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!