## 3 Results and Discussion
### Seismic Design of the Cold-Formed Exoskeleton  
The first instance of damage served as the starting point for the design method, which aimed to determine the thickness of the diagonal members to optimize the retrofitting solution. This initial damage was observed in the longitudinal direction at a displacement of 0.61 cm. The design method, which was based on literature approaches [30,31] and adapted for this specific exoskeleton type [26], began with the assumption that the retrofitted structure should exhibit ductility comparable to that of the unreinforced structure. The process started by setting a target displacement slightly lower than the displacement at the first point of damage, which was fixed at 0.55 cm. Using the equivalent bilinear curve representation of the structure and the elastic spectrum in the acceleration displacement response spectrum (ADRS) format, an iterative procedure defined the ultimate displacement of the retrofitted structure and, consequently, its stiffness. This iterative process continued until the ductility of the retrofitted structure matched that of the unreinforced one (\(\mu\) = 2.16). At the end of the procedure, an equivalent bilinear curve representative of the cold-formed exoskeleton was derived by subtracting the design parameters of the retrofitted structure from those of the existing structure. According to the procedure, the design stiffness of the Resisto 5.9 Tube was determined to be 1.039.603 \(\mathrm{daN/cm}\). This value represents the total stiffness of the exoskeleton in the longitudinal direction and needed be redistributed across each level and module of the Resisto 5.9 Tube. The distribution of stiffness along the building height was achieved using a proportionality factor \(\mathrm{r_{k}}\), which relates the overall stiffness of the existing structure to that of the Resisto 5.9 Tube. This factor was then applied to the floor stiffnesses of the existing structure to determine the floor stiffnesses of the exoskeleton, as detailed in Table 7. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Dir.** & **d\({}_{\text{dam}}\)****1[cm]** & **F\({}_{\text{dam}}\)****1[kN]** & **d\({}_{\text{cu}}\)****1[cm]** & **F\({}_{\text{max}}\)****1[kN]** & **d\({}^{\ast}\)****y** 1[cm]** & **F\({}^{\ast}\)****y** 1[kN]** & **K\({}^{\ast}\)****1[kN/cm]** \\ \hline X & 0.61 & 1323.7 & 3.25 & 4096 & 1.5 & 3008 & 2002 \\ Y & 1.75 & 1100.7 & 17.22 & 3058 & 3.49 & 2102 & 602 \\ \hline \end{tabular}  
* Displacement (d\({}_{\text{dam}}\)) and shear (F\({}_{\text{dam}}\)) corresponding to the first damage, ultimate displacement (d\({}_{\text{cu}}\)) and maximum shear (F\({}_{\text{max}}\)). Asterisked terms indicate the equivalent bilinear yielding displacement (d\({}^{\ast}\)\({}_{\text{y}}\)), shear (F\({}^{\ast}\)\({}_{\text{y}}\)) and stiffness (K\({}^{\ast}\)).  
\end{table} 
Table 6: Results of the pushover analysis on the existing building. Figure 9: Capacity of the existing structure in the most severe condition: pushover curves (**a**) and location of the first point of damage (**b**). To evaluate the regularity in height of the existing structure and determine whether the exoskeleton is needed to redistribute and homogenize the floor stiffnesses, reference to the Italian Code was made. According to the code, for a structure to be considered regular in height, the stiffness of successive floors should fall within the range in Equation (1):  
\[-0.1\ \leq\text{Ks},\text{i}\ \leq 0.3 \tag{1}\]  
The existing buildings met the requirements for regularity in height; therefore, it was not necessary to use the exoskeleton for redistributing the stiffnesses. To simplify the design procedure, the floor stiffness was divided by the number of exoskeleton modules in each direction (n\({}_{\text{Ri}}\)) to determine the stiffness of a single diagonal (kRi,s).