Welcome to section 2. In this section, we’re going to lay the theoretical foundation for everything that follows in this course.

Remember, the fundamental building block for any finite element code is the element stiffness matrix, which captures the relationship between actions applied to an element and the resulting displacements of the element.

In order for us to build this element stiffness matrix, we need two quantities: a constitutive matrix that encodes information about the element material and a strain-displacement matrix that relates the strains in the element to the vector of nodal displacements for the element.

Our aim in this section is to develop these matrices. To do so, we will need to start from a fundamental mechanical model of plate behaviour - this is the Reissner-Mindlin model we touched on in the previous section.

We start by focusing on building the constitutive matrix. We do this by establishing the stress and strain fields for the element. To do this, we build on the core idea behind Ressner-Mindlin theory, the assumption that plane sections remain straight after deformation, but not necessarily perpendicular to the middle plane of the element.

Then, we can turn our attention to developing the strain-displacement matrix. Along the way, we’ll cover several key concepts, such as turning stresses into stress resultants, how shape functions are used to interpolate nodal quantities such as displacements and the role of the Jacobian matrix in mapping from global to local element coordinates. If you’ve completed the prerequisite course, many of these concepts will be familiar, but we’ll be working on the assumption that they’re new.

This section is only required if you’re not already familiar with the underlying mechanics, although I recommend completing this section for the most complete understanding of what’s to follow later in the course.