In this lecture, we'll cover:

• How to use local reference frames to build a transformation matrix for an element
• How to convert element nodal coordinates from a global frame into a local frame using a rotation matrix,
• How to transform a local stiffness matrix into a global stiffness matrix,
• How these pieces fit together as preparation for building a full shell solver.

The lecture focuses on taking a single shell element and working through the full workflow needed for finite element analysis. We start by constructing the transformation matrix from the element’s local axes vectors, then use it to rotate global nodal coordinates back into a local 2D plane, where the element can be treated in terms of x and y coordinates only. Once we have the local coordinates, we feed them into the existing stiffness matrix function to obtain the local element stiffness matrix.

We then move from local back to global using the standard transformation $K_g = T^T K_l T$, which gives us a $24 \times 24$ global stiffness matrix from the $20 \times 20$ local matrix. The key idea is that the local formulation is conveniently obtained using our function from the previous section, but the global formulation is what we need for assembling and solving the full structure.

Next up

In the next section we'll combine the building blocks we've developed so far, the element stiffness and transformation matrices, and build our first complete shell solver.