## 5 Minimum Thickness of Parametric Groin Vaults
### Minimum Thickness of Groin Vaults Assuming Fan-Like Form Diagram  
Figure 5 shows the results of the parametric analysis for the fan-like form diagram (Figure 3a). The minimum thicknesses are presented as functions of the radius over length and are grouped in five curves corresponding to different springing angles \(\beta=[0^{\circ},10^{\circ},20^{\circ},30^{\circ},40^{\circ}]\). Figure 5 shows that each curve presents a singular point (cusp), which is also its global minimum. This minimum represents the optimal \(r/l_{0}\) value for each springing angle, i.e., the best achievable geometry once \(\beta\) has been defined. Overall, increasing \(\beta\) results in lower global values of minimum thickness. However, after a certain \(r/l_{0}\) is reached, lowering the springing angle will provide reduced minimum thickness values. It is also noticeable that the optimum \(r/l_{0}\) decreases with increasing spring angles, meaning that for higher springing angles, the optimum \(r/l_{0}\) is less \"pointy\". This geometric trade-off can be seen by looking at the geometries in Figure 6, where solutions corresponding to optimal \(r/l_{0}\) values are reported. In this figure, the network edges have a thickness proportional to the compressive axial force, while edges carrying zero force are not shown. The points where the thrust network touches the intrados and extrados are highlighted in blue and green, respectively. Figure 6 shows the minimum thickness solutions corresponding to the \(r/l_{0}\) minima per springing angle \(\beta\). Table 1 lists the \(r/l_{0}\) ratio at which this minimum point is achieved, and the corresponding minimum thickness over span calculated (\(t_{\min}/s\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Form Diagram** & \multicolumn{2}{c}{**Fan-Like**} & \multicolumn{2}{c}{**Orthogonal**} & \multicolumn{2}{c}{**Curved**} \\ \hline \(\beta\) & \(r/l_{0}\) & \(t_{\min}/s\) & \(r/l_{0}\) & \(t_{\min}/s\) & \(r/l_{0}\) & \(t_{\min}/s\) \\ \hline \(0^{\circ}\) & 1.14 & 3.3\% & 0.72 & 2.4\% & - & - \\ \(10^{\circ}\) & 0.97 & 2.8\% & 0.68 & 2.0\% & - & - \\ \(20^{\circ}\) & 0.79 & 2.1\% & 0.61 & 1.4\% & 0.65 & 1.5\% \\ \(30^{\circ}\) & 0.69 & 1.5\% & 0.56 & 0.9\% & - & - \\ \(40^{\circ}\) & 0.62 & 1.1\% & 0.53 & 0.5\% & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of minimum thickness obtained for all form diagrams considered. The results are grouped by different springing angles (\(\beta\)), and show the optimal pair radius-over-length (\(r/l_{0}\)) and minimum thickness over span (\(t_{\min}/s\)) calculated. Figure 5: Minimum thickness over span (\(t_{\min}/s\)) computed with the fan-like form diagram. Curves are grouped considering different springing angles \(\beta=[0^{\circ},10^{\circ},20^{\circ},30^{\circ},40^{\circ}]\). When \(\beta=0^{\circ}\), the minimum thickness is 3.3% of the effective span, and the corresponding vault is defined by \(r/l_{0}=1.14\) (Figure 6a/Table 1), which corresponds to a highly pointed vault. Conversely, for the highest springing angle considered, i.e., \(\beta=40^{\circ}\), the minimum thickness corresponds to only 1.1% of the effective span, and this global minimum was found for \(r/l_{0}=0.62\) (Figure 6e/Table 1), which illustrates how this optimal \(r/l_{0}\) reflects in a less pointed vault cross-section. The points where the thrust network touches the envelope suggest the location of potential hinges, as noted in [5]. Figure 6 shows that all the geometries corresponding to optimal \(r/l_{0}\) values show the same global 7-hinge mechanism reported in Figure 6f. Regarding the qualitative distribution of the forces in the structure, the edges connecting the fan segments have zero force, and therefore disappear in Figure 6. This indicates that all the fan lines can be considered as separate arches, spanning from the support to the cross vault's ridge line. In fact, the only non-negligible connection among these arches can be found along the midspan of the vault. These points coincide with the hinge line number 4 (Figure 6f).