# From Stress to Shape: Equilibrium of Cloister and Cross Vaults
## 2 Mathematical Preliminaries
### Geometry and Equilibrium  
Referring to Figure 1 for notation, the geometry of a generic membrane surface \(S\) contained inside a typical shell structure can be described in Monge form in terms of the position vector \(\mathbf{x}\) of the points of \(S\) as follows:  
\[\mathbf{x}=x_{1}\mathbf{e}_{1}+x_{2}\mathbf{e}_{2}+f\mathbf{e}_{3} \tag{1}\]  
where \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}\) is the orthonormal triad coherent with a given Cartesian frame \(\{O;x_{1},x_{2},x_{3}\}\), the couple of coordinates \((x_{1},x_{2})\), which are the parameters of the curvilinear description (1), belongs to a region \(\Omega\) of the plane \(\{O;x_{1},x_{2}\}\) called planform of \(S\), and \(f=f(x_{1},x_{2})\) is the rise of the membrane with respect to the \(x_{1}-x_{2}\) plane. We assume \(f\in\mathbb{C}^{0}(\Omega)\)--that is to say that the considered surface \(S\) is continuous but not necessarily smooth. An efficient way to describe the membrane equilibrium of a membrane under a surface load \(\mathbf{q}\) per unit area, is the formulation proposed by [PERSON] in [38]. [PERSON] analysis is based on the introduction of the projected stress \(\mathbf{S}\), in terms of which two of the three equilibrium equations for the membrane can be made independent of the membrane shape. The membrane stresses are tangent in each point to the membrane surface \(S\). Therefore it is convenient to introduce a curvilinear basis on the membrane, which are computed as  
\[\mathbf{a}_{\alpha}=\frac{\partial\mathbf{x}}{\partial x^{\alpha}},\quad \alpha=1,2 \tag{2}\]  
In addition, it is useful to introduce the contravariant basis \(\mathbf{a}^{\alpha}\). \(\alpha=1,2\) defined such that \(\mathbf{a}_{\alpha}\cdot\mathbf{a}_{\beta}=\delta_{\alpha}^{B}\), being \(\delta_{\alpha}^{B}\) the Kronecker delta. Figure 1: Geometry of a generic membrane, with its planform, the memebrane stress \(\mathbf{T}\) and the Pucher (projected) stress \(\mathbf{S}\). Therefore we can decompose the membrane generalized stress tensor as  
\[\mathbf{T}=T^{\alpha\beta}\mathbf{a}_{\alpha}\otimes\mathbf{a}_{\beta}, \tag{3}\]  
where \(T^{\alpha\beta}\) are the contravariant components of the generalized stress. The projected (Pucher) stress components \(S_{\alpha\beta}\) (see Figure 1) are easily defined in terms of \(T^{\alpha\beta}\) as \(S_{\alpha\beta}=J\,T^{\alpha\beta}\), \(J\) being the ratio between the surface area and the projected area. In terms of Pucher stress components, the vector equilibrium equation, projected into the non-orthonormal vector basis \(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{n}=\mathbf{a}_{3}\), n being the unit normal to \(S\), becomes:  
\[S_{11,1}+S_{12,2}+p_{1}=0,\;\;\;S_{21,1}+S_{22,2}+p_{2}=0,\;\;\;S_{\alpha\beta }f_{\alpha\beta}-p_{\gamma}f_{\gamma}+p_{3}=0. \tag{4}\]  
where \(\{p_{1},p_{2},p_{3}\}\) are the Cartesian components in the given reference of \(\mathbf{p}=J\,\mathbf{q}\), that is the load per unit projected area. By using the projected stress, the first two equilibrium equations do not depend on the membrane shape and are the same as for the plane stress problem. In the case of pure vertical loading, say \(\mathbf{p}=p\mathbf{e}_{3}\), the first two equations can be solved introducing an Airy stress potential \(F(x_{1},x_{2})\)--that here we only assume to be continuous--defined such that  
\[S_{11}=F_{,22}\,,\;\;\;S_{22}=F_{,11}\,,\;\;\;S_{12}=S_{12}=-F_{,12}\,. \tag{5}\]  
The third equation of (4) expresses the balance of the vertical component of the force \(p_{3}=\mathbf{p}\cdot\mathbf{e}_{3}\) with the scalar product between the Pucher stress tensor and the Hessian of the function \(f\), that defines the curvature of the surface. On introducing (5) into (4), we obtain, as transversal equilibrium equation  
\[F_{,11}f_{,22}+F_{,22}f_{,11}-2F_{,12}\,f_{,12}+p_{3}=0 \tag{6}\]  
Equation (6) is a second order partial differential equation, that can be either elliptic, parabolic or hyperbolic, depending on its (usually non-constant) coefficients.