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.

Import validation project into workspace

The shaft has a radius r

$r$= 0.1 m and a length of l

$l$= 0.5 m.

**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 |

Material:

- isotropic: E = 208 GPa, ν

$\nu $= 0.3, G = 80 GPa

Constraints:

- Face A is fixed

Loads:

- Torque T

$T$of 50000 N/m on face B

J=12πr4=1.57⋅10−4m4(1)

$$\begin{array}{}\text{(1)}& J=\frac{1}{2}\pi {r}^{4}=1.57\cdot {10}^{-4}{m}^{4}\end{array}$$

τmax=TrJ=31.847N/mm2(2)

$$\begin{array}{}\text{(2)}& {\tau}_{max}=\frac{Tr}{J}=31.847N/m{m}^{2}\end{array}$$

θ=τmaxlGr=1.9904⋅10−4rad(3)

$$\begin{array}{}\text{(3)}& \theta =\frac{{\tau}_{max}l}{Gr}=1.9904\cdot {10}^{-4}rad\end{array}$$

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 τmax

${\tau}_{max}$and the angle of twist θ

$\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% |