Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

15.2 Unsymmetrical Bending 433

Fig.15.8

Anticlastic bending of a beam section.

15.2 UnsymmetricalBending............................................................................

We have shown that the value of direct stress at a point in the cross section of a beam subjected to
bendingdependsonthepositionofthepoint,theappliedloading,andthegeometricpropertiesofthe
crosssection.Itfollowsthatitisofnoconsequencewhetherthecrosssectionisopenorclosed.We,
therefore, derive the theory for a beam of arbitrary cross section and then discuss its application to
thin-walledopenandclosedsectionbeamssubjectedtobendingmoments.
TheassumptionsareidenticaltothosemadeforsymmetricalbendingandarelistedinSection15.1.1.
However,beforewederiveanexpressionforthedirectstressdistributioninabeamsubjectedtobending,
we shall establish sign conventions for moments, forces, and displacements; investigate the effect of
choiceofsectiononthepositivedirectionsoftheseparameters,anddiscussthedeterminationofthe
componentsofabendingmomentappliedinanylongitudinalplane.

15.2.1 Sign Conventions and Notation

Forces,moments,anddisplacementsarereferredtoanarbitrarysystemofaxesOxyz,ofwhichOzis
parallel to the longitudinal axis of the beam and Oxyare axes in the plane of the cross section. We
assignthesymbolsM,S,P,T,andwtobendingmoment,shearforce,axialordirectload,torque,and
distributedloadintensity,respectively,withsuffixeswhereappropriatetoindicatesenseordirection.
Thus,Mxis a bending moment about thexaxis,Sxis a shear force in thexdirection, and so on.
Figure15.9showspositivedirectionsandsensesfortheaboveloadsandmomentsappliedexternallyto
abeamandalsothepositivedirectionsofthecomponentsofdisplacementu,v,andwofanypointinthe
beamcrosssectionparalleltothex,y,andzaxes,respectively.Afurtherconditiondefiningthesigns
ofthebendingmomentsMxandMyisthattheyarepositivewhentheyinducetensioninthepositivexy
quadrantofthebeamcrosssection.
Ifwereferinternalforcesandmomentstothatfaceofasectionwhichisseenwhenviewedinthe
directionzO,andthen,asshowninFig.15.10,positiveinternalforcesandmomentsareinthesame
directionandsenseastheexternallyappliedloads,whereasontheoppositefacetheyformanopposing