Members in bending (6.2.5)

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:

Cross-section classification

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

1—Legend for the above table: $t_w,\ d$ explained, $\epsilon = \sqrt{235/f_y}$;
2—Before the slender members could develope plastic condition to transmit bending, such members fail due to a local buckling. Only "thick" (class 1, 2) members are able to resist bending in plastic condition
3—Class 4 sections consists of thin plates, these are suspectible to local buckling: only effective area is considered to carry bending

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

$$ \begin{align} M_{pl,R} &= \frac{W_{pl} f_y}{\gamma_{M0}}, \hskip2em &\text{for class 1, 2} \\ M_{el,Rd} &= \frac{W_{el,min}f_y}{\gamma_{M0}}, &\text{for class 3}\\ M_{el,Rd} &= \frac{W_{eff}f_y}{\gamma_{M0}} &\text{for class 4}. \end{align} $$

For class 4 the resistance is being computed with effective values $A_{eff},\ I_{eff},\ W_{eff}$. In general cases the neutral axis is being shifted then.

Plastic section modulus

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).

Holes

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 tension zones, the holes do not need to be consided if the following condition is satisfied:

$$ \begin{equation} \frac{A_{f,net}0.9 f_u}{\gamma_{M2}} \ge \frac{A_f f_y}{\gamma_{M0}} \end{equation} $$

$A_f$ 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).

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