The Polar Area Moment of Inertia

The Second Moment Integral, often just called the Moment of Inertia, can be useful in engineering mechanics calculations for a number of reasons.

For example:

Each of these three types of moments of inertia can be calculated via integration or via composite parts and the parallel axis theorem. On this page we are going to focus calculating the second polar area moment of inertia via integration.

Torsional Stresses and the Second Area Moment

When an object is subjected to a torsional force, that object will experience internal shearing forces as shown in the diagram below. These stresses are oriented in such a way that they counteract the torsional load with a net moment but do not exert any net force on the shaft so that shaft stays in equilibrium.

Torsional Loading in a Shaft
Torsion causes internal shearing stresses that counteract the moment but exert no net force.

As we can see in the diagram, there is a central axis along which there are no shearing forces. This is known as the neutral axis, and if there are no other forces present this will travel through the centroid of the shaft's cross section. As we move out from the neutral axis in any direction the stresses will increase linearly. The moment exerted by the stress at any point will be the stress times the moment arm, which also increases linearly as we move away from the neutral axis. This means that the resistance to torsional loading provided by any one point on the cross section is directly proportional the square of the distance between the point and neutral axis. We can sum up the resistances to torsional loading then using the second polar area moment of inertia, where our distances are measured from the neutral axis (r), a single point in the shaft's cross section.

The Second Polar Area Moment of Interia

Calculating the Second Polar Area Moment of Inertia via Integration

The first step in determining the polar moment of inertia is to identify the point about which we are taking the moment of inertia. In the case of torsional loading, we will usually want to pick the point at which the neutral axis travels through the shaft's cross section. In cases of simple torsional loading, the location of the neutral axis will be the centroid of the cross sectional area of the shaft.

To take the moment of inertia about this point, we will be measuring all distances from neutral axis. Moving radially out from this point, the rate of change of the area will be the circumference within the area at a distance r away time the rate of change of r. This gets multiplied by r squared for the second polar area moment integral.

The Polar Area Moment of Interia
The Polar Moment of Inertia About the Neutral Axis

Worked Problems:

Question 1:

Find the polar moments of inertia for this circular area about its centroid. Leave the answer in terms of the generic radius R.

Problem 1 Diagram

Solution:



Question 2:

Find the polar moment of inertia of this hollow circular shape about its centroid.

Problem 2 Diagram

Solution: