## 5 Proposed Design Procedure
This step can be avoided using a specific software (such as MATLAB or other calculation tools), and the evaluation of the energy can be performed using the real capacity curve from Step 3. Step 5. The MDOF system is converted to an SDOF system by transforming the capacity curve to a capacity spectrum (\(S_{dt}-S_{ab}\)). \[S_{dt}=\frac{D_{t}}{\Gamma\phi_{t}};S_{a}=\frac{F_{S+T}}{\Gamma\cdot L} \tag{9}\]  
where \(\Gamma\) is the participation factor of the modal shape \(\phi\) (\(\Gamma=(\phi^{T}M\Omega)/(\phi^{T}M\phi)\)), and \(L=\phi^{T}MI\). The modal characteristics of the braced structure may change at every iteration owing to new brace characteristics. Therefore, \(\phi\), \(\Gamma\), and \(L\) must be updated with the current configuration. Step 6. The equivalent viscous damping (\(\xi^{*}_{\alpha\beta,S+ADS}\)) of the \(S+ADS\), which is necessary for obtaining the match between the displacement of the equivalent SDOF system and the target spectral displacement (\(S^{*}_{dt}=D^{*}/(\Gamma\phi^{T})\)), was evaluated by imposing the equivalence of the target displacement and performance displacement. According to the capacity spectrum method, the demand spectrum was obtained by reducing the 5% damping response spectrum by multiplying by the damping correction factor (\(\eta\)), which is a function of \(\xi_{tot}\), as follows:  
\[\eta=\sqrt{2+\frac{1}{\xi_{tot}\cdot 10}} \tag{10}\]  
From Equation (10), one obtains \(\xi^{*}_{tot}\), the damping needed to reduce displacement up to the target, \(S^{*}_{dt}\). \[\xi^{*}_{tot}=0.1\bigg{(}\frac{S_{5\%}}{S^{*}_{dt}}\bigg{)}^{2}-0.05 \tag{11}\]  
Step 7. The damping provided by the structure (\(\xi^{*}_{\alpha\beta,S}(D^{*}_{t})\)) can be determined using Equation (7), where \(D_{t}\)* is the top displacement corresponding to \(I_{D,S}\), \(I_{S,S+ADS}\) is the energy dissipated by the \(S\), and the elastic strain energies, \(S+T\)\(I_{D,S}\) and \(I_{S,S+ADS}\), are determined from the capacity curves of the \(S\) and \(S+ADS\), respectively. Step 8. Given \(\xi^{*}_{tot}\) from Equation (11), the equivalent viscous damping required to be supplied by the tower (\(\xi^{*}_{\alpha\beta,ADS}(D^{*}_{t})\)) (the additional equivalent viscous damping contribution owing to the tower) is evaluated from Equations (6) and (8) as follows:  
\[\xi^{*}_{\alpha\beta,ADS}(D^{*}_{t})=\xi^{*}_{tot}(D^{*}_{t})-\xi^{*}_{\alpha \beta,S}(D^{*}_{t})-\xi_{I} \tag{12}\]  
Step 9. Once the additional equivalent viscous damping (\(\xi^{*}_{\alpha\beta,ADS}(D^{*}_{t})\)) (to be provided by the tower) was evaluated using Equation (12), the stiffness and yielding strength required to achieve the desired additional damping can be determined using the same procedure previously adopted for the structure (Step 7). Therefore, the dissipative tower can be designed (e.g., according to the configuration in Figure 1, the extension to the other configurations in Figure 2 is very simple). The energy dissipated by the tower can be expressed as follows:  
\[I^{id}_{D,ADS}=\big{(}F_{Ay}D-F_{Ay}F_{A}(D)\big{)}/0.25 \tag{13}\]  
\(D\) was obtained from the pushover analysis according to the control joint (where \(D\) is the top displacement, \(D_{t}\)). \(D_{Ay}\) is the top displacement corresponding to the yielding of the dissipative devices: \(D_{Ay}\) can be reasonably assumed as being \(D_{Ay}\leq 0.25D^{*}_{t}\). \(F_{Ay}\), once \(D_{Ay}\) has been defined, is consequently determined. A dissipative system usually consists of a dissipative device or a group of devices characterized by \(K_{d}\) and \(F_{dy}\), which are the stiffness in the elastic range and the yielding force of the system, respectively. The tower, except for the dissipative devices, has to be designed to remain elastic and to be as stiff as possible; therefore, the following suggestions should be considered. Designing the tower structure helps to calibrate the stiffness of the dissipative elements; this has to be conducted after the definition of the global parameters of the additional dissipative system.