## 6 Method B1 for Design of the Hysteretic Braces
First, the ultimate displacement capacity (\(d_{u}^{*}=d_{u}/\Gamma\)) was identified as the first point, which, in the softening branch of the reduced response, entails a 15% reduction in the maximum lateral strength (\(F_{bu}^{*}=F_{bu}/\Gamma\)). Afterwards, the yielding force (\(F_{y}^{*}\)) and the corresponding displacement (\(d_{y}^{*}\)) of the ideal elastic-perfectly plastic SDOF system were determined by imposing the equality of the areas subtended by the two plots (see Figure 9a), i.e., equal energy criterion. _Step 2: Selection of the target displacement_  
The target displacement at roof level, or performance point (\(d_{PP}\)), of the frame implementing the dissipative braces can be established by employing the inter-storey displacement thresholds (\(d_{PP,1}\) and \(d_{PP,2}\)) defined in Section 2. For the sake of uniformity and simplicity, a common target displacement for both horizontal directions is established as follows:  
\[d_{PP}=d_{PP,X}=d_{PP,Y}=(d_{PP,L1}+d_{PP,L2})=27.0 mm \tag{9}\]  
\[\left\{ \begin{array}{l}d_{PP,L1}\leq min\big{(}\Delta_{w,1};\ \Delta_{w,2}\big{)}\ \ \rightarrow\ \ d_{PP,L1}=14.0 mm\\ \Delta_{w,1}=min\Big{[}\Big{(}\frac{d_{w,w,i}+d_{w,j}}{2}\Big{)}\cdot cos\theta _{i}\Big{]}=14.5 mm\\ \Delta_{w,2}=0.004h_{1}=17 mm\end{array} \right. \tag{10}\]  
\[\left\{ \begin{array}{l}d_{PP,L2}\leq min\big{(}\Delta_{g,1};\ \Delta_{g,2};\ \Delta_{w,1};\ \Delta_{w,2}\big{)}\ \ \rightarrow\ \ d_{PP,L2}=13.0 mm\\ \Delta_{g,1}=13 mm\\ \Delta_{g,2}=2c_{g}\cdot\Big{(}1+\frac{h_{g}}{h_{g}}\Big{)}\cdot\frac{h_{2}} {h_{g}}=22.3 mm\\ \Delta_{w,1}=min\Big{[}\Big{(}\frac{d_{w,w,i}+d_{w,j}}{2}\Big{)}\cdot cos\theta _{i}\Big{]}=32.2 mm\\ \Delta_{w,2}=0.004h_{2}=13 mm\end{array} \right. \tag{11}\]  
where \(h_{1}=4250 mm\) and \(h_{2}=3250 mm\) are the two \"inter-storey\" heights. At level 2 (suffix L2), for the definition of the displacement threshold of the glazed windows \(\Delta_{g,2}\) the following geometric parameters are adopted: \(c_{g}=3 mm;h_{g}=2000 mm\); and \(b_{g}=1350 mm\). Moreover, it is worth noting that the different displacement thresholds of the masonry infills (i.e., \(\Delta_{w,1}\) and \(\Delta_{w,2}\)) at the two levels L1 and L2 reflect the different associated geometric and mechanical parameters (see Table 1). Eventually, with respect to the equivalent SDOF system of the braced frame, the displacement target established above can be converted as follows:  
\[d_{PP}^{*}=d_{PP,X}^{*}=d_{PP,Y}^{*}=d_{PP}/max\ (T_{X};I_{Y})=24.5 mm \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**dir.** & **Distr.** & \(\Gamma\) **(-)** & \(\mathbf{T_{1,F}^{*}}\) & \(\mathbf{m^{*}}\) & \(\mathbf{F_{y,F}^{*}}\) & \(d_{y,F}^{*}\) & \(k_{e,F}^{*}\) & \(d_{u,F}^{*}\) \\  & & **(s)** & **(t)** & **(kN)** & **(mm)** & **(kN/mm)** & **(mm)** \\ \hline X & modal & 1.09 & 0.63 & 500.4 & 624.5 & 12.4 & 50.4 & 55.9 \\ \hline Y & modal & 1.10 & 0.48 & 603.0 & 556.6 & 5.4 & 103.1 & 51.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the equivalent bilinear SDOF system of the initial unbraced frame in the longitudinal (X) and transversal (Y) directions. Although quite obvious, it is important to check that \(d_{PP}^{*}<d_{u,F,X}^{*}\) and \(d_{PP}^{*}<d_{u,F,Y}^{*}\) in order to avoid structural collapse. Moreover, it is worth noting that the established performance point corresponds to a specified design ductility of the main RC frame (\(\mu_{F}^{*}\)):  
\[\mu_{F}^{*}=d_{PP}^{*}/d_{y,F}^{*} \tag{13}\]  
where \(\mu_{F}^{*}=2.0\) in the longitudinal (X) and \(\mu_{F}^{*}=4.5\) in the transversal (Y) direction. _Step 3: Selection of the design ductility of the dissipative braces_  
With reference to the overall response of the dissipative brace (DB), which is composed of the dissipative device (D) plus a stiff beam (B) for the connection to the main frame, the ductility of the equivalent SDOF of the dissipative brace (\(\mu_{DB}^{*}\)) can be selected within the range \(\mu_{DB}^{*}=4\div 12\)[28; 29].