Fixed Beam Under Changing Temperature
Overview
The aim of this test case is to validate the following functions:
- thermomechanical analysis
The simulation results of SimScale were compared to the analytical results in [Roark]. The meshes used in (A) and (B) were created with the parametrized-tetrahedralization-tool on the SimScale platform and the meshes used in (C) and (D) were meshed with Salome.
Import validation project into workspace
Geometry
Geometry of the beam
The bar has a cross section of 0.05 x 0.05 m2 and a length of l = 1.0 m.
Analysis type and Domain
Tool Type : Calculix, Code_Aster
Analysis Type : Thermomechanical
Mesh and Element types :
| Case | Mesh type | Number of nodes | Element type | Tool type |
|---|---|---|---|---|
| (A) | linear tetrahedral | 84 | 3D isoparametric | Calculix |
| (B) | quadratic tetrahedral | 369 | 3D isoparametric | Calculix |
| (C) | linear hexahedral | 369 | 3D isoparametric | Calculix |
| (D) | quadratic hexahedral | 1221 | 3D isoparametric | Calculix |
| (E) | linear tetrahedral | 84 | 3D isoparametric | Code_Aster |
| (F) | quadratic tetrahedral | 369 | 3D isoparametric | Code_Aster |
| (G) | linear hexahedral | 369 | 3D isoparametric | Code_Aster |
| (H) | quadratic hexahedral | 1221 | 3D isoparametric | Code_Aster |
Mesh used for the SimScale case (A)
Mesh used for the SimScale case (C)
Simulation Setup
Material:
- isotropic: E = 205 GPa, ν = 0.3
- thermal expansion coefficient γ = 12E-6 1/K
Constraints (Boundary Conditions):
- Fixed x-translation of face ABCD and face A’B’C’D’
- Additional points were fixed to restrain the remaining translations and rotations
Loads:
- Temperature change ΔT = 10 K on the outer surface
Reference Solution
$$\sigma = \Delta T \gamma E$$
The equation (1) used to solve the problem is derived in [Roark]. Inserting the values described in the previous chapter results in the unit stress σ = 24.6 Mpa in the whole beam.
Results
Comparison of the unit stress σ in the beam obtained with SimScale with the results derived from the equations presented in [Roark].
| Case | [Roark] | SimScale | Error (%) |
|---|---|---|---|
| (A) | 24.6 | 24.6 | 0.00% |
| (B) | 24.6 | 24.6 | 0.00% |
| (C) | 24.6 | 24.599 | 0.004% |
| (D) | 24.6 | 24.599-24.601 | ± 0.004% |
| (E) | 24.6 | 24.6 | 0.00% |
| (F) | 24.6 | 24.6 | 0.00% |
| (G) | 24.6 | 24.6 | 0.00% |
| (H) | 24.6 | 24.6 | 0.00% |
Note: The worst results was obtained with mesh (D). This was due to an interpolation error and is shown in the picture below.
Stress filed obtained with mesh (D)
References
| [Roark] | (1, 2, 3, 4) (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh |