Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

CHAPTER 5 Energy Methods..........................................................................

In Chapter 2, we saw that the elasticity method of structural analysis embodies the determination of
stressesand/ordisplacementsbyusingequationsofequilibriumandcompatibilityinconjunctionwith
therelevantforce–displacementorstress–strainrelationships.Inaddition,inChapter4,weinvestigated
the use of virtual work in calculating forces, reactions, and displacements in structural systems. A
powerful alternative but equally fundamental approach is the use of energy methods. These, while
providingexactsolutionsformanystructuralproblems,findtheirgreatestuseintherapidapproximate
solutionofproblemsforwhichexactsolutionsdonotexist.Also,manystructureswhicharestatically
indeterminate—inotherwords,theycannotbeanalyzedbytheapplicationoftheequationsofstatical
equilibriumalone—maybeconvenientlyanalyzedusinganenergyapproach.Further,energymethods
providecomparativelysimplesolutionsfordeflectionproblemswhicharenotreadilysolvedbymore
elementarymeans.
Generally,asweshallsee,modernanalysis[Ref.1]usesthemethodsoftotalcomplementaryenergy
andtotalpotentialenergy(TPE).Eithermethodmaybeusedtosolveaparticularproblem,althoughas
ageneralruledeflectionsaremoreeasilyfoundusingcomplementaryenergyandforcesbypotential
energy.
Although energy methods are applicable to a wide range of structural problems and may even
be used as indirect methods of forming equations of equilibrium or compatibility [Refs. 1, 2], we
shallbeconcernedinthischapterwiththesolutionofdeflectionproblemsandtheanalysisofstatically
indeterminatestructures.Weshallalsoincludesomemethodsrestrictedtothesolutionoflinearsystems:
theunitloadmethod,theprincipleofsuperposition,andthereciprocaltheorem.

5.1 StrainEnergyandComplementaryEnergy.......................................................

Figure 5.1(a) shows a structural member subjected to a steadily increasing loadP. As the member
extends,theloadPdoeswork,andfromthelawofconservationofenergy,thisworkisstoredinthe
member asstrain energy. A typical load–deflection curve for a member possessing nonlinear elastic
characteristicsisshowninFig.5.1(b).ThestrainenergyUproducedbyaloadPandcorresponding
extensionyisthen

U=

∫y

0

Pdy (5.1)

Copyright©2010,T.H.G.Megson. PublishedbyElsevierLtd. Allrightsreserved.
DOI:10.1016/B978-1-85617-932-4.00005-1 111