## What is the Parallel Axis Theorem?

The parallel axis theorem can be used to determine the moment of inertia of a rigid body around any axis. Oftentimes the moment of intertia of a rigid body is not taken around the centroid, rather some arbitrary point. A good example of this is an I-Beam. You may need to use the parallel axis theorem to determine the Moment of Inertia of an I-Beam around it's centroid because the top and bottom flange will not be acting through the centroid of the shape (see the Example Below).

 Using the Parallel Axis Theorem

### How can I calculate a moment of inertia using the Parallel Axis Theorem

Figure 1: Variables used for using the Parallel Axis Theorem

 Once the centroid of the shape is found, the parallel axis theorem can be used around any axis by taking: I_{A} =\sum ( I_{x} + Ad^2)

where:

• IA = The moment of inertia taken about the A-A axis (in4)
• Ix = The moment of inertia taken through the centroid, the x-x axis (in4)
• A = The area of the rigid body (in2)
• d = the perpendicular distance between the A-A axis and the x-x axis (in)

Note: Looking closely at the Parallel Axis Theorem you can see that the moment of inertia of a shape will increase rapidly the further the Centroid of the area is from the axis being checked.

### Using the Parallel Axis Theorem in an Example

Figure 2: Parallel Axis Theorem Example

For the above example, take h = 12", h1 = 10", b = 9 " and tw = 1". Solve for Ix using the Parallel Axis Theorem.

1) Solve for Ix of the center section (feel free to use the shortcuts here?:

{ I_x}_{web} = \frac{bh^3}{12} = \frac{1" * 10"^3}{12} = 83 in^4
{I_x}_{flange} = \frac{bh^3}{12} = \frac{9" * 1"^3}{12} = 0.75 in^4

2) Solve the increase in the moment of inertia for A1 using the parallel axis theorem:

A_1 = 1in * 9" = 9 in^2
d = \frac{12in - 10in}{2*2} + \frac{10in}{2} = 0.5in + 5in = 5.5 in
Ad^2 = (9 in^2)(5.5in)^2 = 272 in^4

3) Using the Parallel Axis Theorem:

I_{x} =\sum ( I_{x} + Ad^2) = (83 in^4 + 2*0.75 i^4) + 2*(272 in^4) = 628.5 in^4

As you can see a majority of the Section Modulus (87%) comes from the parallel axis theorem and not from the moment of inertia calculation.

## Resources:

1. Paul A. Tipler, "Physics for Scientists and Engineers (4th Edition)", 1990

Main