Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

98 CHAPTER 4 Virtual Work and Energy Methods

Forastructurecomprisinganumberofloadpositions,thetotalexternalvirtualworkdoneisthen

We=

∑

⎡

⎣W (^) v,y+P (^) v,x+Mθv+Tφv+

∫

L

w(x)v,ydx

⎤

⎦ (4.24)

InEq.(4.24),thereneednotbeacompletesetofexternalloadsappliedateveryloadingpoint,soin
fact the summation is for the appropriate number of loads. Further, the virtual displacements in the
preceding are related to forces and moments applied in a vertical plane. We could, of course, have
forcesandmomentsandcomponentsofthevirtualdisplacementinahorizontalplane,inwhichcase
Eq.(4.24)wouldbeextendedtoincludetheircontribution.
TheinternalvirtualworkequivalentofEq.(4.24)foralinearsystemis,fromEqs.(4.12),(4.17),
(4.21),and(4.22)

Wi=

∑

⎡

⎣

∫

L

NANv

EA

dx+β

∫

L

SASv

GA

dx+

∫

L

MAMv

EI

dx+

∫

L

TATv

GJ

dx+MAθv

⎤

⎦

(4.25)

inwhichthelasttermontheright-handsideisthevirtualworkproducedbyanactualinternalmoment
atahinge(seepreceding).NotethatthesummationinEq.(4.25)istakenoverallthemembersofthe
structure.

4.2.6 Use of Virtual Force Systems

Sofar,inallthestructuralsystemswehaveconsidered,virtualworkhasbeenproducedbyactualforces
movingthroughimposedvirtualdisplacements.However,theactualforcesarenotrelatedtothevirtual
displacementsinanyway,since,aswehaveseen,themagnitudesanddirectionsoftheactualforcesare
unchangedbythevirtualdisplacementssolongasthedisplacementsaresmall.Thus,theprincipleof
virtualworkappliesforanysetofforcesinequilibriumandanysetofdisplacements.Equally,therefore,
wecouldspecifythattheforcesareasetofvirtualforcesinequilibriumandthatthedisplacementsare
actualdisplacements.Therefore,insteadofrelatingactualexternalandinternalforcesystemsthrough
virtualdisplacements,wecanrelateactualexternalandinternaldisplacementsthroughvirtualforces.
Ifweapplyavirtualforcesystemtoadeformablebody,itwillinduceaninternalvirtualforcesystem
whichwillmovethroughtheactualdisplacements;internalvirtualworkwillthereforebeproduced.In
thiscase,forexample,Eq.(4.10)becomes

wi,N=

∫

L

NvεAdx

inwhichNvistheinternalvirtualnormalforceandεAistheactualstrain.Then,foralinearsystem,in
whichtheactualinternalnormalforceisNA,εA=NA/EA,sothatforastructurecomprisinganumber
ofmembers,thetotalinternalvirtualworkduetoavirtualnormalforceis

Wi,N=

∑∫

L

NvNA

EA

dx

whichisidenticaltoEq.(4.12).Equations(4.17),(4.21),and(4.22)maybeshowntoapplytovirtual
forcesystemsinasimilarmanner.