ASTM E1049-85(2017)
(Practice)Standard Practices for Cycle Counting in Fatigue Analysis
Standard Practices for Cycle Counting in Fatigue Analysis
SIGNIFICANCE AND USE
4.1 Cycle counting is used to summarize (often lengthy) irregular load-versus-time histories by providing the number of times cycles of various sizes occur. The definition of a cycle varies with the method of cycle counting. These practices cover the procedures used to obtain cycle counts by various methods, including level-crossing counting, peak counting, simple-range counting, range-pair counting, and rainflow counting. Cycle counts can be made for time histories of force, stress, strain, torque, acceleration, deflection, or other loading parameters of interest.
SCOPE
1.1 These practices are a compilation of acceptable procedures for cycle-counting methods employed in fatigue analysis. This standard does not intend to recommend a particular method.
1.2 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.
1.3 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
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Standards Content (Sample)
This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E1049 − 85 (Reapproved 2017)
Standard Practices for
Cycle Counting in Fatigue Analysis
This standard is issued under the fixed designation E1049; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 3.1.3 mean crossings—in fatigue loading, the number of
times that the load-time history crosses the mean-load level
1.1 These practices are a compilation of acceptable proce-
with a positive slope (or a negative slope, or both, as specified)
duresforcycle-countingmethodsemployedinfatigueanalysis.
during a given length of the history (see Fig. 1).
This standard does not intend to recommend a particular
3.1.3.1 Discussion—For purposes related to cycle counting,
method.
a mean crossing may be defined as a crossing of the reference
1.2 This standard does not purport to address all of the
load level.
safety concerns, if any, associated with its use. It is the
3.1.4 mean load, P —in fatigue loading, the algebraic
responsibility of the user of this standard to establish appro- m
average of the maximum and minimum loads in constant
priate safety and health practices and determine the applica-
amplitude loading, or of individual cycles in spectrum loading,
bility of regulatory limitations prior to use.
1.3 This international standard was developed in accor-
P 5 P 1P /2 (1)
~ !
m max min
dance with internationally recognized principles on standard-
or the integral average of the instantaneous load values or
ization established in the Decision on Principles for the
the algebraic average of the peak and valley loads of a spec-
Development of International Standards, Guides and Recom-
trum loading history.
mendations issued by the World Trade Organization Technical
Barriers to Trade (TBT) Committee. 3.1.5 peak—in fatigue loading, the point at which the first
derivative of the load-time history changes from a positive to
2. Referenced Documents
a negative sign; the point of maximum load in constant
amplitude loading (see Fig. 1).
2.1 ASTM Standards:
E912 Definitions of Terms Relating to Fatigue Loading;
3.1.6 range—in fatigue loading, the algebraic difference
Replaced by E 1150 (Withdrawn 1988)
between successive valley and peak loads (positive range or
increasing load range), or between successive peak and valley
3. Terminology
loads (negative range or decreasing load range); see Fig. 1.
3.1 Definitions:
NOTE 2—In spectrum loading, range may have a different definition,
3.1.1 constant amplitude loading—in fatigue loading,a
depending on the counting method used; for example, “overall range” is
loading in which all of the peak loads are equal and all of the
defined by the algebraic difference between the largest peak and the
valley loads are equal.
smallest valley of a given load-time history.
3.1.2 cycle—in fatigue loading, under constant amplitude
3.1.6.1 Discussion—In cycle counting by various methods,
loading, the load variation from the minimum to the maximum
it is common to employ ranges between valley and peak loads,
and then to the minimum load.
or between peak and valley loads, which are not necessarily
successive events. In these practices, the definition of the word
NOTE 1—In spectrum loading, definition of cycle varies with the
counting method used.
“range” is broadened so that events of this type are also
included.
These practices are under the jurisdiction ofASTM Committee E08 on Fatigue
3.1.7 reversal—in fatigue loading, the point at which the
and Fracture and are the direct responsibility of Subcommittee E08.04 on Structural
first derivative of the load-time history changes sign (see Fig.
Applications.
1).
Current edition approved June 1, 2017. Published June 2017. Originally
ɛ1
approved in 1985. Last previous edition approved in 2011 as E1049–85(2011) .
NOTE 3—In constant amplitude loading, a cycle is equal to two
DOI: 10.1520/E1049-85R17.
reversals.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
3.1.8 spectrum loading—in fatigue loading, a loading in
Standards volume information, refer to the standard’s Document Summary page on
which all of the peak loads are not equal or all of the valley
the ASTM website.
loadsarenotequal,orboth.(Alsoknownasvariableamplitude
The last approved version of this historical standard is referenced on
www.astm.org. loading or irregular loading.)
