# Bending and axial force (6.2.9)

1. For class 1, 2 sections: such sections are stable thus plastic conditions can be achieved;
2. For class 3 sections: plasticity might not occur due to local buckling $\implies$ we have to allow only elastic range;
3. For class 4 sections: only elastic conditions can be taken into consideration.

## Bending + axial forces—class 1, 2 sections

We can work in plastic part of the working diagram. The design plastic resistance $M_{pl,Rd}$ for bending is used but is lowered for the effect of the normal load $N_{Ed}$ which acts in combination with bending.

\begin{align} M_{Ed} &\le M_{N,Rd}, \hskip2em \text{where} \\ M_{N,Rd} &= M_{pl,Rd} (1- (\frac{N_{Ed}}{N_{pl,Rd}})^2) \label{BendAndAx:eq2} \end{align}
 $N_{Ed}$ design axial force from load $N_{pl,Rd}$ design plastic resistance for normal load $M_{pl,Rd}$ design plastic resistance for bending $M_{N,Rd}$ reduced design plastic resistance for bending

Other cases with holes for fasteners are in the code.

### Bi-axial bending and axial force

When a member is loaded by a moment against axis $y$, $z$ then the following criterion is used:

$$$$\left( \frac{M_{y, Ed}}{M_{N,y,Rd}}\right)^{\alpha} + \left(\frac{M_{z,Ed}}{M_{N,z,Rd}}\right)^\beta \le 1$$$$

Both $\alpha,\ \beta$ depend on cross-section and are described in details in the code. $M_{N,y,Rd}$ and $M_{N,z,Rd}$ are design resistance moments $M_{pl,y,Rd}$ and $M_{pl,z,Rd}$ lowered due to axial loading (see $(\ref{BendAndAx:eq2})$).

## Bending + axial forces—class 3, 4 sections

Criteria for allowance is

$$$$\sigma_{x,Ed} \lt \frac{f_y}{\gamma_{M0}}$$$$

where $\sigma_{x,Ed}$ is design value of stress found by Mechanics of elastic solids from both moment + axial force (taking into account holes for fasteners where relevant).

For class 4 additional criteria from code has to be followed.

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