Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

4.2 Principle of Virtual Work 87

4.2.1 Principle of Virtual Work for a Particle

InFig.4.2,aparticle,A,isactedonbyanumberofconcurrentforces,F 1 ,F 2 ,...,Fk,...,Fr;theresultant

oftheseforcesisR.Supposethattheparticleisgivenasmallarbitrarydisplacement, (^) v,toA′insome
specified direction; (^) vis an imaginary orvirtualdisplacement and is sufficiently small so that the
directionsofF 1 ,F 2 ,andsoonareunchanged.LetθRbetheanglethattheresultant,R,oftheforces
makeswiththedirectionof (^) vandθ 1 ,θ 2 ,...,θk,...,θrtheanglesthatF 1 ,F 2 ,...,Fk,...,Frmakewith
thedirectionof (^) v,respectively.Then,fromeitherofEqs.(4.1)or (4.2),thetotalvirtualwork,WF,
donebytheforcesFastheparticlemovesthroughthevirtualdisplacement, (^) v,isgivenby
WF=F 1 vcosθ 1 +F 2 vcosθ 2 +···+Fk (^) vcosθk+···+Fr (^) vcosθr
Thus,

WF=

∑r

k= 1

Fk (^) vcosθk
or,since (^) visafixed,althoughimaginarydisplacement,
WF= (^) v
∑r
k= 1
Fkcosθk (4.4)
InEq.(4.4),
∑r
k= 1 Fkcosθkisthesumofallthecomponentsoftheforces,F,inthedirectionof
(^) vandthereforemustbeequaltothecomponentoftheresultant,R,oftheforces,F,inthedirection
of (^) v;thatis,
WF= (^) v
∑r
k= 1
Fkcosθk= (^) vRcosθR (4.5)
Fig.4.2
Virtual work for a system of forces acting on a particle.