In this lecture, we'll cover:

• A quick review of the local element stiffness matrix and the force–displacement relationship,
• How to distinguish between nodal actions and equivalent nodal actions from distributed loading,
• How the element stiffness matrix is formed using the $B^T D B$ integral over the element area,
• How equivalent nodal actions are obtained from distributed forces and moments using shape functions,
• How the stiffness matrix can be split into membrane, bending, and shear contributions,

In this lecture, we revisit the local element stiffness matrix equation, mainly to refresh your memory before moving on to coding. We start with the familiar force equals stiffness times displacement relationship, where the local stiffness matrix acts on the local nodal displacement vector and the right-hand side includes both directly applied nodal actions and equivalent nodal actions arising from distributed loads such as self-weight. We then restate the stiffness matrix as the area integral of $B^T D B$, noting that the thickness integration has already been handled when deriving the constitutive matrix.

We then break the formulation into its membrane, bending, and shear parts. Because the constitutive matrix is effectively block diagonal for the homogeneous material considered here, the element stiffness matrix can be written as a simple superposition of independent membrane, bending, and shear stiffness matrices, with no coupling at the element level. We also identify the equivalent nodal force vector as the area integral of the shape function matrix multiplied by the vector of distributed actions.

Next up

In the next lecture, we will look at how the stiffness matrix integral is actually computed in practice, using the Jacobian and Gauss numerical integration to evaluate the expression at a set of sampling points.