CHAPTER 17 Torsion of Beams........................................................................
In Chapter 3, we developed the theory for the torsion of solid sections using both the Prandtl stress
function approach and the St. Venant warping function solution. From that point we looked, via the
membraneanalogy,atthetorsionofanarrowrectangularstrip.Weshallusetheresultsofthisanalysis
to investigate the torsion of thin-walled open section beams, but first we shall examine the torsion
of thin-walled closed section beams, since the theory for this relies on the general stress, strain, and
displacementrelationshipswhichweestablishedinChapter16.
17.1 TorsionofClosedSectionBeams..................................................................
AclosedsectionbeamsubjectedtoapuretorqueTasshowninFig.17.1doesnot,intheabsenceofan
axialconstraint,developadirectstresssystem.ItfollowsthattheequilibriumconditionsofEqs.(16.2)
and(16.3)reduceto∂q/∂s=0and∂q/∂z=0,respectively.Theserelationshipsmayonlybesatisfied
simultaneouslybyaconstantvalueofq.Wededuce,therefore,thattheapplicationofapuretorquetoa
closedsectionbeamresultsinthedevelopmentofaconstantshearflowinthebeamwall.However,the
shearstressτmayvaryaroundthecrosssection,sinceweallowthewallthicknessttobeafunction
ofs. The relationship between the applied torque and this constant shear flow is simply derived by
considering the torsional equilibrium of the section shown in Fig. 17.2. The torque produced by the
Fig.17.1
Torsion of a closed section beam.
Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00017-8 503