# Yield Behaviour of Welded I-Shaped Steel Cross-Sections
## 2 Materials and Methods
### Simultaneous Presence of \(T=T_{p}\) and \(N\)\((M=0)\)  
The presence of \(T_{p}\) as the acting shear force does not exclude the simultaneous presence of a limited axial force. In such a case, all along the web, \(\tau_{xz}=\tau_{0}=\sigma_{0}/\sqrt{3}\) occurs, so the presence of normal stresses is excluded. Instead, in the flanges, both \(\tau_{xy}\) and \(\tau_{xz}\) are present, assuming values lesser than \(\tau_{0}\), so that the presence of normal stresses is possible. Referring to Figure 3a, in the ranges \(\frac{h}{2}-t_{f}\leq z\leq\frac{h}{2}\) and \(\frac{t_{w}}{2}\leq y\leq\frac{h}{2}\), the expression of \(\tau_{xy}\) is given by the following relation:  
\[\big{|}\tau_{xy}(y)\big{|}=\frac{T_{p}\Big{(}\frac{b}{2}-y\Big{)}\Big{(}h-t_{ f}\Big{)}}{2I_{y}} \tag{7}\]  
The expression of \(\tau_{xz}\), in the ranges \(\frac{h}{2}-t_{f}\leq z\leq\frac{h}{2}\) and \(0\leq y\leq\frac{h}{2}\) (Figure 3b), is given by:  
\[\big{|}\tau_{xz}(z)\big{|}=\frac{T_{p}\Big{(}\frac{b^{2}}{4}-z^{2}\Big{)}}{2I_ {y}} \tag{8}\]By imposing the full yielding condition on all the points of the cross-section in the ranges \(\frac{h}{2}-t_{f}\leq z\leq\frac{h}{2}\) and \(\frac{t_{w}}{2}\leq y\leq\frac{h}{2}\), the normal stresses acting in the flange must respect Equation (1) as equality; that is, taking into account Equations (1), (7), and (8), the following relation is obtained:  
\[\sigma_{x}(y,z)=\sqrt{\sigma_{0}^{2}-3\frac{T_{p}^{2}}{4I_{y}^{2}}\left[\left( \frac{b}{2}-y\right)^{2}\left(h-t_{f}\right)^{2}+\left(\frac{h^{2}}{4}-z^{2} \right)^{2}\right]} \tag{9}\]  
By imposing the full yielding condition on all the points of the cross-section where \(\tau_{xy}=0\) (i.e., in the ranges \(\frac{h}{2}-t_{f}\leq z\leq\frac{h}{2}\) and \(0\leq y\leq\frac{t_{w}}{2}\)), the normal stresses acting in the flange must respect Equation (1) as equality; that is, taking into account Equations (1) and (8), the following relation is obtained:  
\[\sigma_{x}(z)=\sqrt{\sigma_{0}^{2}-3\frac{T_{p}^{2}\left(\frac{h^{2}}{4}-z^{2 }\right)^{2}}{4I_{y}^{2}}} \tag{10}\]  
It follows that the limit axial force acting when \(T=T_{p}\) is given by:  
\[\begin{split} N_{\ell im}\big{(}T_{p}\big{)}=4\frac{\frac{b}{2}}{ \frac{t_{w}}{2}}\left[\frac{\frac{h}{2}}{\frac{h}{2}-t_{f}}\sqrt{\sigma_{0}^{ 2}-3\frac{T_{p}^{2}}{4I_{y}^{2}}\left[\left(\frac{b}{2}-y\right)^{2}\left(h-t _{f}\right)^{2}+\left(\frac{h^{2}}{4}-z^{2}\right)^{2}\right]}dz\right]dy\\ +2t_{w}\frac{\frac{h}{2}}{\frac{h}{2}-t_{f}}\sqrt{\sigma_{0}^{2}- 3\frac{T^{2}\left(\frac{h^{2}}{4}-z^{2}\right)^{2}}{4I_{y}^{2}}dz}\end{split} \tag{11}\]  
The pair of values \(N_{\ell im}\big{(}T_{p}\big{)},T_{p}\) represents the internal force condition, which determines the full yielding of the whole cross-section. It is worth noting that all the pairs of values \(N\), \(T_{p}\), with \(0\leq N<N_{\ell im}\big{(}T_{p}\big{)}\), always represent limit conditions, but only for shear; indeed, in this latter condition, the shear force reaches its limit yield value, but the  
Figure 3: Portion of the cross-section considered: (**a**) in Equation (7); (**b**) in Equation (8).  
web suffers plastic strains due to the shear, while the flanges behave elastically. It follows that the segment \(\left[0,T_{p};N_{\ell im}\left(T_{p}\right),T_{p}\right]\) in the plane \(N\), \(T\) represents the upper plateau of the relevant yield domain boundary. The entire yield domain boundary is defined in Section 2.4 and is represented in the application stage for two representative cross-sections.