Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

148 CHAPTER 5 Energy Methods

ofalltheloadsis

V=

∑n

r= 1

Vr=

∑n

r= 1

(−Pr (^) r)
andtheTPEofthesystemisgivenby

TPE=U+V=U+

∑n

r= 1

(−Pr (^) r) (5.22)

5.8 ThePrincipleoftheStationaryValueoftheTotalPotentialEnergy...........................

Letusnowconsideranelasticbodyinequilibriumunderaseriesofexternalloads,P 1 ,P 2 ,...,Pn,and
supposethatweimposesmallvirtualdisplacementsδ 1 ,δ 2 ,...,δninthedirectionsoftheloads.
Thevirtualworkdonebytheloadsisthen

∑n

r= 1

Prδr

This work will be accompanied by an increment of strain energyδUin the elastic body, since by
specifying virtual displacements of the loads we automatically impose virtual displacements on the
particlesofthebodyitself,asthebodyiscontinuousandisassumedtoremainso.Thisincrementin
strainenergymayberegardedasnegativevirtualworkdonebytheparticlessothatthetotalworkdone
duringthevirtualdisplacementis

−δU+

∑n

r= 1

Prδr

The body is in equilibrium under the applied loads so that by the principle of virtual work the
precedingexpressionmustbeequaltozero.Hence

δU−

∑n

r= 1

Prδr= 0 (5.23)

TheloadsPrremainconstantduringthevirtualdisplacement;therefore,Eq.(5.23)maybewritten

δU−δ

∑n

r= 1

Pr (^) r= 0
or,fromEq.(5.22)
δ(U+V)= 0 (5.24)
Thus,thetotalpotentialenergyofanelasticsystemhasastationaryvalueforallsmalldisplacements
ifthesystemisinequilibrium.