Page 363 - Aircraft Stuctures for Engineering Student

344 Open and closed, thin-walled beams

and

(9.90)

Similar expressions are obtained for a closed section beam from Eq. (9.80).

Example 9.15
Calculate the deflection of the free end of a cantilever 2000 mm long having a channel
section identical to that in Example 9.13 and supporting a vertical, upward load of
4.8 kN acting through the shear centre of the section. The effective direct stress carry-
ing thickness of the skin is zero while its actual thickness if 1 mm. Young's modulus E
and the shear modulus G are 70 000 N/mm2 and 30 000 N/mm2 respectively.

The section is doubly symmetrical (i.e. the direct stress carrying area) and supports
a vertical load producing a vertical deflection. Thus we apply a unit load through the
shear centre of the section at the tip of the cantilever and in the same direction as the
applied load. Since the load is applied through the shear centre there is no twisting
of the section and the total deflection is given, from Eqs (9.86), (9.88)' (9.89) and
(9.90), by

where Mxp = -4.8 x 103(2000 - z), Mx,l = -1(2000 - z)

and z is measured from the built-in end of the cantilever. The actual shear flow dis-
tribution has been calculated in Example 9.13. In this case the q1 shear flows may
be deduced from the actual distribution shown in Fig. 9.52, i.e.
41 = qo/4.8 x lo3
Evaluating the bending deflection, we have

4.8 x 103(2000 - z)'dz
= 3.81 ~~lfn
70000 x 48 x lo6
The shear deflection As is given by
1
(62 x 200 + 122 x 400 + 62 x 200 dz

= 1.Omm

The total deflection A is then AM + As = 4.81 mm in a vertical upward direction.