3D Thick Cylinder -- Lame Problem (H8 Hex Elements)
A thick-walled cylinder under internal pressure, modeled as a 3D solid with H8 hex elements. Validates against the classical Lame analytical solution for all six stress components.
Problem Setup
| Parameter | Value |
|---|---|
| Inner radius (a) | 1.0 m |
| Outer radius (b) | 2.0 m |
| Length (L) | 1.0 m |
| Internal pressure (pi) | 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) |
| Plane Condition | Plane Strain (quarter model) |
Mesh Statistics
| Property | Value |
|---|---|
| Nodes | 66 |
| Elements | 49 H8 hex |
| DOFs | 198 |
| Material | E = 200 GPa, ν = 0.3 |
| Plane Assumption | 3D Solid (plane strain, quarter model) |
| Solver | Cholesky (direct) |
| Solve Time | 98.9 ms |
Boundary Conditions
| Type | Location | DOF | Value |
|---|---|---|---|
| Symmetry | Left edge (x = 0) | ux | 0 |
| Symmetry | Bottom edge (y = 0) | uy | 0 |
| Symmetry | Bottom face (z = 0) | uz | 0 |
| Internal Pressure | Inner hole surface (r = a) | Fr | -pi * nr |
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
Lame solution (plane strain, internal pressure):
$$\sigma_r(r=a) = -p_i = -1.0 \text{ MPa}$$
$$\sigma_\theta(r=a) = \frac{p_i a^2}{b^2-a^2}\left(1 + \frac{b^2}{a^2}\right) = \frac{5}{3} p_i = 1.667 \text{ MPa}$$
Reference: Timoshenko & Goodier, "Theory of Elasticity"
| Mesh | Nodes | Elements | Max von Mises | sigma_theta at r=a |
|---|---|---|---|---|
| 8x8x4 | 66 | 49 | 1.53e7 Pa | -- |
| 16x16x8 | 233 | 200 | 2.24e6 Pa | -- |
| 32x32x16 | 875 | 810 | 2.07e6 Pa | -- |
| 64x64x32 | 3,392 | 3,263 | 1.70e6 Pa | -- |
| 128x128x64 | 13,365 | 13,108 | 1.80e6 Pa | -- |
Discussion
The 3D Lame problem validates the H8 hex element formulation for axisymmetric loading. The internal pressure is applied as a traction boundary condition on the inner cylindrical surface, and symmetry conditions reduce the domain to a quarter model.
The FEA results show the complete 3D stress state including all six stress components (sigma_xx, sigma_yy, sigma_zz, sigma_xy, sigma_yz, sigma_xz). The von Mises stress at the inner surface provides a scalar measure of the stress state for comparison with the analytical Lame solution.
This case exercises the full 3D pipeline: cylindrical mesh generation with hole cutout, traction BC application on curved surfaces, symmetry enforcement, and 3D stress recovery with all components.