Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

4.2 Principle of Virtual Work 91

whichgives

RC=W

a

L

whichistheresultthatwouldhavebeenobtainedfromaconsiderationofthemomentequilibriumof
thebeamaboutA.RAfollowsinasimilarmanner.Supposenowthatinsteadofthesingledisplacement

(^) v,C,thecompletebeamisgivenaverticalvirtualdisplacement, (^) v,togetherwithavirtualrotation,
θv,aboutAasshowninFig.4.4(c).Thetotalvirtualwork,Wt,donebytheforcesactingonthebeam
isnowgivenby
Wt=RA (^) v−W(v+aθv)+RC(v+Lθv)=0(iv)
sincethebeamisinequilibrium.RearrangingEq.(iv)
(RA+RC−W)v+(RCL−Wa)θv=0(v)
Equation(v)isvalidforallvaluesof (^) vandθvsothat
RA+RC−W= 0 RCL−Wa= 0
whicharetheequationsofequilibriumwewouldhaveobtainedbyresolvingforcesverticallyandtaking
momentsaboutA.
Itisnotbeingsuggestedherethattheapplicationoftheprinciplesofstaticsshouldbeabandoned
infavoroftheprincipleofvirtualwork.ThepurposeofExample4.1istoillustratetheapplicationof
avirtualdisplacementandthemannerinwhichtheprincipleisused.
4.2.3 Virtual Work in a Deformable Body
In structural analysis, we are not generally concerned with forces acting on a rigid body. Structures
andstructuralmembersdeformunderload,whichmeansthatifweassignavirtualdisplacementtoa
particularpointinastructure,notallpointsinthestructurewillsufferthesamevirtualdisplacementas
wouldbethecaseifthestructurewererigid.Thismeansthatthevirtualworkproducedbytheinternal
forcesisnotzeroasitisintherigidbodycasesincethevirtualworkproducedbytheself-equilibrating
forcesonadjacentparticlesdoesnotcancelout.Thetotalvirtualworkproducedbyapplyingavirtual
displacement to a deformable body acted on by a system of external forces is therefore given by
Eq.(4.6).
Ifthebodyisinequilibriumundertheactionoftheexternalforcesystem,theneveryparticleinthe
bodyisalsoinequilibrium.Therefore,fromtheprincipleofvirtualwork,thevirtualworkdonebythe
forcesactingontheparticleiszeroirrespectiveofwhethertheforcesareexternalorinternal.Itfollows
that, since the virtual work is zero for all particles in the body, it is zero for the complete body and
Eq.(4.6)becomes
We+Wi= 0 (4.8)
Notethatintheprecedingargument,onlytheconditionsofequilibriumandtheconceptofworkare
employed.Therefore,Eq.(4.8)doesnotrequirethedeformablebodytobelinearlyelastic(i.e.,itneed
notobeyHooke’slaw)sothattheprincipleofvirtualworkmaybeappliedtoanybodyorstructurethat
is rigid, elastic, or plastic. The principle does require that displacements, whether real or imaginary,