## 3 Numerical Example
### Nonlinear Frame Structures  
The three structures are modeled by fiber beam elements using an elastoplastic material law with kinematic hardening, as shown in Figure 10. We chose a Young's modulus of \(E_{1}=210\) GPa and a material density of \(\rho=7850\frac{\text{kg}}{\text{m}^{3}}\). The yielding limit of the material was set as a value of \(\sigma_{y}=235\) MPa. The post-yielding stiffness was chosen as \(E_{2}=21\) GPa, which is 10% of initial Young's modulus. The beams and columns of the frames were modeled using the same beam element formulation, shown in Figure 10. For the fiber-beam elements, we used hollow cross sections with different widths, heights, and thicknesses.  
Figure 9: Three-dimensional selection strategy: Selection of samples from (a) the full set \(\mathcal{I}\) using the velocity spectrum intensity \(\int_{0.1}^{2.5}S_{u}\), the spectral displacement at the natural period \(S_{d}(T_{1})\) and the spectral acceleration at the natural period \(S_{a}(T_{1})\), and (b) selected samples for the training set \(\mathcal{I}_{t,sel}\).  
The geometrical inputs for all structures used for the numerical experiments are summarized in Tables 2 and 3. Additionally, the structures are presented in Figure 11. To account for a realistic mass distribution, all structures have additional point masses. The first natural period of the structures A, B and C are \(T_{1,A}=0.74\) s, \(T_{1,B}=0.40\) s and \(T_{1,C}=0.88\) s, respectively.  
The PSDR for the first two stories is depicted in Figure 11 for structure C. For these particular problems, we observed that a story drift ratio of \(\sim\)4.5% corresponds to a full plastification of the frame corners in the first stories. These assumptions agree with the estimation of the collapse of steel structures [50]. However, lower story drift ratios can already cause damage to the structure, i.e., story drift ratios above 2% lead already to plastification of the frame corners. More elaborate models could also include damage and softening effects and may improve the failure criterion [51], which is material for future research. However, in this study, we focus on the general approach of the strategy with the PSDR as representative output quantity and an elastoplastic material law from the underlying finite element model.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Dimension** & **Structure A** & **Structure B** & **Structure C** \\ \hline \(w_{bay}\) & 6 m & 5.5 m & 6 m \\ \(h_{s1}\) & 4.5 m & 3.5 m & 4.5 m \\ \(h_{s2}\) & 4 m & 3.0 m & 4 m \\ \(h_{s3}\) & 4 m & & 4 m \\ \(h_{s4}\) & & & 4 m \\ \(h_{s5}\) & & & 4 m \\ \hline \end{tabular}
\end{table}
Table 2: Dimensions of the structures.  
Figure 10: Beam elements with a hollow cross section and elastoplastic material behavior with kinematic hardening.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Dimension** & **P\({}_{1}\)** & **P\({}_{2}\)** & **P\({}_{3}\)** & **P\({}_{4}\)** \\ \hline \(w_{b}\) & 0.3 m & 0.35 m & 0.3 m & 0.35 m \\ \(h_{b}\) & 0.3 m & 0.35 m & 0.4 m & 0.45 m \\ \(t_{b}\) & 0.03 m & 0.03 m & 0.03 m & 0.03 m \\ \hline \end{tabular}
\end{table}
Table 3: Dimensions of the cross sections used for the beams and columns.