In solid mechanics, torsion is the twisting of an object that is result of an applied torque. In circular sections, the resultant shearing stress is perpendicular to the radius.
The shear stress at a point on a shaft is:
T is the applied torque, r is the distance from the center of rotation, and J is the polar moment of inertia.
The angle of twist can be found by using:
Where:
Exclusively for circular and for closed circular ring cross sections, the torsional moment of inertia is equal to the polar area moment of inertia 
:
For other cross-sections, the calculation of the torsional moment of inertia is only possible in closed form in special cases.
In addition, when determining the torsional moment of inertia, it is often important to know whether the cross-sections are warp-free or not, and whether the warping is impeded or not.
For closed sections whose products of wall thickness 
and distance 
from the axis of rotation are constant laterally ( 
), shear stresses are produced in the case of torsion, but no normal stresses in the longitudinal direction and thus no warping of the cross-section. These conditions are met, for example, by a cylindrical tube of constant wall thickness. This case of torsion is called Neuber's shell.
However, it should be noted that the linear elasticity theory applies, i.e. only small distortions and deformations are permitted, but no plastic deformations. In addition, the load should be applied in the form of the torsional moment on the longitudinal axis.
The shear stress τ 
in the member is given by the torsional moment 
divided by the polar section modulus 
:
The maximum shear stress occurs at the edge or at the maximum radius of the cross-section under consideration. When dimensioning, care must be taken to ensure that this shear stress does not exceed the maximum permissible shear stress τ 
of the material to be used:
Otherwise, the deformation of a shaft, for example, passes from the elastic range into the plastic range and finally leads to fracture.
A: Torsion is the twisting of an object that results from an applied torque.
A: In circular sections, the resultant shearing stress is perpendicular to the radius.
A: The equation for calculating shear stress at a point on a shaft is τθz = Tr/J, where T is the applied torque, r is the distance from the center of rotation, and J is the polar moment of inertia.
A: The equation for finding angle of twist is θ = TL/JG, where L represents length and G represents modulus of rigidity.
A: In both equations, "T" represents applied torque.
A: In the equation for shear stress, "r" represents distance from center of rotation.
A:"J" represents polar moment of inertia in both equations.
