## 6 Method B1 for Design of the Hysteretic Braces
In the considered case study, \(\mu_{DB}^{*}=8\) was adopted for both the longitudinal (X) and transversal (Y) frames. In this way, the only remaining unknown design parameter is the yielding strength of the damper (\(F_{y,DB}^{*}\)), as the relevant yielding displacement (\(d_{y,DB}^{*}\)) can be estimated as  
\[d_{y,DB}^{*}=d_{PP}^{*}/\mu_{DB}^{*} \tag{14}\]  
leading to \(d_{y,DB}^{*}=3.1 mm\) for devices installed along both the longitudinal (X) and transversal (Y) directions. _Step 4: Iterative selection of the yielding force of the dampers_  
After selecting the target displacement (\(d_{PP}^{*}=24.5 mm\) for both the X and Y directions) and the ductility of the dissipative braces (\(\mu_{DB}=8\) for both cases), the only unknown variable is the relevant yielding force (\(F_{y,DB}^{*}\)), which can be calculated through an iterative procedure. At the generic _j_th iteration, a yielding force of the elasto-plastic damper F\({}_{y,DB\dot{\beta}}^{*}\) is assumed, resulting in the following effective vibration period of the braced frame:  
\[T_{eff,BF,j}^{*}=2\pi\sqrt{\frac{m^{*}}{k_{eff,BF,j}^{*}}} \tag{15}\]  
where \(k_{eff,BF,j}^{*}=\left[\left(F_{PP,F}^{*}+F_{y,DB,j}^{*}\right)/d_{PP}^{*}\right]\) is the secant or effective stiffness of the braced frame. The overall equivalent viscous damping of the braced frame (\(\xi_{eq,BF,j}^{*}\)) is then estimated through the weighted average proposed by [PERSON] [30]:  
\[\xi_{eq,BF,j}^{*}=\xi_{el,F}^{*}+\frac{\xi_{h,F,j}^{*}\cdot F_{PP,F}^{*}+\xi_ {h,DB,j}^{*}\cdot F_{y,DB,j}^{*}}{F_{PP,F}^{*}+F_{y,DB,j}^{*}} \tag{16}\]  
where \(\xi_{el,F}^{*}=0.05\) is the conventional elastic damping of the frame; \(\xi_{h,F,j}^{*}\) is the hysteretic damping developed by the initial unbraced frame estimated through Equation (6) for \"large Takeda\" systems (i.e., with \(a\) = 65 and \(b\) = 50; see Section 3); \(\xi_{h,DB,j}^{*}\) is the hysteretic damping provided by the hysteretic braces and calculated according to Equation (6) for EPP systems (i.e., with \(a\) = 85 and \(b\) = 60; see Section 3); and \(F_{PP,F}^{*}\) is the maximum forces developed by the initial unbraced frame. In this regard, it is worth noting that when \(d_{PP}^{*}\leq d_{y,F}^{*}\), as in the case shown in Figure 10a, \(F_{PP,F}^{*}=k_{el,F}^{*}\cdot d_{PP}^{*}\); otherwise, when \(d_{PP}^{*}>d_{y,F}^{*}\), as in the case shown in Figure 10b, \(F_{PP,F}^{*}=F_{y,F}^{*}\). _Step 5: Calculation of the seismic displacement of \"BF\"_  
With reference to the ADRS representation of the damped/reduced design spectrum (i.e., \(S_{a}(\mathrm{T})-S_{d}(\mathrm{T})\); see Figure 10), at the generic _j_th iteration the seismic displacement of the braced frame (\(d_{BF,j}^{*}\)) is calculated as  
\[d_{BF,j}^{*}=\eta_{j}\cdot S_{De}\left(T_{j}^{*}\right)=S_{d}\left(T_{j}^{*}\right) \tag{17}\]  
where \(\eta_{j}=\eta_{j}\left(\xi_{eq,BF,j}^{*}\right)\) is the spectral reduction factor calculated through [PERSON]'s formula, i.e., Equation (8). _Step 6: Convergence criterion to end the iterative procedure_The iterative procedure is terminated when the discrepancy between the \(j\)th spectral displacement (\(d^{*}_{BF,j}\)) and the design performance point (\(d^{*}_{PP}\)) is smaller than a certain tolerance threshold \(\varepsilon\); in this study, \(\varepsilon=0.05d^{*}_{PP}\):  
\[\left|d^{*}_{BF,j}-d^{*}_{PP}\right|\leq\varepsilon \tag{18}\]  
The resulting yielding strength (\(F^{*}_{y,DB}\)) of the dampers at convergence in \"Method-B1\" are reported in Table 3; it can be noted that the required damping force (\(F^{*}_{y,DB}=280 kN\)) is lower along the longitudinal (X) direction (\(-38.5\%\)) than that along the transversal (Y) frame (\(F^{*}_{y,DB}=455\)\(kN\)). This reflects the fact that, as reported in Table 2, the second equivalent SDOF system features a higher reduced mass \(m\)* (603 tons vs. 500 tons, i.e., \(+20.6\%\)) with a lower effective stiffness \(k^{*}_{eff,F}=F^{*}_{y,F}/d^{*}_{PP}\) (22.3 kN/mm vs. 25.0 kN/mm, i.e., \(-12.1\%\)).