This validation case belongs to solid mechanics. The aim of this test case is to validate the following parameter:

- Centrifugal force

The simulation results of the cloud-based simulation platform SimScale were compared to the results presented in [HPLA100]\(^1\).

The geometry used for the case is as follows:

The solid body is created by rotating face ABCD by 45° and due to the symmetry of the cylinder, only one-quarter of the cylinder is modeled.

The coordinates of points A, B, C, and D can be seen below:

Point | X | Y | Z |

A | 0.0195 | 0 | 0.01 |

B | 0.0205 | 0 | 0.01 |

C | 0.0205 | 0 | 0 |

D | 0.0195 | 0 | 0 |

**Tool Type**: Code_Aster

**Analysis Type**: Dynamic

**Mesh and Element Types**:

The mesh was created with the *Standard* meshing algorithm in SimScale.

Case | Element Type | Number of Nodes | Element Technology |

A | 1st Order Tetrahedral | 3242 | Standard |

B | 2nd Order Tetrahedral | 20965 | Reduced Integration |

**Material**:

- Steel (linear elastic)
- Isotropic: \(E\) = 200 \(GPa\)
- \(\nu\) = 0.3
- \(\rho\) = 8000 \(kg/m^3\)

**Initial and/or Boundary Conditions**:

- Constraints:
- \(d_z\) = 0:
- Face AA’BB’
- Face CC’DD’

- Symmetry:
- Face ABCD
- Face A’B’C’D’

- \(d_z\) = 0:
- Load:
- Centrifugal force with a rotational speed \(\omega\) of 1 \(rad/s\) around the z-axis applied to the whole body.

The reference solutions for stress and displacement is calculated with the equations below:

$$ u(r)=\frac{−(1+\nu)(1−2\nu)}{(1−\nu)E}\rho\Omega^2\frac{r^3}{8}+Ar+\frac{B}{r}\tag{1} $$

$$\sigma_{zz}(r) = \frac{-\nu}{1-\nu}\rho\Omega^2\frac{r^2}{2}+\frac{2 \nu E}{(1+\nu)(1-2\nu)}A\tag{2}$$

$$A = \frac{(3-2\nu)(1+\nu)(1-2\nu)}{4(1-\nu)E}\rho\Omega^2R^2(1-x^2) = 7.13588\times 10^{-12} \ mm^2\tag{3}$$

$$B = \frac{(3-2\nu)(1+\nu)(1-2\nu)}{8(1-\nu)E}\rho\Omega^2R^4(1-x^2)^2 = 3.561258\times 10^{-15} \ mm^2\tag{4}$$

$$x = \frac{h}{2R} =\frac{0.001 \ m}{2\times0.02 \ m} = 0.025\tag{5}$$

Term \(h\) is the thickness of the cross-section and \(R\) is the radius of the middle surface of the cylinder and both are in meters (\(m\)).

The rotational force is validated by comparing the displacement \(u_r\) in meters \(m\) and Cauchy stresses \(\sigma_{zz}\) in \(N/m^2\) obtained from SimScale against the reference results obtained from [HPLA100] is given below:

Case | Quantity | HPLA-100 | SimScale | Error [%] |

A | \(u_r(r\) = 0.0195 \(m)\) | 2.9424e-13 | 2.939e-13 | -0.109 |

A | \(u_r(r\) = 0.0205 \(m)\) | 2.8801e-13 | 2.877e-13 | -0.113 |

A | \(\sigma_{zz}(r\) = 0.0195 \(m)\) | 0.99488 | 0.990826 | -0.407 |

A | \(\sigma_{zz}(r\) = 0.0205 \(m)\) | 0.92631 | 0.931388 | +0.548 |

B | \(u_r(r\) = 0.0195 \(m)\) | 2.9424e-13 | 2.942E-13 | -0.001% |

B | \(u_r(r\) = 0.0205 \(m)\) | 2.8801e-13 | 2.880E-13 | -0.001% |

B | \(\sigma_{zz}(r\) = 0.0195 \(m)\) | 0.99488 | 0.995056 | +0.018 |

B | \(\sigma_{zz}(r\) = 0.0205 \(m)\) | 0.92631 | 0.926469 | +0.017 |

The stress experienced by the cylinder under the rotational force can be seen below:

Note

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Last updated: September 24th, 2021

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