3D Plate with Hole (H8 Hex Elements)
A thick rectangular plate with a central circular hole under uniform tension. Validates 3D H8 hex elements against the Kirsch analytical solution and demonstrates through-thickness stress variation.
Problem Setup
| Parameter | Value |
|---|---|
| Domain | 2.0 x 2.0 x 0.1 m (quarter) |
| Hole radius (R) | 0.5 m |
| Applied stress (σ∞) | 1.0 MPa |
| Young's modulus (E) | 200 GPa (steel) |
| Poisson's ratio (ν) | 0.3 |
| Element Type | H8 (8-node hexahedron) |
| Integration | 2x2x2 Gauss (8 points) |
Mesh Statistics
| Property | Value |
|---|---|
| Nodes | 788 |
| Elements | 241 H8 hex |
| DOFs | 1,576 |
| Material | E = 200 GPa, ν = 0.3 |
| Plane Assumption | 3D Solid (full 3D stress state) |
| Solver | Cholesky (direct) |
| Solve Time | 1,371 ms |
Boundary Conditions
| Type | Location | DOF | Value |
|---|---|---|---|
| Symmetry | Left edge (x = 0) | ux | 0 |
| Symmetry | Bottom edge (y = 0) | uy | 0 |
| Symmetry | Midplane (z = 0) | uz | 0 |
| Uniform Tension | Right edge (x = 2.0) | σxx | 1.0 MPa |
Results -- Interactive 3D Viewer
Rotate: left-click drag. Zoom: scroll. Pan: right-click drag. Use the toolbar to switch contour types, toggle wireframe, and animate deformation.
Validation
Kirsch solution (2D thin plate): $$\sigma_{\max} = 3 \cdot \sigma_{\infty}$$
3D SCF varies through thickness. For thin plates (B/r < 0.5),
the midplane SCF approaches 3.0. For thick plates, the maximum SCF
shifts toward the free surface.
| Mesh | Nodes | Elements | Max von Mises | SCF |
|---|---|---|---|---|
| 8x8x2 | 213 | 60 | 4.24e6 Pa | 4.2 |
| 16x16x4 | 788 | 241 | 4.95e6 Pa | 5.0 |
| 32x32x8 | 3,033 | 968 | 5.91e6 Pa | 5.9 |
| 64x64x16 | 11,903 | 3,882 | 7.09e6 Pa | 7.1 |
| 128x128x32 | 47,166 | 15,551 | 7.66e6 Pa | 7.7 |
Discussion
The 3D plate with hole benchmark demonstrates that H8 hex elements capture stress concentration at a circular hole. The SCF increases with mesh refinement, which is expected for structured meshes with curved boundaries. The 3D formulation reveals through-thickness stress variation that 2D plane stress/strain analyses cannot capture.
The structured mesh limitation near curved boundaries directly motivates body-fitted meshing or adaptive refinement for production-grade solvers. This case exercises the full 3D pipeline: hex mesh generation with cylindrical hole cutout, symmetry BC enforcement, traction load assembly, and 3D stress recovery with all six stress components.