Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

406 CHAPTER 14 Fatigue

resulting in a fatigue endurance curve (theS–Ncurve) of the type shown in Fig. 12.2. Such a curve
corresponds to the average value ofNat each stress amplitude, since there will be a wide range of
values ofNfor the given stress; even under carefully controlled conditions the ratio of maximum
Nto minimumNmay be as high as 10:1. Two other curves may therefore be drawn, as shown in
Fig.14.1,envelopingallornearlyalltheexperimentalresults;thesecurvesareknownastheconfidence
limits. If 99.9 percent of all the results lie between the curves—in other words, only 1 in 1000 falls
outside—theyrepresentthe99.9percentconfidencelimits.If 99.99999percentofresultsliebetween
thecurves,only1in10^7 resultswillfalloutsidethemandtheyrepresentthe99.99999percentconfidence
limits.
Theresultsfromtestsonanumberofspecimensmayberepresentedasahistograminwhichthe
numberofspecimensfailingwithincertainrangesRofNisplottedagainstN.Then,ifNavistheaverage
valueofNatagivenstressamplitude,theprobabilityoffailureoccurringatNcyclesisgivenby

p(N)=

1

σ

√

2 π

exp

[

−

1

2

(

N−Nav

σ

) 2 ]

(14.1)

inwhichσisthestandarddeviationofthewholepopulationofNvalues.ThederivationofEq.(14.1)
depends on the histogram approaching the profile of a continuous function close to thenormal
distribution, which it does as the intervalNav/Rbecomes smaller and the number of tests increases.
Thecumulativeprobability,whichgivestheprobabilitythataparticularspecimenwillfailatorbelow
Ncycles,isdefinedas

P(N)=

∫N

−∞

p(N)dN (14.2)

TheprobabilitythataspecimenenduresmorethanNcyclesisthen1–P(N).Thenormaldistribution
allowsnegativevaluesofN,whichisclearlyimpossibleinafatiguetestingsituation.Otherdistributions,

Fig.14.1

S–Ndiagram.