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.
When a member is loaded by a moment against axis $y$, $z$ then the following criterion is used:
$$ \begin{equation} \left( \frac{M_{y, Ed}}{M_{N,y,Rd}}\right)^{\alpha} + \left(\frac{M_{z,Ed}}{M_{N,z,Rd}}\right)^\beta \le 1 \end{equation} $$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 $(\ref{BendAndAx:eq2})$).
Criteria for allowance is
$$ \begin{equation} \sigma_{x,Ed} \lt \frac{f_y}{\gamma_{M0}} \end{equation} $$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.