Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

102 CHAPTER 4 Virtual Work and Energy Methods

TheactualsignofFBAisthengovernedbythesignoftheexternalvirtualwork.Thus,ifthe30kN

load had been in the opposite direction to (^) C, the external work done would have been negative, so
thatFBAwould be negative and therefore compressive. This situation can be verified by inspection.
Alternatively,fortheloadingasshowninFig.4.9(a),acontractioninABwouldhaveimpliedthatFBA
wascompressive.Inthiscase,DCwouldhaverotatedinananticlockwisesense,and (^) Cwouldhave
beenintheoppositedirectiontothe30kNloadsothattheexternalvirtualworkdonewouldbenegative,
resultinginanegativevalueforthecompressiveforceFBA;FBAwouldthereforebetensileasbefore.
Notealsothatthe10kNloadatDdoesnowork,sinceDremainsundisplaced.
Weshallnowconsiderproblemsinvolvingtheuseofvirtualforces.Generally,weshallrequirethe
displacementofaparticularpointinastructure,sothatifweapplyavirtualforcetothestructureatthe
pointandinthedirectionoftherequireddisplacement,theexternalvirtualworkdonewillbetheproduct
ofthevirtualforceandtheactualdisplacement,whichmaythenbeequatedtotheinternalvirtualwork
producedbytheinternalvirtualforcesystemmovingthroughactualdisplacements.Sincethechoice
ofthevirtualforceisarbitrary,wemaygiveitanyconvenientvalue;thesimplesttypeofvirtualforce
isthereforeaunitload,andthemethodthenbecomestheunitloadmethod(seealsoSection5.5).
Example 4.4
DeterminetheverticaldeflectionofthefreeendofthecantileverbeamshowninFig.4.10(a).
LetussupposethattheactualdeflectionofthecantileveratBproducedbytheuniformlydistributed
loadisυBandthataverticallydownwardvirtualunitloadwasappliedatBbeforetheactualdeflection
Fig.4.10
Deflection of the free end of a cantilever beam using the unit load method.