524 CHAPTER 17 Torsion of Beams
Calculatetherateoftwistinrad/mm.Giveasketchillustratingthedistributionofwarpingdisplacementinthe
crosssectionandquotevaluesatpoints1and4.
Ans.dθ/dz=29.3× 10 −^6 rad/mm, w 3 =−w 4 =−0.19mm,
w 2 =−w 1 =−0.056mm.
Fig. P.17.6
P.17.7 AuniformbeamwiththedoublysymmetricalcrosssectionshowninFig.P.17.7,hashorizontalandvertical
wallsmadeofdifferentmaterialswhichhaveshearmoduliGaandGb,respectively.Ifforanymaterialtheratio
massdensity/shearmodulusisconstant,findtheratioofthewallthicknessestaandtbsothatforagiventorsional
stiffnessandgivendimensionsa,bthebeamhasminimumweightperunitspan.AssumetheBredt–Bathotheory
oftorsionisvalid.
Ifthisthicknessrequirementissatisfied,findthea/bratio(previouslyregardedasfixed),whichgivesminimum
weightforgiventorsionalstiffness.
Ans. tb/ta=Ga/Gb,b/a=1.
Fig. P.17.7 Fig. P.17.8
P.17.8 Thecold-formedsectionshowninFig.P.17.8issubjectedtoatorqueof50Nm.Calculatethemaximum
shearstressinthesectionanditsrateoftwist.G=25000N/mm^2.
Ans. τmax=220.6N/mm^2 ,dθ/dz=0.0044rad/mm.