Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

7.2Plates Subjected to Bending and Twisting 223

7.2 PlatesSubjectedtoBendingandTwisting........................................................

Ingeneral,thebendingmomentsappliedtotheplatewillnotbeinplanesperpendiculartoitsedges.Such
bendingmoments,however,mayberesolvedinthenormalmannerintotangentialandperpendicular
components,asshowninFig.7.4.TheperpendicularcomponentsareseentobeMxandMyasbefore,
whilethetangentialcomponentsMxyandMyx(againthesearemomentsperunitlength)producetwisting
oftheplateaboutaxesparalleltothexandyaxes.Thesystemofsuffixesandthesignconventionforthese
twistingmomentsmustbeclearlyunderstoodtoavoidconfusion.Mxyisatwistingmomentintensityin
averticalxplaneparalleltotheyaxis,whereasMyxisatwistingmomentintensityinaverticalyplane
paralleltothexaxis.Notethatthefirstsuffixgivesthedirectionoftheaxisofthetwistingmoment.We
alsodefinepositivetwistingmomentsasbeingclockwisewhenviewedalongtheiraxesindirections
parallel to the positive directions of the correspondingxoryaxis. In Fig. 7.4, therefore, all moment
intensitiesarepositive.
Since the twisting moments are tangential moments or torques, they are resisted by a system of
horizontalshearstressesτxy,asshowninFig.7.6.Fromaconsiderationofcomplementaryshearstresses
(seeFig.7.6),Mxy=−Myx,sothatwemayrepresentageneralmomentapplicationtotheplateinterms
ofMx,My,andMxyasshowninFig.7.5(a).Thesemomentsproducetangentialandnormalmoments,
MtandMn,onanarbitrarilychosendiagonalplaneFD.Wemayexpressthesemomentintensities(in
ananalogousfashiontothecomplexstresssystemsofSection1.6)intermsofMx,My,andMxy.Thus,
forequilibriumofthetriangularelementABCofFig.7.5(b)inaplaneperpendiculartoAC

MnAC=MxABcosα+MyBCsinα−MxyABsinα−MxyBCcosα

giving

Mn=Mxcos^2 α+Mysin^2 α−Mxysin2α (7.10)

Similarly,forequilibriuminaplaneparalleltoCA

MtAC=MxABsinα−MyBCcosα+MxyABcosα−MxyBCsinα

Fig.7.4

Plate subjected to bending and twisting.