In this section, we'll cover the following:

• Identifying and eliminating zero-energy (spurious) displacement modes in the model.
• Deriving the substitute transverse strain–displacement matrix.
• Modifying the element stiffness matrix within the code.
• Validating the corrected model against an OpenSeesPy benchmark.
• Extending the model to include point loads and patch loading.

In this section, we focus on resolving the issue of zero-energy displacement modes by introducing a correction to the element formulation. We begin by carefully deriving the substitute transverse strain–displacement matrix, ensuring we fully understand the theoretical basis for the modification before applying it in code. This derivation underpins the adjustments made to the element stiffness matrix, which are key to eliminating the spurious behaviour observed earlier.

We then implement these changes in the solver and validate the updated model against an OpenSeesPy benchmark, confirming that our custom finite element code produces accurate and reliable results. Finally, we round off the implementation by incorporating point loads and patch loading, completing the development of a fully functional and validated finite element analysis tool.

Next up

In the next lecture, we will begin deriving the substitute transverse shear strain matrix, starting with the assumed strain field and its polynomial representation.