## 4 Case Study
### Remarks  
The analyses performed on the retrofitted church configurations showed that the EDR values determined using FEMA 356 or ATC-40 were generally higher than those calculated with the TH method. The influence of the hysteretic model (SD and SSD) adopted for the inelastic hinges of the roof is clear, with a lower damping coefficient in the case of the SSD model. This seems justified by the lower amount of energy dissipated by the roof connections during the seismic event due to the narrower shape of the hysteretic  
Figure 11: Status of the yielding of the roof’s hinges (on the **left**) and the walls’ hinges (on the **right**) at the end of one of the accelerograms. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Mode** & **Period** & \multicolumn{2}{c}{**tan-x**} & \multicolumn{2}{c}{**tan-y**} & \multicolumn{2}{c}{**rotn-z**} \\ \hline
**n.\({}^{\circ}\)** & **[s]** & **Mass (\%)** & **Sum (\%)** & **Mass (\%)** & **Sum (\%)** & **Mass (\%)** & **Sum (\%)** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 3: Case study: the features of the first twenty vibrational mode shapes (equivalent FEM). Figure 12: Average values of the displacement ratio (on the **left**) and force ratio (on the **right**) by varying the hysteretic variable \(\beta_{\text{-}HYS}\). loops that characterize the SSD model, regardless of the trends of the accelerograms. In the EDR evaluation, the value of the hysteretic variable \(\beta_{\textit{-HYS-opt}}\), corresponding to the maximum EDR, is close to its limit value of \(\beta_{\textit{-HYS-opt}}\) = 1.5. Beyond this value, the EDR curve is essentially flat and, indeed, towards the value of \(\beta_{\textit{-HYS-opt}}\) = 2.1, the EDR begins to decrease. For the case study, in the design phase, \(\beta_{\textit{-HYS-opt}}\) = 2.1 represents an extremely rigid configuration of the connections. On the other hand, in the naive transversal response of the case study analyzed in terms of the damped rocking by the introduction of the displacement ratio and force ratio, the optimal value of the hysteretic variable is about \(\beta_{\textit{-HYS-opt}}\) = 1.40. Since the evaluation of the EDR and the damped rocking analyses gave very similar results in terms of the hysteretic variable, it is possible to compare the EDR values for \(\beta_{\textit{-HYS-opt}}\) = 1.40, calculated by FEMA 356, ATC-40, and the TH. In correspondence of \(\beta_{\textit{-HYS-opt}}\), including 5% of the inherent damping, the EDR evaluated with FEMA 356 is in the range between 16.00% and 16.80%, while, when considering ATC-40, it is in the range between 16.3% and 25%, for the SSD and SD models, respectively. The EDR values obtained for FEMA 356 and ATC-40 in correspondence of \(\beta_{\textit{-HYS-opt}}\) are listed in Table 4.  
When FEMA 356 is adopted, the difference in the EDR values for the SD and SSD models is \(\Delta_{\textit{FEMA-SD-SSD}}\) = 5%, while, when considering ATC-40, the difference is \(\Delta_{\textit{ATC-SD-SSD}}\) = 53%. This could be a limit for these codes, which is due to the different approaches (the coefficient method and equivalent linearization method, respectively). When the TH method is adopted, the computed EDR values are much more homogeneous. In fact, when the cycle extracted in the TH method refers to FEMA 356, the difference between the EDR with the SD model and the one with the SSD model is \(\Delta_{\textit{TH-FEMA-SD-SSD}}\) \(\cong\) 21%, whereas, if the TH refers to ATC-40, the corresponding difference is \(\Delta_{\textit{TH-ATC-SD-SSD}}\) \(\cong\) 22%. Moreover, when comparing the estimated EDR values obtained from the TH analyses and related to the selected cycles identified by the two codes for the same hysteretic cycle model, the following differences can be highlighted: (i) for the SD model, the difference between the EDR values based on FEMA 356 and ATC-40 is about 2.7%, while (ii), for the SSD model, the difference between the EDR values calculated using FEMA 356 and ATC-40 is about 3.3%.