Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

480 CHAPTER 16 Shear of Beams

Fig.16.1

(a) General stress system on element of a closed or open section beam; (b) direct stress and shear flow system
on the element.

inclosedsectionbeams,byinternalpressure.Althoughwehavespecifiedthattmayvarywiths,this
variationissmallformostthin-walledstructuressothatwemayreasonablymaketheapproximation
thattisconstantoverthelengthδs.Also,fromEq.(1.4),wededucethatτzs=τsz=τ,say.However,
weshallfinditconvenienttoworkintermsofshearflowq—thatis,shearforceperunitlengthrather
thanintermsofshearstress.Hence,inFig.16.1(b),

q=τt (16.1)

andisregardedasbeingpositiveinthedirectionofincreasings.
Forequilibriumoftheelementinthezdirectionandneglectingbodyforces(seeSection1.2)
(
σz+

∂σz

∂z

δz

)

tδs−σztδs+

(

q+

∂q

∂s

δs

)

δz−qδz= 0

whichreducesto

∂q

∂s

+t

∂σz

∂z

= 0 (16.2)

Similarly,forequilibriuminthesdirection

∂q

∂z

+t

∂σs

∂s

= 0 (16.3)

Thedirectstressesσzandσsproducedirectstrainsεzandεs,whiletheshearstressτinducesashear
strainγ(=γzs=γsz).Weshallnowproceedtoexpressthesestrainsintermsofthethreecomponentsof
thedisplacementofapointinthesectionwall(seeFig.16.2).Ofthesecomponents,vtisatangential
displacementinthexyplaneandistakentobepositiveinthedirectionofincreasings;vnisanormal
displacementinthexyplaneandispositiveoutward;andwisanaxialdisplacementwhichhasbeen
definedpreviouslyinSection15.2.1.Immediately,fromthethirdofEqs.(1.18),wehave

εz=

∂w

∂z

(16.4)