Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

114 CHAPTER 5 Energy Methods

Thefirsttermintheprecedingexpressionisthenegativevirtualworkdonebytheparticlesintheelastic
body,whilethesecondtermrepresentsthevirtualworkoftheexternallyappliedvirtualforces.From
theprincipleofvirtualwork,

−

∫

vol

ydP+

∑n

r= 1

(^) rδPr= 0 (5.6)
ComparingEq.(5.6)withEq.(5.2),weseethateachtermrepresentsanincrementincomplementary
energy—the first, of the internal forces, and the second, of the external loads. Equation (5.6) may
thereforeberewrittenas
δ(Ci+Ce)= 0 (5.7)
where
Ci=

∫

vol

∫P

0

ydP and Ce=−

∑n

r= 1

(^) rPr (5.8)
Weshallnowcallthequantity(Ci+Ce)thetotalcomplementaryenergyCofthesystem.
ThedisplacementsspecifiedinEq.(5.6)arerealdisplacementsofacontinuouselasticbody;they
thereforeobeytheconditionofcompatibilityofdisplacementsothatEqs.(5.6)and(5.7)areequations
ofgeometricalcompatibility.Theprincipleofthestationaryvalueofthetotalcomplementaryenergy
maythenbestatedasfollows:
For an elastic body in equilibrium under the action of applied forces the true internal forces (or
stresses)andreactionsarethoseforwhichthetotalcomplementaryenergyhasastationaryvalue.
Inotherwords,thetrueinternalforces(orstresses)andreactionsarethosewhichsatisfythecondition
ofcompatibilityofdisplacement.Thispropertyofthetotalcomplementaryenergyofanelasticsystem
isparticularlyusefulinthesolutionofstaticallyindeterminatestructures,inwhichaninfinitenumber
ofstressdistributionsandreactiveforcesmaybefoundtosatisfytherequirementsofequilibrium.

5.3 APPLICATION TO DEFLECTION PROBLEMS

Generally,deflectionproblemsaremostreadilysolvedbythecomplementaryenergyapproach,although
forlinearlyelasticsystemsthereisnodifferencebetweenthemethodsofcomplementaryandpotential
energy,since,aswehaveseen,complementaryandstrainenergythenbecomecompletelyinterchange-
able.Weshallillustratethemethodbyreferencetothedeflectionsofframesandbeamswhichmayor
maynotpossesslinearelasticity.
Let us suppose that we want to find the deflection 2 of the loadP 2 in the simple pin-jointed
framework consisting, say, ofkmembers and supporting loadsP 1 ,P 2 ,...,Pn, as shown in Fig. 5.3.
FromEqs.(5.8),thetotalcomplementaryenergyoftheframeworkisgivenby

C=

∑k

i= 1

∫Fi

0

λidFi−

∑n

r= 1

(^) rPr (5.9)