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DTD Handbook

Handbook for Damage Tolerant Design

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    • About
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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
      • 3. Damage Size Characterizations
      • 4. Residual Strength
      • 5. Analysis Of Damage Growth
      • 6. Examples of Damage Tolerant Analyses
      • 7. Damage Tolerance Testing
      • 8. Force Management and Sustainment Engineering
      • 9. Structural Repairs
        • 0. Structural Repairs
        • 1. Required Analysis
        • 3. Spectrum Analysis for Repair
          • 0. Spectrum Analysis for Repair
          • 1. Definition of Stress Histories
          • 2. Spectra Descriptions
            • 0. Spectra Descriptions
            • 1. Exceedance Curve Descriptions
            • 2. RMS Descriptions
          • 3. Crack Growth Analysis
        • 4. Life Sensitivity for Stress Effects
        • 5. Life Sensitivity Analysis for Hole Repair
        • 6. Blend-Out Repairs
        • 7. Residual Strength Parametric Analysis
        • 8. References
      • 10. Guidelines for Damage Tolerance Design and Fracture Control Planning
      • 11. Summary of Stress Intensity Factor Information
    • Examples

Section RMS Descriptions

The presentation of complicated variable amplitude stress histories can be simplified by defining average or RMS values of the stress event characteristics, i.e. the maximum stresses and positive stress ranges of the history.  The difference between the average value and the RMS value of a given characteristic is normally not more than 3 percent when one is considering stress histories with more than 1000 stress events.  For average stress analysis, one uses


while for RMS analysis, one uses


where si is the characteristic (maximum stress or stress range) for the ith stress event and N is the total number of stress events.

Similar analysis schemes have also been employed where the slope (p) of the crack growth rate power law expression (Equation 9.3.1) is used to calculate a representative stress, i.e.


Experience has shown that such schemes (Equation 9.3.4) are not appreciably of more value than the average or RMS determined characteristics.

The RMS equation (Equation 9.3.3) was applied to the three transport wing stress histories to obtain RMS values for the maximum stress and stress range.  The results are summarized in Table 9.3.3.

Table 9.3.3.  Per Cycle Root Mean Square (RMS)Representative Stress Values

for the Three Wing Stress Histories

Stress History

Maximum Stress (ksi)

Stress Range (ksi)

Cycles per 100 Flights

Center Wing (BL-70)




Inner Wing (WS-733)




Outer Wing





Based on the RMS analyses presented in Table 9.3.3, it appears as if the three stress histories are quite similar on a per cycle basis (the stress ranges are within five (5) percent, and the maximum stresses are within ten (10) percent).  Based on a constant amplitude analysis of these stress conditions, the damage per cycle would be expected also to be similar.  From Table 9.3.3, one can note the number of cycles applied per 100 flight block differs substantially from stress history to stress history.  If the RMS stresses are similar and the number of stresses per flight differ, then one would expect that the damage per flight would favor the stress history with the most stress events per flight.

One of the reasons that the RMS representative stresses can not be blindly used in a constant amplitude equation to accurately estimate crack growth behavior is because the damage is a non-linear function of the different events in the history.  The analyst must understand where the damage is coming from and isolate on those events.  For example, a transport wing stress history generates damage as a result of both GAG cycle loading and gust/maneuver cycle loading.  A second analysis was therefore conducted on the three wing histories to obtain per flight characteristics for the GAG and gust/maneuver cycles.  This analysis is presented in Table 9.3.4.

Table 9.3.4.  Per Flight Root Mean Square Representative Stress Values for

the Three Wing Stress Histories

Stress History

GAG Max Stress (ksi)

GAG Min Stress (ksi)

Gust/Manu. Max Stress (ksi)

Gust/Manu. Stress Range (ksi)

Number of Gust/Maneuver Cycles

Center Wing (BL-70)






Inner Wing (WS-733)






Outer Wing







Relative to the per flight RMS representative stress values for GAG and gust/maneuver cycles, the three stress histories are shown to be relatively similar.  The magnitude of the GAG cycle appears to be increasing as the location moves outboard; this would indicate that the GAG cycle causes more damage per flight in the outboard wing than at the inner and center wing locations.  We note that the largest number of gust/maneuver cycles occur at the outer wing location and this would also favor more damage per flight (due to gust/maneuver cycles) than the other two locations.