In this section, we’ll start by clearly defining the types of structures we want to study, and outline the assumptions and limitations that will apply to everything we discuss throughout the course.

With the limits established, we’ll briefly consider the role of plate and shell structures in the built environment - in particular, we’ll touch on the long history and evolution of shell structures throughout the centuries.

This historical context really emphasises how foundational shell structures have been to the built environment for the last 2000 years…from the Pantheon in Rome to today’s large elaborate roof shells.

Next, we’ll consider the fundamental load resistance mechanisms of plates and shells and how, when designed efficiently, to best utilise their geometry, they both carry loads is fundamentally different ways.

The second half of this section focuses on geometry and curvature in particular. Without a clear understanding of curvature, we can’t further our understanding of the underlying mechanics since it’s through the study of geometry that we acquire the tools and vocabulary to work with curved surface structures.

By the end of this section, you’ll have a conceptual understanding of how plates and shells resist transverse loading, you’ll be familiar with the relevant terminology used in their analysis and you’ll be able to classify shell structures based on their principal curvatures.

With the foundational concepts established in the previous section, in this section, we move on and start analysing our first structures - circular plates. Circular plates are a great starting point since they provide a relatively straightforward structure from which we can develop our mechanical model.

We’ll focus on axi-symmetrically supported and loaded plates, which means the plate support conditions will be uniform, and the loading if it varies, will only vary as a function of the plate radius. Again, the relative simplicity here allows us to focus on developing an analytical model of the behaviour without getting bogged down dealing with edge cases.

This section can be divided into two halves; in the first, we derive the governing differential equation for plate bending. We’ll build up this equation by taking small, easily digestible steps, starting with a basic description of the deformed plate geometry.

Once we’ve established the governing differential equation - we’ll set about solving it to obtain the general solution. We’ll then discuss the influence of boundary conditions, which allow us to determine an analytical solution for our specific plate support conditions.

In this section, we’ll make use of Python to help us with the heavy lifting involved in analytically solving the differential equation. If you’re new to Python - I’ll signpost you to resources to get you up and running. Using Python in this way dramatically speeds up our workflow and is a skill you can directly map onto other areas of engineering mathematics.

In the second half of this section, we’ll focus on putting our model of plate behaviour to work. We’ll use it to analyse various case studies and see how we can handle different plate supports and loading conditions.

By the end of this section, you’ll have developed some really important skills:

- You’ll see how we can develop a differential equation that describes a complex mechanical behaviour, starting from basic descriptions of the behaviour - the same playbook can be deployed in many areas of engineering analysis. You’ll probably recognise many of the steps involved.

- You’ll see how we can make use of open-source Python libraries to help shortcut the tedious work of manipulating and solving equations.

- And finally, you’ll have worked through enough case study examples to be able to confidently tackle unseen axisymmetric plate analyses.

This section will leave you in a good position to tackle the analysis of rectangular plates in the next section.

In this section, we’ll turn our attention to rectangular plates. Although the circular plates we analysed in the previous section certainly have a lot of practical uses, it’s fair to say that the need to analyse rectangular plates occurs much more frequently in the civil and structural engineering world, most commonly as floor structures.

We begin our exploration of rectangular plates just as we did in the previous section by developing the governing differential equation for plate bending. We’ll build up from first principles using the same roadmap we saw previously - analyse the deformed geometry, jump from strains to stresses to stress resultants and then evaluate the equilibrium equations for the element.

From here, we’ll briefly discuss various aspects of rectangular plate behaviour, from boundary conditions to the corner uplift induced by torsional moments at plate corners. After this, we introduce an elegant solution for simply supported rectangular plates - Navier’s solution, first proposed by Claud-Louis Navier in 1820.

This elegant solution involves the clever application of the Fourier Series to approximate the solution of the governing differential equation. We’ll work our way through the logic behind this solution before spending our remaining time in this section working through some analysis case studies.

Case studies will give you a good understanding of how to apply Navier’s solution to various loading conditions. Along the way, we’ll sprinkle in some Python, where it’s helpful and generate some visualisations of predicted plate behaviour.

By the end of this section, you’ll understand:

- The origins of the governing differential equation of rectangular plate bending

- What makes Navier’s approximate solution such an elegant approach

- How to apply Navier’s solution for simply supported rectangular plates

Once complete, you’ll have taken the analytical modelling of plates about as far as is practical - of course, we could always cover more, diving deeper and deeper into the calculus, but for more complex geometries, restraints and loading conditions, it typically makes sense to explore computational approaches - which we’ll cover elsewhere!