## 7 Distribution in Elevation of the Damping Force  
In this section, the overall damping force of the (equivalent SDOF) hysteretic bracing system, i.e., \(F^{*}_{y,DB}=280 kN\) for the longitudinal (X) and \(F^{*}_{y,DB}=455 kN\) for the transversal (Y) direction, is distributed among the physical dampers installed in the frame. In this regard, it is worth noting that each real/physical DB system is composed of a steel hysteretic damper (D) arranged in series to a beam that acts as a rigid link (RL) to the RC frame. For sake of simplicity, the RLs, in addition to operating far from their yield threshold, are hereinafter assumed to ensure negligible deformations in the overall lateral response of the DB. **X Direction**: in the longitudinal direction, as shown in Figure 10(a), the upper windowed band interrupting the integrity of the masonry infills generates a sharp decrease in the lateral stiffness of the frame. To compensate for this irregularity, the vertical distribution of the overall (equivalent SDOF) damping force is articulated on two levels (see Figure 11(a),b). This means that despite the absence of any intermediate storeys, all columns along the X direction have been virtually divided in two levels (see Figures 7(e) and 10(a)): level 1 from the foundation to the lower RC edge beam (i.e., from z = 0 m to z = 4.25 m) and level 2 up to the roof edge beam (i.e., from z = 4.25 m to z = 7.50 m). Figure 11: Post-processing of the pushover analysis of the bare frame along the X direction: (**a**) ultimate lateral deformation capacity before structural collapse; (**b**) storey shear vs. inter-storey displacement; (**c**) storey elastic stiffnesses; and (**d**) yielding strengths. Figure 12: Final layout of building after implementing the hysteretic dampers: (**a**) overall 3D representation; (**b**) details of the longitudinal; and (**c**) transversal bracing systems. The [PERSON] criterion [28; 29] is enrolled within an iterative procedure that aims to identify the dampers' distribution in elevation that fulfil the two inter-storey displacement targets previously identified (\(d_{PP,L1}=14.0 mm\) and \(d_{PP,L2}=13.0 mm\)). As shown in Figure 11b, the pushover analysis on the bare frame (implementing the propaedeutically interventions; see Section 4) is plotted in terms of the shear force vs. the inter-storey displacement at both levels in order to identify their elastic stiffness (\(k_{el,F,i}\)) and the yielding strength (\(F_{y,F,i}\)). In this regard, \(k_{el,F,i}\) has been calculated as the secant stiffness from 0% to 60% of \(F_{u,F,i}\) while \(F_{y,F,i}\) been identified as the storey shear force associated with the yielding displacement of the MDOF frame (i.e., \(d_{y,F}=\Gamma\cdot d_{y,F}^{*}\)). The obtained level distributions of \(k_{el,F,i}\) and \(F_{y,F,i}\) are shown in Figure 11c,d, respectively. To trigger (iteration \(j=1\)) the iterative procedure for the distribution of the dampers on the two levels (subscripts L1 and L2) of the longitudinal (X) frame, these two ratios are calculated as follows [28; 29]:  
\[\left\{ \begin{array}{l}r_{k}^{(j=1)}=r_{k,L1}^{(j=1)}=r_{k,L2}^{(j=1)}=k_{el,DB}^{*}/k_{el,F}^{*}\\ r_{F}^{(j=1)}=r_{F,L1}^{(j=1)}=r_{F,L2}^{(j=1)}=r_{y,DB}^{*}/F_{y,F}^{*}\end{array} \right. \tag{19}\]  
Then, the tentative elastic stiffnesses (\(k_{el,DB,L1}^{(j)}\) and \(k_{el,DB,L2}^{(j)}\)) and damping forces (\(F_{y,DB,L1}^{(j)}\) and \(F_{y,DB,L2}^{(j)}\)) of each single dissipating trace installed at the two levels (L1 and L2) are calculated based on the number of devices (\(n_{DB,L1}\) and \(n_{DB,L2}\)) and the relevant inclination angle (\(\phi_{s,L1}\) and \(\phi_{s,L2}\)) provided in Table 4:  
\[\left\{ \begin{array}{l}k_{el,DB,L1}^{(j)}=(r_{k,L1}^{(j)}.k_{el,F}^{*})/\left( n_{DB,L1}\cdot cos^{2}\ \phi_{s,L1}\right)\\ k_{el,DB,L2}^{(j)}=(r_{k,L2}^{(j)}.k_{el,F}^{*})/\left(n_{DB,L2}\cdot cos^{2} \ \phi_{s,L2}\right)\end{array} \right. \tag{20}\]  
\[\left\{