CHAPTER 4 Virtual Work and Energy Methods.....................................................
Manystructuralproblemsarestaticallydeterminate;inotherwords,thesupportreactionsandinternal
forcesystemsmaybefoundusingsimplestaticswherethenumberofunknownsisequaltothenumberof
equationsofequilibriumavailable.Incaseswherethenumberofunknownsexceedsthepossiblenumber
ofequationsofequilibrium—forexample,aproppedcantileverbeam—othermethodsofanalysisare
required.
Themethodsfallintotwocategoriesandarebasedontwoimportantconcepts;thefirst,whichis
presentedinthischapter,istheprincipleofvirtualwork.Thisisthemostfundamentalandpowerful
tool available for the analysis of statically indeterminate structures and has the advantage of being
abletodealwithconditionsotherthanthoseintheelasticrange.Thesecond,basedonstrainenergy,
canprovideapproximatesolutionsofcomplexproblemsforwhichexactsolutionsdonotexistandis
discussedinChapter5.Insomecases,thetwomethodsareequivalent,since,althoughthegoverning
equationsdiffer,theequationsthemselvesareidentical.
Inmodernstructuralanalysis,computer-basedtechniquesarewidelyused;theseincludetheflexi-
bilityandstiffnessmethods(seeChapter6).However,theformulationof,say,stiffnessmatricesforthe
elementsofacomplexstructureisbasedononeoftheprecedingapproachessothataknowledgeand
understandingoftheirapplicationisadvantageous.
4.1 Work...................................................................................................
Beforeweconsidertheprincipleofvirtualworkindetail,itisimportanttoclarifyexactlywhatismeant
bywork. The basic definition of work in elementary mechanics is that “work is done when a force
movesitspointofapplication”.However,weshallrequireamoreexactdefinition,sinceweshallbe
concernedwithworkdonebybothforcesandmomentsandwiththeworkdonebyaforcewhenthe
body on which it acts is given a displacement, which is not coincident with the line of action of the
force.
Considertheforce,F,actingonaparticle,A,inFig.4.1(a).Iftheparticleisgivenadisplacement, ,
bysomeexternalagencysothatitmovestoA′inadirectionatanangleαtothelineofactionofF,the
work,WF,donebyFisgivenby
WF=F( cosα) (4.1)or
WF=(Fcosα) (4.2)Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00004-X 85