Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

118 CHAPTER 5 Energy Methods

andthehorizontalmovementofDis

(^) D,h=

880 × 106

1800 × 200000

=2.44mm

whichagreewiththevirtualworksolution(Example4.6).Thepositivevaluesof (^) B,vand (^) D,hindicate
thatthedeflectionsareinthedirectionsofPB,fandPD,f.
Theanalysisofbeamdeflectionproblemsbycomplementaryenergyissimilartothatofpin-jointed
frameworks,exceptthatweassumeinitiallythatdisplacementsarecausedprimarilybybendingaction.
Shearforceeffectsarediscussedlaterinthechapter.Figure5.5showsatip-loadedcantileverofuniform
crosssectionandlengthL.ThetiploadPproducesaverticaldeflection (^) vwhichwewanttofind.
ThetotalcomplementaryenergyCofthesystemisgivenby

C=

∫

L

∫M

0

dθdM−P (^) v (5.12)
inwhich

∫M

0 dθdMisthecomplementaryenergyofanelementδzofthebeam.Thiselementsubtends
anangleδθatitscenterofcurvatureduetotheapplicationofthebendingmomentM.Fromtheprinciple
ofthestationaryvalueofthetotalcomplementaryenergy,

∂C

∂P

=

∫

L

dθ

dM

dP

− (^) v= 0
or
(^) v=

∫

L

dθ

dM

dP

(5.13)

Equation(5.13)isapplicabletoeitheranonlinearoralinearelasticbeam.Toproceedfurther,therefore,
werequiretheload–displacement(M–θ)andbendingmoment–load(M–P)relationships.Itisimma-
terialforthepurposesofthisillustrativeproblemwhetherthesystemislinearornonlinearsincethe

Fig.5.5

Beam deflection by the method of complementary energy.