In this lecture, we'll cover the following:

• Why the displacement patterns from the previous lecture occur.
• How selective (reduced) integration for shear introduces zero-energy modes.
• The role of the strain–displacement matrix and single-point sampling.
• What it means for a stiffness matrix to be rank deficient.
• Physical interpretation of zero-energy (hourglass) displacement patterns.

In this lecture, we explore the origin of the unusual zigzag displacement patterns observed previously and identify them as zero-energy modes caused by selective integration. We revisit how the shear stiffness matrix is constructed and show that using a one-point Gauss integration scheme effectively samples the strain–displacement matrix only at the element centre. This assumption forces the transverse shear strain to appear constant across the element, even though in reality it can vary, leading to a loss of critical information.

We then see that this under-integration causes the element stiffness matrix to become rank deficient, meaning it admits non-zero displacement patterns without any applied force. These non-physical modes, also known as hourglass modes, correspond to deformations with no associated strain energy. Through simple element and multi-element examples, we build intuition for how these patterns arise and why they manifest as characteristic zigzag shapes. We conclude that although selective integration alleviates shear locking, it introduces these spurious modes, motivating the need for an improved integration strategy in the next lecture.

Next up

In the next lecture, we will develop a detailed plan for fixing the Reissner–Mindlin element by addressing the root cause of shear locking without reintroducing zero-energy modes.