Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.4Application to the Solution of Statically Indeterminate Systems 133

Fig.5.15

Distribution of bending moment in a doubly symmetric ring.

normalforceNA.Weproceed,asinthepreviousexample,bywritingdownthetotalcomplementary
energyCofthesystem.Then,neglectingshearstrains

C=

∫

ring

∫M

0

dθdM−P (i)

inwhich isthedeflectionofthepointofapplicationofPrelativetothetopoftheframe.NotethatMA
andNAdonotcontributetothecomplementofthepotentialenergyofthesystem,since,bysymmetry,
therotationandhorizontaldisplacementsatAarezero.Fromtheprincipleofthestationaryvalueof
thetotalcomplementaryenergy,

∂C

∂MA

=

∫

ring

dθ

∂M

∂MA

=0(ii)

and

∂C

∂NA

=

∫

ring

dθ

∂M

∂NA

=0(iii)

Thebendingmomentataradialsectioninclinedatanangleθtotheverticaldiameteris,fromFig.5.16(c),

M=MA+NAR( 1 −cosθ)+

∫θ

0

qBDRdα

or

M=MA+NAR( 1 −cosθ)+

∫θ

0

P

πR

sinα[R−Rcos(θ−α)]Rdα