Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

504 CHAPTER 17 Torsion of Beams

Fig.17.2

Determination of the shear flow distribution in a closed section beam subjected to torsion.

shearflowactingonanelementδsofthebeamwallispqδs.Hence,

T=

∮

pqds

or,sinceqisconstantand

∮

pds= 2 A(seeSection16.3)

T= 2 Aq (17.1)

NotethattheoriginOoftheaxesinFig.17.2maybepositionedinoroutsidethecrosssectionof
thebeam,sincethemomentoftheinternalshearflows(whoseresultantisapuretorque)isthesame
aboutanypointintheirplane.Foranoriginoutsidethecrosssection,theterm

∮

pdswillinvolvethe
summationofpositiveandnegativeareas.Thesignofanareaisdeterminedbythesignofp,whichitself
isassociatedwiththesignconventionfortorqueasfollows.Ifthemovementofthefootofpalongthe
tangentatanypointinthepositivedirectionofsleadstoananticlockwiserotationofpabouttheorigin
ofaxes,pispositive.Thepositivedirectionofsisinthepositivedirectionofq,whichisanticlockwise
(correspondingtoapositivetorque).Thus,inFig.17.3ageneratorOA,rotatingaboutO,willinitially
sweepoutanegativearea,sincepAisnegative.AtB,however,pBispositivesothattheareasweptout
bythegeneratorhaschangedsign(atthepointwherethetangentpassesthroughOandp=0).Positive
andnegativeareascanceleachotheroutastheyoverlap,soasthegeneratormovescompletelyaround
thesection,startingandreturningtoA,say,theresultantareaisthatenclosedbytheprofileofthebeam.
ThetheoryofthetorsionofclosedsectionbeamsisknownastheBredt–Bathotheory,andEq.(17.1)
isoftenreferredtoastheBredt–Bathoformula.

17.1.1 Displacements Associated with the Bredt–Batho Shear Flow

TherelationshipbetweenqandshearstrainγestablishedinEq.(16.19),namely,

q=Gt

(

∂w

∂s

+

∂vt

∂z

)