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.