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1049 − 85 (2017)
FIG. 1 Basic Fatigue Loading Parameters
3.1.9 valley—in fatigue loading, the point at which the first 5.1.2 In practice, restrictions on the level-crossing counts
derivative of the load-time history changes from a negative to are often specified to eliminate small amplitude variations
a positive sign (also known as trough); the point of minimum which can give rise to a large number of counts. This may be
load in constant amplitude loading (see Fig. 1). accomplished by filtering small load excursions prior to cycle
counting. A second method is to make no counts at the
3.2 Definitions of Terms Specific to This Standard:
reference load and to specify that only one count be made
3.2.1 load—used in these practices to denote force, stress,
between successive crossings of a secondary lower level
strain, torque, acceleration, deflection, or other parameters of
associated with each level above the reference load, or a
interest.
secondary higher level associated with each level below the
3.2.2 reference load—for spectrum loading, used in these
reference load. Fig. 2(b) illustrates this second method. A
practices to denote the loading level that represents a steady-
variation of the second method is to use the same secondary
state condition upon which load variations are superimposed.
level for all counting levels above the reference load, and
The reference load may be identical to the mean load of the
another for all levels below the reference load. In this case the
history, but this is not required.
levels are generally not evenly spaced.
3.3 For other definitions of terms used in these practices
5.1.3 The most damaging cycle count for fatigue analysis is
refer to Definitions E912.
derived from the level-crossing count by first constructing the
largest possible cycle, followed by the second largest, etc.,
4. Significance and Use
until all level crossings are used. Reversal points are assumed
to occur halfway between levels. This process is illustrated by
4.1 Cycle counting is used to summarize (often lengthy)
Fig. 2(c). Note that once this most damaging cycle count is
irregularload-versus-timehistoriesbyprovidingthenumberof
obtained, the cycles could be applied in any desired order, and
times cycles of various sizes occur. The definition of a cycle
this order could have a secondary effect on the amount of
varieswiththemethodofcyclecounting.Thesepracticescover
damage. Other methods of deriving a cycle count from the
the procedures used to obtain cycle counts by various methods,
level-crossings count could be used.
including level-crossing counting, peak counting, simple-range
counting, range-pair counting, and rainflow counting. Cycle
5.2 Peak Counting:
counts can be made for time histories of force, stress, strain,
5.2.1 Peak counting identifies the occurrence of a relative
torque, acceleration, deflection, or other loading parameters of
maximum or minimum load value. Peaks above the reference
interest.
load level are counted, and valleys below the reference load
level are counted, as shown in Fig. 3(a). Results for peaks and
5. Procedures for Cycle Counting
valleys are usually reported separately. A variation of this
method is to count all peaks and valleys without regard to the
5.1 Level-Crossing Counting:
reference load.
5.1.1 Results of a level-crossing count are shown in Fig.
5.2.2 To eliminate small amplitude loadings, mean-crossing
2(a). One count is recorded each time the positive sloped
peak counting is often used. Instead of counting all peaks and
portion of the load exceeds a preset level above the reference
valleys, only the largest peak or valley between two successive
load, and each time the negative sloped portion of the load
mean crossings is counted as shown in Fig. 3(b).
exceeds a preset level below the reference load. Reference load
crossings are counted on the positive sloped portion of the 5.2.3 The most damaging cycle count for fatigue analysis is
loading history. It makes no difference whether positive or derived from the peak count by first constructing the largest
negative slope crossings are counted. The distinction is made possible cycle, using the highest peak and lowest valley,
only to reduce the total number of events by a factor of two. followed by the second largest cycle, etc., until all peak counts
E1049 − 85 (2017)
(a)—Level Crossing Counting
(b)—Restricted Level Crossing Counting
FIG. 2 Level-Crossing Counting Example
are used.This process is illustrated by Fig. 3(c). Note that once counted, then each is counted as one cycle. If both positive and
thismostdamagingcyclecountisobtained,thecyclescouldbe negative ranges are counted, then each is counted as one-half
applied in any desired order, and this order could have a
cycle. Ranges smaller than a chosen value are usually elimi-
secondary effect on the amount of damage. Alternate methods
nated before counting.
of deriving a cycle count, such as randomly selecting pairs of
5.3.2 When the mean value of each range is also counted,
peaks and valleys, are sometimes used.
the method is called simple range-mean counting. For the
5.3 Simple-Range Counting:
example of Fig. 4, the result of a simple range-mean count is
5.3.1 For this method, a range is defined as the difference
given in X1.1 in the form of a range-mean matrix.
