Page 469 - Aircraft Stuctures for Engineering Student

450 Structural constraint

't

isplacement due to

z

Fig. 11.5 Idealized rectangular section beam built-in at one end and subjected to a torque at the other.

In Fig. 1 1.5 suppose that the shear flows in the covers and webs at any section are qa
and qb respectively; from antisymmetry the shear flows in both covers will be qa and in
both webs qb. The resultant of these shear flows is equivalent to the applied torque so
that
b U
qpds = 2qaa-+2qbb-
2 2
or

= ab(% + qb) (11.4)
We now use Eq. (9.39), i.e.
(E E)
q=Gt -+-

to determine qa and qb. Since the beam cross-section is doubly symmetrical the axis of
twist passes through the centre of symmetry at any section so that, from Eq. (9.28)

(11.5)
Therefore for the covers of the beam

(11.6)

and for the webs
aq - n de
dz 2 dz (11.7)