## 2 Materials and Methods
### Design of Exoskeletons  
To improve the seismic behavior of the factory, both orthogonal and parallel exoskeletons are designed and placed along the building transverse direction (y-direction). From the elastic response spectrum, a target displacement \(\Delta^{*}_{\text{tar}}\) is defined, so that the retrofitted structure remains in the elastic field. In particular, \(\Delta^{*}_{\text{tar}}\leq\Delta^{*}_{\text{y}}\), \(\Delta^{*}_{\text{y}}\) being the yielding displacement of the structure.  
To evaluate the global lateral stiffness of the retrofitted structure \(\text{K}_{\text{d}}\), the following equation is used:  
\[\text{K}_{\text{d}}=\frac{\text{F}_{\text{e}}^{*}}{\Delta^{*}_{\text{tar}}}= \frac{\text{m}^{*}\cdot\text{S}_{\text{ADRS}}(\Delta^{*}_{\text{tar}})}{\Delta^ {*}_{\text{tar}}} \tag{1}\]  
where \(\text{m}^{*}\) is the mass of the structure and \(\text{S}_{\text{ADRS}}\) is the elastic spectral acceleration for the displacement \(\Delta^{*}_{\text{tar}}\). The formulation above is valid under the hypothesis that the equivalent mass and the modal participation factor of the existing construction and the retrofitted ones remain the same. Furthermore, it is valid under the hypothesis that the yielding displacement of the exoskeleton corresponds to the yielding displacement of the existing structure. Under the hypothesis of parallel coupling, the global lateral stiffness of the exoskeleton Ke can be evaluated by the following equation:  
\[\text{K}_{\text{e}}=\text{K}_{\text{d}}-\text{K}_{\text{EX}} \tag{2}\]  
where \(\text{K}_{\text{EX}}\) is the global lateral stiffness of the existing structure, that can be evaluated as  
\[\text{K}_{\text{EX}}=\frac{\text{F}_{\text{y}}^{*}}{\text{d}_{\text{y}}^{*}} \tag{3}\]  
where \(\text{F}_{\text{y}}^{*}\) and \(\text{d}_{\text{y}}^{*}\) are yielding force and displacement of the system, respectively [24; 25; 26; 27].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Distribution** & **du,d* (cm)** & **du,b* (cm)** & **dr* (cm)** & **IR,d** & **IR,b** \\ \hline Modal\_x & 19.2 & 16.2 & 9.3 & 2.1 & 1.7 \\ \hline Accel\_x & 18.8 & 15.8 & 7.7 & 2.4 & 2.1 \\ \hline Modal\_y & 5.0 & 6.0 & 8 & 0.6 & 0.8 \\ \hline Accel\_y & 10.5 & 12.5 & 8.9 & 1.2 & 1.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Safety indices.  
Figure 6: Seismic checks in the ADRS plane: (**a**) modal_x (**b**) accel_x (**c**) modal_y (**d**) accel_y.  
With regard to the typological choice, based on the seismic-resistant scheme, and the dimensional choice, related to the first attempt for system design, concentric bracing frames (CBS) and circular hollow sections (CHS) are chosen to comply with the previous provisions related to global lateral stiffness. Columns have diameter of 193.7 mm with thickness from 5 to 10 mm on the longest sides (north and south) and of 219.1 mm with thickness from 8 to 16 mm on the shortest sides (east and west). CHS profiles of 168.3 \(\times\) 8 mm are used for beams. Bracings along the east and west sides are made of CHS sections with diameter of 114.3 mm and thickness ranging from 3.2 to 8 mm. Instead, bracings of the exoskeletons placed on the building north and south sides have diameter of either 88.9 mm (thickness of 3.2 mm) or 101.6 mm (thickness from 3.2 to 6.3 mm).  
In Figure 7, the disposition of exoskeleton is shown. The orthogonal exoskeletons are connected to horizontal St. Andrew's cross bracings at every floor.