# From Stress to Shape: Equilibrium of Cloister and Cross Vaults
## 4 Application
### Cloister Vault
In Figures 6, we show the the stress potential for two representative values of \(r_{0}\). Adopting these two shapes as the stress potential, and taking as boundary condition for the membrane geometry \(f(x_{1},x_{2})\) the value of the rise of the middle surface between the intrados and the extrados of the vault, Equation (6) can be solved using LSM, as described in Section 3.2. By constructing a structured triangulation based on a regular node distribution over the planform \(\Omega\), the membrane shapes corresponding to the stress potentials of Figure 6, are represented in Figure 7.  
Figure 7 gives an idea on how the choice of the parameter \(r_{0}\) (and thus, the extension of the central zone characterized by a biaxial stress state) influences the final shape of the membrane geometry. In the second case (\(r_{0}=1\)), the membrane geometry resembles the shape of a cloister vault more than in the case of \(r_{0}=0.5\). It is also evident that the height of the two shapes (in the direction of the \(x_{3}\)-axis) are not comparable with the height of the cloister vault defined from Equations (23) and (24), which is \(2.13\) m.  
The minimization of the error function (20) with respect to the parameters \(p_{0}\) and \(r_{0}\) is performed as follows: first, \(r_{0}\) is given a finite number of values in the range \([0,L]\), i.e., \(r_{0,k}=1.1+k0.01\), \(k=0,\ldots,20\). Then, for each value of \(r_{0,k}\), we search the \(p_{0,k}\) that minimizes \(I_{k}=I_{k}(r_{0,k},p_{0,k})\). In Figure 8 we show the considered values of the \(r_{0,k}\) on the horizontal axis, and on the vertical axes we show the corresponding optimal value of \(I_{k}\). The lowest value of \(I_{k}\) is reached when \(r_{k}=1.14\), which correspond to \(p_{0,k}=3.45\times 10^{3}\). Therefore the parameters of the stress potential are fixed to \(r_{0}=1.14\) and \(p_{0}=3.45\times 10^{3}\). The best-fitting membrane geometry is then represented in Figure 4.0  
In Figure 10 we also show two sections of the membrane and of the intrados and extrados, the first one taken at \(x_{2}=0\), and the second one taken along the diagonal of the domain, that is for \(x_{2}=-x_{1}\). Figure 4.1 Representation of the stress potential assuming (**a**) \(r_{0}=0.5\) and \(p_{0}=1\); (**b**) \(r_{0}=1\) and \(p_{0}=1\). Figure 4.2 Membrane surfaces in equilibrium with the stress potential of Figure 4.3 (**a**) \(r_{0}=0.5\) and \(p_{0}=1\); (**b**) \(r_{0}=1\) and \(p_{0}=1\). Figure 4.4 Plot of \(I_{k}\) as function of \(r_{k}\). Figure 4.5 Sections of the vault and of the membrane surface represented in Figure 9 along \(x_{2}=0\) (**a**) and along the diagonal \(x_{2}=-x_{1}\) (**b**), after optimization with respect to the shape of the stress potential. Figure 4.6 Best fitting membrane shape after the first optimization phase, for the problem of the cloister vault. We notice from Figure 10b that some points of the membrane fall outside the volume of the vault; this is confirmed also by Figure 11, where we show the color map of the function \(g(x_{1},x_{2})\) defined as  
\[g(x_{1},x_{2})= \begin{cases}f-f_{intr}&f<f_{intr}\\ 0&f_{intr}<f<f_{extr}\\ f-f_{extr}&f>f_{extr}\end{cases}  \tag{30}\]  
for which positive values indicate points in which the membrane shape is above the extrados of the vault, negative values indicate when the membrane is below the intrados, and vanishing values indicate nodes that are comprised within the volume of the vault. We then perform the last step of the optimization, with respect to the boundary values of the membrane shape, obtaining a new membrane geometry whose sections for \(x_{2}=0\) and \(x_{2}=-x_{1}\) are shown in Figure 12. In this case, we notice that the points that were falling outside the volume of the vault before this optimization step, are now comprised between the intrados and the extrados. Figure 11: Colormap of \(g\) with respect to \(x_{1}\) and \(x_{2}\). Yellow points represent regions of the domain for which the membrane is above the extrados.