Idealy a beam subjected to an axial compression force would be compressed but since there are always imperfections, it is not compression what determines strength of such members:

- load is not exactly on the axis of beam,
- the geometry of the beam is imperfect,
- the material is not ideally homogeneous,
- imperfections due to real state of connections (imperfections at the time of manufacturing), ...

Slender (slim) members in compression are subjected to buckling. The assessment is

$$ \begin{equation} N_{Ed} \leq N_{b,Rd}, \end{equation} $$where $N_{Ed}$ is design value of the compression force (from load) and
$N_{b,Rd}$ is design buckling resistance of the compression member. The
resistance is being found as a resistance in pure compression which is
*lowered* by reduction factor $\chi$ (effect of buckling):

For class 4 sections in $(\ref{eq:ass_buck})$ $A_{eff}$ instead of $A$ is used. Nonsymmetric class 4 members need allowance also for additional moment $\Delta M_{Ed}$ from eccentricity.

Note: holes for fasteners at the columns' ends need not to be considered to determine $A, \ A_{eff}$.

Reduction factor $\chi$ is found from the relevant buckling curve (according to cross-section) or from the formula:

$$ \begin{eqnarray} \chi &=& \frac{1}{\Phi + \sqrt{\Phi^2 - \overline{\lambda}^2}} \hskip2em \chi \leq 1.0 \label{eq:buck_chi} \\ \Phi &=& 0.5(1+\alpha(\overline{\lambda} -0.2) +\overline{\lambda}^2) \label{eq:buck_fi} \\ \overline{\lambda} &=& \sqrt{\frac{A f_y}{N_{cr}}}, \end{eqnarray} $$Symbol $\lambda$ is non-dimensional slenderness (for class 4: $A_{eff}$ is used instead of $A$), $\alpha$ is an imperfection factor (according to the profile taken from table 6.2 of the Eurocode), $N_{cr}$ is elastic critical force for the relevant buckling mode.

Buckling curve | $a_0$ | $a$ | $b$ | $c$ | $d$ |

$\alpha$ | 0.13 | 0.21 | 0.34 | 0.49 | 0.76 |

The elastic critical force comes from the Euler theory (1757):

$$ \begin{equation} N_{cr} = \pi^2 \frac{EI}{L^2_{cr}}, \label{eq:buckl_cL}\\ L_{cr} = \beta L, \end{equation} $$letter $\beta$ is coefficient of the length for buckling according to Euler and the basic cases are listed on the picture. In practice we determine non-dimensional slenderness as

$$ \begin{eqnarray} \overline\lambda &=& \frac{\lambda}{\lambda_1}, \\ \lambda_1 &=& \pi \sqrt{\frac{E}{f_y}} = 93.9 \sqrt\frac{235}{f_y} = 93.9 \epsilon,\\ \lambda &=& \frac{L_{cr}}{i}. \end{eqnarray} $$Above $i = \sqrt{{I}/{A}}$ is radius of gyration about relevant axis and for standard sections can be found in tables.

Then

$$ \begin{equation} \overline\lambda = \sqrt {A f_y / N_{cr}} \end{equation} $$is a comparisson between full strength with no buckling against the critical force. Sidesway movement is expected when critical force is reached. For slenderness $\overline{\lambda} \leq 0.2$ (or for $N_{Ed}/N_{cr} \leq 0.04$) the buckling effect can be ignored.

In steel design a beam is usually subjected to buckling against *both* axis $y,\ z$ at the same time. Therefore we need to know a way how to combine such cases.

Closed sections | Flexural buckling | $\bot y$ or |

$\bot z$ | ||

Doubly symmetrical sections | Flexural buckling | $\bot y$ or |

$\bot z$ or | ||

Torsional buckling | $\omega$ | |

Symmetrical sections | Flexural buckling | $\bot y$ or |

Flexural torsional buckling | $\bot z + \omega$ | |

Non-symmetrical sections | Flexural torsional buckling | $\bot y + \bot z + \omega$ |

Critical length $L_{cr,\omega}$ in $(\ref{eq:buck2})$ is established as an analogy to Euler's method:

rotation allowed: | no constraint |

rotation prevented: | simple support |

deplanation prevented: | fixed support |

deplanation allowed: | simple support |

$I_\omega$ is torsion warping constant, $I_t$ is torsion section constant. The polar moment of inertia $I_p$ is being found as

$$ \begin{equation} I_p = I_y + I_z + Aa^2, \end{equation} $$where $I_y$ and $I_z$ are moments of inertia to the main axes, $A$ is section area, $a$ is the distance of shear center $C_s$ from the centroid $C_g$.

We have to combine

- $\lambda_y$ and $\lambda_\omega $ into $\lambda_{y \omega}$ or
- $\lambda_z$ and $\lambda_\omega $ into $\lambda_{z \omega}$.

Let us focus on the case $\bot z+ \omega$ only, since the other direction is an analogy: $\lambda_{z\omega} = f(\lambda_z,\ \lambda_\omega)$.

are slendernesses $\lambda_z$, $\lambda_\omega$ ordered from the greatest to the smallest.

$$ \begin{equation} \alpha = \left(\frac{a_z}{i_p}\right)^2\\ i_p = \sqrt{I_p/A}. \end{equation} $$Length $a_z$ is ordinate of $C_s$ to $C_g$ to axis $z$.

We have to combine $\lambda_y$, $\lambda_z$ and $\lambda_\omega $ into $\lambda_{yz\omega}$ or $\lambda_{yz\omega} = f(\lambda_y,\ \lambda_z,\ \lambda_\omega)$

are $\lambda_y$, $\lambda_z$, $\lambda_\omega$ are ordered from the greatest to the smallest. As $\alpha_1 \geq \alpha_2$ are either $\alpha_y$ or $\alpha_z$:

\begin{equation} \alpha_y=\left(\frac{\alpha_y}{i_p}\right)^2, \hskip2em \alpha_z=\left(\frac{\alpha_z}{i_p}\right)^2 \end{equation}