# From Stress to Shape: Equilibrium of Cloister and Cross Vaults
## 4 Application  
### Cloister Vault  
Consider the case of a cloister vault under its self-weight, whose geometry is described by the volume enclosed in-between two surfaces, the intrados and the extrados, defined canonically inside the domain \(\Omega=\{(x_{1},x_{2})\in[-L,L]\times[-L,L]\}\) as follows:  
\[f_{intr}(x_{1},x_{2})=\left\{ \begin{aligned} &-h\Bigg{(}\frac{x_{1}}{L}\Bigg{)}^{2} \quad(x_{1},x_{2})\in(\Omega_{1}\cup\Omega_{3})\\ &-h\Bigg{(}\frac{x_{2}}{L}\Bigg{)}^{2}\quad(x_{1},x_{2})\in( \Omega_{2}\cup\Omega_{4})\end{aligned} \right. \tag{23}\]\[f_{extr}(x_{1},x_{2})=\left\{ \begin{aligned} &-h\Bigg{(}\frac{x_{1}}{L+t}\Bigg{)}^{2}+t \quad(x_{1},x_{2})\in(\Omega_{1}\cup\Omega_{3})\\ &-h\Bigg{(}\frac{x_{2}}{L+t}\Bigg{)}^{2}+t\quad(x_{1},x_{2})\in( \Omega_{2}\cup\Omega_{4})\end{aligned} \right. \tag{24}\]  
where \(\Omega_{1}\), \(\Omega_{2}\), \(\Omega_{3}\), and \(\Omega_{4}\) are the subregions of \(\Omega\) depicted in Figure 3.  
A perspective representation of the midsurface of this vault is represented in Figure 4.  
The values adopted in Equations (23) and (24) are \(L=2.5\) m, \(h=2.13\) m, and \(t=0.23\) m. The values have been taken by the geometry of the vault analyzed in [21], and located in the interior of Palazzo Caracciolo di Avellino in Naples. From [21] we also assume \(p=8\)\(\mathrm{kN}/\mathrm{m}^{2}\) as distributed load per unit projected area. By following the methodology described in Section 3, the transverse equilibrium Equation (6) can be solved for the geometry \(f(x_{1},x_{2})\), by assuming a special parametric form for the stress potential. For the case at hand, inspired by the analysis described in [21], we identify a central zone \(\Omega^{*}\) of the domain \(\Omega\), in which the projected stress is a plane pressure. \(\Omega^{*}\) is delimited by a curve \(\Gamma\) defined as  
\[r_{c}(\theta)=r_{0}\sqrt{2}(1-\alpha(cos(2\theta))^{P}) \tag{25}\]  
where \(r_{0}\) is a parameter, \(\alpha\) is set to \(0.05\) and \(P\) to \(40\). In Figure 5 we show the contour \(r_{c}\) obtained for two representative values of \(r_{0}\), namely \(r_{0}=0.5\) and \(r_{0}=1\). Figure 4: Three-dimensional view of the midsurface of the vault described by Equations (23) and (24). Figure 3: Partition of \(\Omega\) into four portions. Inside \(\Omega^{*}\), the Pucher stress is assumed as isotropic, that is  
\[\mathbf{S}=-p_{0}(\mathbf{e}_{1}\otimes\mathbf{e}_{1}+\mathbf{e}_{2}\otimes \mathbf{e}_{2})\quad. \tag{26}\]  
The corresponding stress potential is therefore easily computed as  
\[F=-\frac{1}{2}p_{0}(x_{1}^{2}+x_{2}^{2})\quad, \tag{27}\]  
where \(p_{0}\) is a parameter defining the intensity of the stress potential (i.e., of the stress) that will be subsequently optimized. In the external zone \(\Omega_{ext}\) the stress is assumed to be quasi-uniaxial in the radial direction. This means that, on defining \(\hat{\mathbf{r}}\) as the direction of the radii emanating from the geometrical center of the domain and directed toward the external boundary, the stress in \(\Omega_{ext}\) can be written as  
\[\mathbf{S}=-\sigma(s)\hat{\mathbf{r}}\otimes\hat{\mathbf{r}}+\varepsilon \sigma_{0}\hat{\mathbf{t}}\otimes\hat{\mathbf{t}} \tag{28}\]  
where \(\sigma(s)\) is related to \(p_{0}\) and denotes the stress intensity along the radius, and \(\varepsilon\sigma_{0}\) is a small stress which has the function to give a small extra-curvature to the potential in the direction \(\hat{\mathbf{t}}\) orthogonal to \(\hat{\mathbf{r}}\), in order to preserve the ellipticity of Equation (6) when solved for the shape \(f(x_{1},x_{2})\). The stress potential associated to the stress distribution (28) in the external zone is obtained by imposing a linear growth in the radial direction, enforcing the continuity of the potential \(F\) and of its slope across the curve \(\Gamma\):  
\[F_{ext}=\
abla F_{int}|_{\Gamma}\cdot\hat{\mathbf{r}}\,s+F|_{\Gamma}- \varepsilon\frac{\sigma_{0}}{2}s^{2}\quad, \tag{29}\]  
where \(s\) is a parameter indicating the distance from \(\Gamma\) along the radius.