Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

138 CHAPTER 5 Energy Methods

Fig.5.18

Distribution of bending moment in frame of Example 5.6.

Infact,thequestionofwhetherastructurepossesseslinearornonlinearcharacteristicsarisesonly
after the initial step of writing down expressions for the total potential or complementary energies.
However,agreatnumberofstructuresarelinearlyelasticandpossessuniquepropertieswhichenable
solutions,insomecases,tobemoreeasilyobtained.Theremainderofthischapterisdevotedtothese
methods.

5.5 UnitLoadMethod...................................................................................

InSection5.3,wediscussedthedummyorfictitiousloadmethodofobtainingdeflectionsofstructures.
Foralinearlyelasticstructure,themethodmaybestreamlinedasfollows.Considertheframeworkof
Fig.5.3inwhichwerequire,say,tofindtheverticaldeflectionofthepointC.Followingtheprocedure
ofSection5.3,wewouldplaceaverticaldummyloadPfatCandwritedownthetotalcomplementary
energyoftheframework,thatis,

C=

∑k

i= 1

∫Fi

0

λidFi−

∑n

r= 1

(^) rPr (seeEq.(5.9))
ForastationaryvalueofC,

∂C

∂Pf

=

∑k

i= 1

λi

∂Fi

∂Pf

− (^) C= 0 (5.18)