# Yield Behaviour of Welded I-Shaped Steel Cross-Sections
## 3 Numerical Application  
In order to test the reliability of the domain formulations defined in the foregoing section, in the following section, numerical applications to two welded sections are presented and compared with the domain defined in the EC3 standard. The selected sections possess the geometrical characteristics reported in Table 1. As can be easily observed, with the aim of performing the necessary comparison, the geometry of section W1 is chosen with width, depth, and thickness equal to those of HEA300, while the geometry of section W2 is chosen with width, depth, and thickness equal to those of IPE360. The equations defining the proposed domains were written in Mathematica(r) 14.0 software to obtain the corresponding graphs. For the plain domains, the investigated range for \(z\) was divided into 20 intervals, leading to satisfactory results, which are reported in the following section. For 3D domains, the investigated range for \(T\) was divided into 10 intervals, and for each of these intervals, 100 points were calculated. The overall computing time for each section was equal to 5 min on HP I7-32 GB RAM workstation. As the first application, the case of the W1 section was considered. In Figure 4, the \(N,M\) domain obtained by Equations (13) and (14), the \(N,T\) domain obtained by Equations (21) and (27), and the \(M,T\) domain obtained by Equations (28) and (29) are sketched, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Section** & **Width \(b\)** & **Depth \(h\)** & **Web Thickness \(t_{w}\)** & **Flange Thickness \(t_{f}\)** \\  & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline W1 & 300 & 290 & 8.5 & 14 \\ W2 & 170 & 360 & 8 & 12.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical characteristics of the welded sections under examination. In Figure 5, the \(N,T,M\) domain is sketched both with and without a bounding surface. In Figure 5, the plane domains reported in Figure 4 are also reported to enhance the overall coherence of the proposed approach. It is worth noting that the boundary of the upper plateau of the domain in Figure 5b represents the yield conditions for the whole cross-section, while all the points within the  
Figure 4: Yield domains for W1 section: (**a**) \(N,M\) domain; (**b**) \(N,T\) domain; (**c**) \(M,T\) domain. Figure 5: Proposed yield domain in \(N,M,T\) space for section W1: (**a**) 3D domain with plane domains evidenced; (**b**) 3D domain with bounding surface. relevant boundary define the limit conditions related to the shear forces; in other words, for this plane domain, just the web of the cross sections is fully plasticized, while the flanges remain in the elastic regime. On the contrary, all the points of the lateral surface represent yield conditions for the whole cross-section. To check the affordability of the proposed approach, the plane domains reported in Figure 4 are compared with those arising from the EC3 international standard and sketched in Figure 6. It is important to emphasize that the EC3 code does not provide any information about the interaction between axial and shear force. Two outcomes can be inferred from the last remark: the first is that no comparison can be made for the \(N,T\) domain sketched in Figure 4b; the second is that no limit condition is derived from the EC3 for the \(N,T\) domain, and, consequently, the 3D \(N,T,M\) is that reported in Figure 7. An examination of Figures 5-7 allows us to draw the following conclusions: the proposed domains are fully coherent with those proposed by the international standards, but they represent a rigorous definition of the real yield conditions of the cross-section subjected to the presence of axial force, shear, and bending moment; furthermore, the proposed approach can easily be extended to the case of cross-sections subjected to more complex combinations of internal forces (\(N,T_{y},T_{z},M_{x},M_{y},M_{z}\)), but this point will be considered in future studies. As a second application, the case of the W2 section is considered.