In this lecture, we will cover the following:

• How adding an arbitrary drilling stiffness to the local element stiffness matrix improves the stiffness matrix condition,
• We examine why the drilling stiffness does not couple into the other degrees of freedom,
• We reduce the theory to a compact implementation strategy.

We begin by introducing the central fix: adding a small arbitrary stiffness associated with the missing drilling rotation. In the local element stiffness matrix, this stiffness is placed only on the drilling diagonal terms. Because the off-diagonal coupling terms remain zero, the added value prevents the drilling mode from mapping to zero without contaminating the meaningful membrane, bending, or transverse shear behaviour.

We then work through how this local drilling stiffness appears once transformed into the global reference frame. Although the derivation involves expanding the stiffness and transformation matrices, the final implementation is relatively simple: calculate a drilling contribution based on the element normal, place it into the rotational blocks of the element stiffness matrix, and then assemble the structure as usual.

Next up

In the next lecture, we will implement this drilling stabilisation strategy directly in our custom shell solver and check whether it removes the rank deficiency.