Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

408 CHAPTER 14 Fatigue

Therefore,ifrdifferentcyclesareappliedinwhichastressamplitudeSj(j=1,2,...,r)wouldcause
failureinNjcycles,thenumberofcyclesnjrequiredtocausetotalfatiguefailureisgivenby

∑r

j= 1

nj

Nj

= 1 (14.5)

AlthoughS–Ncurvesmaybereadilyobtainedfordifferentmaterialsbytestingalargenumberofsmall
specimens(coupontests),itisnotpracticabletoadoptthesameapproachforaircraftcomponents,since
theseareexpensivetomanufactureandthetestprogramistooexpensivetorunforlongperiodsoftime.
However, such a program was initiated in the early 1950s to test the wings and tailplanes of Meteor
andMustangfighters.Theseweresubjectedtoconstantamplitudeloadinguntilfailurewithdifferent
specimens being tested at different load levels. Stresses were measured at points where fatigue was
expected(andactuallyoccurred)andS–Ncurvesplottedforthecompletestructure.Thecurveshadthe
usualappearanceandatlowstresslevelshadsuchlargeendurancesthatfatiguedidnotoccur;thus,a
fatiguelimitexisted.ItwasfoundthattheaverageS–Ncurvecouldbeapproximatedtobytheequation

Sa=10.3( 1 + 1000 /

√

N) (14.6)

inwhichthemeanstresswas90N/mm^2 .Ingeneralterms,Eq.(14.6)maybewrittenas

Sa=S∞( 1 +C/

√

N) (14.7)

inwhichS∞isthefatiguelimitandCisaconstant.Thus,Sa→S∞asN→∞.Equation(14.7)may
berearrangedtogivetheendurancedirectly:

N=C^2

(

S∞

Sa−S∞

) 2

(14.8)

whichshowsclearlythatasSa→S∞,N→∞.
It has been found experimentally thatNis inversely proportional to the mean stress as the latter
variesintheregionof90N/mm^2 ,whileCisvirtuallyconstant.Thissuggestsamethodofdetermining
a“standard”endurancecurve(correspondingtoameanstresslevelof90N/mm^2 )fromtestscarriedout
onafewspecimensatothermeanstresslevels.SupposethatSmisthemeanstresslevel,not90N/mm^2 ,
intestscarriedoutonafewspecimensatanalternatingstresslevelSa,mwherefailureoccursatamean
numberofcyclesNm.Then,assumingthattheS–NcurvehasthesameformasEq.(14.7),

Sa,m=S∞,m( 1 +C/

√

Nm) (14.9)

inwhichC=1000andS∞,misthefatiguelimitstresscorrespondingtothemeanstressSm.Rearranging
Eq.(14.9),wehave

S∞,m=Sa,m/( 1 +C/

√

Nm) (14.10)

Thenumberofcyclestofailureatameanstressof90N/mm^2 wouldhavebeen,fromtheabove,

N′=

Sm

90

Nm (14.11)