Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

166 CHAPTER 5 Energy Methods

TheframeisloadedinitsownplanebyasystemofpointloadsPwhicharebalancedbyaconstantshearflowq
aroundtheoutside.Determinethedistributionofthebendingmomentintheframeandsketchthebendingmoment
diagram.Intheanalysis,takebendingdeformationsonlyintoaccount.

Ans. Shears only at midpoints of vertical members. On the lower half of the frame,S 43 =0.27Pto right,

S 52 =0.69Ptoleft,S 61 =1.08Ptoleft;thebendingmomentdiagramfollows.

P.5.19 AcircularfuselageframeshowninFig.P.5.19,ofradiusrandconstantbendingstiffnessEI,hasastraight
floorbeamoflengthr

√
2,bendingstiffnessEI,rigidlyfixedtotheframeateitherend.Theframeisloadedbya
coupleTappliedatitslowestpointandaconstantequilibratingshearflowqarounditsperiphery.Determinethe
distributionofthebendingmomentintheframe,illustratingyouranswerbymeansofasketch.
Intheanalysis,deformationsduetoshearandendloadmaybeconsiderednegligible.Thedepthoftheframe
crosssectionincomparisonwiththeradiusrmayalsobeneglected.

Ans. M 14 =T(0.29sinθ−0.16θ),M 24 =0.30Tx/r,M 43 =T(0.59sinθ−0.16θ).

Fig. P.5.19

P.5.20 Athin-walledmemberBCDisrigidlybuilt-inatDandsimplysupportedatthesamelevelatC,asshown
inFig.P.5.20.

Fig. P.5.20

FindthehorizontaldeflectionatBduetothehorizontalforceF.Fullaccountmustbetakenofdeformations
duetoshearanddirectstrains,aswellastobending.
Thememberisofuniformcrosssection,ofareaA,relevantsecondmomentofareainbendingI=Ar^2 / 400
and“reduced”effectiveareainshearingA′=A/4.Poisson’sratioforthematerialisν=1/3.
GivetheanswerintermsofF,r,A,andYoung’smodulusE.
Ans. 448 Fr/EA.