In this lecture, we'll cover the following:

• How shear locking is addressed by modifying Gauss quadrature sampling.
• Implementing separate sampling parameters for bending and shear contributions.
• Updating the stiffness matrix formulation to use adjustable integration points.
• Re-running validation and parameter sweeps to assess model accuracy.
• Comparing results across custom, OpenSeesPy, and PyNite models.

In this lecture, we revisit the shear locking issue and implement a practical remedy by reducing the number of integration points used in the shear component of the stiffness matrix. We modify our existing code to introduce separate parameters for bending and shear sampling, taking advantage of the earlier decision to split these contributions. This allows us to selectively apply reduced integration to the shear term while maintaining accuracy in the bending response.

We then re-run our simulations and parameter sweep to evaluate the impact of this change. The results show a significant improvement, with near-perfect agreement between the custom finite element model and reference solutions across the full range of plate thicknesses. This confirms both the effectiveness of the reduced integration approach and the correctness of our implementation, giving us confidence in the solver as we prepare to extend it further with mesh generation capabilities.

Next up

With the shear locking issue resolved, the next section introduces meshing with Gmsh, enabling us to move beyond simple rectangular meshes to more realistic geometries.