# Fixed Beam Under Gravitational Load¶

## Overview¶

The aim of this test case is to validate the following functions:

- gravitational load

The simulation results of SimScale were compared to the analytical results derived from [Roark]. The meshes used in (A) and (B) were created with the parametrized-tetrahedralization-tool on the SimScale platform and then downloaded. Both meshes were then rotated and uploaded again. These were used in case (C) and (D). With this method we achieve the exact same mash for the rotated and non-rotated case.

## Analysis type and Domain¶

**Tool Type** : CalculiX/Code_Aster

**Analysis Type** : Static

**Mesh and Element types** :

Case | Mesh type | Number of nodes | Element type |
---|---|---|---|

(A) | linear tetrahedral | 6598 | 3D isoparametric |

(B) | quadratic tetrahedral | 44494 | 3D isoparametric |

(C) | linear tetrahedral | 6598 | 3D isoparametric |

(D) | quadratic tetrahedral | 44494 | 3D isoparametric |

## Simulation Setup¶

Material:

- isotropic: E = 205 GPa, \(\nu\) = 0.3, \(\rho\) = 7870 kg/m³

Constraints:

- Face ABCD is fixed: \(d_x=d_y=d_z=0.0\)

Loads:

- Case (A) and (B): gravitational load \(g\) = 9.81 m/s² pointing in the negative z-direction: \((0,0,-1)\)
- Case (C) and (D): gravitational load \(g\) = 9.81 m/s² now rotated +45° around the x-axis: \((0,1,-1)\)

## Reference Solution¶

The equation \(\eqref{ref4}\) used to solve the problem is derived in [Roark]. In equation \(\eqref{ref1}\) the gravitational load is converted in a line load \(w_a\).

## Results¶

Comparison of the displacement at the free end in the direction of the gravitation vector.

Case | Tool Type | [Roark] | SimScale | Error |
---|---|---|---|---|

(A) | CalculiX | -2.2597E-4 | -2.1340E-4 | 5.56% |

(B) | CalculiX | -2.2597E-4 | -2.2559E-4 | 0.17% |

(C) | CalculiX | -2.2597E-4 | -2.1340E-4 | 5.56% |

(D) | CalculiX | -2.2597E-4 | -2.2560E-4 | 0.16% |

(A) | Code_Aster | -2.2597E-4 | -2.1340E-4 | 5.56% |

(B) | Code_Aster | -2.2597E-4 | -2.2559E-4 | 0.17% |

(C) | Code_Aster | -2.2597E-4 | -2.1340E-4 | 5.56% |

(D) | Code_Aster | -2.2597E-4 | -2.2560E-4 | 0.16% |