In this lecture, we will cover the following:

• We will define a drilling degree of freedom as rotation about the element normal,
• We will see why co-planar connected elements can leave drilling rotations unrestrained,
• We will start linking drilling degrees of freedom with stiffness matrix singularity.

We start by returning to the local element reference frame, where each shell node has five degrees of freedom: membrane displacements, transverse displacement, and two rotations. When the element is placed into the global structure, rotations are expressed relative to the global axes, and this leads to a sixth rotational degree of freedom at every node.

We then identify this extra rotation as the drilling degree of freedom: a rotation about the element normal. Because the local shell formulation does not include stiffness associated with this rotation, the element provides no resistance to it. When co-planar elements share nodes, these drilling rotations can remain unrestrained, creating the conditions for a singular stiffness matrix.

Next up

In the next lecture, we will make this connection more precise by looking directly at what stiffness matrix singularity means.