between two successive reversals, the range being positive
5.4 Rainflow Counting and Related Methods:
when a valley is followed by a peak and negative when a peak
5.4.1 A number of different terms have been employed in
is followed by a valley. The method is illustrated in Fig. 4.
the literature to designate cycle-counting methods which are
Positive ranges, negative ranges, or both, may be counted with
this method. If only positive or only negative ranges are similar to the rainflow method. These include range-pair
E1049 − 85 (2017)
(a)—Peak Counting
(b)—Mean Crossing Peak Counting
(c)—Cycles Derived from Peak Count of (a)
FIG. 3 Peak Counting Example
counting (1, 2), the Hayes method (3), the original rainflow 5.4.2 The various methods similar to the rainflow method
method (4-6), range-pair-range counting (7), ordered overall may be used to obtain cycles and the mean value of each cycle;
range counting (8), racetrack counting (9), and hysteresis loop they are referred to as two-parameter methods.When the mean
counting (10). If the load history begins and ends with its value is ignored, they are one-parameter methods, as are
maximum peak, or with its minimum valley, all of these give simple-range counting, peak counting, etc.
identical counts. In other cases, the counts are similar, but not
5.4.3 Range-Pair Counting—The range-paired method
generally identical. Three methods in this class are defined
counts a range as a cycle if it can be paired with a subsequent
here: range-pair counting, rainflow counting, and a simplified
loading in the opposite direction. Rules for this method are as
method for repeating histories.
follows:
5.4.3.1 Let X denote range under consideration; and Y,
previous range adjacent to X.
The boldface numbers in parentheses refer to the list of references appended to
these practices. (1) Read next peak or valley. If out of data, go to Step 5.
E1049 − 85 (2017)
FIG. 4 Simple Range Counting Example—Both Positive and Negative Ranges Counted
(2) If there are less than three points, go to Step 1. Form (8) End of counting. See the table in Fig. 5 for a summary
ranges X and Y using the three most recent peaks and valleys of the cycles counted in this example, and see Appendix X1.2
that have not been discarded. for this cycle count in the form of a range-mean matrix.
(3) Compare the absolute values of ranges X and Y. 5.4.4 Rainflow Counting:
(a)If X < Y, go to Step 1. 5.4.4.1 Rules for this method are as follows: let X denote
(b)If X ≥ Y, go to Step 4. rangeunderconsideration; Y,previousrangeadjacentto X;and
(4) Count range Y as one cycle and discard the peak and S, starting point in the history.
valley of Y; go to Step 2. (1) Read next peak or valley. If out of data, go to Step 6.
(5) The remaining cycles, if any, are counted by starting at (2) If there are less than three points, go to Step 1. Form
the end of the sequence and counting backwards. If a single ranges X and Y using the three most recent peaks and valleys
range remains, it may be counted as a half or full cycle. that have not been discarded.
5.4.3.2 The load history in Fig. 4 is replotted as Fig. 5(a) (3) Compare the absolute values of ranges X and Y.
and is used to illustrate the process. Details of the cycle (a)IfX
counting are as follows: (b)If X ≥ Y, go to Step 4.
(1) Y =|A-B |; X =|B-C|; andX>Y. Count |A-B|asone (4) If range Y contains the starting point S, go to Step 5;
cycle and discard points A and B. (See Fig. 5(b). Note that a otherwise, count range Y as one cycle; discard the peak and
cycle is formed by pairing range A-B and a portion of range valley of Y; and go to Step 2.
B-C.) (5) Count range Y as one-half cycle; discard the first point
(2) Y = |C-D|; X = |D-E|; andX
(3) Y = |D-E|; X = |E-F|; andX
(4) Y = |E-F|; X = |F-G|; andX>Y. Count |E-F|asone (6) Count each range that has not been previously counted
cycle and discard points E and F. (See Fig. 5(c).) as one-half cycle.
(5) Y = |C-D|; X = |D-G|; andX>Y. Count |C-D| as one 5.4.4.2 The load history of Fig. 4 is replotted as Fig. 6(a)
cycle and discard points C and D. (See Fig. 5(d).) and is used to illustrate the process. Details of the cycle
(6) Y = |G-H|; X = |H-I|; andX
count backwards. (1)S=A;Y = |A-B|;X = |B-C|;X>Y.
...
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: E1049 − 85 (Reapproved 2017)
Standard Practices for
Cycle Counting in Fatigue Analysis
This standard is issued under the fixed designation E1049; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 3.1.3 mean crossings—in fatigue loading, the number of
times that the load-time history crosses the mean-load level
1.1 These practices are a compilation of acceptable proce-
with a positive slope (or a negative slope, or both, as specified)
dures for cycle-counting methods employed in fatigue analysis.
during a given length of the history (see Fig. 1).
This standard does not intend to recommend a particular
3.1.3.1 Discussion—For purposes related to cycle counting,
method.
a mean crossing may be defined as a crossing of the reference
1.2 This standard does not purport to address all of the
load level.
safety concerns, if any, associated with its use. It is the
3.1.4 mean load, P —in fatigue loading, the algebraic
responsibility of the user of this standard to establish appro- m
average of the maximum and minimum loads in constant
priate safety and health practices and determine the applica-
amplitude loading, or of individual cycles in spectrum loading,
bility of regulatory limitations prior to use.
1.3 This international standard was developed in accor-
P 5 ~P 1P !/2 (1)
m max min
dance with internationally recognized principles on standard-
or the integral average of the instantaneous load values or
ization established in the Decision on Principles for the
the algebraic average of the peak and valley loads of a spec-
Development of International Standards, Guides and Recom-
trum loading history.
mendations issued by the World Trade Organization Technical
3.1.5 peak—in fatigue loading, the point at which the first
Barriers to Trade (TBT) Committee.
derivative of the load-time history changes from a positive to
2. Referenced Documents
a negative sign; the point of maximum load in constant
amplitude loading (see Fig. 1).
2.1 ASTM Standards:
E912 Definitions of Terms Relating to Fatigue Loading;
3.1.6 range—in fatigue loading, the algebraic difference
Replaced by E 1150 (Withdrawn 1988)
between successive valley and peak loads (positive range or
increasing load range), or between successive peak and valley
3. Terminology
loads (negative range or decreasing load range); see Fig. 1.
3.1 Definitions:
NOTE 2—In spectrum loading, range may have a different definition,
3.1.1 constant amplitude loading—in fatigue loading, a
depending on the counting method used; for example, “overall range” is
loading in which all of the peak loads are equal and all of the
defined by the algebraic difference between the largest peak and the
valley loads are equal.
smallest valley of a given load-time history.
3.1.2 cycle—in fatigue loading, under constant amplitude
3.1.6.1 Discussion—In cycle counting by various methods,
loading, the load variation from the minimum to the maximum
it is common to employ ranges between valley and peak loads,
and then to the minimum load.
or between peak and valley loads, which are not necessarily
NOTE 1—In spectrum loading, definition of cycle varies with the successive events. In these practices, the definition of the word
counting method used.
“range” is broadened so that events of this type are also
included.
These practices are under the jurisdiction of ASTM Committee E08 on Fatigue 3.1.7 reversal—in fatigue loading, the point at which the
and Fracture and are the direct responsibility of Subcommittee E08.04 on Structural
first derivative of the load-time history changes sign (see Fig.
Applications.
1).
Current edition approved June 1, 2017. Published June 2017. Originally
ɛ1
approved in 1985. Last previous edition approved in 2011 as E1049–85(2011) .
NOTE 3—In constant amplitude loading, a cycle is equal to two
DOI: 10.1520/E1049-85R17.
reversals.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
3.1.8 spectrum loading—in fatigue loading, a loading in
Standards volume information, refer to the standard’s Document Summary page on
which all of the peak loads are not equal or all of the valley
the ASTM website.
3 loads are not equal, or both. (Also known as variable amplitude
The last approved version of this historical standard is referenced on
www.astm.org. loading or irregular loading.)
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1049 − 85 (2017)
FIG. 1 Basic Fatigue Loading Parameters
3.1.9 valley—in fatigue loading, the point at which the first 5.1.2 In practice, restrictions on the level-crossing counts
derivative of the load-time history changes from a negative to are often specified to eliminate small amplitude variations
a positive sign (also known as trough); the point of minimum which can give rise to a large number of counts. This may be
load in constant amplitude loading (see Fig. 1). accomplished by filtering small load excursions prior to cycle
counting. A second method is to make no counts at the
3.2 Definitions of Terms Specific to This Standard:
reference load and to specify that only one count be made
3.2.1 load—used in these practices to denote force, stress,
between successive crossings of a secondary lower level
strain, torque, acceleration, deflection, or other parameters of
associated with each level above the reference load, or a
interest.
secondary higher level associated with each level below the
3.2.2 reference load—for spectrum loading, used in these
reference load. Fig. 2(b) illustrates this second method. A
practices to denote the loading level that represents a steady-
variation of the second method is to use the same secondary
state condition upon which load variations are superimposed.
level for all counting levels above the reference load, and
The reference load may be identical to the mean load of the
another for all levels below the reference load. In this case the
history, but this is not required.
levels are generally not evenly spaced.
