Torsion is when a member is subjected to twisting couples, or torques, T and T'. These couples have a common magnitude T, and opposite senses. They are vector quantities and can be represented either by curved arrows or by couple vectors as shown:
Fig1.
Fig2.
represented using curved arrows
represented using couple vectors
Consider the following system which consists of a turbine and an electric generator connected by a transmission shaft. By breaking the system into its three component parts, it can be seen that the turbine exerts a twisting couple or torqueT on the shaft and that the shaft exerts an equal torque on the generator. The generator reacts by exerting the equal and opposite torque T' on the shaft, and the shaft by exerting the torque T' on the turbine.
Any moment vector that is collinear with an axis of a mechanical element is called a torque vector, because the moment causes the element to be twisted about that axis. A bar subjected to such a moment is also said to be in torsion.
As shown in Fig3, the torque T applied to a bar can be designated by drawing arrows on the surface of the bar to indicate direction or by drawing torque-vector arrows along the axes of twist of the bar. Torque vectors are the hollow arrows shown on the x axis in Fig3. Note that they conform to the right-hand rule for vectors.
The angle of twist, in radians, for a solid round bar is:
(Eq1)
θ =
TL
GJ
Shear stresses develop throughout the cross section. For a round bar in torsion, these stresses are proportional to the radius ρ and are given by:
(Eq2)
τ =
Tr
J
Designating rmax as the radius to the outer surface:
(Eq3)
τmax =
Trmax
J
The assumptions used in the analysis are:
• The bar is acted upon by a pure torque, and the sections under consideration are remote from the point of application of the load and from a change in diameter.
• Adjacent cross sections originally plane and parallel remain plane and parallel after twisting, and any radial line remains straight.
• The material obeys Hooke's law.
Eq3 applies only to circular sections. For a solid round section:
(Eq4)
J =
πd 4
32
where d is the diameter of the bar. For a hollow round section:
(Eq5)
J =
π
32
(do4 − di4)
where the subscripts o and i refer to the outside and inside diameters, respectively.
