In this lecture, we'll cover the following:

• Separating the element stiffness matrix into bending and shear components.
• Using different numbers of Gauss sampling points for bending and shear.
• Constructing a reusable function to compute the stiffness matrix.
• Implementing bending and shear strain–displacement matrices independently.
• Verifying equivalence with the previous full stiffness matrix formulation.

In this lecture, we build on the previous formulation of the element stiffness matrix by splitting it into two distinct contributions: bending and shear. We follow the same underlying Gauss quadrature process but modify the implementation so that each component can use a different number of sampling points. This is achieved by forming separate strain–displacement matrices and constitutive matrices for bending and shear, computing their respective stiffness contributions, and then summing them to recover the full $12\times 12$ element stiffness matrix.

We implement this approach in a dedicated function that accepts independent sampling choices for bending and shear, making the formulation more flexible. Although the mathematical operations remain largely unchanged, this restructuring allows us to control integration schemes more precisely. We confirm that the new function reproduces the same results as the original formulation when identical sampling is used, and we prepare it for reuse throughout the remainder of the course by placing it in a utilities file.

Next up

With the stiffness matrix now split into bending and shear components, the next lecture turns to computing the equivalent nodal force vector for distributed loads.