3.3 For other definitions of terms used in these practices
5.1.3 The most damaging cycle count for fatigue analysis is
refer to Definitions E912.
derived from the level-crossing count by first constructing the
largest possible cycle, followed by the second largest, etc.,
4. Significance and Use
until all level crossings are used. Reversal points are assumed
to occur halfway between levels. This process is illustrated by
4.1 Cycle counting is used to summarize (often lengthy)
Fig. 2(c). Note that once this most damaging cycle count is
irregular load-versus-time histories by providing the number of
obtained, the cycles could be applied in any desired order, and
times cycles of various sizes occur. The definition of a cycle
this order could have a secondary effect on the amount of
varies with the method of cycle counting. These practices cover
damage. Other methods of deriving a cycle count from the
the procedures used to obtain cycle counts by various methods,
level-crossings count could be used.
including level-crossing counting, peak counting, simple-range
counting, range-pair counting, and rainflow counting. Cycle
5.2 Peak Counting:
counts can be made for time histories of force, stress, strain,
5.2.1 Peak counting identifies the occurrence of a relative
torque, acceleration, deflection, or other loading parameters of
maximum or minimum load value. Peaks above the reference
interest.
load level are counted, and valleys below the reference load
level are counted, as shown in Fig. 3(a). Results for peaks and
5. Procedures for Cycle Counting
valleys are usually reported separately. A variation of this
method is to count all peaks and valleys without regard to the
5.1 Level-Crossing Counting:
reference load.
5.1.1 Results of a level-crossing count are shown in Fig.
5.2.2 To eliminate small amplitude loadings, mean-crossing
2(a). One count is recorded each time the positive sloped
peak counting is often used. Instead of counting all peaks and
portion of the load exceeds a preset level above the reference
valleys, only the largest peak or valley between two successive
load, and each time the negative sloped portion of the load
mean crossings is counted as shown in Fig. 3(b).
exceeds a preset level below the reference load. Reference load
crossings are counted on the positive sloped portion of the 5.2.3 The most damaging cycle count for fatigue analysis is
loading history. It makes no difference whether positive or derived from the peak count by first constructing the largest
negative slope crossings are counted. The distinction is made possible cycle, using the highest peak and lowest valley,
only to reduce the total number of events by a factor of two. followed by the second largest cycle, etc., until all peak counts
E1049 − 85 (2017)
(a)—Level Crossing Counting
(b)—Restricted Level Crossing Counting
FIG. 2 Level-Crossing Counting Example
are used. This process is illustrated by Fig. 3(c). Note that once counted, then each is counted as one cycle. If both positive and
this most damaging cycle count is obtained, the cycles could be
negative ranges are counted, then each is counted as one-half
applied in any desired order, and this order could have a
cycle. Ranges smaller than a chosen value are usually elimi-
secondary effect on the amount of damage. Alternate methods
nated before counting.
of deriving a cycle count, such as randomly selecting pairs of
5.3.2 When the mean value of each range is also counted,
peaks and valleys, are sometimes used.
the method is called simple range-mean counting. For the
5.3 Simple-Range Counting:
example of Fig. 4, the result of a simple range-mean count is
5.3.1 For this method, a range is defined as the difference
given in X1.1 in the form of a range-mean matrix.
between two successive reversals, the range being positive
5.4 Rainflow Counting and Related Methods:
when a valley is followed by a peak and negative when a peak
5.4.1 A number of different terms have been employed in
is followed by a valley. The method is illustrated in Fig. 4.
Positive ranges, negative ranges, or both, may be counted with the literature to designate cycle-counting methods which are
this method. If only positive or only negative ranges are similar to the rainflow method. These include range-pair
E1049 − 85 (2017)
(a)—Peak Counting
(b)—Mean Crossing Peak Counting
(c)—Cycles Derived from Peak Count of (a)
FIG. 3 Peak Counting Example
counting (1, 2), the Hayes method (3), the original rainflow 5.4.2 The various methods similar to the rainflow method
method (4-6), range-pair-range counting (7), ordered overall may be used to obtain cycles and the mean value of each cycle;
range counting (8), racetrack counting (9), and hysteresis loop they are referred to as two-parameter methods. When the mean
counting (10). If the load history begins and ends with its value is ignored, they are one-parameter methods, as are
maximum peak, or with its minimum valley, all of these give simple-range counting, peak counting, etc.
identical counts. In other cases, the counts are similar, but not
5.4.3 Range-Pair Counting—The range-paired method
generally identical. Three methods in this class are defined
counts a range as a cycle if it can be paired with a subsequent
here: range-pair counting, rainflow counting, and a simplified
loading in the opposite direction. Rules for this method are as
method for repeating histories.
follows:
5.4.3.1 Let X denote range under consideration; and Y,
4 previous range adjacent to X.
