Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

456 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams

Atthefreeendofthecantilever(z=L),thehorizontalcomponentofdeflectionis

uf.e.=

WIxyL^3

3 E(IxxIyy−I^2 xy)

(vi)

Similarly,theverticalcomponentofthedeflectionatthefreeendofthecantileveris

vf.e.=

−WIyyL^3

3 E(IxxIyy−Ixy^2 )

(vii)

Theactualdeflectionδf.e.atthefreeendisthengivenby

δf.e.=(u^2 f.e.+v^2 f.e.)

(^12)
atanangleoftan−^1 uf.e./vf.e.tothevertical.
NotethatifeitherCxorCywereanaxisofsymmetry,Ixy=0andEqs.(vi)and(vii)reduceto
uf.e.= 0 vf.e.=

−WL^3

3 EIxx

thewell-knownresultsforthebendingofacantileverhavingasymmetricalcrosssectionandcarrying
aconcentratedverticalloadatitsfreeend(seeExample15.5).

15.4 CalculationofSectionProperties..................................................................

Itwillbehelpfulatthisstagetodiscussthecalculationofthevarioussectionpropertiesrequiredinthe
analysisofbeamssubjectedtobending.Initially,however,twousefultheoremsarequoted.

15.4.1 Parallel Axes Theorem

ConsiderthebeamsectionshowninFig.15.25andsupposethatthesecondmomentofarea,IC,about
anaxisthroughitscentroidCisknown.Thesecondmomentofarea,IN,aboutaparallelaxis,NN,a
distancebfromthecentroidalaxisisthengivenby

IN=IC+Ab^2 (15.33)

Fig.15.25

Parallel axes theorem.