It can be noted from Table 9.3.6 that the ratios of crack growth lives for the two stress magnification factors are nearly the same (within 2 percent) for the three stress histories.  The reason for this happening can be justified on the basis of the crack growth rate behavior.  Consider Figure 9.3.8 where both the crack growth life and crack growth rate behavior associated with the scaled inner wing stress histories are described.  Figure 9.3.8 shows that while the life behavior is different, the crack growth rate behavior can be described by a common curve.  If the common crack growth rate curve is a power law equation (Equation 9.3.1) then its integral form, i.e.

can be written, using Equations 9.3.10 and 9.3.11, as

Figure 9.3.8.  Flight-by-Flight Crack Growth Behavior Exhibited for the Inner Wing (WS733) Stress History Scaled to two Different Stress Levels

If all the stresses in a stress history are scaled, then the s_max characterizing stress will be scaled by the same factor.  So, if the crack growth interval remains the same, the life ratio (L_0.903 / L₁ where L₁ = F and L_0.903 = F with lower stress) is given by:

Since all other factors in Equation 9.3.13 are constant, note that the integral is only a function of geometry and once the geometry is defined the stress level does not influence its value.

Using Equation 9.3.14 and the power law exponents given in Table 9.3.5, the life ratio for the scaled stress histories is noted to vary between 1.32 and 1.35 (lowest value of exponent yield lowest life ratio).  The life ratio estimate based on the crack growth rate power law exponent is noted to closely approximate the life ratios given in Table 9.3.6.  Thus, if one can obtain an estimate of the crack growth rate power law exponent, then one can closely approximate the effect of stress scaling on the crack growth life behavior.  Section 9.4 provides additional information on the use of this analysis approach for estimating the lives of structural repairs.

Independent of the above remarks, Equation 9.3.12 has an important application for directly estimating the structural life of cracked components.  As an example of its use for conducting such analysis, we compared the results of the computer analysis with life estimates made using the data presented in Table 9.3.5 and Equation 9.3.13.  These results are presented in Table 9.3.7, where it is seen that the power law life prediction ratios, which are conservative relative to the least squares procedure, result in estimates which more closely approximate the estimates for all three stress histories.

Table 9.3.7.  Ratio of Power Law Life Predictions (L_PL) to Life Predictions (L_CG)

(Ratio > 1, Unconservative)

Because the least squares determined coefficients are insensitive to the accuracy with which the crack growth rate data are described, it is suggested that the analyst comparatively review the least squares results in a graphical format such as Figure 9.3.7.  One reason for choosing the graphical method is to emphasize the (log-log) lower portion of the crack growth rate behavior.  (The least squares procedure results in a "best" fit to all the data).

When flight-by-flight crack growth rate behavior is shown to be independent of stress scaling effects, the behavior will also be independent of the geometry used to collect the crack growth life data.  This has been shown for a number of aircraft stress histories similar to those analyzed in this section.

One cautionary remark must be made relative to geometrical effects - if one reduces crack growth life data using a stress-intensity factor which is substantially in error of the actual stress-intensity factor for the geometry, then the transference of the crack growth rate data from one geometry to another will not be possible.  In other words, take care in reducing crack growth life data from structural geometries where the stress-intensity factor is not well defined.