Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

4.2 Principle of Virtual Work 97

asitslineofaction.However,thevirtualworkdonebytheinternalforce,N(=P),isnegative,sincethe
displacementofBisintheoppositedirectiontoitslineofaction;inotherwords,workisdoneonthe
member.Thus,fromEq.(4.8),weseethatinthiscase

We=Wi (4.23)

Equation(4.23)wouldapplyifthevirtualdisplacementhadbeenacontractionandnotanextension,
inwhichcasethesignsoftheexternalandinternalvirtualworkinEq.(4.8)wouldhavebeenreversed.
Clearly, the preceding applies equally ifPis a compressive load. The previous arguments may be
extendedtostructuralmemberssubjectedtoshear,bending,andtorsionalloads,sothatEq.(4.23)is
generallyapplicable.

4.2.5 Virtual Work due to External Force Systems

So far in our discussion, we have only considered the virtual work produced by externally applied
concentratedloads.Forcompleteness,wemustalsoconsiderthevirtualworkproducedbymoments,
torques,anddistributedloads.
InFig.4.7,astructuralmembercarriesadistributedload,w(x),andataparticularpointaconcentrated
load,W;amoment,M;andatorque,T.Supposethatatthepointavirtualdisplacementisimposedthat

hastranslationalcomponents, (^) v,yand (^) v,x,paralleltotheyandxaxes,respectively,androtational
components,θvandφv,intheyxandzyplanes,respectively.
Ifweconsiderasmallelement,δx,ofthememberatthepoint,thedistributedloadmayberegarded
asconstantoverthelengthδxandacting,ineffect,asaconcentratedloadw(x)δx.Thevirtualwork,
we,donebythecompleteexternalforcesystemisthereforegivenby
we=W (^) v,y+P (^) v,x+Mθv+Tφv+

∫

L

w(x)v,ydx

Fig.4.7

Virtual work due to externally applied loads.