486 CHAPTER 16 Shear of Beams
Fig.16.7
Shear flow distribution in Z-section of Example 16.1.
WenoteinEq.(iv)thattheshearflowisnotzerowhens 2 =0butequaltothevalueobtainedbyinserting
s 1 =h/2inEq.(iii)—thatis,q 2 =0.42Sy/h.IntegrationofEq.(iv)yields
q 23 =Sy
h^3(
0.42h^2 +3.42hs 2 −3.42s^22)
(v)ThisdistributionissymmetricalaboutCxwithamaximumvalueats 2 =h/ 2 (y=0),andtheshearflow
ispositiveatallpointsintheweb.Theshearflowdistributionintheupperflangemaybededucedfrom
antisymmetrysothatthecompletedistributionisoftheformshowninFig.16.7.
16.2.1 Shear Center
Wehavedefinedthepositionoftheshearcenterasthatpointinthecrosssectionthroughwhichshear
loadsproducenotwisting.Itmaybeshownbyuseofthereciprocaltheoremthatthispointisalsothe
centeroftwistofsectionssubjectedtotorsion.Thereare,however,someimportantexceptionstothis
generalrule.Clearly,inthemajorityofpracticalcases,itisimpossibletoguaranteethatashearload
willactthroughtheshearcenterofasection.Equallyapparentisthefactthatanyshearloadmaybe
represented by the combination of the shear load applied through the shear center and a torque. The
stressesproducedbytheseparateactionsoftorsionandshearmaythenbeaddedbysuperposition.It
is,therefore,necessarytoknowthelocationoftheshearcenterinalltypesofsectionortocalculateits
position.Whereacrosssectionhasanaxisofsymmetry,theshearcentermust,ofcourse,lieonthis
axis.ForcruciformoranglesectionsofthetypeshowninFig.16.8,theshearcenterislocatedatthe
intersectionofthesides,sincetheresultantinternalshearloadsallpassthroughthesepoints.
Example 16.2
Calculatethepositionoftheshearcenterofthethin-walledchannelsectionshowninFig.16.9. The
thicknesstofthewallsisconstant.