Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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)