# OS-V: 0080 Buckling of Shells and Composites with Offset

A test of influence of offset on buckling solution for shells, including composite with offset Z0 and element offset ZOFFS.

## Model Files

Before you begin, copy the file(s) used in this problem
to your working directory.

## Benchmark Model

Here, you solve several problems to calculate the critical load on different conditions. The model is a simply supported beam of height 1 mm, breadth 2 mm and length 100 mm with one end constrained in all DOFs and an axial load applied on the other end.

The material properties for the beam are:

**MAT1**- Young's Modulus
- 1 x 10
^{6}N/mm^{2} - Poisson's Ratio
- 0.0
- Density
- 2 kg/mm
^{3} - Thermal Expansion Coefficient
- 1 x 10
^{-4}ºC^{-1} - Reference Temperature for Thermal Loading
- 300ºC

The different case description of the problem are:

- Buckling without offset.
- Buckling with moment equivalent to offset.
- Buckling with offset created by a frame.
- Buckling with offset applied through ZOFFS.
- Buckling of composite with non-symmetrical layup.
- Buckling of composite with offset.

The theoretical critical buckling load is calculated using the Euler Buckling equation:

$${f}_{crit}=\pi \frac{EI}{{\left(KL\right)}^{2}}$$

Where,

- ${f}_{crit}$
- Maximum or critical force
- $E$
- Modulus of elasticity
- $I$
- Area moment of inertia (second moment of area)
- $L$
- Unsupported length of the beam
- $K$
- Column effective length factor (for one end fixed and the other end free, $K$ =2)

## Results

Quantity | Theoretical | No-offset | Normalized |
---|---|---|---|

$\lambda $
_{cr}^{(1)} |
4.1123 | 4.1208 | 0.997937 |

$\lambda $
_{cr}^{(2)} |
16.449 | 16.513 | 0.996124 |

$\lambda $
_{cr}^{(3)} |
37.011 | 37.701 | 0.981698 |

$\lambda $
_{cr}^{(4)} |
102.81 | 108.19 | 0.950273 |

Quantity | Theoretical | No-offset + Moment | Normalized |
---|---|---|---|

$\lambda $
_{cr}^{(1)} |
4.1123 | 4.1208 | 0.997937 |

$\lambda $
_{cr}^{(2)} |
16.449 | 16.513 | 0.996124 |

$\lambda $
_{cr}^{(3)} |
37.011 | 37.701 | 0.981698 |

$\lambda $
_{cr}^{(4)} |
102.81 | 108.19 | 0.950273 |

Quantity | Theoretical | C-Frame | Normalized |
---|---|---|---|

$\lambda $
_{cr}^{(1)} |
4.1123 | 4.1208 | 0.997937 |

$\lambda $
_{cr}^{(2)} |
16.449 | 16.513 | 0.996124 |

$\lambda $
_{cr }^{(3)} |
37.011 | 37.700 | 0.981724 |

$\lambda $
_{cr}^{(4)} |
102.81 | 108.19 | 0.950273 |

Quantity | Theoretical | ZOFFS | Normalized |
---|---|---|---|

$\lambda $
_{cr}^{(1)} |
4.1123 | 4.1208 | 0.997937 |

$\lambda $
_{cr}^{(2)} |
16.449 | 16.513 | 0.996124 |

$\lambda $
_{cr}^{(3)} |
37.011 | 37.700 | 0.981724 |

$\lambda $
_{cr}^{(4)} |
102.81 | 108.19 | 0.950273 |

Quantity | Theoretical | Non-symmetric Layup | Normalized |
---|---|---|---|

$\lambda $
_{cr}^{(1)} |
4.1123 | 4.1203 | 0.998058 |

$\lambda $
_{cr}^{(2)} |
16.449 | 16.510 | 0.996305 |

$\lambda $
_{cr}^{(3)} |
37.011 | 37.663 | 0.982689 |

$\lambda $
_{cr}^{(4)} |
102.81 | 107.89 | 0.952915 |

Quantity | Theoretical | Offset Composite | Normalized |
---|---|---|---|

$\lambda $
_{cr}^{(1)} |
4.1123 | 4.1203 | 0.998058 |

$\lambda $
_{cr}^{(2)} |
16.449 | 16.510 | 0.996305 |

$\lambda $
_{cr}^{(3)} |
37.011 | 37.663 | 0.982689 |

$\lambda $
_{cr}^{(4)} |
102.81 | 107.89 | 0.952915 |