Torsion

Abstract

As we established in Sect. 4.3, the shear stress τ in a thin-walled circular tube of mean radius r subject to a torque T can be determined independently of the material properties and is given (at least far enough away from the ends, in accordance with Saint-Venant’s principle) by Eq. (4.15),

$$ \displaystyle{ \tau = \frac{T} {2\pi {r}^{2}t}.}$$

(7.1)

A circular shaft, whether hollow (that is, a thick-walled tube) or solid, may be treated as a large set of concentric thin-walled tubes, as shown (in cross-section) in Fig. 7.1. If the inner and outer radii of the shaft are b and c, respectively (with the special case b = 0 representing a solid shaft), then a differential element of thickness dr at a mean radius r (b < r < c) may be regarded as a thin-walled tube carrying a torque dT, such that, in accord with Eq. (7.1),

$$\displaystyle{ dT\ =\ 2\pi \tau (r){r}^{2}dr\ , }$$

(7.2)

where τ(r) is the average shear stress in the given differential element. The total torque is consequently given by

$$\displaystyle{ T\ = 2\pi \int _{b}^{c}\tau (r){r}^{2}dr.}$$

(7.3)