Time to Deploy our Models

In this section, we’ll start to deploy our equations and see how they can be used to describe membrane forces and displacements. We’ll start by focussing on a specific and very common geometry - the sphere or spherical dome.

Spherical Shells/Domes

Spherical domes will serve as an excellent case-study structure and test-bed for us to get comfortable with our new toolbox of equations and the associated analysis techniques.

Membrane Forces

Our first analysis will consider the membrane force distribution for a dome under its own weight. We’ll tailor our general force distribution equations for the particular geometry of the sphere. Then, we’ll jump into a Jupyter Notebook and start evaluating and visualising the membrane force equations.

SymPy to unlock the math

A feature of this type of analysis is the use of symbolic math—you’ll see this extensively throughout the rest of the course. We’ll be using SymPy, the Python Scientific Computing library, to do much of the procedural math. Then, we’ll evaluate the resulting equations over the relevant parameter space.

This is a really powerful workflow and provides so much insight into the structural behaviour. It’s amazing how much of this analysis can feel locked away behind what looks like impenetrable math - SymPy is our key to unlocking this.

Membrane Displacements

After considering the force distribution due to self-weight, we’ll evaluate and plot the membrane displacements. This first force and displacement analysis will be key in helping you move from the theory-heavy derivations in the previous section to seeing the practical utility in these equations.

Expanding beyond self-weight

From here, we’ll expand to consider loading applied over the dome’s projected area—the most readily imaginable example would be snow loading, falling vertically and evenly distributing over the shell’s projected area.

After this, we move on to consider truncated domes that have openings at their crown - again, a common interpretation of this would be a dome with a symmetrical skylight.

Getting comfortable with variation

The underlying objective of these different analyses is to get comfortable adapting our equations to different situations. The aim is not to give an example of every conceivable loading and geometry configuration - but to help you get comfortable adapting to whatever analysis scenario you encounter.

By the end of this section, you should see that, regardless of the specific structure or loading, the analysis process follows the same repeatable pattern or sequence of steps. This is your roadmap for independently tackling analytical shell modelling.