In this lecture, we'll cover:

• How to relate local element displacements and forces to global coordinates,
• How the transformation matrix is built from direction cosines,
• How the element stiffness matrix is transformed from local to global form,
• How to recover local stress resultants from global nodal displacements through post-processing.

We begin by establishing that individual element stiffness matrices are first defined in each element’s local reference frame, but assembly into a global stiffness matrix requires these to be expressed in one common global reference frame. We therefore introduce a transformation matrix that handles this conversion.

We also explain that for a four-noded element, the local displacement vector has 20 components, whilst the global version has 24 because an extra rotational degree of freedom, $\theta_z$, appears at each node in the global frame. The transformation matrix is assembled from direction cosines, which are simply cosines of angles between local and global axes.

We then shift to the practical issue of results post-processing: nodal displacements and rotations are obtained directly in the global frame, but stress resultants such as membrane forces, bending moments, torsion, and shear forces are needed in the local frame for design. Therefore, we combine the local constitutive matrix, the local strain-displacement matrix, and the transformation matrix to express local stress resultants as a function of global nodal displacements. This gives us a clear workflow: solve globally for displacements first, then post-process those results to recover local stress resultants for structural design.

Next up

In the next lecture, you will look at how to construct the local element axes and derive the direction cosines that populate the transformation matrix.