Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

290 CHAPTER 8 Columns

Fig. P.8.9

where

α=P/Pe, Pe=π^2 EI/l^2

Assumetissmallcomparedwithdsothatthefollowingrelationshipsareapplicable:

• Cross-sectionalareaoftube=πdt.


• Secondmomentofareaoftube=πd^3 t/8.

P.8.10 FigureP.8.10illustratesanidealizedrepresentationofpartofanaircraftcontrolcircuit.Auniform,straight
baroflengthaandflexuralstiffnessEIisbuiltinattheendAandhingedatBtoalinkBC,oflengthb,whose
otherendCispinnedsothatitisfreetoslidealongthelineABCbetweensmooth,rigidguides.A,B,andCare
initiallyinastraightline,andthesystemcarriesacompressionforceP,asshown.

Fig. P.8.10

AssumingthatthelinkBChasasufficientlyhighflexuralstiffnesstopreventitsbucklingasapin-endedstrut,
show,bysettingupandsolvingthedifferentialequationforflexureofAB,thatbucklingofthesystem,ofthetype
illustratedinFig.P.8.10,occurswhenPhassuchavaluethat

tanλa=λ(a+b)

where

λ^2 =P/EI

P.8.11 Apin-endedcolumnoflengthlhasitscentralportionreinforced,thesecondmomentofitsareabeingI 2 ,
whilethatoftheendportions,eachoflengtha,isI 1 .Usetheenergymethodtodeterminethecriticalloadofthe
column, assuming that its center-line deflects into the parabolav=kz(l−z)and taking the more accurate of the
twoexpressionsforthebendingmoment.
In the case whereI 2 =1.6I 1 anda=0.2l, find the percentage increase in strength due to reinforcement, and
compareitwiththepercentageincreaseinweightonthebasisthatthesection’sradiusofgyrationisnotaltered.

Ans. PCR=14.96EI 1 /l^2 ,52%,36%.