## 5 Benchmark Numerical Example
### Finite Element Modeling and Validation  
In order to establish a strong foundation and ensure reliable results, eight distinct FE setups were created and validated, all based on consistent modeling principles. These configurations are classified into two main categories, in accordance with the previous section: flush and extended types. The experimental setup used for validation is illustrated  
To develop the FE models, 8-node linear brick elements with reduced integration (C3D8R) were used. The mesh size was assigned individually to each component to optimize computational efficiency while maintaining accuracy. Specifically, a mesh size of 8 mm was applied to the end-plate, stiffeners, and the beam and column plates in the immediate vicinity of the connection. A finer mesh of 2 mm was used for the bolts, while a coarser mesh size of 30 mm was adopted for regions farther from the connection. An essential aspect of achieving accurate results in FE simulations is the proper application of material models. In line with the optimization framework, the elasto-plastic quad-linear material model for steel is employed in this study, following the approach of [PERSON] and [PERSON] [45]. The corresponding stress-strain relationship, illustrated in Figure 6, consists of four distinct phases, which are described as follows:  
\[\sigma= \begin{cases}\begin{array}{c}E\varepsilon,\;\;if\;\varepsilon< \varepsilon_{y}\\ f_{y},\;\;if\;\varepsilon_{y}<\varepsilon\leq\varepsilon_{h}\\ f_{y}+E_{sh}(\varepsilon-\varepsilon_{sh}),\;\;if\;\varepsilon_{sh}< \varepsilon\leq C_{1}\varepsilon_{u}\\ f_{C_{1}\varepsilon_{u}}+\frac{f_{u}-f_{C_{1}\varepsilon_{u}}}{\varepsilon_{u }-C_{1}\varepsilon_{u}}(\varepsilon-C_{1}\varepsilon_{u}),\;\;if\;C_{1} \varepsilon_{u}<\varepsilon\leq\varepsilon_{u}\end{array}\end{cases}  \tag{20}\]  
where \(\sigma\) and \(\varepsilon\) denote the steel stress and its corresponding strain. The yield stress and yield strain are represented by \(f_{y}\) and \(\varepsilon_{y}\), respectively, while \(f_{u}\) and \(\varepsilon_{u}\) refer to the ultimate tensile stress and the associated strain. The strain hardening modulus is indicated by \(E_{sh}\), and \(\varepsilon_{sh}\) marks the strain at the onset of strain hardening. The strain and stress at the intersection between the third segment of the idealized model and the actual stress-strain curve are denoted by \(C_{1}\varepsilon_{u}\) and \(f_{C_{1}\varepsilon_{u}}\), respectively, where \(C_{1}\) is a material-dependent coefficient. These parameters can be determined using the following equations:  
\[\varepsilon_{u}=0.6\bigg{(}1-\frac{f_{y}}{f_{u}}\bigg{)}\geq 0.06 \tag{21}\]  
\[\varepsilon_{sh}=0.1\frac{f_{y}}{f_{u}}-0.055\;and\;0.015\leq\varepsilon_{sh} \leq 0.03 \tag{22}\]  
\[C_{1}=\frac{\varepsilon_{sh}+0.25(\varepsilon_{u}-\varepsilon_{sh})}{ \varepsilon_{u}} \tag{23}\]  
\[E_{sh}=\frac{f_{u}-f_{y}}{0.4(\varepsilon_{u}-\varepsilon_{y})} \tag{24}\]  
Figure 6: The quad-linear material model of steel. The main material properties of steel were defined based on the experimental results reported by [PERSON] et al. (2018), whose measured values were also adopted in the study by [PERSON] et al. (2018). These parameters are summarized in Table 3.  
In both the experiments and the simulations, the bolts were pretensioned. The pretension force was set to 155 kN for 20 mm diameter bolts and 225 kN for 24 mm diameter bolts, in accordance with the specifications provided in [PERSON] et al. (2018). In the simulations, bolt pretension was applied during the initial load step, prior to the application of the nodal force \(F\), which generated the moment in the connection. To accurately model the real-life behavior of the bolted connection, it is essential to account for frictional interactions between the contacting parts. To achieve this, a _surface-to-surface_ contact formulation was adopted, utilizing the penalty method to define the tangential behavior with a friction coefficient of 0.44, as presented in [PERSON] et al. (2018).