## 6 Numerical Analysis of the Structural Unit and Assessment of Strengthening Interventions

Table 3: Model parameters. load cases LC2 and LC4 (loads along the x-axis and the negative y-direction). As for load case LC3, (positive y-direction), the identified collapse mode involved a combination between the out-of-plane overturning of the upper part of the facade and, in some cases (Figure 14b), out-of-plane bending with a vertical cylindrical plastic hinge. As for the second retrofitting intervention (R2), additional quoins (red blocks in Figure 13c) were inserted at the level of the second floor (Figures 12, 13, 14 and 15c). In such a case, similar but more limited collapse mechanisms were observed in the load cases LC1 and LC3 (along the x-axis and positive y-direction). Conversely, the analysis for LC2 and LC4 (negative x-, y-axes) revealed a quite different behavior of the construction, which tended to experience global collapse mechanisms. As for the last retrofitting intervention (R3), in addition to the aforementioned quoins, a set of ties was applied. Those were modeled in LiABlock_3D through the insertion of washer-bearing plates (green blocks in Figure 12d) modeled as special rigid blocks of null volume and active vertical contacts  
Figure 12: Collapse mechanisms and participating masses (in blue) provided by LiABlock_3D for the load case LC1 (live loads/\(f_{L}\) along the positive x-axis): (**a**) the castle wing in its present conditions; (**b**) retrofitting intervention n.1 (R1); (**c**) retrofitting intervention n.2 (R2); (**d**) retrofitting intervention n.3 (R3). Quoins in red; washer-bearing plates in green. at the interface between the facade and the plate itself. Constant horizontal forces, equal to the ties' failure load, were applied to the bearing plates, so as to simulate the effect of tie forces. Figures 12d and 13d report the collapse mechanisms for the load cases LC1 and LC2 (along the x-axis) where a single tie force was applied to the washer-bearing plates located on the F2 and F4 facades. Figure 14d and d highlights the failure modes for the load cases LC3 and LC4 (live loads along the y-axis), considering the effects of two ties applied to the four plates located on the F1 and F3 facades. In such case, global mechanisms were obtained with a damage distribution along the whole structure. Figure 13: Collapse mechanisms and participating masses (in blue) provided by LiABlock_3D for the load case LC1 (live loads \(f_{\mathrm{L}}\) along the positive x-axis): (**a**) the castle wing in its present condition; (**b**) retrofitting intervention n.1 (R1); (**c**) retrofitting intervention n.2 (R2); (**d**) retrofitting intervention n.3 (R3). Quoins in red; washer-bearing plates in green. In Table 4, the collapse load multipliers for each load case and retrofitting interventions are reported. In the same table, the percentage increments are also listed, proving the effectiveness of the applied devices. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Load** & **Loading** & **Present** & \multicolumn{4}{c}{**Retrofit Interventions**} \\ \cline{4-9}
**Case** & **Axis** & **Conditions** & \multicolumn{2}{c}{**R1**} & \multicolumn{2}{c}{**R2**} & \multicolumn{2}{c}{**R3**} \\ \hline LC1 & +x & 0.13460 & 0.14634 & (+9\%) & 0.22059 & (+64\%) & 0.22308 & (+66\%) \\ LC2 & \(-\)x & 0.06625 & 0.09721 & (+47\%) & 0.20033 & (+202\%) & 0.22073 & (+233\%) \\ LC3 & +y & 0.03794 & 0.07432 & (+96\%) & 0.09333 & (+146\%) & 0.10774 & (+184\%) \\ LC4 & \(-\)y & 0.08496 & 0.09638 & (+13\%) & 0.12455 & (+47\%) & 0.13443 & (+58\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Collapse load multipliers for the four structural configurations under investigation. The percentage increases respective to the present conditions are reported in brackets.