## 4 Examples  
### Three-Bay Two-Story Framework  
The three-bay two-story structure shown in Figures 6-8 has been previously studied by [PERSON] and [PERSON] [32], [PERSON] et al. [33], and [PERSON] and [PERSON] [21]. Figure 6 shows the geometry of the structure and the loads considered. The load factors that have been considered are \(\gamma_{\text{G}}=1.3\) for self-weight loads (\(q_{sw}\)) and \(\gamma_{\text{Q}}=1.5\) for imposed (\(q_{il}\)) and wind (\(q_{w}\)) loads.  
Figure 7 shows the numbering used for the nodes and members. The expressions from 6 to 19 have been considered following the requirements of Eurocode 3 [24]. The columns and beams have HEB and IPE steel shapes, respectively.  
Figure 5: Function \(\mu^{k+1}(\pi_{z})\), estimated maximun value \(\mu_{Z}^{k+1}\), and obtained parameter \(\pi_{int}^{k+1}\).  
Figure 6: Geometry and loads for the three-bay two-story framework.  
The material considered for the steel shapes in members and steel end-plates in semi-rigid connections was steel S275 with [PERSON]'s modulus \(E\) = 210 GPa, \(f_{y}\) = 275 MPa yield strength, and a resistance factor of \(\gamma_{m}\) = 1.0. The ultimate tensile strength and the resistance factor considered for bolts in the semi-rigid connections were \(f_{ub}\) = 800 MPa and \(\gamma_{m}\) = 1.25, respectively. The material cost of the steel considered was \(C\) = 1.6 EUR/kg. The cost of the rigid connections in the frame was calculated assuming a fivity factor of 0.964 [21]. To model the behavior of the extended end-plates with stiffened column flange semi-rigid connections (EEP1) and extended end-plates without stiffened column flange semi-rigid connections (EEP2), the Component Method was used.  
Figure 8 shows the numbering used for the design variables. The design variables are included in Table 2.  
Figure 8: Number of variables for the three-bay two-story framework.  
Figure 7: Node and member numbers for the three-bay two-story framework.  
The PFAM algorithm was run 100 times using different random seeds to guarantee that the best-generated solution would be considered the optimal design solution. Ten subpopulations were considered in all the runs. The values of the parameters in the PFAM algorithm were the following: \(\beta_{0}\) = 1, \(m\) = 2, \(\gamma_{0}\) = 10, \(\alpha_{0}\) = 1, \(nt\) = 200, \(k_{\max}\) = 100, and \(t_{c}\) = 0.1. Figure 9 shows the evolutions of the objective functions obtained in one of the 100 executions considering rigid connections and both EEP1 and EEP2 semi-rigid connections. The optimal design solutions obtained by [PERSON] and [PERSON] [32], [PERSON] et al. [33], and the PFAM algorithm proposed in this work are included in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Members** & **Design Variable** & **Lower Bound** & **Upper Bound** \\ \hline columns & \(X_{1}\) (area), \(X_{2}\) (area) & 2.604 mm\({}^{2}\) (HEB100) &
\begin{tabular}{c} 40.000 mm\({}^{2}\) \\ (HEB1000) \\ \end{tabular} \\ \hline beams & \(X_{3}\) (area), \(X_{4}\) (area) & 764 mm\({}^{2}\) (IPE80) & 15.600 mm\({}^{2}\) (IPE600) \\ \hline beams & \(X_{5}\) (\(d_{b}\)), \(X_{7}\) (\(d_{b}\)), \(X_{9}\) & 10 mm & 36 mm \\  & \((d_{b})\), \(X_{11}\) (\(d_{b}\)) & & \\ \hline beams & \(X_{6}\) (\(t_{ep}\)), \(X_{8}\) (\(t_{ep}\)), \(X_{10}\) & 8 mm & 25 mm \\  & \((t_{ep})\), \(X_{12}\) (\(t_{ep}\)) & & \\ \hline \multicolumn{3}{c}{\(d_{b}\) = bolt diameter; \(t_{ep}\) = end-plate thickness} \\ \end{tabular}
\end{table}
Table 2: Design variables for the three-bay two-story framework.  
Figure 9: Objective function evolutions using PFAM for the three-bay two-story framework and considering rigid and semi-rigid connections EEP1 and EEP2.