Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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15.3 Deflections due to Bending 455

Fig.15.24


Determination of the deflection of a cantilever.


Example 15.12
Determine the horizontal and vertical components of the tip deflection of the cantilever shown in
Fig.15.24.ThesecondmomentsofareaofitsunsymmetricalsectionareIxx,Iyy,andIxy.


FromEqs.(15.29)

u′′=

MxIxy−MyIxx
E(IxxIyy−Ixy^2 )

(i)

Inthiscase,Mx=W(L−z),My=0sothatEq.(i)simplifiesto


u′′=

WIxy
E(IxxIyy−Ixy^2 )

(L−z) (ii)

IntegratingEq.(ii)withrespecttoz,


u′=

WIxy
E(IxxIyy−Ixy^2 )

(

Lz−
z^2
2

+A

)

(iii)

and


u=

WIxy
E(IxxIyy−Ixy^2 )

(

L

z^2
2


z^3
6

+Az+B

)

(iv)

in whichu′denotesdu/dzand the constants of integrationAandBare found from the boundary
conditions;thatis,u′=0andu=0whenz=0.FromthefirstoftheseandEq.(iii),A=0,whilefrom
thesecondandEq.(iv),B=0.Hence,thedeflectedshapeofthebeaminthexzplaneisgivenby


u=

WIxy
E(IxxIyy−Ixy^2 )

(

L

z^2
2


z^3
6

)

(v)
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