# Hollow cylinder in plain strain condition¶

## Overview¶

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

- distributed pressure
- symmmetry BC
- nodal displacements
- strains and nodal stresses

The simulation results of SimScale were compared to the analytical results in [SSLV04_A] and the numerical results in [SSLV04]. 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, resulting in four different modelizations.

## Geometry¶

A | B | E | F | A’ | B’ | E’ | F’ | |
---|---|---|---|---|---|---|---|---|

x [m] | 0.1 | 0.2 | 2/2 | 2 | 0.1 | 0.2 | 2/2 | 2 |

y [m] | 0 | 0 | 2/2 | 2 | 0 | 0 | 2/2 | 2 |

z [m] | 0 | 0 | 0 | 0 | 0.01 | 0.01 | 0.01 | 0.01 |

## Analysis type and Domain¶

**Tool Type** : Code_Aster

**Analysis Type** : Static (linear elastic)

**Mesh and Element types** :

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

(A) | linear tetrahedral | 1542 | 3D isoparametric |

(B) | quadratic tetrahedral | 9623 | 3D isoparametric |

(C) | linear hexahedral | 768 | 3D isoparametric |

(D) | quadratic hexahedral | 2720 | 3D isoparametric |

## Simulation Setup¶

Material:

- isotropic: E = 200 GPa, \(\nu\) = 0.3

Constraints:

- Face EFE’F’ zero normal-displacement
- Face ABA’B’ zero y-displacement
- Face ABEF and face A’B’E’F’ zero z-displacements fixed

Loads:

- Pressure of 60 MPa on Face AEA’E’

## Results¶

Comparison of the displacements and stresse at point A obtained with SimScale with the results presented in [SSLV04]. The Error was calculated with respect to [SSLV04_A].

Case | Quantity | [SSLV04_A] | [SSLV04] | SimScale | Error (%) |
---|---|---|---|---|---|

(A) | \(d_X\) [m] | 5.90E-005 | 5.89E-005 | 5.72E-005 | -3.10 |

(B) | \(d_X\) [m] | 5.90E-005 | 5.90E-005 | 5.72E-005 | -3.05 |

(C) | \(d_X\) [m] | 5.90E-005 | 5.90E-005 | 5.71E-005 | -3.14 |

(D) | \(d_X\) [m] | 5.90E-005 | 5.89E-005 | 5.72E-005 | -3.05 |

(A) | \(\sigma_{XX}\) [Mpa] | -6.00E+001 | -5.72E+001 | -5.61E+001 | -6.45 |

(B) | \(\sigma_{XX}\) [Mpa] | -6.00E+001 | -6.04E+001 | -6.00E+001 | -0.02 |

(C) | \(\sigma_{XX}\) [Mpa] | -6.00E+001 | -5.92E+001 | -5.12E+001 | -14.63 |

(D) | \(\sigma_{XX}\) [Mpa] | -6.00E+001 | -5.94E+001 | -5.99E+001 | -0.19 |

(A) | \(\sigma_{YY}\) [Mpa] | 1.00E+002 | 9.79E+001 | 1.02E+002 | 1.51 |

(B) | \(\sigma_{YY}\) [Mpa] | 1.00E+002 | 9.92E+001 | 1.00E+002 | 0.00 |

(C) | \(\sigma_{YY}\) [Mpa] | 1.00E+002 | 1.00E+002 | 1.03E+002 | 3.49 |

(D) | \(\sigma_{YY}\) [Mpa] | 1.00E+002 | 9.96E+001 | 9.99E+001 | -0.06 |

(A) | \(\epsilon_{XX}\) | -4.50E-004 | -4.33E-004 | -4.53E-004 | 0.71 |

(B) | \(\epsilon_{XX}\) | -4.50E-004 | -4.49E-004 | -4.68E-004 | 3.98 |

(C) | \(\epsilon_{XX}\) | -4.50E-004 | -4.47E-004 | -4.35E-004 | -3.36 |

(D) | \(\epsilon_{XX}\) | -4.50E-004 | -4.47E-004 | -4.67E-004 | 3.86 |

(A) | \(\epsilon_{YY}\) | 5.90E-004 | 5.75E-004 | 5.71E-004 | -3.14 |

(B) | \(\epsilon_{YY}\) | 5.90E-004 | 5.88E-004 | 5.72E-004 | -3.05 |

(C) | \(\epsilon_{YY}\) | 5.90E-004 | 5.90E-004 | 5.71E-004 | -3.26 |

(D) | \(\epsilon_{YY}\) | 5.90E-004 | 5.87E-004 | 5.72E-004 | -3.13 |