Torsional Shear Stress Formula

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  • 0:03 What Is Torsional…
  • 0:38 Formula for Torsional…
  • 1:30 Formula for Polar…
  • 2:05 Example
  • 2:51 Shear Stree for a…
  • 3:30 Lesson Summary
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Lesson Transcript
Instructor: Hassan Alsaud

Earned my B.S. in Civil Engineering back in 2011. Have two years of experience in oil and gas fields and two year as a graduate research assistant. Earned my Master degree in Engineering from Tennessee State University in 2016.

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:

torsional shear stress

In this formula:

τ = 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:

polar moment of inertia of a circle around the center point

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

Polar moment of inertia of a rectangle


Let's work through an example:

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