The Theory of Torsion

Abstract

Consider a cantilevered beam with a stiff crossbar at its outer tip, as illustrated in Fig. 10.1. At either end the crossbar is subject to a transverse force $F$. This will cause a twisting $\theta \left( x \right)$ along the entire length of the beam. The force effect that causes this twisting is the torsion moment $M_{x}$, which in this particular case is equal to $2Fa$. The way a line-like beam type of system will carry a torsion moment depend on the type of cross section, as illustrated in Fig. 10.2. In a circular or rectangular tube, the torsion moment will cause a constant shear stress flow around the entire cross section, as illustrated in Fig. 10.2.a. In a thin walled open cross section, the torsion moment will cause two shear stress effects, as illustrated in Fig. 10.2.b. First, there will be a shear stress flow in each of the plate elements within the cross section. This shear flow varies linearly across the thickness of each of the plate elements. This effect is called St. Venant torsion. Secondly, there will be a constant shear flow across each of the outer flange elements, an effect which will cause the flanges to bend in opposite directions as illustrated in Fig. 10.3. This effect is called warping torsion. As can be seen, the effect of warping is not only shear, but also normal stresses due to bending in the flanges. In general, the torsion moment stress effects will be a combination of St. Venant torsion and warping. We shall in the following assume elastic, homogeneous and isotropic material behaviour. We shall take it for granted that any cross section is thin walled or else compact.