In this lecture, we'll cover the following:

• Recap of the four-node plate element and its degrees of freedom.
• Definition of bending and transverse shear strains in the Reissner–Mindlin formulation.
• Construction and structure of the constitutive matrix (bending and shear components).
• Interpretation of stress resultants (moments and shear forces).
• Role and formulation of the strain–displacement matrix $B$ and the Jacobian matrix.
• How all components combine to form the element stiffness matrix $K$.

In this lecture, we revisit the full formulation of a four-node plate element and bring together the key components developed throughout the section. We begin by restating the element’s geometry, material properties, and nodal degrees of freedom, before reviewing how strains are defined. In particular, we distinguish between bending strains, which vary linearly through the thickness, and transverse shear strains, which are assumed constant in the Reissner–Mindlin theory to account for thick plate behaviour.

We then consolidate how material behaviour is captured through the constitutive matrix, split into bending and shear parts, and how this links stresses to strains. Building on this, we reinterpret stresses as stress resultants (moments and shear forces) obtained by integrating through the plate thickness, introducing the generalised constitutive matrix. Finally, we connect the strain–displacement matrix, the Jacobian, and material properties to show how they collectively enable the computation of the element stiffness matrix via integration, completing the theoretical framework needed before moving on to implementation.

Next up

With the theory now consolidated, the next section introduces the principle of virtual work and shows how it leads to the element stiffness matrix $K$.