Building a Toolbox of Equations - Our Models

In this section, we’re going to focus on developing the equations that describe membrane behaviour. These equations are the tools we’ll use in later sections for analysing shells - so it’s important that we develop a solid understanding at this early stage. What we cover in this section is essentially the foundation upon which everything that follows is built.

The equations we develop are our models of membrane behaviour in shells. We’ll develop two groups of equations — one pair that describes membrane forces and another set that describes membrane displacements.

Theory-heavy But Hang In There

This is the most theory-dense section of the course—we’ll be going through quite a few derivations. I’ll try to keep signposting where we’re headed so you don’t feel like you're getting too bogged down in the weeds. Just keep reminding yourself of the overall objective if you find yourself drifting during this section — the payoff will be a shiny new toolbox of equations to deploy in the following sections!

The Plan

Okay, so let’s summarise the roadmap before we dive in. We’ll start with a description of the membrane state of stress and the conditions that must be satisfied for an assumption of membrane stress to be valid — you can think of these as our limiting assumptions.

We start with strains

Over the following three lectures, we’ll examine shell displacement geometry. We’ll consider the linear displacement of a point in the meridional and normal directions as well as the angular displacement. The end result of this analysis is an expression for the meridional strain, hoop strain and angular displacement. These will be used a little later in the section to determine the expressions for displacement that I mentioned earlier.

Equilibrium of the infinitesimal element

Then, we temporarily put the discussion of displacements to one side. Over the following three lectures, we move on to consider the force equilibrium of an infinitesimal shell element. The objective of this discussion is to determine expressions for the membrane forces, the so-called meridional and hoop forces.

From strains to displacements

Then, in the final two lectures, we’ll return to our earlier equations for linear strain and rotation and expand on these to determine the final expressions for membrane displacements.

At this point, we’ll be more or less finished deriving the fundamental equations. Our derivations up to this point will have been somewhat abstract in the sense that we will have generally been discussing infinitesimal shell elements. In the next section, after this one, we’ll start to look at how this theory applies to one very common geometry - the spherical shell.

All will become clear through application!

If you find the derivations in this section a bit challenging or struggle to see the immediate application of the equations we derive - don’t worry too much; it will all become much clearer when you see the equations in action in the upcoming sections.