In this lesson, you’ll learn about torsional shear stress, how it’s distributed and what formulas are used to calculate it. To fully understand the lesson, you should already be familiar with shear stress and the difference between shear stress and normal stress.

## What Is Torsional Shear Stress?

**Torsional shear** is shear formed by torsion exerted on a beam. **Torsion** occurs when two forces of similar value are applied in opposite directions, causing torque. For example, picture a traffic sign mounted on a single column on a windy day. The wind causes the sign to twist, and this twist causes shear stress to be exerted along the cross section of the structural member. Therefore, when designing the traffic sign, it is very important to accurately estimate the value of the shear stress caused by torsion in order to design bolts that can resist that stress.

## Formula for Torsional Shear Stress

The value of the torsional shear stress at any point in the structural member’s cross-sectional area is calculated using the following formula:

In this formula:

*;* = shear stress (‘lbf/ft2‘ for the English unit system)

*T* = torque (‘lbf.ft’ for the English unit system)

*r* = radius (or distance to the center point of the cross sectional area) (‘ft’ for the English unit system)

*J* = polar moment of inertia (‘ft4‘ for the English unit system)

From this formula, we can see that shear stress at the center is 0 because r equals 0. Furthermore, as the point of interest moves away from the center towards the edge of the cross section, the shear stress increases in a linear fashion.

## Formula for Polar Moment of Inertia

To calculate the polar moment of inertia, we need to do some further calculations. The **polar moment of inertia** of an area about a given point is equal to the sum of the moment of inertia of that particular area about any two perpendicular axes that are passing through that very point.

The polar moment of inertia of a circle around the center point is given by:

The polar moment of inertia of a rectangle of dimensions *b* by *h* around the center point is given by:

## Example

Let’s work through an example:

A 100 N.m torque is applied to a 20 cm diameter circular cantilever at a given section. What is the value of the shear stress at the circumference of the section?

First, let’s find the first moment of inertia of the cross-sectional area:

*J* = 3.14 * 0.14 / 2 = 1.6 x 10-4 m-4

Therefore, the shear stress can be calculated by the given formula:

*; = T * r / J*

*;* = 100 N.m * 0.1 m /(1.6 x 10-4 m-4)

*;* = 62.5 kPa

## Formula for Shear Stress of a Cantilevered Beam

If you need to calculate torsional shear stress for a cantilever beam, you’ll need to use a different formula:

Where, G is the **shear modulus**, or modulus of rigidity, and ϕ is the angle of twist in radians. ϕ is calculated by the following formula:

*T* = torque (‘lbf.ft’ for the English unit system and ‘N.m’ for the international system)

*r* = radius

*l* = length of the member (ft in the English system)

## Lesson Summary

Under **torsion**, structural members are exposed to shear stress that starts at zero in the center of the cross sectional area and reaches the maximum value at the edge of the cross-sectional area. Two formulas are typically used to calculate the value of the shear stress at a given cross section: one involves torque and the polar moment of inertia, and the other involves the angle of twist and the length at which the torque applies. There is an additional formula that is used when you want to determine shear stress for a cantilevered beam.