128 CHAPTER 5 Energy Methods
whereRjand (^) jarethereactionandknowndeflection(atleastintermsofRj)ofthejthsupportpoint
inatotalofmsupports.ThestationaryvalueofCgives
∂C
∂Rj
=
∫
Ldθ∂M
∂Rj− (^) j= 0 (j=1,2,...,m)
producingmsimultaneousequationsforthemunknownreactions.
Theintentionhereisnottosuggestthatcontinuousbeamsarebestormostreadilysolvedbythe
energymethod;themomentdistributionmethodproducesamorerapidsolution,especiallyforbeams
inwhichthedegreeofredundancyislarge.Instead,thepurposeistodemonstratetheversatilityand
power of energy methods in their ready solution of awiderangeof structural problems. A complete
investigationofthisversatilityisimpossiblehereduetorestrictionofspace;infact,wholebookshave
beendevotedtothistopic.Wethereforelimitouranalysistoproblemspeculiartothefieldofaircraft
structures with which we are primarily concerned. The remaining portion of this section is therefore
concernedwiththesolutionofframesandringspossessingvaryingdegreesofredundancy.
The frameworks we considered in the earlier part of this section and in Section 5.3 comprised
members capable of resisting direct forces only. Of a more general type are composite frameworks
in which some or all of the members resist bending and shear loads in addition to direct loads. It
is usual, however, except for the thin-walled structures in Part B of this book, to ignore deflections
produced by shear forces. We only consider, therefore, bending and direct force contributions to the
internalcomplementaryenergyofsuchstructures.Themethodofanalysisisillustratedinthefollowing
example.
Example 5.5
ThesimplysupportedbeamABCshowninFig.5.12isstiffenedbyanarrangementofpin-jointedbars
capableofsustainingaxialloadsonly.Ifthecross-sectionalareaofthebeamisABandthatofthebars
isA,calculatetheforcesinthemembersoftheframeworkassumingthatdisplacementsarecausedby
bendinganddirectforceactiononly.
Fig.5.12
Analysis of a trussed beam by the method of complementary energy.