In this lecture, we will cover the following:

• We highlight the importance of an invertible stiffness matrix,
• We define stiffness matrix singularity in structural analysis terms,
• We connect null-space vectors to zero-energy displacement modes,
• We will distinguish rigid body modes from spurious zero-energy modes,
• We relate unrestrained drilling rotations to rank deficiency.

We start with the basic structural relationship between force, stiffness, and displacement. Solving the model requires the stiffness matrix to be invertible; if the matrix is singular, the displacement solution cannot be obtained. The lecture explains singularity in physical terms by describing a non-zero displacement pattern that produces no internal force.

We then learn that these displacement patterns are null-space vectors, and that each independent null-space vector corresponds to a zero-energy mode. Some zero-energy modes are expected, such as rigid body translations and rotations before restraints are applied. The problematic cases are spurious modes that remain after the structure is restrained, so we focus on unresisted drilling rotations at shared nodes as the relevant mechanism.

Next up

In the next lecture, we will apply this understanding to our specific shell structure that produced the unexpected solver behaviour.