Polar Moment of Inertia

The polar moment of inertia is used to compute the maximal shear stress when an element is subject to torsional load. Consequently, it is important to know how to calculate the polar moment of inertia for simple geometries.

Consider a circle shown in the figure below.

The polar moment of inertia is defined as

(1)  

where is a polar coordinate shown in the figure, and is loosely speaking, an area of an infinitely small segment shown in the figure. Our first goal is to express dA using polar coordinates and associated differentials . Consider the next figure

The length of the arc segment is

(2)  

where is measured in radians. Similarly, the length of the arc segment is

(3)  

On the other hand, since is very small, we have that .

Furthermore, since the angle and the length are small, the segment PQLM can be approximated by a rectangle shown in Fig. 3 below.

From Fig. 2, we can see that can be approximated by an area of the rectangle, that is,

(4)  

By substituting this approximation in (1), we have

(5)  

In (5), we have substituted the surface bounds. The variable goes from to . On the other hand, the variable goes from to . Next, we solve the integral

(6)  

Practice example:

Using double integrals compute the polar moment of inertia for the cross-section shown below. Express the results using and .