Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

94 CHAPTER 4 Virtual Work and Energy Methods

Shear Force

Theshearforce,S,actingonthemembersectioninFig.4.5producesadistributionofverticalshearstress
whichdependsonthegeometryofthecrosssection.However,sincetheelement,δA,isinfinitesimally
small, we may regard the shear stress,τ, as constant over the element. The shear force,δS,onthe
elementisthen

δS=τδA (4.13)

Suppose that the structure is given an arbitrary virtual displacement which produces a virtual shear
strain,γv, at the element. This shear strain represents the angular rotation in a vertical plane of the
elementδA×δxrelativetothelongitudinalcentroidalaxisofthemember.Theverticaldisplacement
atthesectionbeingconsideredis,therefore,γvδx.Theinternalvirtualwork,δwi,S,donebytheshear
force,S,ontheelementallengthofthememberisgivenby

δwi,S=

∫

A

τdAγvδx

Auniformshearstressthroughthecrosssectionofabeammaybeassumedifweallowfortheactual
variation by including a form factor,β[Ref. 1]. The expression for the internal virtual work in the
membermaythenbewrittenas

δwi,S=

∫

A

β

(

S

A

)

dAγvδx

or

δwi,S=βSγvδx (4.14)

Hence,thevirtualworkdonebytheshearforceduringthearbitraryvirtualstraininamemberoflength
Lis

wi,S=β

∫

L

Sγvdx (4.15)

Foralinearlyelasticmember,asinthecaseofaxialforce,wemayexpressthevirtualshearstrain,γv,
intermsofanequivalentvirtualshearforce,Sv:

γv=

τv

G

=

Sv

GA

sothatfromEq.(4.15)

wi,S=β

∫

L

SASv

GA

dx (4.16)

Forastructurecomprisinganumberoflinearlyelasticmembersthetotalinternalwork,Wi,S,doneby
theshearforcesis

Wi,S=

∑

β

∫

L

SASv

GA

dx (4.17)