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
  1. The Equation
  2. Show me an Example
  3. References


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

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

Parallel Axis Theorem 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

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