CHAPTER 6 Matrix Methods...........................................................................
Actualaircraftstructuresconsistofnumerouscomponentsgenerallyarrangedinanirregularmanner.
These components are usually continuous and therefore theoretically possess an infinite number of
degreesoffreedomandredundancies.Analysisisthenonlypossibleiftheactualstructureisreplaced
by an idealized approximation or model. This procedure is discussed to some extent in Chapter 19,
wherewenotethatthegreaterthesimplificationintroducedbytheidealization,thelesscomplexbut
more inaccurate the analysis becomes. In aircraft design, where structural weight is of paramount
importance,anaccurateknowledgeofcomponentloadsandstressesisessentialsothatatsomestage
inthedesignthesemustbecalculatedasaccuratelyaspossible.Thisaccuracymayonlybeachievedby
consideringanidealizedstructurewhichcloselyrepresentstheactualstructure.Standardmethodsof
structuralanalysisareinadequateforcopingwiththenecessarydegreeofcomplexityinsuchidealized
structures.Itwasthissituationwhichled,inthelate1940sandearly1950s,tothedevelopmentofmatrix
methodsofanalysisandatthesametimetotheemergenceofhigh-speed,electronic,digitalcomputers.
Conveniently,matrixmethodsareideallysuitedforexpressingstructuraltheoryandforexpressingthe
theoryinaformsuitablefornumericalsolutionbycomputer.
Astructuralproblemmaybeformulatedineitheroftwodifferentways.Oneapproachproceedswith
thedisplacementsofthestructureastheunknowns,theinternalforcesthenfollowfromthedetermination
ofthesedisplacements,whileinthealternativeapproach,forcesaretreatedasbeinginitiallyunknown.
Inthelanguageofmatrixmethods,thesetwoapproachesareknownasthestiffness(ordisplacement)
methodandtheflexibility(orforce)method,respectively.Themostwidelyusedofthesetwomethods
isthestiffnessmethod,andforthisreason,weshallconcentrateonthisparticularapproach.Argyris
andKelsey[Ref.1],however,showedthatcompletedualityexistsbetweenthetwomethodsinthatthe
formofthegoverningequationsisthesamewhethertheyareexpressedintermsofdisplacementsor
forces.
Generally, actual structures must be idealized to some extent before they become amenable to
analysis. Examples of some simple idealizations and their effect on structural analysis are presented
inChapter19foraircraftstructures.Outsidetherealmsofaeronauticalengineering,therepresentation
ofatrussgirderbyapin-jointedframeworkisawell-knownexampleoftheidealizationofwhatare
knownas“skeletal”structures.Suchstructuresareassumedtoconsistofanumberofelementsjoined
atpointscallednodes.Thebehaviorofeachelementmaybedeterminedbybasicmethodsofstructural
analysis, and hence, the behavior of the complete structure is obtained by superposition. Operations
suchastheseareeasilycarriedoutbymatrixmethods,asweshallseelaterinthischapter.
Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00006-3 169