Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

132 CHAPTER 5 Energy Methods

butthedirectloadshaveavalueP/2.Thetotalcomplementaryenergyofthesystemis(againignoring
shearstrains)

C=

∫

ring

∫M

0

dθdM− 2

(

P

2

)

taking the bending moment as positive when it increases the curvature of the ring. In the preceding
expressionforC, isthedisplacementofthetop,A,oftheringrelativetothebottom,B.Assigninga
stationaryvaluetoC,wehave

∂C

∂MA

=

∫

ring

dθ

∂M

∂MA

= 0

orassuminglinearelasticityandconsidering,fromsymmetry,halfthering

∫πR

0

M

EI

∂M

∂MA

ds= 0

Thus,since

M=MA−

P

2

Rsinθ

∂M

∂MA

= 1

andwehave

∫π

0

(

MA−

P

2

Rsinθ

)

Rdθ= 0

or
[
MAθ+

P

2

Rcosθ

]π

0

= 0

fromwhich

MA=

PR

π

Thebendingmomentdistributionisthen

M=PR

(

1

π

−

sinθ

2

)

andisshowndiagrammaticallyinFig.5.15.
Letusnowconsideramorerepresentativeaircraftstructuralproblem.Thecircularfuselageframe
ofFig.5.16(a)supportsaloadPwhichisreactedbyashearflowq(i.e.,ashearforceperunitlength:
seeChapter15),distributedaroundthecircumferenceoftheframefromthefuselageskin.Thevalue
anddirectionofthisshearflowarequotedherebutarederivedfromtheoryestablishedinSection15.3.
From our previous remarks on the effect of symmetry, we observe that there is no shear force at the
sectionAontheverticalplaneofsymmetry.TheunknownsarethereforethebendingmomentMAand