Example: Plate with Hole
A tensile plate with a circular hole — stress concentration and mesh convergence in practice.
A plate with a central hole under tension is a classic FEM validation example. It has a known stress concentration factor and no singularity — which makes it ideal for learning mesh convergence.
Understanding stress concentration
A hole in a tensile plate creates a stress concentration at the hole edge. The theoretical stress concentration factor for a small hole in a wide plate is:
Kt ≈ 3
The maximum stress at the hole edge is approximately 3× the nominal (average) stress in the cross-section.
The maximum stress at the hole edge is about 3× the nominal stress. FEM should reproduce this factor well, which makes it a reliable validation case.
Meshing strategy
- Fine mesh at the hole edge (at least 12 elements around the circumference)
- Coarse mesh in the far-field regions
- Use symmetry (quarter model) to reduce element count
- Run a mesh convergence study
Why there's no singularity here
Unlike a sharp corner, a circular hole does not produce a singularity. The stress at the hole edge converges to a finite value as the mesh is refined — this is what makes the example suitable for convergence studies.
If your mesh convergence study shows the stress at the hole edge keeps growing without limit, something is wrong with the model (possibly a sharp edge adjacent to the hole).
Validation criterion
Your FEM result should give a stress concentration factor within 5% of the theoretical Kt ≈ 3. If it does, your meshing approach and workflow are correct.
Use this example as a reference benchmark. Once you can reproduce Kt ≈ 3 reliably, you have confidence in your meshing workflow for other geometries.