## 6 Stability Domain Analysis of Groin Vaults  
After having shown a minimum thickness parametric analysis on different groin vault geometries, this section is devoted to computing the stability domain for some of them in order to provide a methodology that can give further information on their structural behaviour and robustness. A meaningful measure of masonry vaults' stability is provided by the ratio among their minimum and maximum thrusts [27]. The larger this ratio, the more robust the structure, since it indicates that a larger space of admissible stress states (in the present case, admissible thrust networks) exists. At the limit state, where only one equilibrium solution is possible, minimum and maximum thrust coincide, and this factor drops to 1.0. In [25] a diagram is proposed to observe how the distance between these extreme thrusts evolves towards the collapse state for a given force flow. This diagram together with the GSF provides meaningful information about the robustness, and stability of masonry vaults which is not possible to fully describe just by looking at the GSF [2], but neither by only computing the extreme thrusts for the initial state. In fact, by plotting minimum and maximum thrust curves, the area among the curves measures the space of internal admissible stress states. To obtain such diagrams, the optimisation problem of Equation (4) is modified with specific objective functions that minimise/maximise the horizontal thrust for a given form diagram and structural thickness. This optimisation is performed recursively for reduced structural thicknesses until the limit state is attained. To illustrate the proposed methodology, the structural geometries corresponding to points A, B, and C of Figure 12 are selected and assessed, considering a thickness over span \(t/s=0.050\). Point A corresponds to the minimum of the orthogonal diagram (\(r/l_{0}=0.61\)), where the curved and fan-like diagrams are outperformed in terms of minimum thickness. Point B corresponds to the intersection of the orthogonal and fan-like diagrams (i.e., \(r/l_{0}=0.71\)), where the curved diagram performs better than the others. Finally, point C represents the fan-like minimum point and occurs for \(r/l_{0}=0.79\). To gather further information about the structure's robustness and its extremes of thrust, the stability domain was drawn for these three points. The stability domains corresponding to these three points are presented in Figure 15, where on the right the perspective of the vaults' geometry is reported. The stability domain plots the extremes of maximum and minimum thrust as percentages of the vault's total weight for each form diagram. The curves approach until they coincide at the point which the minimum thickness is found and the GSF is calculated. The stability domain for each vault corresponds to the convex envelope of the different form diagrams analysed. The stability domains of Figure 15 give a full picture of the stability of the three groin vaults. They also show that a trans-topology analysis, based on different form diagrams, is required to model the intricate structural behaviour of cross vaults. The stability domain of structure A shows that the maximum and minimum thrust are obtained with the curved and orthogonal diagrams, respectively. This indicates that for the extreme thrust analyses, both diagrams have to be considered. Moreover, the fan-like  
Figure 14: Comparison of (**a**) the three dimensional analysis, and (**b**) the equivalent 2D problem for the first arch of the network analysed considering the curved form diagram with \(\beta=20^{\circ}\). Hinges observed are highlighted and numbered. diagram is found to be entirely contained within the bounds of the other diagrams. Finally, the orthogonal arrangement achieves an overall minimum thickness value, resulting in a GSF of 3.6. Looking at structure B, both the orthogonal and fan-like diagrams present similar values of minimum thickness (i.e., GSF = 2.1).