What is the “second moment of area”? I understand length, area and volume. But how can a 3D object have a 4D property?

Question

I get the math and application, but I can’t physically grasp the concept of mm^4 or inch^4.

In: Mathematics

Answer 1

It’s not even a 3D object, it’s a 2D area that has this property.

Consider a beam’s resistance to bending. If you start with a shape (say a square cross section), and double the width, it is as if you have two beams sharing the load, so you expect the resistance to be proportional to width.

If you make the beam twice as thick, you can see the improvement will be greater than making it twice as wide. The material at the top and bottom will be stretched/squished more to reach the same curve, and so will resist harder. It turns out the resistance is proportional to thickness cubed.

So you have a width term, and a thickness^3 term, giving units of length^4

I’m trying to think of a simple explanation of why the resistance is proportional to thickness cubed, rather than squared, but am coming up blank. Hopefully you can see why it should be a power higher than 1.