## 5 Minimum Thickness of Parametric Groin Vaults
### Minimum Thickness of Groin Vaults Assuming an Orthogonal Form Diagram  
Figure 8 shows the results for the parametric analysis with the orthogonal form diagram (Figure 3b). As in the results for the first pattern, the minimum thickness is presented as a function of radius over length (\(r/l_{0}\)), and is grouped into five curves corresponding to different springing angles \(\beta=[0^{\circ},10^{\circ},20^{\circ},30^{\circ},40^{\circ}]\). Figure 8: Minimum thickness over span (\(t_{\min}/s\)) computed with the orthogonal form diagram. Curves are grouped considering different springing angles \(\beta=[0^{\circ},10^{\circ},20^{\circ},30^{\circ},40^{\circ}]\). Figure 7: A comparison of (**a**) the three dimensional analysis, and (**b**) the equivalent 2D problem for the first arch of the network analysed considering the fan-like form diagram, and \(\beta=20^{\circ}\). Hinges observed are highlighted and numbered. The results present a trend similar to the ones obtained for the fan-like diagram. Each curve has a singular global minimum which identifies the optimal ratio \(r/l_{0}\) for each springing angle. Similarly to the results for the first diagram, the optimal \(r/l_{0}\) decreases, for larger springing angles. Moreover, one can observe that after a certain point, decreasing the springing angle is preferred to obtain reduced values of minimum thickness. The solutions for optimal \(r/l_{0}\) points obtained with the orthogonal form diagram are presented in Figure 9. The points where the thrust network touches intrados and extrados, are highlighted. The values of minimum thickness obtained, and the optimal \(r/l_{0}\) ratios which connect to each of the plotted solutions, are reported in Table 1.  
A comparison of the results shows that the orthogonal form diagram achieves the global minimum for lower \(r/l_{0}\), i.e., for vaults that are less pointy (Figure 9a-e). As a comparison to the first diagram, for \(\beta=0^{\circ}\), the minimum thickness corresponds to 2.4% of the span (Figure 9a/Table 1), which represents a reduction of 27% in the global minimum thickness achieved with the first diagram; this minimum was achieved for a vault with \(r/l_{0}=0.72\), instead of the 1.14 observed for the fan-like diagram. For the highest springing angle, \(\beta=40^{\circ}\), the minimum thickness obtained corresponds to 0.5% of the span (Figure 9e/Table 1), which occurred for a ratio \(r/l_{0}=0.53\); this value represents a reduction of 54% in the value of minimum thickness obtained with the first diagram. A different hinge configuration was found for the results with the orthogonal diagram (Figure 9f). Since this flow of forces behaves as a series of arches that thrust to the main diagonals, and then towards the supports, these arches have different spans, and do not present the same individual hinge behaviour. Instead, a global 7-hinge mechanism appears in the vault, which is highlighted in Figure 9f. Figure 9f shows that two hinges formed at the supports (1, 7), two hinges points were found in the diagonals (2, 6), two hinge-lines extended through the diagonals (3, 5), and a hinge point was found in the apex of the arch in the boundary (4). In Figure 10, the solution obtained for the springing angle \(\beta=20^{\circ}\) is compared with the equivalent problem in the 2D arch after the adjustment of  
Figure 9: Perspective of the solutions obtained with the orthogonal diagram for each minimum for \(\beta\) equals (**a**) \(0^{\circ}\), (**b**) \(10^{\circ}\), (**c**) \(20^{\circ}\), (**d**) \(30^{\circ}\), and (**e**) \(40^{\circ}\); (**f**) the 7-hinge mechanism observed for all solutions. the 3D self-weight. Unlike the case of the fan-like diagram (Figure 7), the solution for the sole-standing last arch is not the same as for the vault (Figure 10), and different minimum thicknesses (\((t_{\min}/s)_{\text{vault}}=1.38\%\), \((t_{\min}/s)_{\text{arch}}=1.34\%\)) and hinge locations are observed, which reinforces that a global mechanism is encountered in the vault, instead of a series of identical local mechanisms.