## 4 OpenSees Numerical Model  
The case study structures are modelled as 3D buildings in the OpenSees software program version 3.6.0 [47; 48], adopting the modelling procedure presented in [49] and validated in [1; 5; 6; 41; 43]. Beams and columns are implemented through the _forceBeamColumn_ element object [50], assigning a non-linear behavior to the external sections and a linear elastic behavior to the internal section of the element, as shown in Figure 9. Since the case study structures are designed to fail in flexure in accordance with the requirements of the IBC [30], the external sub-elements correspond to the dissipative areas in which plastic flexural mechanisms are expected to be activated and concrete is defined with a well-detailed confinement model, correspondent to the _Concrete04_ material model, which is based on the model proposed by [PERSON] [51]. In particular, concrete is assumed nonresistant under tension, and the core region is characterized by concrete strength \(f_{cc}\) and deformations \(\varepsilon_{cc}\) and \(\varepsilon_{cc}\) defined according to Equations (A.6)-(A.8) of Part 3 of Eurocode 8 [52]. The length of dissipative zones \(L_{pl}\) (plastic hinge length) is defined according to Equation (A.9) of Eurocode 8 [52]. Regarding the reinforcement, the single steel bar is modelled as a fiber defined through the material model _Steel02_ with isotropic strain hardening [53; 54], assuming a strain-hardening ratio \(b\) equal to 0.005 [40] and the parameters that control the transition from the elastic to the plastic branch as \(R_{0}=18\), \(C_{R1}=0.925\), \(C_{R2}=0.15\). An effective area moment of inertia, \(I_{eq}=0.5I_{g}\), where \(I_{g}\) is the area moment of inertia of the gross section, is assigned to the internal elastic section to account for concrete cracking.  
The floor slabs are defined as rigid diaphragms where the masses of the structural members are concentrated in the center of mass of each story. Since the rigid diaphragms constrains the nodes belonging to the same floor to have the same displacement, a fictitious axial force is generated in the beams, as demonstrated in [55]. Therefore, it is necessary to add an axial release in the form of an axial buffer (defined through a _zeroLength_ element object [56]) between one end of each beam and the adjacent node.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{\(\mu_{\text{D}}\) (-)} & \multicolumn{2}{c}{\(K_{\text{B}}/K_{\text{D}}\) = 2} & \multicolumn{2}{c}{\(K_{\text{B}}/K_{\text{D}}\) = 3} & \multicolumn{2}{c}{\(K_{\text{B}}/K_{\text{D}}\) = 4} & \multicolumn{2}{c}{\(K_{\text{B}}/K_{\text{D}}\) = 5} \\ \cline{2-9}  & \(\mu_{\text{DB}}\) (+) & \(\xi_{DB}\) (\%) & \(\mu_{\text{DB}}\) (\%) & \(\xi_{DB}\) (\%) & \(\mu_{\text{DB}}\) (-) & \(\xi_{DB}\) (\%) & \(\mu_{\text{DB}}\) (-) & \(\xi_{DB}\) (\%) \\ \hline
4 & 3.0 & 42.4 & 3.3 & 44.1 & 3.4 & 44.9 & 3.5 Figure 9: Modelling approach of beams and columns.  
With the aim of obtaining results independent from the directionality of the excitations, two analyses were carried out for each bidirectional ground motion by switching the two components of the seismic input with respect to the main plan directions (X and Z) of the building.