Statics

We use the capital letter I as the symbol for moment of inertia. Here are the integrals that define the (area) moment of inertia about an axis. Note that the origin of the coordinate system is at the centroid of the cross-section. This detail is critical if you want to calculate the centroidal moments of inertia. Also, note that the distance we want to square is perpendicular to the axis of interest.

This infographic of the parallel axis theorem is our guide for computing the centroidal moments of inertia for a composite (built-up) shape (Ix and Iy). Inspect it carefully.

Here is an example problem (and solution) for the parallel axis theorem. You must calculate y bar as the first step (if you need to review Lesson 6 Centroids, please do so).

In the answer, the solution is written as 5.18 E6 mm^4. That is the best way to express that answer. We would not add a prefix to it (e.g. we would not write "M mm^4").

Sometimes, it is advantageous to use solids and voids when using the parallel axis theorem. Here is an example that shows how that works. It's similar to the logic we used when we learned in the centroids lesson.

Some engineers prefer to approach these calculations in a tabular format. This is an alternate solution format to the same problem as above. The results are exactly the same.

The product of inertia is always equal to zero when one or both centroidal axes are a line of symmetry. 

The significance of the product of inertia, Ixy, is explained in Section 16.9.

Before tackling that, you need to learn a little about rotating axes in Section 16.7 and principal moments of inertia in Section 16.8.

❏ Mohr's Circle: it's a visual representation of the moment of inertia

I like to describe Mohr's Circle as a tool. It helps us transform moments of inertia from one pair of axes (x and y) to another pair of axes (u and v).

Mohr's Circle is one of those concepts in engineering that may not "click" for you the first time through. You'll need to spend a little time with Mohr's Circle before it starts making sense. With time and effort, it will make sense, I promise. Its application is not limited to Statics - you'll also see it in other courses.

Here is the procedure to create a Mohr's Circle:

• Plot Ix and Iy on the horizontal axis; plot Ixy on the vertical axis

• Create a circle by plotting two ordered pairs:

  • • • (Ix, Ixy)

      • (Iy, -Ixy)

• Use fundamental geometry to solve for the center of the circle and the radius of the circle.

• The principal moments of inertia lie at the left and right quadrants of the circle (where Ixy = 0)

• The angle from the input xy axes to the output principal axes is:

  • • equal to 2θp in the circle

    • equal to θp in real life

Did you notice that all angles in Mohr’s Circle are twice the change in angle in real life?

• If you rotate a coordinate system 30 degrees clockwise in real life, then you need to travel 60 degrees clockwise in Mohr’s Circle.

• The x and y axes are separated by 90 degrees in real life, and 180 degrees apart in Mohr’s Circle.

We can use Mohr's Circle to transform the moments of inertia from any input pairs of axes to any output pairs of axes (such as the angle in section 16.7).

The most practical use of Mohr's Circle is to determine the principal moments of inertia and to sketch the principal axes on the cross-sectional geometry.