The ability to calculate the second moment of area – also called the area moment of inertia or second moment of inertia – of a structural element becomes necessary when needing to determine the load-bearing capacity of the element in a given dimension. I covered the basics on what the second moment of area is for and the various types of cross section shapes, characteristics, and their advantages in my article Second Moment of Area made Simple.
In this article however, I will show you how to calculate this value so you can actually use it. It’s really not all that difficult once you understand the basic formula. First of all, you have to know that there are two axes around which the given cross section will tend to rotate – the x and y axes. So unless the member in question has reflectional symmetry around both the x and y axes, you’ll need two different equations for both axes.
Simple Rectangular Cross Section
Let’s take a look at this rectangular cross section. As you can see, its height and width are different. This means that there will be one second moment of area for the rotation about the x axis, and another for that of the y axis. The equations go as follows:
This is a fairly self-explanatory equation, where I is the second moment of area, and b and h are the respective base and height of the rectangle. The equations show that the second moment of area about the y axis is considerably higher than that for x – for obvious reasons. You can also see how it would simplify to the following equation if both dimensions were equal, where s = side:
Now that wasn’t too difficult was it? Just remember that it’s important to specify the x or y subscript when a distinction between the two must be made, as without it, it would be ambiguous as to which axis the quotient is referring to. Let’s work out a simple example to solidify your understanding of this concept. Say you want to know the second moment of area of a cross section of 0.09 by 0.09 meters. This is what it would look like.
Second moment of area is usually expressed using m4, cm4, or mm4. Just remember though, that when plugged into other equations, all units must be the same for it to yield the correct answer. This means that the units you decide to use will probably depend on what you plan to use it for – in other words, the units of the other variables in a second equation that you would plug I into.
Circular Cross Section
- DO = outside diameter
- DI = inside diameter
- rO = outside radius
- rI = inside radius
Visit Wikipedia’s List of Area Moments of Inertia for a more comprehensive list.