In this lecture, we'll cover:

• How to define a local element axis system for a shell element,
• How to construct a local x-axis from nodal coordinates and normalise it to a unit vector,
• How to obtain a local z-axis using a vector cross product of two in-plane vectors,
• How to determine the local y-axis so that the three axes form a right-handed orthogonal basis,
• How the local vector basis relates to the direction cosines used in the transformation matrix,

We begin by building a local element reference frame in three steps. First, we choose the local x-axis to lie along, or parallel to, one edge of the element, and we form its unit vector from the nodal coordinate differences divided by the edge length. Next, we define a second in-plane vector and take a cross product to obtain a unit vector perpendicular to the element plane, giving the local z-axis. Finally, we use another cross product to find the local y-axis, ensuring the axes are mutually orthogonal and consistent with the right-hand rule.

We then connect this vector basis to the direction cosines that appear in the transformation matrix. Using the dot product definition, we see that the direction cosine between a local axis and a global axis is simply the corresponding component of the local unit vector, since the global basis vectors are unit vectors aligned with the coordinate axes. In this way, the components of the transformation matrix can be read directly from the local basis vectors.

Next up

In the next lecture, we'll put this into practice by coding an edge-aligned local axis system for a shell element and visualising it in 3D.