# 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.

## 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 |

## Simulation Setup¶

Material:

- isotropic: E = 205 GPa, \(\nu\) = 0.3
- thermal expansion coefficient \(\gamma\) = 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 \(\Delta T\) = 10 K on the outer surface

## Reference Solution¶

The equation \(\eqref{ref1}\) used to solve the problem is derived in [Roark]. Inserting the values described in the previous chapter results in the unit stress \(\sigma\) = 24.6 Mpa in the whole beam.

## Results¶

Comparison of the unit stress \(\sigma\) 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 | \(\pm\) 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.