Page 305 - Aircraft Stuctures for Engineering Student

286 Open and closed, thin-walled beams

X

z

Fig. 9.8 Determination of the deflection of a cantilever.

In this case Mx = W(L - z), My = 0 so that Eq. (i) simplifies to
I1
u= Wr, (L - 2) (ii)
mxzyy - I$)
Integrating Eq. (ii) with respect to z
2
ut = wzxy (LZ-T+A) (iii)
WxxIyy - z.$)
and
U= mxzyy - ey) (L$-$+Rz+B 1 (iv)

6
2
wzxy
in which u' denotes du/& and the constants of integration A and B are found from
the boundary conditions, viz. u' = 0 and u = 0 when z = 0. From the first of these and
Eq. (iii), A = 0, while from the second and Eq. (iv), B = 0. Hence the deflected shape
of the beam in the xz plane is given by
(LpJ
U= wIxy (VI
EVxx4Jy -
At the free end of the cantilever (z = L) the horizontal component of deflection is

Uf.e. = WIXYL3 (vi)
3mCJyy - z:y)
Similarly, the vertical component of the deflection at the free end of the cantilever is
- WIYYL3 (vii)
%e. =
3WJyy - z:y)
The actual deflection Sf., at the free end is then given by

6f.e. = (de. + &e.>'
at an angle of tan-' uf.e,/vf..e. to the vertical.