Page 328 - Aircraft Stuctures for Engineering Student

9.5 Torsion of closed section beams 309
The theory of the torsion of closed section beams is known as the Bredt-Batho
rlteory and Eq. (9.49) is often referred to as the Bredt-Batho formula.

9.5.1 Displacements associated with the Bredt-Batho shear flow

The relationship between q and shear strain established in Eq. (9.39), namely
q=Gt (E 2)
-+-

is valid for the pure torsion case where q is constant. Differentiating this expression
with respect to z we have

or

(9.50)

In the absence of direct stresses the longitudinal strain div/az( = E,) is zero so that

Hence from Eq. (9.27)
d28 d’u d%
p- + -cos + + -sin @ = 0 (9.51)
dz2 dz2 dz2
For Eq. (9.51) to hold for all points around the section wall, in other words for all
values of +

d2 8 d2u d2v
dz2
7=0, -- &2-0, -- -
dz-
v
It follows that 8 = Az + B, u = Cz + D, = Ez + F, where A, B, C, D, E and F are
unknown constants. Thus 8, w and v are all linear functions of z.
Equation (9.42), relating the rate of twist to the variable shear flow qs developed in
a shear loaded closed section beam, is also valid for the case qs = q = constant. Hence
d6’

which becomes, on substituting for q from Eq. (9.49)

(9.52)

The warping distribution produced by a varying shear flow, as defined by Eq. (9.45)
for axes having their origin at the centre of twist, is also applicable to the case of a