# From Stress to Shape: Equilibrium of Cloister and Cross Vaults
## 4 Application
### Cross Vault of the Anterior Portico of the Church of St. Pietro in Vineis in Anagni  
In this section we consider the case of the anterior portico of the church of San Pietro in Vineis in Anagni. The portico consists of three similar cross vaults: the geometrical description of each vault, referring for notation to Figures 13 and 14, is given by:  
\[f_{intr}(x_{1},x_{2})= \begin{cases} 4\dfrac{h_{i,1}}{b_{i}^{2}} \Bigg{(}\dfrac{b_{i}^{2}}{4}-x_{2}^{2}\Bigg{)}+4\dfrac{h_{i}-h_{i,1}}{a_{i}^{2 }}\Bigg{(}\dfrac{a_{i}^{2}}{4}-x_{1}^{2}\Bigg{)}&(x_{1},x_{2})\in(\Omega_{1} \cup\Omega_{3})\\ 4\dfrac{h_{i,2}}{b_{i}^{2}}\Bigg{(}\dfrac{b_{i}^{2}}{4}-x_{2}^{2}\Bigg{)}+4 \dfrac{h_{i}-h_{i,2}}{a_{i}^{2}}\Bigg{(}\dfrac{a_{i}^{2}}{4}-x_{1}^{2}\Bigg{)}& (x_{1},x_{2})\in(\Omega_{2}\cup\Omega_{4})\end{cases}  \tag{31}\] \[f_{extr}(x_{1},x_{2})= \begin{cases} 4\dfrac{h_{e,1}}{b_{e}^{2}} \Bigg{(}\dfrac{b_{e}^{2}}{4}-x_{2}^{2}\Bigg{)}+4\dfrac{h_{e}-h_{e,1}}{a_{e}^{2 }}\Bigg{(}\dfrac{a_{e}^{2}}{4}-x_{1}^{2}\Bigg{)}&(x_{1},x_{2})\in(\Omega_{1} \cup\Omega_{3})\\ 4\dfrac{h_{e,2}}{b_{e}^{2}}\Bigg{(}\dfrac{b_{e}^{2}}{4}-x_{2}^{2}\Bigg{)}+4 \dfrac{h_{e}-h_{e,2}}{a_{e}^{2}}\Bigg{(}\dfrac{a_{e}^{2}}{4}-x_{1}^{2}\Bigg{)}& (x_{1},x_{2})\in(\Omega_{2}\cup\Omega_{4})\end{cases}  \tag{32}\]  
The values given to the geometrical parameters introduced in Equations (31) and (32), are reported in Table 1.  
Figure 14: Perspective of the cross vault with main geometrical parameters. Figure 13: Partition of the domain \(\Omega\). For this kind of vault a concave membrane surface comprised between the intrados and the extrados does not exist; as a consequence, using Equation (6) with a known shape and unknown stress potential would lead to a non-elliptic problem, that cannot be solved using a FEM-like procedure. The inverse problem, consisting in the search of the shape corresponding to a given concave stress potential, is elliptic instead, and the FEM-like procedure proposed in [21] can be applied. It is a well-known result (see [27]) that the stress potential corresponding to a standard cross vault has the shape of a cloister vault. The case at hand, however, is not a standard cross vault since there is a curvature not only orthogonally to each median of the plan-form, but also in the median direction. We expect, therefore, a biaxial stress field, with components both along the direction of the medians, and in its orthogonal direction. We propose therefore the following stress potential \(F(x_{1},x_{2})\)  
\[F(x_{1},x_{2})=\left\{ \begin{aligned} &-\alpha_{0}\Bigg{[}\Bigg{(}x_{1}^{2}- \frac{a_{i}^{2}}{4}\Bigg{)}+\beta\left(x_{2}^{2}-\frac{b_{i}^{2}}{4}\right) \Bigg{]}&&(x_{1},x_{2})\in(\Omega_{1}\cup\Omega_{3})\\ &-\alpha_{0}\Bigg{[}\Bigg{(}x_{2}^{2}-\frac{b_{i}^{2}}{4}\Bigg{)} +\beta\left(x_{1}^{2}-\frac{a_{i}^{2}}{4}\right)\Bigg{]}&&(x_ {1},x_{2})\in(\Omega_{2}\cup\Omega_{4})\end{aligned} \right. \tag{33}\]  
For sake of example, in Figure 15 we present three different stress potentials obtained for \(\beta=0\), \(\beta=0.3\), and \(\beta=0.7\) and \(\alpha_{0}=1\). \begin{table}
\begin{tabular}{c c c c c} \hline \(a_{i}\) [m] & \(b_{i}\) [m] & \(h_{i}\) [m] & \(h_{i,1}\) [m] & \(h_{i,2}\) [m] \\ \hline
4.68 & 4.59 & 2.16 & 1.84 & 2.05 \\ \hline \(a_{e}\) [m] & \(b_{e}\) [m] & \(h_{e}\) [m] & \(h_{e,1}\) [m] & \(h_{e,2}\) [m] \\ \hline
4.88 & 4.79 & 2.36 & 2.04 & 2.25 \\ \hline \end{tabular}
\end{table} 
Table 1: Intrados and extrados data referring to notation in Figure 14.  
We assume the same definition of the error introduced in Equation (20) and evaluate it for different values of \(\beta\), and then minimizing \(I_{k}\) with respect to \(\alpha_{0}\). The values of \(\beta\) considered in this case are  
\[\beta_{k}=0.3+k0.02\,,\qquad k=0,...,10 \tag{34}\]  
The corresponding plot of the error \(I_{k}\) versus \(\beta\) is reported in Figure 16 in a logarithmic scale along the error axis. The minimum occurs at \(\beta=0.4\), and corresponds to an error of \(3.94\times 10^{-2}\).