Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.2 Unsymmetrical Bending 435

15.2.2 Resolution of Bending Moments

A bending momentMapplied in any longitudinal plane parallel to thezaxis may be resolved into
componentsMxandMybythenormalrulesofvectors.However,avisualappreciationofthesituation
isoftenhelpful.ReferringtoFig.15.11,weseethatabendingmomentMinaplaneatanangleθto
Oxmayhavecomponentsofdifferingsigndependingonthesizeofθ.Inbothcases,forthesenseof
Mshown

Mx=Msinθ

My=Mcosθ

which give, forθ<π/2,MxandMypositive (Fig. 15.11(a)) and forθ>π/2,Mxpositive andMy
negative(Fig.15.11(b)).

15.2.3 Direct Stress Distribution due to Bending

ConsiderabeamhavingthearbitrarycrosssectionshowninFig.15.12(a).Thebeamsupportsbending
momentsMxandMyandbendsaboutsomeaxisinitscrosssectionwhichisthereforeanaxisofzero
stressoraneutralaxis(NA).LetussupposethattheoriginofaxescoincideswiththecentroidCofthe
crosssectionandthattheneutralaxisisadistancepfromC.Thedirectstressσzonanelementofarea
δAatapoint(x,y)andadistanceξfromtheneutralaxisis,fromthethirdofEq.(1.42)

σz=Eεz (15.13)

If the beam is bent to a radius of curvatureρabout the neutral axis at this particular section then,
sinceplanesectionsareassumedtoremainplaneafterbending,andbyacomparisonwithsymmetrical
bendingtheory

εz=

ξ

ρ

Fig.15.11

Resolution of bending moments: (a)θ< 90 ◦and (b)θ> 90 ◦.