# Rainflow-counting algorithm

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Rainflow-counting algorithm

The rainflow-counting algorithm (also known as the "rain-flow counting method") is used in the analysis of fatigue data in order to reduce a spectrum of varying stress into a set of simple stress reversals. Its importance is that it allows the application of Miner's rule in order to assess the fatigue life of a structure subject to complex loading. The algorithm was developed by Tatsuo Endo and M. Matsuiski in 19681. Though there are a number of cycle-counting algorithms for such applications, the rainflow method is, as of 2004, the most popular.

Downing and Socie2 created one of the more widely referenced and utilized rainflow cycle-counting algorithms in 1982, which was included as one of many cycle-counting algorithms in ASTM E 1049-853. This algorithm is used in Sandia National Laboratories LIFE24 code for the fatigue analysis of wind turbine components.

Igor Rychlik5 gave a mathematical definition for the rainflow counting method, thus enabling closed-form computations from the statistical properties of the load signal.

For simple periodic loadings, such as Figure 1, rainflow counting is unnecessary. That sequence clearly has 10 cycles of amplitude 20 MPa and a structure's life can be estimated from a simple application of the relevant S-N curve.

Compare this with Figure 2 which cannot be assessed in terms of simply-described stress reversals.

The algorithm

#Reduce the time history to a sequence of (tensile) peaks and (compressive) troughs.
#Imagine that the time history is a template for a rigid sheet (pagoda roof).
#Turn the sheet clockwise 90° (earliest time to the top).
#Each "tensile peak" is imagined as a source of water that "drips" down the pagoda.
#Count the number of half-cycles by looking for terminations in the flow occurring when either:
##It reaches the end of the time history;
##It merges with a flow that started at an earlier "tensile peak"; or
##It flows opposite a "tensile peak" of greater magnitude.
#Repeat step 5 for "compressive troughs".
#Assign a magnitude to each half-cycle equal to the stress difference between its start and termination.
#Pair up half-cycles of identical magnitude (but opposite sense) to count the number of complete cycles. Typically, there are some residual half-cycles.

Example

*The stress history in Figure 2 is reduced to peaks and troughs in Figure 3.
*Half-cycle (A) starts at tensile peak (1) and terminates opposite a greater tensile stress, peak (2). Its amplitude is 16 MPa.
*Half-cycle (B) starts at tensile peak (4) and terminates where it is interrupted by a flow from an earlier peak, (3), Its amplitude is 17 MPa.
*Half-cycle (C) starts at tensile peak (5) and terminates at the end of the time history.
*Similar half-cycles are calculated for compressive stresses (Figure 4) and the half-cycles are then matched.

References

#Matsuiski, M. & Endo, T. (1969) Fatigue of metals subjected to varying stress, "Japan Soc. Mech. Engrg".
#Downing, S. D., Socie, D. F. (1982). Simple rainflow counting algorithms. International Journal of Fatigue, Volume 4, Issue 1, January, 31-40.
#ASTM E 1049-85. (Reapproved 2005). Standard practices for cycle counting in fatigue analysis. ASTM International.
#Schluter, L. (1991). Programmer's Guide for LIFE2's Rainflow Counting Algorithm. Sandia Report SAND90-2260.
#Rychlik, I. (1987) A New Definition of the Rainflow Cycle Counting Method, " Int. J. Fatigue 9:2, 119-121".

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