Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

96 CHAPTER 4 Virtual Work and Energy Methods

sothatforastructurecomprisinganumberofmembersthetotalinternalvirtualwork,Wi,M,produced
bybendingis

Wi,M=

∑∫

L

MAMv

EI

dx (4.21)

Torsion

Theinternalvirtualwork,wi,T,duetotorsionintheparticularcaseofalinearlyelasticcircularsection
barmaybefoundinasimilarmannerandisgivenby

wi,T=

∫

L

TATv

GIo

dx (4.22)

inwhichIoisthepolarsecondmomentofareaofthecrosssectionofthebar(seeExample3.1).For
beamsofnoncircularcrosssection,Ioisreplacedbyatorsionconstant,J,which,formanypractical
beamsectionsisdeterminedempirically.

Hinges

In some cases, it is convenient to impose a virtual rotation,θv, at some point in a structural member
where,say,theactualbendingmomentisMA.TheinternalvirtualworkdonebyMAisthenMAθv(see
Eq.(4.3));physicallythissituationisequivalenttoinsertingahingeatthepoint.

Sign of Internal Virtual Work
So far we have derived expressions for internal work without considering whether it is positive or
negativeinrelationtoexternalvirtualwork.Supposethatthestructuralmember,AB,inFig.4.6(a)is,
say,amemberofatrussandthatitisinequilibriumundertheactionoftwoexternallyappliedaxial
tensileloads,P;clearlytheinternalaxial,thatisnormal,forceatanysectionofthememberisP.
Supposenowthatthememberisgivenavirtualextension,δv,suchthatBmovestoB′.Thenthe
virtualworkdonebytheappliedload,P,ispositive,sincethedisplacement,δv,isinthesamedirection

Fig.4.6

Sign of the internal virtual work in an axially loaded member.