# Stability Analysis of Steel Column According to EN 1993-1-1

### Technical Article

#### Structure

The column cross-section is a square hollow section. The structural system and the design load are shown in Figure 01.

#### Design

The steel columns displayed in Figure 1 are subjected to specific loads, and their flexural buckling will be analyzed. As in the present case there is axial compression, the design can be carried out according to EN 1993-1-1 Clause 6.3.1 The design according to the second-order analysis with precamber would be an option as well.

The following table shows the corresponding designs with manual calculation.

Cross-Section | QHP 260x8 | QHP 300x6 | QHP 250x6.3 |

Material | S 235 | S 235 | S 550 |

Classification | |||
---|---|---|---|

c/t | 28.5 | 46 | 35.68 |

Cross-Section Area [cm²] | 79.95 | 70.17 | 60.99 |

Stress [kN/cm²] | -12.51 | -14.25 | -16.40 |

Stress Ratio | 1.0 | 1.0 | 1.0 |

Material Factor | 1.0 | 1.0 | 1.0 |

max. c/t Class 1 | 33 | 33 | 21.57 |

max. c/tClass 2 | 38 | 38 | 24.84 |

max. c/t Class 3 | 42 | 42 | 27.45 |

Cross-Section Class | 1 | 4 | 4 |

Effective Cross-Section Properties | |||

${\mathrm\lambda}_\mathrm p\;=\;\frac{\displaystyle\frac{\mathrm b}{\mathrm t}}{28.4\;\cdot\;\mathrm\varepsilon\;\cdot\;\sqrt{{\mathrm k}_\mathrm\sigma}}$ | 0.81 | 0.97 | |

$\mathrm\rho\;=\;\frac{{\mathrm\lambda}_\mathrm p\;-\;0.188}{\mathrm\lambda_\mathrm p^2}\;\leq\;1.0$ | 0.89 | 0.8 | |

b_{eff} = ρ ∙ b [cm] | 24.81 | 17.98 | |

A and A_{eff} [cm²] | 79.95 | 63.51 | 49.79 |

Flexural Buckling Design | |||

${\mathrm N}_\mathrm{cr}\;=\;\frac{9.87\;\cdot\;\mathrm E\;\cdot\;{\mathrm I}_\mathrm z}{\mathrm L^2}$ [kN] | 1,745.66 | 2,089.14 | 1,246.46 |

N_{pl} = A ∙ f_{y} bzw. A_{eff} ∙ f_{y} [kN] | 1,878.83 | 1.492.49 | 2,738.45 |

$\mathrm\lambda\;=\;\sqrt{\frac{{\mathrm N}_\mathrm{pl}}{{\mathrm N}_\mathrm{cr}}}$ | 1.04 | 0.85 | 1.48 |

Imperfection Factor α | 0.21 (BC a) | 0.21 (BC a) | 0.13 (BC a_{0}) |

Φ = 0.5 ∙ [1 + α (λ - 0.2) + λ²] | 1.13 | 0.93 | 1.68 |

$\mathrm\chi\;=\;\frac1{\mathrm\Phi\;+\;\sqrt{\mathrm\Phi²\;-\;\mathrm\lambda²}}$ | 0.639 | 0.769 | 0.404 |

${\mathrm N}_{\mathrm b,\mathrm{Rd}}\;=\;\frac{\mathrm\chi\;\cdot\;{\mathrm A}_\mathrm{eff}\;\cdot\;{\mathrm f}_\mathrm y}{{\mathrm\gamma}_{\mathrm M1}}$ [kN] | 1,091.4 | 1,043.4 | 1,005.8 |

$\mathrm\eta\;=\;\frac{{\mathrm N}_\mathrm{Ed}}{{\mathrm N}_{\mathrm b,\mathrm{Rd}}}$ | 0.92 | 0.96 | 0.99 |

All three columns are capable of bearing the specified load. The middle column has larger external dimensions, but is slender with regard to the cross‑section dimensions. The cross-section is therefore classified in cross-section class 4 and the design has to be carried out with the effective cross-section area according to EN 1993-1-5. It results in a reduction of 11 % due to local buckling. However, the larger external dimensions have a positive effect on the critical load N_{cr}. As a result, this column has a more favorable slenderness ratio for flexural buckling.

In the case of the middle column (member 2), about 12% of the cross‑sectional area can be saved compared to the left column which has the same steel grade.

In the case of the right column (member 3), about 13 % of the cross‑sectional area can still be reduced compared to the middle column. The smaller dimensions have a negative effect on the critical load. About 20% of the cross‑sectional area is also ineffective due to the local buckling. The slenderness ratio of this column is significantly worse than in the case of the other columns, although it is allowed to calculate N_{cr }with the real cross-section area. However, the design is fulfilled due to the larger yield strength.

The design results with RF-/STEEL EC3 are shown in Figure 02.

#### Keywords

flexural buckling, stability, steel structure

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