Theoretical and finite element analysis results of the rotating hollow disk problem are compared. The free and open-source finite element analysis software CalculiX is used. The hollow disk is modelled using axisymmetric elements in CalculiX.
| File | Contents |
|---|---|
| rotating_disk_axisymmetric_pre.fbd | Pre-processing script for CalculiX GraphiX |
| rotating_disk_axisymmetric.inp | CalculiX input |
| rotating_disk_axisymmetric_post.fbd | CalculiX GraphiX post-processing script |
| run_rotating_disk_axisymmetric_py.py | Python script to run CalculiX files |
| load_calculix_data.m | MATLAB file that loads CalculiX results to MATLAB workspace |
| rotating_disk_analytical_vs_calculix_matlab.m | Main MATLAB file |
| rotating_disk_axisymmetric_tex.tex | LaTeX file for the study report |
| rotating_disk_axisymmetric_tex.pdf | LaTeX output pdf for the study report |
| Parameter | Value | Unit | Description |
|---|---|---|---|
r_i |
28 | mm | Inner radius of the disk |
r_e |
125 | mm | Outer radius of the disk |
h |
4 | mm | Thickness of the disk |
E |
2.1e5 | MPa | Young's modulus of the disk material |
nu |
0.3 | Poisson's ration of the disk material | |
rho |
7.85e-9 | tonne/mm^3 | Density of the disk material |
Omega |
14e3 | rpm | Rotational speed of the disk |
Etyp |
qu8cr | Element type of the beam (CAX8R) |
Rotating hollow disk problem is shown in the following Figure.
Figure Rotating hollow disk.
Theory of elasticity gives the following results for the rotating hollow disk problem. Radial stress, hoop stress, and radial displacement are given in the following Figure in Eqs. 1, 2, and 3, respectively. See reference book.
Figure Theory of elasticity solution for thin rotating hollow disk.
The general purpose quadratic axisymmetric element with reduced integration (CAX8R) is used. One node at inner radius, r_i, is constrained in the axial direction. The finite element model of the rotating hollow disk problem is shown in the following Figure.
Figure Finite element model of the rotating hollow disk.
Dimensionless stresses and displacement are defined in the following Figure in Eqs. 4, 5, and 6.
Figure Dimensionless stress and displacement definitions.
Comparison between the analytical results obtained from MATLAB and finite element results obtained from CalculiX are presented in the following Figure.
Figure Rotating hollow disk: Analytical solution vs. CalculiX results.
The following table compares the minimum and maximum stresses, and displacements at inner and outer radii between analytical results and finite element solution.
| Analytical Solution, MATLAB | Finite Element Results, CalculiX | |
|---|---|---|
| Maximum radial stress | 65 MPa | 65 MPa |
| Minimum radial stress | 0 MPa | 0 MPa |
| Maximum hoop stress | 218 MPa | 218 MPa |
| Minimum hoop stress | 0 MPa | 0 MPa |
| Radial displacement at the inner radius | 0.029119 mm | 0.029116 mm |
| Radial displacement at the outer radius | 0.033738 mm | 0.033731 mm |
The radial stress, hoop stress, and radial displacements from CalculiX is following figures. Finite element results are extracted from the midline of the 2D axisymmetric disk from inner radius, r_i, to the outer radius, r_e.
Figure CalculiX radial stress plot.
Figure CalculiX hoop stress plot.
Figure CalculiX radial displacement plot.