What is a Moment of Inertia?

The moment of inertia of a plane area about a given axis describes how difficult it is to change its angular motion about that axis (another way to put it is how resistant the object is to bending and torsional? stresses) . Therefore, it encompasses not just how much mass the object has overall, but how far each bit of mass is from the axis. The farther out the object's mass is, the more rotational inertia the object has, and the more force is required to change its rotation rate. This concept is made very clear in the parallel axis theorem.

In mathematical terms, the moment of inertia can be defined about any given axis as:

I_x = \int y^2dA

I_y = \int x^2dA

where:

  • x & y = the coordinates of the differential element dA multiplied by the square of the distance from a designated reference axis (normally the reference axis will be taken about the edge or through the centroid of the shape).
  • dA = a very small element of area

Note: Moments of inertia can also be known as second moments of area.

Example of How to Calculate a Moment of Inertia

Finding Ix

Moment of Inertia of a Rectangular Solid
Figure 1: Variables used for calculating Moment of Inertia

Figure 1 shows an example of a rectangle with width b and height h. The x and y axis are based off of the origin at the centroid of the rectangle (C is at \frac{h}{2} and \frac{b}{2} from the bottom right corner of the rectangle respectively). A hypothetical thin strip of area can be taken with width b and height dy (therefore dA = b * dy). Therefore Ix can be solved by integrating the following:

I_x = \int y^2dA = \int_{- \frac{h}{2}}^{\frac{h}{2}} y^2 b dy = \frac{bh^3}{12}

Note: Ix has been integrated with respect to the x-axis. The moment of inertia is dependent on what you pick as your axis. See how IBB has been calculated below.

Finding Iy

Similarly we can choose a strip of area around the y-axis by taking dA = h * dx and get the moment of inertia with regard to the y-axis:

I_y = \int x^2dA = \int_{- \frac{b}{2}}^{\frac{b}{2}} x^2 h dx = \frac{hb^3}{12}

Finding IBB

Although the most common axis' are through the centroid of the area, the moment of inertia can be taken around any arbitrary axis. For example IBB can be solved by taking:

I_{BB} = \int y^2dA = \int_{0}^{h} y^2 b dy = \frac{bh^3}{3}

Common Moment of Inertia Shapes

As you can see from the examples above, it is not always easy to calculate the moment of inertia of shapes. For that reason tables have been created to speed up the process for common shapes.

The Moment of Inertia for various Structural Shapes can be found here?.

Short Excerpt on the Parallel Axis Theorem

See a full analysis of the Parallel Axis Theorem here.

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.


Resources

  1. H.E. Murdock, "Strength of Materials (1st Edition)", 1911
  2. R.C. Hibbeler, "Mechanics of Materials (7th Edition)", 2007

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