How to Find Moment of Inertia - I Beam
Now let's look at a more complex case of where the cross section is an I beam, with different flange dimensions. The concept is the same, however the approach in this case is quite different. Basically, we need to look at the I beam as a combination of different rectangles and sum the different parts to get the sections full Moment of Inertia. In short, we need to follow these three steps:
- Calculate the Neutral Axis for the entire section
- Calculate the MOI of each part
- Calculate the moment of inertia using the Parallel Axis Theorem - which is essentially the sum of individual moment of inertias
So let's consider the following section:
The The Neutral Axis (NA) is located at the centroid. This is essentially a weighted average of the area and distance from bottom for each segment. We simply need to use the centroid equation for calculating the vertical (y) centroid of a multi-segment shape.
We will take the datum or reference line from the bottom of the beam section. Now let's find Ai and yi for each segment of the I-beam section shown above so that the vertical or y centroid can be found.
Now we have the centroid. We can continue to calculate the moment of inertia. To calculate the total moment of inertia of the section we need to use the "Parallel Axis Theorem" as defined below:
Since we have split it into three rectangular parts, we must calculate the moment of inertia of each of these sections. We can now use the simplified rectangular moment of inertia formula:
Now we have all the information we need to use the "Parallel Axis Theorem" and find the total moment of inertia of the I-beam section. In our moment of inertia example:
Once again, we can compare this result with that of the free moment inertia calculator to compare the results of both the centroid and moment of inertia, where both the centroid (216.29 in) and Moment of Inertia (4.74 x 10^8 in^4) match:
Moment of Inertia Equations
Simple equations can also be used to calculate the Moment of Inertia of common shapes and sections. These are quick moment of inertia equations that provide quick values and are a great way to cross reference or double check your results. Focusing on simple shapes only, the below diagram shows some of these equations: