Composite Parts for Moments of Inertia and the Parallel Axis Theorem

As an alternative to integration, both area and mass moments of inertia can be calculated by breaking down a complex shape into simple, common parts, looking up the moments of inertia for these parts in a table, adjusting the moments of inertia for position, and adding them together to find the overall moment of inertia. This method is known as the method of composite parts.

A key part to this process that was not present in centroid calculations is the adjustment for position. Moments of inertia for the parts of the body can only be added if they all have the same axis of rotation. The moments of inertia are in the table are generally listed relative to that shape's centroid though. Since each part has it's own individual centroid coordinate, we cannot simply add these numbers up. We will use something called the Parallel Axis Theorem to adjust the moments of inertia so that they are all taken about some standard point. Once the moments of inertia are adjusted with the Parallel Axis Theorem, then we can add them together using the method of composite parts.

The Parallel Axis Theorem

When we calculated the area and mass moments of inertia via integration, one of the first things we had to do was to select a point or axis we were going to take the moment of inertia about. We then measured all distances from that point or axis, where the distances were the moment arms in our moment integrals. If we pick a different point or axis to serve as the center all of these distances will be different, which means that we will get a different moment of inertia.

The distances used in our moment integrals depends on the point or axis chosen.

For the instance on the right however, each of these distance vectors can be broken down into a vector from the origin to the centroid, and then a vector out from the centroid to all the points in the shape. Similarly, we can find the overall moment of inertia by adding two sets of moment integrals. The first moment integral will add up all the distance vectors from the origin to the centroid (This will be like a point mass on a massless stick, so we get a mass times distance squared term), the second will be the moment integral about the centroid (which is what is listed in the tables). By adding these two terms together we can find the moment of inertia about the given origin point.

Parallel Axis Theorem
The Parallel Axis Theorem states that a bodies moment of inertia about any given axis is the moment of inertia about the centroid plus the mass of the body times the distance between the point and the centroid squared.

This works for both mass and area moments of inertia as well as for both rectangular and polar moments of inertia. Above the mass moment of inertia is listed, but if we substitute in areas instead of masses we can use it for area moments of inertia.

Parellel Axis Theorem
For mass moments of inertia we will use the mass of the body and for area moments of inertia we will use the area of the shape.

For rectangular area moments of inertia and for 3D mass moments of inertia the distances in the equation will be the distance between the axis or rotation and the centroid while for 2D polar moments of inertia we will measure the distances from the point of rotation to the centroid.

Parallel Axis Theorem

With these equations we can see that the moment of inertia of a body is always lowest about it's centroid (where d = 0), and that the further we move away from the centroid the larger the moment of inertia will become.

Using the Method of Composite Parts to Find the Moment of Inertia

To find the moment of inertia of a body using the method of composite parts, we must go the following steps.

  1. First, we need to break the complex shape down into simple shapes. These should be shapes that have moments of inertia listed in moment of inertia tables.
  2. For each of the individual shapes we will want to identify the area or mass (where holes or cavities count as negative areas or masses), the coordinates of the centroid, and the shape's moment of inertia about it's centroid. It is often useful to list these values in a table in order to more easily keep track of the values.
  3. Next we will want to identify the common point we will take the overall moment of inertia about. Sometimes this will be given to us and other times it will need to be calculated (it is often the centroid of the overall shape, in which case you use the method of composite parts to calculate that).
  4. Once you have identified the point you are taking the moment of inertia about, you will need to measure the distances between this point and the centroids of each shape (the way you measure these distances will depend on the type of moment or inertia, see the figure above for details). Add these distances to your table as the d values.
  5. Next use the distances and the area or mass to calculate the correction for your moments of inertia (m d squared or A d squared). Add these corrections to your the moments of inertia about the centroids to get the corrected moments of inertia.
  6. Add the corrected moments of inertia to find the total moment of inertia for the combined shape.

The diagram below shows a shape that has been broken down and the table used to calculate the overall moment of inertia.

Moment of Interia Table
It is easiest to list areas, centroid coordinates, moments of inertia, distance corrections, and corrected moments of inertia in a table. The overall moments of will be the sums of the corrected moment of inertia columns. 

Worked Problems:

Question 1:

Use the parallel axis theorem to find the mass moment of inertia of this slender rod with mass m and length L about the z axis at it's end point.

Problem 1 Diagram

Solution:



Question 2:

A beam is made by connecting two 2" x 4" beams in a T pattern with the cross section as shown below. Determine the location of the centroid of this combined cross section and then find the rectangular area moment of inertia about the x axis through the centroid point.

Problem 2 Diagram

Solution:



Question 3:

A dumbbell consists of two .2 meter diameter spheres, each with a mass of 40 kg spheres attached to the ends of a .6 meter long, 20 kg slender rod. Determine the mass moment of inertia of the dumbbell about the y axis shown in the diagram.

Problem 3 Diagram

Solution: