Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

100 CHAPTER 4 Virtual Work and Energy Methods

ifweimaginethatthebeamishingedatBandthatthelengthsABandBCarerigid,avirtualdisplacement,

(^) v,B,atBwillresultinthedisplacedshapeshowninFig.4.8(b).
NotethatthesupportreactionsatAandCdonoworkandthattheinternalmomentsinABandBC
donoworkbecauseABandBCarerigidlinks.FromFig.4.8(b)
(^) v,B=aβ=bα (i)
Hence,
α=
a
b
β
andtheangleofrotationofBCrelativetoABisthen
θB=β+α=β

(

1 +

a

b

)

=

L

b

β (ii)

Now equating the external virtual work done byWto the internal virtual work done byMB(see
Eq.(4.23)),wehave

W (^) v,B=MBθB (iii)
SubstitutinginEq.(iii)for (^) v,BfromEq.(i)andforθBfromEq.(ii),wehave
Waβ=MB

L

b

β

whichgives

MB=

Wab

L

whichistheresultwewouldhaveobtainedbycalculatingthemomentofRC(=Wa/LfromExample4.1)
aboutB.

Example 4.3
DeterminetheforceinthememberABinthetrussshowninFig.4.9(a).

WearerequiredtocalculatetheforceinthememberAB,sothatagainweneedtorelatethisinternal
forcetotheexternallyappliedloadswithoutinvolvingtheinternalforcesintheremainingmembersof

thetruss.Wethereforeimposeavirtualextension, (^) v,B,atBinthememberAB,suchthatBmoves
toB′.Ifweassumethattheremainingmembersarerigid,theforcesinthemwilldonowork.Further,
thetriangleBCDwillrotateasarigidbodyaboutDtoB′C′DasshowninFig.4.9(b).Thehorizontal
displacementofC, (^) C,isthengivenby
(^) C= 4 α
while
(^) v,B= 3 α