Page 323 - Aircraft Stuctures for Engineering Student

304 Open and closed, thin-walled beams

The last two terms in Eq. (9.44) represent the effect of relating the warping displace-
ment to an arbitrary origin which itself suffers axial displacement due to warping. In
the case where the origin coincides with the centre of twist R of the section then
Eq. (9.44) simpliiies to

(9.45)

In problems involving singly or doubly symmetrical sections, the origin for s may be
taken to coincide with a point of zero warping which will occur where an axis of sym-
metry and the wall of the section intersect. For unsymmetrical sections the origin for s
may be chosen arbitrarily. The resulting warping distribution will have exactly the
same form as the actual distribution but will be displaced axially by the unknown
warping displacement at the origin for s. This value may be found by referring to
the torsion of closed section beams subject to axial constraint (see Section 11.3). In
the analysis of such beams it is assumed that the direct stress distribution set up by
the constraint is directly proportional to the free warping of the section, i.e.
u = constant x w
Also, since a pure torque is applied the resultant of any internal direct stress system
must be zero, in other words it is self-equilibrating. Thus
f
Resultant axial load = utds

where u is the direct stress at any point in the cross-section. Then, from the above
assumption
f
o= wtds

or

so that

(9.46)

9.4.2 Shear centre

The shear centre of a closed section beam is located in a similar manner to that
described in Section 9.3 for open section beams. Therefore, to determine the coordi-
nate ts (referred to any convenient point in the cross-section) of the shear centre S of
the closed section beam shown in Fig. 9.25, we apply an arbitrary shear load S,
through S, calculate the distribution of shear flow qs due to S,, and then equate
internal and external moments. However, a difficulty arises in obtaining qs,o since,