Rainflow Cycle Counting

Cycle counting is used to extract discrete simple "equivalent" constant amplitude cycles from a random loading sequence.

Note: For Random Response Fatigue and Sine-Sweep Fatigue, the traditional rainflow counting method mentioned in this section is not conducted. Instead, the concept of stress range and number of cycles is inherently taken into account as part of the fatigue calculation. For more information, refer to Random Response Fatigue Analysis and Sine Sweep Fatigue Analysis.

One way to understand "cycle counting" is as a changing stress-strain versus time signal. Cycle counting will count the number of stress-strain hysteresis loops and keep track of their range/mean or maximum/minimum values.

Rainflow cycle counting is the most widely used cycle counting method. It requires that the stress time history be rearranged so that it contains only the peaks and valleys and it starts either with the highest peak or the lowest valley (whichever is greater in absolute magnitude). Then, three consecutive stress points (1, 2, and 3) will define two consecutive ranges as ΔS12=|S1−S2| MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaikdaaeqaaaGccaGLhWUaayjcSdaaaa@428A@ and ΔS23=|S2−S3| MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@ |. A cycle from 1 to 2 is only extracted if ΔS12≤ΔS23 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGHKjYOcqGHuoarcaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaaa@3F7C@ . Once a cycle is extracted, the two points forming the cycle are discarded and the remaining points are connected to each other. This procedure is repeated until the remaining data points are exhausted.
  • Simple Load History:


    Figure 1. Continuous Load History
    Since this load history is continuous, it is converted into a load history consisting of peaks and valleys only.


    Figure 2. Peaks and Valleys for Rainflow Counting. 1, 2, 3, and 4 are the four peaks and valleys
    It is clear that point 4 is the peak stress in the load history, and it will be moved to the front during rearrangement (Figure 3). After rearrangement, the peaks and valleys are renumbered for convenience.


    Figure 3. Load History after Rearrangement and Renumbering

    Next, pick the first three stress values (1, 2, and 3) and determine if a cycle is present.

    If Si MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBaaaleaacaWGPbaabeaaaaa@37E9@ represents the stress value, point i MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaaiaadMgaaeqaaaaa@3711@ then:(1)
    ΔS12=|S1−S2| MathType@MTEF@5@5@+=feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaikdaaeqaaaGccaGLhWUaayjcSdaaaa@428A@
    (2)
    ΔS23=|S2−S3| MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@
    As you can see from Figure 3, ΔS12≥ΔS23 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIXaGaaGOmaaqabaGccqGHLjYScqGHuoarcaWGtbWaaSbaaSqaaiaaikdacaaIZaaabeaaaaa@3F8D@ ; therefore, no cycle is extracted from point 1 to 2. Now consider the next three points (2, 3, and 4).(3)
    ΔS23=|S2−S3| MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@
    (4)
    ΔS34=|S3−S4| MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@
    ΔS23≤ΔS34 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGHKjYOcqGHuoarcaWGtbWaaSbaaSqaaiaaiodacaaI0aaabeaaaaa@3F80@ , hence a cycle is extracted from point 2 to 3. Now that a cycle has been extracted, the two points are deleted from the graph.


    Figure 4. Delete and Reconnect Remaining Points
    The same process is applied to the remaining points:(5)
    ΔS14=|S1−S4| MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@
    (6)
    ΔS45=|S4−S5| MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaabdaqaaiaadofadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGtbWaaSbaaSqaaiaaiodaaeqaaaGccaGLhWUaayjcSdaaaa@428F@

    In this case, ΔS14=ΔS45 MathType@MTEF@5@5@+=feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdqKaam4uamaaBaaaleaacaaIXaGaaGinaaqabaGccqGH9aqpcqGHuoarcaWGtbWaaSbaaSqaaiaaisdacaaI1aaabeaaaaa@3ED3@ , so another cycle is extracted from point 1 to 4. After these two points are also discarded, only point 5 remains; therefore, the rainflow counting process is completed.

    Two cycles (2→3 and 1→4) have been extracted from this load history. One of the main reasons for choosing the highest peak/valley and rearranging the load history is to guarantee that the largest cycle is always extracted (in this case, it is 1→4). If you observe the load history prior to rearrangement, and conduct the same rainflow counting process on it, then clearly, the 1→4 cycle is not extracted.

  • Complex Load History
    The rainflow counting process is the same regardless of the number of load history points. However, depending on the location of the highest peak/valley used for rearrangement, it may not be obvious how the rearrangement process is conducted. Figure 7 shows just the rearrangement process for a more complex load history. The subsequent rainflow counting is just an extrapolation of the process mentioned in the simple example above, and is not repeated here.


    Figure 5. Continuous Load History
    Since this load history is continuous, it is converted into a load history consisting of peaks and valleys only:


    Figure 6. Peaks and Valleys for Rainflow Counting
    Clearly, load point 11 is the highest valued load and therefore, the load history is now rearranged and renumbered.


    Figure 7. Load History After Rearrangement and Renumbering

    The load history is rearranged such that all points including and after the highest load are moved to the beginning of the load history and are removed from the end of the load history.

    Parameters affecting rainflow cycle counting may be defined on a FATPARM Bulk Data Entry. The appropriate FATPARM Bulk Data Entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information Entry.