Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

86 CHAPTER 4 Virtual Work and Energy Methods

Fig.4.1

Work done by a force and a moment.

Therefore,weseethattheworkdonebytheforce,F,astheparticlemovesfromAtoA′mayberegarded
aseithertheproductofFandthecomponentof inthedirectionofF(Eq.(4.1))orastheproductof
thecomponentofFinthedirectionof and (Eq.(4.2)).
Now,considerthecouple(puremoment)inFig.4.1(b)andsupposethatthecoupleisgivenasmall
rotationofθradians.TheworkdonebyeachforceFisthenF(a/2)θsothatthetotalworkdone,WC,
bythecoupleis

WC=F

a

2

θ+F

a

2

θ=Faθ

Itfollowsthattheworkdone,WM,bythepuremoment,M,actingonthebarABinFig.4.1(c)asitis
givenasmallrotation,θ,is

WM=Mθ (4.3)
Notethatinthepreceding,theforce,F,andmoment,M,areinpositionbeforethedisplacements
take place and are not the cause of them. Also, in Fig. 4.1(a), the component of parallel to the
directionofFisinthesamedirectionasF;ifithadbeenintheoppositedirection,theworkdonewould
havebeennegative.Thesameargumentappliestotheworkdonebythemoment,M,whereweseein
Fig. 4.1(c) that the rotation,θ,isinthesamesenseasM. Note also that if the displacement, ,had
beenperpendiculartotheforce,F,noworkwouldhavebeendonebyF.
Finally,itshouldberememberedthatworkisascalarquantitysinceitisnotassociatedwithdirection
(inFig.4.1(a)theforceFdoesworkiftheparticleismovedinanydirection).Thus,theworkdoneby
aseriesofforcesisthealgebraicsumoftheworkdonebyeachforce.

4.2 PrincipleofVirtualWork...........................................................................

Theestablishmentoftheprinciplewillbecarriedoutinstages.Firstweshallconsideraparticle,thena
rigidbody,andfinallyadeformablebody,whichisthepracticalapplicationwerequirewhenanalyzing
structures.