The boldface numbers in parentheses refer to the list of references appended to
these practices. (1) Read next peak or valley. If out of data, go to Step 5.
E1049 − 85 (2017)
FIG. 4 Simple Range Counting Example—Both Positive and Negative Ranges Counted
(2) If there are less than three points, go to Step 1. Form (8) End of counting. See the table in Fig. 5 for a summary
ranges X and Y using the three most recent peaks and valleys of the cycles counted in this example, and see Appendix X1.2
that have not been discarded. for this cycle count in the form of a range-mean matrix.
(3) Compare the absolute values of ranges X and Y. 5.4.4 Rainflow Counting:
(a) If X < Y, go to Step 1. 5.4.4.1 Rules for this method are as follows: let X denote
(b) If X ≥ Y, go to Step 4. range under consideration; Y, previous range adjacent to X; and
(4) Count range Y as one cycle and discard the peak and S, starting point in the history.
valley of Y; go to Step 2. (1) Read next peak or valley. If out of data, go to Step 6.
(5) The remaining cycles, if any, are counted by starting at (2) If there are less than three points, go to Step 1. Form
the end of the sequence and counting backwards. If a single ranges X and Y using the three most recent peaks and valleys
range remains, it may be counted as a half or full cycle. that have not been discarded.
5.4.3.2 The load history in Fig. 4 is replotted as Fig. 5(a) (3) Compare the absolute values of ranges X and Y.
and is used to illustrate the process. Details of the cycle (a) If X < Y, go to Step 1.
counting are as follows: (b) If X ≥ Y, go to Step 4.
(1) Y = |A-B |; X = |B-C|; and X > Y. Count |A-B| as one (4) If range Y contains the starting point S, go to Step 5;
cycle and discard points A and B. (See Fig. 5(b). Note that a otherwise, count range Y as one cycle; discard the peak and
cycle is formed by pairing range A-B and a portion of range valley of Y; and go to Step 2.
B-C.) (5) Count range Y as one-half cycle; discard the first point
(2) Y = |C-D|; X = |D-E|; and X < Y. (peak or valley) in range Y; move the starting point to the
(3) Y = |D-E|; X = |E-F|; and X < Y. second point in range Y; and go to Step 2.
(4) Y = |E-F|; X = |F-G|; and X > Y. Count |E-F| as one (6) Count each range that has not been previously counted
cycle and discard points E and F. (See Fig. 5(c).) as one-half cycle.
(5) Y = |C-D|; X = |D-G|; and X > Y. Count |C-D| as one 5.4.4.2 The load history of Fig. 4 is replotted as Fig. 6(a)
cycle and discard points C and D. (See Fig. 5(d).) and is used to illustrate the process. Details of the cycle
(6) Y = |G-H|; X = |H-I|; and X < Y. Go to the end and counting are as follows:
count backwards. (1) S = A; Y = |A-B |; X = |B-C|; X > Y. Y contains S, that is,
(7)
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
´1
Designation: E1049 − 85 (Reapproved 2011) E1049 − 85 (Reapproved 2017)
Standard Practices for
Cycle Counting in Fatigue Analysis
This standard is issued under the fixed designation E1049; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Reference (12) was editorially corrected in October 2011.
1. Scope
1.1 These practices are a compilation of acceptable procedures for cycle-counting methods employed in fatigue analysis. This
standard does not intend to recommend a particular method.
1.2 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory
limitations prior to use.
1.3 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
E912 Definitions of Terms Relating to Fatigue Loading; Replaced by E 1150 (Withdrawn 1988)
3. Terminology
3.1 Definitions:
3.1.1 constant amplitude loading—in fatigue loading, a loading in which all of the peak loads are equal and all of the valley
loads are equal.
3.1.2 cycle—in fatigue loading, under constant amplitude loading, the load variation from the minimum to the maximum and
then to the minimum load.
NOTE 1—In spectrum loading, definition of cycle varies with the counting method used.
3.1.3 mean crossings—in fatigue loading, the number of times that the load-time history crosses the mean-load level with a
positive slope (or a negative slope, or both, as specified) during a given length of the history (see Fig. 1).
