## 5 Benchmark Numerical Example
### Finite Element Modeling and Validation
This value was selected based on an earlier experimental study [PERSON] et al. (2018), where blasted surface preparation was used to achieve the required slip resistance. Additionally, normal behavior was defined as _hard_ contact, which prevents the transfer of tensile stresses across the interface. Finally, the load and boundary conditions were applied to replicate those of the physical experiment, as illustrated in Figure 7 for a typical connection. As previously described, a geometrical and material nonlinear analysis (GMNA) was carried out on imperfect geometry. To establish the initial imperfections, LBA was first performed, and the resulting mode shapes were examined. Two types of eigenmodes were considered: one involving local buckling of the column's web plate, and the other affecting the beam's flange plate as shown in Figure 8. These correspond to the first and second critical buckling modes, respectively, and were selected based on their relevance to the observed structural behavior. Since these modes represent the most critical local instabilities--where stress concentrations and plastic deformations are expected--they were considered sufficient to capture the dominant imperfection-sensitive behavior. Including these specific imperfection shapes helps to realistically predict the load-bearing capacity and failure mechanisms of the connection. The imperfection amplitudes \(\omega_{j}\) were defined in accordance with EN1993-1-5 ([PERSON] et al., 2018), using a magnitude of \(a/200\), where \(a\) denotes the shorter side of the subpanel. The final analysis was conducted using the previously mentioned Newton-Raphson iterative solution method to account for the effects of large deformations. It is important to note that, during the subsequent optimization process, the maximum joint rotation was limited to \(\theta_{max}=0.025\) rad. Accordingly, the load magnitude for each of the eight configurations was selected to ensure compliance with this rotational constraint. To maintain consistency with this criterion, the validation was carried out by comparing the moment-rotation curves obtained from the experiments and the numerical simulations. As illustrated in Figure 9, the results show good agreement. A detailed comparison indicates that the maximum deviation in force at a joint rotation of 0.025 rad is less than 6.00%, as summarized in Table 4. This table also presents the volume functions, normalized using the maximum volume \(V_{max}\), defined as the volume of configuration SC7. It should be noted that the same \(V_{max}\) is used throughout the optimization process discussed later in this study. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Part** & \(E\) (N/mm\({}^{2}\)) & \(f_{y}\) (N/mm\({}^{2}\)) & \(f_{u}\) (N/mm\({}^{2}\)) \\ \hline Plates (\(\leq\)16 mm thickness) & 190,707 & 391 & 559 \\ Plates (\(>\)16 mm thickness) & 204,228 & 363 & 537 \\ Bolts & 206,000 & 990 & 1160 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material properties used to define the quad-linear stress–strain relationship. Figure 8: Considered mode shapes introduced as geometric imperfections: (**a**) local buckling of the column’s web plate and (**b**) local buckling of the beam’s flange plate. Figure 7: The FE model of a typical connection. Figure 9: _Cont._