Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.2 The Principle of the Stationary Value 113

Fig.5.2

Load–deflection curve for a linearly elastic member.

Hence,

dU

dy

=P

dU

dP

=

1

n

(

P

b

) 1 /n

=

1

n

y (5.3)

dC

dP

=y

dC

dy

=bnyn=nP (5.4)

Whenn=1,

dU

dy

=

dC

dy

=P

dU

dP

=

dC

dP

=y

⎫

⎪⎪

⎬

⎪⎪

⎭

(5.5)

andthestrainandcomplementaryenergiesarecompletelyinterchangeable.Suchaconditionisfoundin
alinearlyelasticmember;itsrelatedload–deflectioncurveisshowninFig.5.2.Clearly,areaOBD(U)
isequaltoareaOBA(C).
It will be observed that the latter of Eqs. (5.5) is in the form of what is commonly known as
Castigliano’sfirsttheorem,inwhichthedifferentialofthestrainenergyUofastructurewithrespectto
aloadisequatedtothedeflectionoftheload.Tobemathematicallycorrect,however,itisthedifferential
ofthecomplementaryenergyCwhichshouldbeequatedtodeflection(compareEqs.(5.3)and(5.4)).

5.2 ThePrincipleoftheStationaryValueoftheTotalComplementaryEnergy..................

ConsideranelasticsysteminequilibriumsupportingforcesP 1 ,P 2 ,...,Pnwhichproducerealcorre-

spondingdisplacements 1 , 2 ,..., (^) n.IfweimposevirtualforcesδP 1 ,δP 2 ,...,δPnonthesystem
actingthroughtherealdisplacements,thenthetotalvirtualworkdonebythesystem(seeChapter4)is

−

∫

vol

ydP+

∑n

r= 1

(^) rδPr