These practices are under the jurisdiction of ASTM Committee E08 on Fatigue and Fracture and are the direct responsibility of Subcommittee E08.04 on Structural
Applications.
Current edition approved Oct. 1, 2011June 1, 2017. Published October 2011June 2017. Originally approved in 1985. Last previous edition approved in 20052011 as
ɛ1
E1049–85(2005).E1049–85(2011) . DOI: 10.1520/E1049-85R11E01.10.1520/E1049-85R17.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
The last approved version of this historical standard is referenced on www.astm.org.
3.1.3.1 Discussion—
For purposes related to cycle counting, a mean crossing may be defined as a crossing of the reference load level.
3.1.4 mean load, P —in fatigue loading, the algebraic average of the maximum and minimum loads in constant amplitude
m
loading, or of individual cycles in spectrum loading,
P 5 ~P 1P !/2 (1)
m max min
or the integral average of the instantaneous load values or the algebraic average of the peak and valley loads of a spectrum
loading history.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1049 − 85 (2017)
FIG. 1 Basic Fatigue Loading Parameters
3.1.5 peak—in fatigue loading, the point at which the first derivative of the load-time history changes from a positive to a
negative sign; the point of maximum load in constant amplitude loading (see Fig. 1).
3.1.6 range—in fatigue loading, the algebraic difference between successive valley and peak loads (positive range or increasing
load range), or between successive peak and valley loads (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” is defined
by the algebraic difference between the largest peak and the smallest 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 necessarily successive events. In these practices, the definition of the word “range” is broadened so that events
of this type are also included.
3.1.7 reversal—in fatigue loading, the point at which the first derivative of the load-time history changes sign (see Fig. 1).
NOTE 3—In constant amplitude loading, a cycle is equal to two reversals.
3.1.8 spectrum loading—in fatigue loading, a loading in which all of the peak loads are not equal or all of the valley loads are
not equal, or both. (Also known as variable amplitude loading or irregular loading.)
3.1.9 valley—in fatigue loading, the point at which the first derivative of the load-time history changes from a negative to a
positive sign (also known as trough); the point of minimum load 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 of interest.
3.2.2 reference load—for spectrum loading, used in these practices 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 the history, but
this is not required.
3.3 For other definitions of terms used in these practices refer to Definitions E912.
4. Significance and Use
4.1 Cycle counting is used to summarize (often lengthy) irregular load-versus-time histories by providing the number of times
cycles of various sizes occur. The definition of a cycle varies with the method of cycle counting. These practices cover the
procedures used to obtain cycle counts by various methods, including level-crossing counting, peak counting, simple-range
counting, range-pair counting, and rainflow counting. Cycle counts can be made for time histories of force, stress, strain, torque,
acceleration, deflection, or other loading parameters of interest.
5. Procedures for Cycle Counting
5.1 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 sloped portion of
the load exceeds a preset level above the reference load, and each time the negative sloped portion of the load exceeds a preset
level below the reference load. Reference load crossings are counted on the positive sloped portion of the loading history. It makes
no difference whether positive or negative slope crossings are counted. The distinction is made only to reduce the total number
of events by a factor of two.
E1049 − 85 (2017)
(a)—Level Crossing Counting
(b)—Restricted Level Crossing Counting
FIG. 2 Level-Crossing Counting Example
5.1.2 In practice, restrictions on the level-crossing counts are often specified to eliminate small amplitude variations which can
give rise to a large number of counts. This may be accomplished by filtering small load excursions prior to cycle counting. A second
method is to make no counts at the reference load and to specify that only one count be made between successive crossings of a
secondary lower level associated with each level above the reference load, or a secondary higher level associated with each level
below the reference load. Fig. 2(b) illustrates this second method. A variation of the second method is to use the same secondary
level for all counting levels above the reference load, and another for all levels below the reference load. In this case the levels
are generally not evenly spaced.
5.1.3 The most damaging cycle count for fatigue analysis is derived from the level-crossing count by first constructing the
largest possible cycle, followed by the second largest, etc., until all level crossings are used. Reversal points are assumed to occur
halfway between levels. This process is illustrated by Fig. 2(c). Note that once this most damaging cycle count is obtained, the
cycles could be applied in any desired order, and this order could have a secondary effect on the amount of damage. Other methods
of deriving a cycle count from the level-crossings count could be used.
5.2 Peak Counting:
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5.2.1 Peak counting identifies the occurrence of a relative maximum or minimum load value. Peaks above the reference load
level are counted, and valleys below the reference load level are counted, as shown in Fig. 3(a). Results for peaks and valleys are
usually reported separately. A variation of this method is to count all peaks and valleys without regard to the reference load.
