and a portion of range F-G.(3) Y = |D-G|; X = |G-H|; XY. Count |A-B| as one cycleand discard points A and B. (See Fig. 7(d).)(7) Y = |G-H|; X = |H-C|; XY.Count |H-C| as one cycleand discard points H and C. (See Fig. 7(e).)(9) Y = |D-G|; X = |G-D|; X=Y.Count| D-G| as one cycleand discard points D and G. (See Fig. 7(f).)(10) End of counting. See the table in Fig. 7 for a summaryof the cycles counted in this example, and see Appendix X1.4for this cycle count in the form of a range-mean matrix.(c)(d)(e)(f)FIG. 6 Rainflow Counting ExampleE1049 − 85 (2017)7APPENDIX(Nonmandatory Information)X1. RANGE-MEAN MATRIXES FOR CYCLE COUNTING EXAMPLESX1.1 The Tables X1.1-X1.4 which follow correspond to thecycle-counting examples of Figs. 4-7. In each case, the table isa matrix giving the number of cycles counted at the indicatedcombinations of range and mean. Note that these exampl