# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 5 Design of the Seismic Rehabilitation
### Step 4: Determination of the Equivalent Damped Brace System  
In this work, hysteretic dampers with an elastic-perfectly plastic behavior similar to those assumed in other studies [2; 9; 11; 12; 47; 55] were considered. Figure 7 shows the force-displacement curve of the equivalent SDOF damped brace, modeled as an in-parallel system of an elastic steel brace (B) and an equivalent SDOF hysteretic damper (D). The equivalent damping of the system \(\xi_{DB}\) was calculated according to the formula reported in Figure 7, where \(\kappa_{DB}\) is a parameter that accounts for the energy dissipation capacity of the damper and is calibrated on experimental tests [12; 48]. The ductility \(\mu_{DB}\) of the dampers was chosen at the beginning of the retrofit procedure by the designer, considering the characteristics of the devices available on the market. For steel hysteretic dampers, \(\mu_{DB}\) is typically between 4 and 10, providing an equivalent viscous damping ratio between 20 and 40% [9; 10; 39; 89]. In this study, two analyses were performed considering the two boundaries of the range, namely \(\mu_{DB}=4\) and \(\mu_{DB}=10\)[11]. At the first iteration of the procedure, the effective viscous damping ratio \(\xi_{eff}^{1}\) of the combined in-parallel system (F + DB) made of the equivalent SDOF frame and the equivalent SDOF damped brace was evaluated in order to fulfill the displacement equality \(S_{d}\left(\tau_{F}^{*};\,\xi_{eff}^{1}\right)=d_{p}^{*}\). By evoking the damping correction factor according to [20],  
Figure 6: Comparison of the equivalent SDOF bilinear capacity curves of the four-story frame with the relevant demand response spectra in X- and Z-directions. Figure 7: Force-displacement of the equivalent SDOF damped brace. \(V_{y}^{*DB}\) and \(V_{p}^{*DB}\): yield and ultimate force; \(d_{y}^{*DB}\) and \(d_{p}^{*DB}\): yield and ultimate displacement; \(\mu_{DB}\): ductility factor. \[S_{d}\Big{(}T_{F}^{*};\,\xi_{eff}^{1}\Big{)}=S_{d}(T_{F}^{*};\,\xi=3\%)\cdot\sqrt{ \frac{10}{5+\xi_{eff}^{1}}}\,,\,\text{leading to}\,\,\xi_{eff}^{1}=10\cdot\left(\frac{S_{d} \big{(}T_{F}^{*};\,\xi=3\%\big{)}}{d_{p}^{*}}\right)^{2}-5.\,\,\text{Eventually,}\,\,\xi_{eff}^{1}\) can be expressed according to the energetic equivalence of Equation (1), where the yield force \(V_{p,1}^{*^{DB}}\) of the damped bracing system is the unknown.  
\[\xi_{eff}^{1}\cdot V_{p}^{*^{F}}=\xi_{F}\cdot V_{p}^{*^{F}}+\xi_{DB}\cdot V_{p,1}^{*^{DB}} \tag{1}\]  
Once \(V_{p,1}^{*^{DB}}\) was determined from Equation (1), the equivalent bilinear curve of the equivalent SDOF retrofitted frame (F + DB) was constructed and plotted in the ADRS space, Figure 8. The ultimate displacement of the capacity curve of the braced frame is equal again to the target displacement \(d_{p}^{*}\), while the corresponding force at second iteration becomes \(V_{p,2}^{*^{F+DB}}=V_{p}^{*^{F}}+V_{p,1}^{*^{DB}}\). Step 4 was iterated until the displacement at the performance point of the equivalent SDOF braced frame (F + DB) converges to the target displacement \(d_{p}^{*}\). At the \(i^{\text{th}}\) iteration, the equivalent damping ratio of the retrofitted frame \(\xi_{eff}^{i}\) is expressed by Equation (2) where \(V_{p,i}^{*^{DB}}=V_{y,i}^{*^{DB}}\) is the only unknown, as \(V_{p,(i-1)}^{*^{DB}}\) has been determined in the previous iteration. \[\xi_{eff}^{i}\cdot\left(V_{p}^{*^{F}}+V_{p,(i-1)}^{*^{DB}}\right)=\xi_{F}\cdot V _{p}^{*^{F}}+\xi_{DB}\cdot V_{p,j}^{*^{DB}} \tag{2}\]  
As shown in Figure 8, the graphical representation in the ADRS plane shows at each iteration the relationship between demand and capacity. It must be noted that, at the \(i^{\text{th}}\) iteration, the effective stiffness and the effective period of the equivalent SDOF upgraded frame are equal to \(K_{F+DB}^{i}=\left(V_{p}^{*^{F}}+V_{p,i}^{*^{DB}}\right)/d_{p}^{*}\), and \(T_{eff}^{i}=2\pi\sqrt{\frac{\pi\pi}{K_{F+DB}^{i}}}\), respectively. For the three case-study MRFs, the convergence was reached at the third iteration.