## Torsion

Torsion is the twisting in a bar when a force is applied to rotate it. A good example is a turning a screwdriver to screw in a screw.

Of course, there are wider applications of Torsion and this post will go into detail on Torsion from a mechanical engineering point of view. This page will be fairly barebones until I figure out a simple and efficient way of linking images to this site.

### Torque on a shaft

Any force applied to a shaft that does not intersect with the central line of the shaft (i.e. doesn’t point directly at the centre of the shaft) will create a torque, or turning force, on the shaft. This in turn, creates *shearing stresses* on faces *perpendicular to the axis*, that is, on a circular shaft it’d be the circle cross-section.

Consider a very elastic metal shaft (or a very strong person): If the rod is twisted on one end, that end will rotate slightly due to deformation. This rotation is called the *shearing strain* \(\gamma\) (measured in rad) such that:

Where \(c\) is the radius of the shaft and \(\rho\) is the radial distance of the point from the shaft. I’ll include an illustrated graphic later.

#### Within the Elastic range

Given Hooke’s law: \(\tau=G\gamma\), the equation for shearing stress is:

The **Elastic Tension Formula** is:

Where \(J\) is the polar moment of inertia.

The important take-away is how to use the above formulae to calculate the theoretical maximum torque that can be safely applied to a shaft. In the case of a circular shaft:

For a solid circular shaft:

For a hollow circular shaft:

#### Angle of twist in the elastic range

Simply put, the angle of twist is given by:

Where \(\phi\) is the angle of twist in radians.

Extended for a general shaft subjected to varied torques: