136 CHAPTER 5 Energy Methods
Fig.5.17
Determination of bending moment distribution in an antisymmetrical fuselage frame.
carriedbyeachpartoftheframeatthejunctionwiththestraightmember.Deformationsonlydueto
bendingstrainsneedbetakenintoaccount.
Theloadingisantisymmetricalsothattherearenobendingmomentsornormalforcesontheplane
ofantisymmetry;thereremainthreeshearloads:SA,SD,andSC,asshowninFig.5.17(b).Thetotal
complementaryenergyofthehalf-frameisthen(neglectingshearstrains)
C=
∫
half-frame∫M
0dθdM−M 0 αB−M 0
r(^) B (i)
whereαBand (^) BaretherotationanddeflectionoftheframeatBcausedbytheappliedmomentM 0 and
concentratedloadM 0 /r,respectively.Fromantisymmetry,thereisnodeflectionatA,D,orCsothat
SA,SD,andSCmakenocontributiontothetotalcomplementaryenergy.Inaddition,overallequilibrium
ofthehalf-framegives
SA+SD+SC=
M 0
r(ii)Assigning stationary values to the total complementary energy and considering the half-frame only,
wehave
∂C
∂SA=
∫
half-framedθ∂M
∂SA
= 0
and
∂C
∂SD=
∫
half-framedθ