Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

2.6 Bending of an End-Loaded Cantilever 55

Fig.2.5

Displacements produced by rigid body rotation.

2.6 BendingofanEnd-LoadedCantilever............................................................

In his semi-inverse solution of this problem, St. Venant based his choice of stress function on the
reasonableassumptionsthatthedirectstressisdirectlyproportionaltobendingmoment(andtherefore
distancefromthefreeend)andheightabovetheneutralaxis.Theportionofthestressfunctiongiving
shearstressfollowsfromtheequilibriumconditionrelatingσxandτxy.Theappropriatestressfunction
forthecantileverbeamshowninFig.2.6isthen

φ=Axy+

Bxy^3

6

(i)

whereAandBareunknownconstants.Hence

σx=

∂^2 φ

∂y^2

=Bxy

σy=

∂^2 φ

∂x^2

= 0

τxy=−

∂^2 φ

∂x∂y

=−A−

By^2

2

⎫

⎪⎪

⎪⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎭

(ii)

Substitutionforφinthebiharmonicequationshowsthattheformofthestressfunctionsatisfiescom-
patibilityforallvaluesoftheconstantsAandB.TheactualvaluesofAandBarechosentosatisfythe
boundarycondition—thatis,τxy=0—alongtheupperandloweredgesofthebeam,andtheresultant
shearloadoverthefreeendisequaltoP.
Fromthefirstofthese

τxy=−A−

By^2

2

=0aty=±

b

2