If a flight-by-flight stress history is developed for damage tolerance analysis or tests, it will be given as a sequence of load levels.  Each of the cases, a, b, c, and d in Figure 5.5.1, could be considered as a series of details in such a sequence.  Each case is a stress excursion of 8d between levels A and B containing a dip of increasing size from a to d.  In case a, the dip might be so small that for practical purposes it can be neglected.  The cycle then can be considered as a single excursion with a range DK₁ of size 8d.  In cases b through d, the dips are too big to be neglected.  Normal crack growth calculations might consider each of these cases as a sequence of two excursions, for example case b would be made up of two excursions, one with a range DK₂, the other with a range DK₃, each of size 5d.

Figure 5.5.1.  Definition of Cycles

Table 5.5.1.  Calculation of Crack Growth For Figure 5.5.1

If the four cases were treated this way, the calculated crack extension based on range excursions would be as given Table 5.5.1, where, for simplicity, the crack growth equation is taken as da/dN = C(DK)⁴ and the R ratio effect is ignored.  As indicated in this table, the damage estimates for cases b and c are considerably less than the crack damage estimated for case a.  This is very unlikely in practice, since the crack would see one excursion from A to B in each case.  Therefore, cases b, c, and d should be more damaging than case a in view of the extra cycle due to the dip.  Although the effect of cycle ratio was neglected, the small influence of R could not account for the discrepancies.

It seems more reasonable to treat each case as one excursion with a range of DK₁ plus one excursion of a smaller range (e.g., DK₄ in case b) which follows the philosophy of range-pair counting.  If this is done, the ranges considered would be as indicated by the dashed lines in Figure 5.5.1.  The crack growth calculation based on range-pair counting is shown at the bottom of Table 5.5.1, indicating an increasing amount of damage going from a to d.

Another cycle definition is obtained by rainflow counting [VanDÿk, 1972; Dowling, 1972].  The method is illustrated in Figure 5.5.2.  While placing the graphical display of the stress history vertical, it is considered as a stack of roofs.  Rain is assumed to flow from each roof.  If it runs off the roof, it drips down the roof below, etc., with the exception that the rain does not continue on a roof that is already wet.  The range of the rain flow is considered the range of the stress.  The ranges so obtained are indicated by AB, CD, etc., in Figure 5.5.2.  Figure 5.5.3 shows how cycle counting methods may affect a crack growth prediction.

Figure 5.5.2.  Rain Flow Count

Figure 5.5.3.  Calculated Crack Growth Curves for Random Flight-by-Flight Fighter Spectrum [VanDÿk, 1972]

Several other counting methods exist, and they are reviewed in Schijve [1963] and VanDÿk [1972].  Counting methods were originally developed to count measured load histories for establishing an exceedance diagram.  Therefore, the opinions expressed in the literature on the usefulness of the various counting procedures should be considered in that light.  The counting procedure giving the best representation of a spectrum need not necessarily be the best descriptor of fatigue behavior.

It is argued that ranges are more important to fatigue behavior than load peaks.  On this basis, the so-called range-pair count and the rainflow count are considered the most suitable.  However, no crack growth experiments were ever reported to prove this.

The use of counting procedures in crack growth prediction is an entirely new application.  An experimental program is required for a definitive evaluation.  Calculated crack growth curves show that the difference in crack growth life may be on the order of 25-30 percent.  It should be noted that counting is not as essential when the loads are sequenced low-high-low in each flight.  The increasing ranges automatically produce an effect similar to counting. 

For the time being, it seems that a cycle count will give the best representation of fatigue behavior.  Therefore, it is recommended that cycle counting per flight be used for crack growth predictions of random sequences.  Care should be taken that the stress ranges are sequenced properly to avoid different interaction effects (note that K_max determines retardation and not DK).  As an example, consider again Figure 5.5.2.  The proper sequence for integration is:  CD, GH, KL, EF, AB, PQ, MN.  In this way, the maximum stress intensity (at B) occurs at the proper time with respect to its retardation effect, and the maximum stress-intensity of cycle AB will cause retardation for cycles PQ and MN only.