5.2.2 To eliminate small amplitude loadings, mean-crossing peak counting is often used. Instead of counting all peaks and
valleys, only the largest peak or valley between two successive mean crossings is counted as shown in Fig. 3(b).
5.2.3 The most damaging cycle count for fatigue analysis is derived from the peak count by first constructing the largest possible
cycle, using the highest peak and lowest valley, followed by the second largest cycle, etc., until all peak counts are used. This
process is illustrated by Fig. 3(c). Note that once this most damaging cycle count is obtained, the cycles could be applied in any
desired order, and this order could have a secondary effect on the amount of damage. Alternate methods of deriving a cycle count,
such as randomly selecting pairs of peaks and valleys, are sometimes used.
5.3 Simple-Range Counting:
(a)—Peak Counting
(b)—Mean Crossing Peak Counting
(c)—Cycles Derived from Peak Count of (a)
FIG. 3 Peak Counting Example
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5.3.1 For this method, a range is defined as the difference between two successive reversals, the range being positive when a
valley is followed by a peak and negative when a peak is followed by a valley. The method is illustrated in Fig. 4. Positive ranges,
negative ranges, or both, may be counted with this method. If only positive or only negative ranges are counted, then each is
counted as one cycle. If both positive and negative ranges are counted, then each is counted as one-half cycle. Ranges smaller than
a chosen value are usually eliminated before counting.
5.3.2 When the mean value of each range is also counted, the method is called simple range-mean counting. For the example
of Fig. 4, the result of a simple range-mean count is given 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 in the literature to designate cycle-counting methods which are similar
to the rainflow method. These include range-pair counting (1, 2), the Hayes method (3), the original rainflow method (4-6),
range-pair-range counting (7), ordered overall range counting (8), racetrack counting (9), and hysteresis loop counting (10). If the
load history begins and ends with its maximum peak, or with its minimum valley, all of these give identical counts. In other cases,
the counts are similar, but not generally identical. Three methods in this class are defined here: range-pair counting, rainflow
counting, and a simplified method for repeating histories.
5.4.2 The various methods similar to the rainflow method may be used to obtain cycles and the mean value of each cycle; they
are referred to as two-parameter methods. When the mean value is ignored, they are one-parameter methods, as are simple-range
counting, peak counting, etc.
5.4.3 Range-Pair Counting—The range-paired method counts a range as a cycle if it can be paired with a subsequent loading
in the opposite direction. Rules for this method are as follows:
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.
(2) If there are less than three points, go to Step 1. Form ranges X and Y using the three most recent peaks and valleys that
have not been discarded.
The boldface numbers in parentheses refer to the list of references appended to these practices.
FIG. 4 Simple Range Counting Example—Both Positive and Negative Ranges Counted
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(3) Compare the absolute values of ranges X and Y.
(a) If X < Y, go to Step 1.
(b) If X ≥ Y, go to Step 4.
(4) Count range Y as one cycle and discard the peak and valley of Y; go to Step 2.
(5) The remaining cycles, if any, are counted by starting at the end of the sequence and counting backwards. If a single range
remains, it may be counted as a half or full cycle.
5.4.3.2 The load history in Fig. 4 is replotted as Fig. 5(a) and is used to illustrate the process. Details of the cycle counting are
as follows:
(1) Y = |A-B |; X = |B-C|; and X > Y. Count |A-B| as one cycle and discard points A and B. (See Fig. 5(b). Note that a cycle
is formed by pairing range A-B and a portion of range B-C.)
(2) Y = |C-D|; X = |D-E|; and X < Y.
(3) Y = |D-E|; X = |E-F|; and X < Y.
(4) Y = |E-F|; X = |F-G|; and X > Y. Count |E-F| as one cycle and discard points E and F. (See Fig. 5(c).)
(5) Y = |C-D|; X = |D-G|; and X > Y. Count |C-D| as one cycle and discard points C and D. (See Fig. 5(d).)
(6) Y = |G-H|; X = |H-I|; and X < Y. Go to the end and count backwards.
(a) (b)
(c) (d)
FIG. 5 Range-Pair Counting Example
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(7) Y = |H-I|; X = |G-H|; and X > Y. Count |H-I| as one cycle and discard points H and I. (See Fig. 5(e).)
(8) End of counting. See the table in Fig. 5 for a summary of the cycles counted in this example, and see Appendix X1.2 for
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 denote range under consideration; Y, previous range adjacent to X; and S,
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. Form ranges X and Y using the three most recent peaks and valleys that
have not been discarded.
(3) Co
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