When a beam is subjected to bending, in steel design two cases are recognized.

- The stresses are in the range of Hooke's law or
- the stresses are in the range of plasticity (from working diagram) and the ultimate state might be achieved.

Assuming no plasticity is reached, from mechanics of materials then

$$ \sigma = \frac{My}{I_x} = \frac{M}{W_x}, \hskip2em \text{where}\hskip2em W = \frac{I}{y_\max} = \frac{I}{c}, $$$c$ or $y_\max$ is maximum distance from neutral axis (NA), $W_{x}$ is elastic section modulus. According to Eurocode the assessment is

$$ \begin{equation} M_{Ed} \le M_{c,Rd}, \end{equation} $$where $M_{Ed}$ is design value of the bending moment from external load and $M_{c,Rd}$ is the design resistance for bending. It depends on the section class whether ultimate state (plasticity) can be achieved or not. For that purpose we recognize 4 classes:

Class 1 | Plastic | $d/t_w \lt 72 \epsilon$ | Beam is stiff/stable enough to develope a plastic hinge with no reduction of capacity |

Class 2 | Compact | $d/t_w \lt 83 \epsilon$ | Beam can develope a plastic hinge |

Class 3 | Semi-compact | $d/t_w \lt 124 \epsilon$ | Local buckling prevents development of the plastic moment resistance. The section is supposed to work only in elastic range until yield stress is reached |

Class 4 | Slender | thin walled (usually welded) section | Local buckling expected. Plasticity will not develope in bending, only effective (not weakened) cross-section area considered, see picture |

According to the cross-section, $M_{c,Rd}$ is

$$ \begin{align} M_{pl,R} &= \frac{W_{pl} f_y}{\gamma_{M0}}, \hskip2em &\text{for class 1, 2} \\ M_{el,Rd} &= \frac{W_{el,min}f_y}{\gamma_{M0}}, &\text{for class 3}\\ M_{el,Rd} &= \frac{W_{eff}f_y}{\gamma_{M0}} &\text{for class 4}. \end{align} $$Plastic section modulus is computed when the cross-section is being exposed to a
load which exceeds elastic capabilities of a member. Although not strictly
real, it is supposed that the whole section is plasticized. Therefore the
section is divided into two *equal* areas, one which carries compression, the
second one carries tension. These areas are separated by plastic neutral axis
(PNA) which might differ from NA. $W_{pl} = A_C y_C + A_T y_T$ ($A_C$ is area
of section transmitting compression, $y_C$ is its distance from PNA).

If the holes are filled with fasteners in the *compression zone*, the
holes do not need to be consided (the impact of holes might be ignored). In

$A_f$ is the area of the tension zone (that means the tension flange of section or the tension flange + tension zone within web of the section).