Expanding Beyond Positive Gaussian Curvature

So far, we’ve discussed spherical shells, which, as you know, have positive Gaussian curvature since both the principal circular and meridional curvatures are positive.

We’ve been applying the fundamental equations derived in section 2 to this specific geometry, which has given us a good grasp of what the shell analysis workflows look like. We understand how to determine and visualise both membrane forces and displacements.

The Plan

Now, we’re ready to expand and apply what we’ve learned to new geometries - so, in this section, we start to consider shells with zero and negative Gaussian curvature.

Zero Gaussian Curvature - Cylindrical and Conical Shells

We kick things off by focusing on shells with zero Gaussian curvature. Since we’re dealing exclusively with shells of revolution in this course, this means that these shells must have a straight meridian.

Unpacking that statement a little…if the Gaussian curvature is zero and we know the principal circular curvature is positive by definition, the meridional curvature must be zero.

As the meridional curvature approaches zero, the corresponding radius of curvature approaches infinity. Practically, this means the meridian must be a straight line. And if we rotate a straight line about an axis, we end up with a cylindrical or conical shell. So, we’ll spend the first half of this section mapping our fundamental equations to this specific geometry.

Negative Gaussian Curvature - Hyperboloid Shells - The Cooling Tower!

Then, in the second half of the section, we move on to consider negative Gaussian curvature - and by far the most prevalent example of a shell of revolution with negative Gaussian curvature is the Hyperboloid - we know this as the classic cooling tower!

There’s a lot to unpack here in terms of geometry as well as membrane behaviour - but by the end of our discussion, you’ll have unlocked the membrane force distribution, which means you can readily apply what you’ve already learned to calculate the corresponding membrane displacements.

So, we have quite a bit to cover in this section - but once you’ve finished, you’ll have massively expanded the shell geometries you’re comfortable working with.