# Circular Shaft Under Torque¶

## Overview¶

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

- torque
- remote force

The simulation results of SimScale were compared to the numerical results presented in [Roark]. The meshes used in (A) and (B) were created with the parametrized-tetrahedralization-tool on the SimScale platform. The meshes used in (C) and (D) were locally created with Salome.

## 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 | 12940 | 3D isoparametric |

(B) | quadratic tetrahedral | 94919 | 3D isoparametric |

(C) | linear hexahedral | 10325 | 3D isoparametric |

(D) | quadratic hexahedral | 40935 | 3D isoparametric |

## Simulation Setup¶

Material:

- isotropic: E = 208 GPa, \(\nu\) = 0.3, G = 80 GPa

Constraints:

- Face A is fixed

Loads:

- Torque \(T\) of 50000 N/m on face B

## Reference Solution¶

\[\begin{equation}\label{ref1}
J = \frac{1}{2} \pi r^4 = 1.57 \cdot 10^{-4} m^4
\end{equation}\]\[\begin{equation}\label{ref2}
\tau_{max} = \frac {T r} {J} = 31.847 N/mm^2
\end{equation}\]\[\begin{equation}\label{ref3}
\theta = \frac {\tau_{max} l} {G r} = 1.9904 \cdot 10^{-4} rad
\end{equation}\]

## Results¶

Important

- The analytical solution assumes an undeformable surface on the face, wich is subject to the torque bc. This can be modelled in Code_Aster with the option ‘undeformable’ in the remote displacement bc.
- CalculiX has no such option and therefore the stresses in the entitites, which are assigned to the torque bc, are unphysical in CalculiX. So the stresses were computed at a point on the edge of face A.

Comparison of the maximum shear stress \(\tau_{max}\) and the angle of twist \(\theta\) obtained with SimScale and the results derived from [Roark].

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

(A) | CalculiX | 31.847 | 30.212 | 5.13% |

(A) | Code_Aster | 31.847 | 30.212 | 5.13% |

(B) | CalculiX | 31.847 | 31.842 | 0.02% |

(B) | Code_Aster | 31.847 | 31.838 | 0.03% |

(C) | Code_Aster | 31.847 | 32.066 | -0.69% |

(D) | Code_Aster | 31.847 | 31.879 | -0.10% |

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

(A) | CalculiX | 0.0019904 | 0.001953 | 1.88% |

(A) | Code_Aster | 0.0019904 | 0.001969 | 1.08% |

(B) | CalculiX | 0.0019904 | 0.001969 | 1.08% |

(B) | Code_Aster | 0.0019904 | 0.001989 | 0.07% |

(C) | Code_Aster | 0.0019904 | 0.002004 | -0.68% |

(D) | Code_Aster | 0.0019904 | 0.00199 | 0.02% |