The Rectangular Area Moment of Inertia

The Second Moment Integral, often just called the Moment of Inertia, can be useful in engineering mechanics calculations for a number of reasons.

For example:

Each of these three types of moments of inertia can be calculated via integration or via composite parts and the parallel axis theorem. On this page we are going to focus calculating the second rectangular area moment of inertia via integration.

Bending Stresses and the Second Area Moment

When an object is subjected to a bending moment that body will experience both internal tensile stresses and compressive stresses as shown in the diagram below. These stresses exert a net moment to counteract the loading moment, but exert no net force so that the body remains in equilibrium.

Beam Bending and the Second Rectangualr Area Moment of Interia
A bending moment both internal tensile stresses and internal compressive stresses.

As we can see in the diagram, there is some central plane along which there are no tensile or compressive stresses. This is known as the neutral surface, and if there are no other forces present it will run through the centroid of the cross section. As we move up or down from the neutral surface the stresses increase linearly. The moment exerted by this stress at any point will be the stress times moment arm, which also linearly increases as we move away from the neutral surface. This means that the resistance to bending provided by any point in the cross section is directly proportional to the distance from the neutral axis squared. We can sum up the resistances to bending then using the second rectangular area moment of inertia, where our distances are measured from the neutral axis.

Second Rectangualr Area Moment of Interia

Calculating the Second Rectangular Area Moment of Inertia via Integration

The first step in determining the rectangular moment of inertia for an area is to determine what axis we will define as the zero point. In cases on bending stress this will be the neutral surface, which will travel through the centroid of the beam's cross sectional area. It is usually useful at this point to draw the neutral surface onto the shape we are finding the moment of inertia for.

To take the moment of inertia about the x axis we will need to use the distances from the x axis (in this case the y coordinates of each point) and integrate over the area. Moving from bottom to top, the rate of change of the area at any given y value will be the width of the shape at that point times the rate at which we are moving along the y axis. Since the width often changes as we move along the y axis, we will need to find an equation that describes width at any value of y (this will be some function of the variable y). This gets multiplied by y squared in the second moment integral.

Moment of Inertia about the X axis
The rectangular moment of inertia about the x axis.

To find the moment of inertia about the Y axis we will use the distances from the Y axis (in this case the X coordinates of each point). Moving from left to right, the rate of change of the area will be the height of the shape at that point times the rate at which we are moving left to right. Again we will need to describe this with an equation if the height is constantly changing. We will multiply this by x squared for the second moment integral and this will give us the moment of inertia about the y axis.

Moment of Intertia about the Y axis
The rectangular moment of inertia about the y axis.

Worked Problems:

Question 1:

Find the rectangular moments of inertia for this shape about both the X and Y axes though the centroid. Leave the answer in terms of the generic width (b) and height (h) of the rectangle.

Problem 1 Diagram

Solution: