## 2 Description of the Investigated Joints and Relevant Damper Configurations  
The geometrical features of the investigated FREEDAM beam-to-column joints are summarized in Table 1[18], where the following parameters are varied: (1) the number of bolts of the damper \(n_{b}\); (2) the bolt diameter \(d_{b}\); (3) the lever arm of the connection \(z\); (4) the maximum and the minimum beam section; and (5) the maximum and the minimum column section. According to the scheme reported in Figure 2, the design force of the friction damper can be derived as follows:  
\[F_{D,Ed}=\frac{M}{z}=\frac{\overline{m}\times M_{pl,beam}}{z} \tag{1}\]  
where \(M_{pl,beam}\) is the plastic moment of the beam, and \(\overline{m}\) is the bending utilization factor, ranging from 0.3 to 0.6. A thorough investigation of the maximum and minimum values of \(\overline{m}\) has been formerly conducted in [7; 18]. The selection of the minimum value is driven by the consideration that values below this threshold do not satisfy the sliding resistance requirements of the damper for the ultimate limit state in non-seismic conditions. The upper limit of \(\overline{m}\) is derived by the requirement to maintain the beam within the elastic range and enhance the stiffness of the MRF, ensuring compliance with ECS criteria concerning drift and P-delta effects. The friction resistance, on the other hand, is defined as follows:  
\[F_{D,Rd}=\frac{h_{st,k}\cdot n_{s}\cdot n_{b}\cdot F_{p}}{\gamma_{crep}} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Damper**} & \(n_{b}\) & \(d_{b}\) & \(z\) & \multicolumn{2}{c}{**Beam Section**} & \multicolumn{2}{c}{**Column Section**} \\ \cline{2-9}  & [-] & [mm] & [mm] & **Minimum** & **Maximum** & **Minimum** & **Maximum** \\ \hline D1 & 4 & 16 & 170 & IPE 270 & IPE 450 & HE 240 B & HE 240 B \\ D2 & 4 & 20 & 250 & IPE 360 & IPE 600 & HE 240 B & HE 320 M \\ D3 & 6 & 20 & 250 & IPE 400 & IPE 600 & HE 300 B & HE 320 M \\ D4 & 8 & 20 & 250 & IPE 500 & IPE 750 \(\times\) 196+ & HE 280 M & HE 650 M \\ D5 & 8 & 24 & 330 & IPE 600 & IPE 750 \(\times\) 196+ & HE 320 M & HE 650 M \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical features of the investigated FREEDAM joints. Figure 2: FREEDAM beam-to-column joint: mechanical scheme. where \(F_{p}\) is the design bolt preload, \(n_{s}\) is the number of friction surfaces, \(\mu_{st,k}\) is the characteristic static friction coefficient, and \(\gamma_{creep}\) is the safety factor accounting for the loss of initial preload due to short-term and long-term relaxation phenomena. More details regarding the meaning of the terms and the suggested values are reported in [19]. After defining the geometric characteristics, the design force of the friction damper, and the friction resistance, a design chart was created (in collaboration with the PIP MEC company) where the required damper type is expressed based on the beam section and the bending utilization factor, as reported in Table 2.  
The analyzed dampers have been labeled D1A, D2B, D2C, and D3B. In accordance with Tables 1 and 2, associations were made between devices and the beam and column sections of the node. In Table 3, the features of the analyzed dampers, i.e., the length of the column \(L_{column}\) and the half-length of the beam \(L_{beam}/2\), are reported. As reported in Section 1, the simulations involved two configurations of the study setup, characterized by a different haunch, (1) single-slot haunch and (2) multiple-slot haunch, as reported in Figure 3. In Table 4, the main dimensions of the various components are reported (haunch, T-stub, L-stub) with reference to Figure 4.