# Rainflow Counting

The fatigue life of a component which is subjected to Constant Varying (VA) loading can be calculated using faigue curve (S-N curve for instance).  In real world the fatigue loading is very complex.  When the varying loading is random it is difficult to estimate the fatigue life.  The rainflow counting algorithm (also known as the “rainflow counting method”) is used in the analysis of fatigue data in order to reduce a spectrum of varying stresses (strains) into a set of simple stress (strain) 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. Matsuishi in 1968. Downing and Socie 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 1049-85 (Standard Practices for Cycle Counting in Fatigue Analysis).  Igor Rychlik 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 9.5 cycles of amplitude 10 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. ## Rainflow Counting Algorithm

The “rainflow” was named from a comparison of this algorithm to the flow of rain falling on a pagoda and running down the edges of the roof. The rainflow counting algorithm consists of the following steps:

1. Reduce the time history to a sequence of (tensile) peaks and (compressive) valleys.
2. Imagine that the time history is a template for a rigid sheet like a pagoda roof.
3. Turn the sheet clockwise 90° (earliest time to the top), and now it looks like a pagoda roof (Figure 3).
4. Each tensile peak is imagined as a source of water that “drips” down the pagoda.
5. 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 when an opposite tensile peak has greater magnitude.
6. Repeat step 5 for compressive valleys.
7. Assign a stress range Δσ = σmax – σmin to each half-cycle equal to the stress difference between its start and termination.
8. Pair up half-cycles of identical magnitude but opposite sense to count the number of complete cycles. Unmatched half-cycles are residual half-cycles. Figure 3:  Pagoda roof

A given half cycle may contain smaller half cycles.  As a general rule, large stress cycles must not be fragmented into smaller ones as this will lead to underestimation of fatigue damage.  Smaller stress cycles should be treated as temporary interruptions of larger stress reversals.

## Rainflow Counting Example

• The stress history in Figure 2 is reduced to peaks and valleys in Figure 4.
• Half-cycle (A) starts at tensile peak (1) and terminates opposite a greater tensile stress, peak (2); the stress range Δσ = 2 – (-14) = 16 MPa.
• Half-cycle (B) starts at tensile peak (4) and terminates where it is interrupted by a flow from an earlier peak, (3); the stress range Δσ = 8 – (-9) = 17 MPa.
• Half-cycle (C) starts at tensile peak (5) and terminates at the end of the time history; the stress range Δσ = 15 – (-4) = 19 MPa.
• Similar half-cycles are calculated for compressive stresses (Figure 5) and the half-cycles are then matched.  A pair of matched half cycles (one tension, one compression) is counted as a full cycle (or 1 cycle); an unmatched half-cycle is counted as a half cycle (or 0.5 cycle). Figure 4:  Rainflow for tensile peaks Figure 5:  Rainflow for compressive valleys

The results are summarized in the following table:

 Stress Range Δσ (MPa) Full Cycles Half Cycles 10 2 0 13 0 1 16 0 2 17 0 2 19 1 0 20 0 1 22 0 1 24 0 1 27 0 1

## ASTM E1049-85 (2017) Rainflow Counting Example Figure 6: Rainflow Counting Example

The following is an example of SFAT software.  The unit of the stresses is MPa.  The stress history is taken from FIG. 6 of ASTM E1049, Standard Practices for Cycle Counting in Fatigue Analysis, amplified by 10 times.   