88 CHAPTER 4 Virtual Work and Energy Methods
Iftheparticle,A,isinequilibriumundertheactionoftheforces,F 1 ,F 2 ,...,Fk,...,Fr,theresultant,R,
oftheforcesiszero.ItfollowsfromEq.(4.5)thatthevirtualworkdonebytheforces,F,duringthe
virtualdisplacement, (^) v,iszero.
Wecan,therefore,statetheprincipleofvirtualworkforaparticleasfollows:
Ifaparticleisinequilibriumundertheactionofanumberofforces,thetotalworkdonebytheforces
forasmallarbitrarydisplacementoftheparticleiszero.
Itispossibleforthetotalworkdonebytheforcestobezeroeventhoughtheparticleisnotinequilibrium
ifthevirtualdisplacementistakentobeinadirectionperpendiculartotheirresultant,R.Wecannot,
therefore,statetheconverseoftheprecedingprincipleunlesswespecifythatthetotalworkdonemust
bezeroforanyarbitrarydisplacement.Thus:
Aparticleisinequilibriumundertheactionofasystemofforcesifthetotalworkdonebytheforces
iszeroforanyvirtualdisplacementoftheparticle.
Notethatinthepreceding, (^) visapurelyimaginarydisplacementandisnotrelatedinanywaytothe
possibledisplacementoftheparticleundertheactionoftheforces,F. (^) vhasbeenintroducedpurelyas
adeviceforsettingupthework–equilibriumrelationshipofEq.(4.5).Theforces,F,thereforeremain
unchangedinmagnitudeanddirectionduringthisimaginarydisplacement;thiswouldnotbethecase
ifthedisplacementwerereal.
4.2.2 Principle of Virtual Work for a Rigid Body
Consider the rigid body shown in Fig. 4.3, which is acted on by a system of external forces,
F 1 ,F 2 ,...,Fk,...,Fr. These external forces will induce internal forces in the body, which may be
regardedascomprisinganinfinitenumberofparticles;onadjacentparticles,suchasA 1 andA 2 ,these
Fig.4.3
Virtual work for a rigid body.