# ASTM E1049-85 (Reapproved 2017)

Designation E1049 85 Reapproved 2017Standard 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 s employed in fatigue analysis.This standard does not intend to recommend a particular.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 Referenced Documents2.1 ASTM Standards2E912 Definitions of Terms Relating to Fatigue Loading;Replaced by E 1150 Withdrawn 198833. Terminology3.1 Definitions3.1.1 constant amplitude loadingin fatigue loading,aloading in which all of the peak loads are equal and all of thley loads are equal.3.1.2 cyclein fatigue loading, under constant amplitudeloading, the load variation from the minimum to the maximumand then to the minimum load.NOTE 1In spectrum loading, definition of cycle varies with thecounting used.3.1.3 mean crossingsin 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 specifiedduring a given length of the history see Fig. 1.3.1.3.1 DiscussionFor purposes related to cycle counting,a mean crossing may be defined as a crossing of the referenceload level.3.1.4 mean load, Pmin fatigue loading, the algebraicaverage of the maximum and minimum loads in constantamplitude loading, or of individual cycles in spectrum loading,Pm5 Pmax1Pmin/2 1or 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 peakin 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 rangein 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 2In spectrum loading, range may have a different definition,depending on the counting 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 DiscussionIn cycle counting by various s,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 reversalin fatigue loading, the point at which thefirst derivative of the load-time history changes sign see Fig.1.NOTE 3In constant amplitude loading, a cycle is equal to tworeversals.3.1.8 spectrum loadingin 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 E10498520111.DOI 10.1520/E1049-85R17.2For 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 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 valleyin 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 Standard3.2.1 loadused in these practices to denote force, stress,strain, torque, acceleration, deflection, or other parameters ofinterest.3.2.2 reference loadfor 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 lengthyirregular load-versus-time histories by providing the number oftimes cycles of various sizes occur. The definition of a cyclevaries with the of cycle counting. These practices coverthe procedures used to obtain cycle counts by various s,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 Counting5.1.1 Results of a level-crossing count are shown in Fig.2a. 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 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. 2b illustrates this second . Avariation of the second 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. 2c. 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 s of deriving a cycle count from thelevel-crossings count could be used.5.2 Peak Counting5.2.1 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. 3a. Results for peaks andvalleys are usually reported separately. A variation of this 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. 3b.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 20172are used. This process is illustrated by Fig. 3c. 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 sof deriving a cycle count, such as randomly selecting pairs ofpeaks and valleys, are sometimes used.5.3 Simple-Range Counting5.3.1 For this , 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 is illustrated in Fig. 4.Positive ranges, negative ranges, or both, may be counted withthis . 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 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 of a range-mean matrix.5.4 Rainflow Counting and Related s5.4.1 A number of different terms have been employed inthe literature to designate cycle-counting s which aresimilar to the rainflow . These include range-pairaLevel Crossing CountingbRestricted Level Crossing CountingFIG. 2 Level-Crossing Counting ExampleE1049 85 20173counting 1, 2,4the Hayes 3, the original rainflow 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 s in this class are definedhere range-pair counting, rainflow counting, and a simplified for repeating histories.5.4.2 The various s similar to the rainflow may be used to obtain cycles and the mean value of each cycle;they are referred to as two-parameter s. When the meanvalue is ignored, they are one-parameter s, as aresimple-range counting, peak counting, etc.5.4.3 Range-Pair CountingThe range-paired counts a range as a cycle if it can be paired with a subsequentloading in the opposite direction. Rules for this are asfollows5.4.3.1 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.aPeak CountingbMean Crossing Peak CountingcCycles Derived from Peak Count of aFIG. 3 Peak Counting ExampleE1049 85 201742 If there are less than three points, go to Step 1. ranges 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.aIfX Y.Count |A-B|asonecycle and discard points A and B. See Fig. 5b. Note that acycle is ed 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. 5c.5 Y |C-D|; X |D-G|; and XY.Count |C-D| as onecycle and discard points C and D. See Fig. 5d.6 Y |G-H|; X |H-I|; and XY.Count |H-I|asonecycle and discard points H and I. See Fig. 5e.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 of a range-mean matrix.5.4.4 Rainflow Counting5.4.4.1 Rules for this 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. ranges 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.aIfXY.Ycontains S, that is,point A. Count |A-B| as one-half cycle and discard point A;SB.See Fig. 6b.FIG. 4 Simple Range Counting ExampleBoth Positive and Negative Ranges CountedE1049 85 201752 Y |B-C|; X |C-D|; XY.Ycontains S, that is, pointB. Count| B-C| as one-half cycle and discard point B; SC.See Fig. 6c.3 Y |C-D|; X |D-E|; XY.Count |E-F| as one cycleand discard points E and F. See Fig. 6d. Note that a cycle ised 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. SD.See Fig. 6e.7 Y |D-G|; X |G-H|; XY. Count | E-F| as one cycleand discard points E and F. See Fig. 7c. Note that a cycleis ed 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. 7d.7 Y |G-H|; X |H-C|; XY.Count |H-C| as one cycleand discard points H and C. See Fig. 7e.9 Y |D-G|; X |G-D|; XY.Count| D-G| as one cycleand discard points D and G. See Fig. 7f.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 of a range-mean matrix.cdefFIG. 6 Rainflow Counting ExampleE1049 85 20177APPENDIXNonmandatory InationX1. 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