Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

482 CHAPTER 16 Shear of Beams

Inadditiontotheassumptionsspecifiedintheearlierpartofthissection,wefurtherassumethatduring
any displacement, the shape of the beam cross section is maintained by a system of closely spaced
diaphragms which are rigid in their own plane but are perfectly flexible normal to their own plane
(CSRDassumption).Thereis,therefore,noresistancetoaxialdisplacementw,andthecrosssection
movesasarigidbodyinitsownplane,thedisplacementofanypointbeingcompletelyspecifiedby
translationsuandvandarotationθ(seeFig.16.4).
Atfirstsightthisappearstobearathersweepingassumption,butforaircraftstructuresofthethin
shell type described in Chapter 11 whose cross sections are stiffened by ribs or frames positioned at
frequentintervalsalongtheirlengths,itisareasonableapproximationfortheactualbehaviorofsuch
sections.ThetangentialdisplacementvtofanypointNinthewallofeitheranopenorclosedsection
beamisseenfromFig.16.4tobe

vt=pθ+ucosψ+vsinψ (16.7)

whereclearlyu,v,andθarefunctionsofzonly(wmaybeafunctionofzands).
TheoriginOoftheaxesinFig.16.4hasbeenchosenarbitrarily,andtheaxessufferdisplacements
u,v,andθ.Thesedisplacements,inaloadingcasesuchaspuretorsion,areequivalenttoapurerotation
aboutsomepointR(xR,yR)inthecrosssectionwhereRisthecenteroftwist.Therefore,inFig.16.4,

vt=pRθ (16.8)

and

pR=p−xRsinψ+yRcosψ

whichgives

vt=pθ−xRθsinψ+yRθcosψ

Fig.16.4

Establishment of displacement relationships and position of center of twist of beam (open or closed).