4.2 Principle of Virtual Work 89internalforceswillbeequalandopposite,inotherwordsself-equilibrating.Supposenowthattherigid
bodyisgivenasmall,imaginary—thatis,virtual—displacement, (^) v(orarotationoracombinationof
both),insomespecifieddirection.Theexternalandinternalforcesthendovirtualwork,andthetotal
virtualworkdone,Wt,isthesumofthevirtualwork,We,donebytheexternalforcesandthevirtual
work,Wi,donebytheinternalforces.Thus,
Wt=We+Wi (4.6)
Sincethebodyisrigid,alltheparticlesinthebodymovethroughthesamedisplacement, (^) v,sothatthe
virtualworkdoneonalltheparticlesisnumericallythesame.However,forapairofadjacentparticles,
suchasA 1 andA 2 inFig.4.3,theself-equilibratingforcesareinoppositedirections,whichmeansthat
theworkdoneonA 1 isoppositeinsigntotheworkdoneonA 2 .Therefore,thesumofthevirtualwork
doneonA 1 andA 2 iszero.Theargumentcanbeextendedtotheinfinitenumberofpairsofparticlesin
thebodyfromwhichweconcludethattheinternalvirtualworkproducedbyavirtualdisplacementin
arigidbodyiszero.Equation(4.6)thenreducesto
Wt=We (4.7)
Since the body is rigid and the internal virtual work is therefore zero, we may regard the body as
a large particle. It follows that if the body is in equilibrium under the action of a set of forces,
F 1 ,F 2 ,...,Fk,...,Fr, the total virtual work done by the external forces during an arbitrary virtual
displacementofthebodyiszero.
Example 4.1
CalculatethesupportreactionsinthesimplysupportedbeamshowninFig.4.4.
Onlyaverticalloadisappliedtothebeamsothatonlyverticalreactions,RAandRC,areproduced.
SupposethatthebeamatCisgivenasmallimaginary—thatis,avirtual—displacement, (^) v,C,in
thedirectionofRCasshowninFig.4.4(b).Sinceweareconcernedheresolelywiththeexternalforces
actingonthebeam,wemayregardthebeamasarigidbody.Therefore,thebeamrotatesaboutAso
thatCmovestoC′andBmovestoB′.Fromsimilartriangles,weseethat
(^) v,B=
a
a+b
(^) v,C=
a
L
(^) v,C (i)
Thetotalvirtualwork,Wt,donebyalltheforcesactingonthebeamisthengivenby
Wt=RC (^) v,C−W (^) v,B (ii)
Notethattheworkdonebytheload,W,isnegative,since (^) v,Bisintheoppositedirectiontoitslineof
action.Notealsothatthesupportreaction,RA,doesnoworksincethebeamonlyrotatesaboutA.Now
substitutingfor (^) v,BinEq.(ii)fromEq.(i),wehave
Wt=RC (^) v,C−W
a
L
(^) v,C (iii)
Sincethebeamisinequilibrium,Wtiszerofromtheprincipalofvirtualwork.Hence,fromEq.(iii)
RC (^) v,C−W
a
L
(^) v,C= 0