Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

316 CHAPTER 9 Thin Plates

Further,thedirectstressσηontheplaneFD(Fig.9.16(c))whichisperpendiculartotheplaneofthe
buckleisfoundfromtheequilibriumoftheelementFED.Then,

σηFD+σmbEDsinθ+τmEFsinθ+τmDEcosθ= 0

DividingbyFDandrearranginggive

ση=−σmbsin^2 θ−τmsin2θ (9.49)

Notethattheshearstressonthisplaneformsacomplementaryshearstresssystemwithτηξ.
Thefailureconditionisreachedbyaddingσt(max)toσξandusingthevonMisestheoryofelastic
failure(see[Ref.15]),thatis,

σy^2 =σ 12 +σ 22 −σ 1 σ 2 + 3 τ^2 (9.50)

whereσyis the yield stress of the material,σ 1 andσ 2 are the direct stresses acting on two mutually
perpendicularplanes,andτistheshearstressactingonthesametwoplanes.Hence,whentheyield
stressinthewebisσyw,failureoccurswhen

σyw^2 =(σξ+σt(max))^2 +ση^2 −ση(σξ+σt(max))+ 3 τηξ^2 (9.51)

Eqs.(9.47),(9.48),(9.49),and(9.51)maybesolvedforσt(max),whichisthengivenby

σt(max)=−

1

2

A+

1

2

[A^2 − 4 (σmb^2 + 3 τm^2 −σyw^2 )]

1

(^2) (9.52)
where
A= 3 τmsin2θ+σmbsin^2 θ− 2 σmbcos^2 θ (9.53)
Theseequationshavebeenderivedforapointontheedgeofthepanelbutareapplicabletoanypoint
withinitsboundary.Therefore,theresultantforceFwcorrespondingtothetensionfieldinthewebmay
becalculatedanditslineofactiondetermined.
Iftheaveragestressesinthecompressionandtensionflangesareσcfandσtfandtheyieldstressof
theflangesisσyf,thereducedplasticmomentsintheflangesare(see[Ref.15])
Mpc′ =Mpc

[

1 −

(

σcf

σyf

) 2 ]

(compressionflange) (9.54)

Mpt′ =Mpt

[

1 −

(

σtf

σyf

)]

(tensionflange) (9.55)

Thepositionofeachplastichingemaybefoundbyconsideringtheequilibriumofalengthofflange
andusingtheprincipleofvirtualwork.InFig.9.17,thelengthWXoftheupperflangeofthebeamis
givenavirtualdisplacementφ.TheworkdonebytheshearforceatXisequaltotheenergyabsorbed
bytheplastichingesatXandWandtheworkdoneagainstthetensionfieldstressσt(max).Suppose
thattheaveragevalueofthetensionfieldstressisσtc—thatis,thestressatthemidpointofWX.