92 CHAPTER 4 Virtual Work and Energy Methods
mustbesmall,sothatwemayassumethatexternalandinternalforcesareunchangedinmagnitudeand
directionduringthedisplacements.Inaddition,thevirtualdisplacementsmustbecompatiblewiththe
geometryofthestructureandtheconstraintsthatareapplied,suchasthoseatasupport.Theexceptionis
thesituationwehaveinExample4.1,whereweapplyavirtualdisplacementatasupport.Thisapproach
isvalidsinceweincludetheworkdonebythesupportreactionsinthetotalvirtualworkequation.
4.2.4 Work Done by Internal Force Systems
The calculation of the work done by an external force is straightforward in that it is the product of
theforceandthedisplacementofitspointofapplicationinitsownlineofaction(Eqs.(4.1),(4.2),or
(4.3)),whereasthecalculationoftheworkdonebyaninternalforcesystemduringadisplacementis
muchmorecomplicated.Generally,nomatterhowcomplexaloadingsystemis,itmaybesimplified
toacombinationofuptofourloadtypes:axialload,shearforce,bendingmoment,andtorsion;these
inturnproducecorrespondinginternalforcesystems.Weshallnowconsidertheworkdonebythese
internalforcesystemsduringarbitraryvirtualdisplacements.
Axial Force
Considertheelementallength,δx,ofastructuralmemberasshowninFig.4.5andsupposethatitis
subjected to a positive internal force system comprising a normal force (i.e., axial force),N; a shear
Fig.4.5
Virtual work due to internal force system.