Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

112 CHAPTER 5 Energy Methods

Fig.5.1

(a) Strain energy of a member subjected to simple tension; (b) load–deflection curve for a nonlinearly elastic
member.

andisclearlyrepresentedbytheareaOBDundertheload–deflectioncurve.Engesser(1889)calledthe
areaOBAabovethecurvethecomplementaryenergyC,andfromFig.5.1(b),

C=

∫P

0

ydP (5.2)

Complementaryenergy,asopposedtostrainenergy,hasnophysicalmeaning,beingpurelyaconvenient
mathematical quantity. However, it is possible to show that complementary energy obeys the law of
conservationofenergyinthetypeofsituationusuallyarisinginengineeringstructuressothatitsuse
asanenergymethodisvalid.
DifferentiationofEqs.(5.1)and(5.2)withrespecttoyandP,respectively,gives
dU
dy

=P

dC

dP

=y

Bearing these relationships in mind, we can now consider the interchangeability of strain and
complementaryenergy.SupposethatthecurveofFig.5.1(b)isrepresentedbythefunction

P=byn

wherethecoefficientbandexponentnareconstants.Then,

U=

∫y

0

Pdy=

1

n

∫P

0

(

P

b

) 1 /n

dP

C=

∫P

0

ydP=n

∫y

0

byndy