Second moment of plane area (Moment of inertia)

Moment of inertia is given by the integrals: in which the x and y coordinates are those of a differential area, dA (as pictured here). The name second moment of area is derived from the multiplication of a differential area, dA, by the square of the distance from each reference axis. Since both factors of multiplication are in squared units, the resulting value is always positive (unlike first moment of area).

In order to familiarise ourselves with the formula thoroughly, lets examine each component more closely:

Now let's look at 2 examples to help us understand why moments of inertia are important:

Example 1

Find I_x and I_y of the following shape:

Solution

This is a straight forward shape where we can immediately apply the integrals discussed above:

Similarly,

Example 2

Find I_x and I_y of the following shape:

Solution

Now we have the same shape as in example 1, but rotated through 90 degrees clockwise. Again, we can immediately apply the equations for moments of inertia to obtain the solution:

And similarly,

Examples 1 & 2 show us is that I_x = I_y respectively for the same shape when it has been turned through 90º CW. In other words, we can compare which alignment will give us the best rigidity for our shape if we know how to calculate moments of inertia.

Now lets explore this topic further by taking a different reference axis:

Example 3

Find the moment of inertia from axis BB.

Solution

We must define y as the coordinate distance from BB to the differential area dA:

In the last example, we will calculate the moment of inertia of a composite shape:

Example 4

Find I_x of the dotted frame:

Solution

The process of finding moment of inertia is same as before, except this time we need to take away the moment of inertia of the inner rectangle.

Although the solution looks complex, and the integration may be slightly confusing due to constants and limits having more than one term in them, the method used is very simple.