3D Cantilever Beam (H8 Hex Elements)
A 3D cantilever beam clamped at the left face with a point load at the right tip. Validates the H8 hexahedral element formulation against the Euler-Bernoulli beam analytical solution. Demonstrates 3D solid element capability with full 3D stress recovery.
Problem Setup
| Parameter | Value |
|---|---|
| Length (L) | 1.0 m |
| Height (H) | 0.1 m |
| Thickness (t) | 0.1 m |
| Young's modulus (E) | 200 GPa (steel) |
| Poisson's ratio (ν) | 0.3 |
| Tip load (P) | -1000 N (downward) |
| Element Type | H8 (8-node hexahedron) |
| Integration | 2x2x2 Gauss (8 points) |
Mesh Statistics
| Property | Value |
|---|---|
| Nodes | 81 |
| Elements | 32 H8 hex |
| DOFs | 243 |
| Material | E = 200 GPa, ν = 0.3 |
| Plane Assumption | 3D Solid (full 3D stress state) |
| Solver | Cholesky (direct) |
| Solve Time | 15.7 ms |
Boundary Conditions
| Type | Location | DOF | Value |
|---|---|---|---|
| Fixed Face | Left face (x = 0, all nodes) | ux, uy, uz | 0 |
| Point Load | Tip centroid (top-right-front node) | Fy | -1000 N |
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
Analytical tip deflection (Euler-Bernoulli): $$\delta = \frac{PL^3}{3EI}$$
Analytical max stress: $$\sigma = \frac{|P| \cdot H}{2I}$$
Note: H8 with full 2x2x2 Gauss integration exhibits shear locking in bending.
The error decreases with mesh refinement as the element resolves the bending mode.
| Mesh | Nodes | Elements | Tip Deflection | Error |
|---|---|---|---|---|
| 8x2x2 | 81 | 32 | 1.23e-4 m | 38.6% |
| 16x4x4 | 405 | 256 | -- | -- |
| 32x8x8 | 2,025 | 2,048 | -- | -- |
Discussion
The 3D cantilever beam validates the H8 hexahedral element formulation. With full 2x2x2 Gauss integration, H8 elements exhibit shear locking in bending-dominated problems -- the same phenomenon observed in Q4 elements. The error on the coarse 8x2x2 mesh (38.6%) is expected because the element cannot represent the quadratic displacement field needed for pure bending.
As the mesh is refined, the solution converges toward the Euler-Bernoulli analytical solution. The 3D formulation captures all stress components (sigma_xx, sigma_yy, sigma_zz, sigma_xy, sigma_yz, sigma_xz), providing a complete 3D stress state that 2D plane stress/strain analyses cannot.
This case exercises the full 3D pipeline: structured hex mesh generation, 3D boundary condition enforcement (face constraints), point load assembly, Cholesky factorization, and 3D stress recovery with principal stress computation.