Torsion of Beams - Structural and Stress Analysis - page 290

y

Area, A

T

O

z q

δ A

p

F IGURE 11.12 Torque-shear
flow relationship in a
thin-walled closed section beam

δ s

or, since q = constant

q p d s

T = (11.19)

We have seen in Section 10.5 that p d s

= 2 A where A is the area enclosed by the

midline of the beam wall. Hence

T = 2 Aq (11.20)

The theory of the torsion of thin-walled closed section beams is known as the Bredt-
Batho theory and Eq. (11.20) is often referred to as the Bredt-Batho formula .

It follows from Eq. (11.20) that

q
t

T
2 At

τ = = (11.21)

and that the maximum shear stress in a beam subjected to torsion will occur at the
section where the torque is a maximum and at the point in that section where the
thickness is a minimum. Thus

T max
2 At min

τ max = (11.22)

In Section 10.5 we derived an expression (Eq. (10.28)) for the rate of twist, d θ /d x ,
in a shear-loaded thin-walled closed section beam. Equation (10.28) also applies to
the case of a closed section beam under torsion in which the shear flow is constant
if it is assumed that, as in the case of the shear-loaded beam, cross sections remain
undistorted after loading. Thus, rewriting Eq. (10.28) for the case q s = q = constant,

we have

d s
Gt

d θ
d x =

q
2 A

(11.23)

Substituting for q from Eq. (11.20) we obtain

d s
Gt

T
4 A 2

d θ
d x

=

(11.24)