# Graphical and Analytical Quantitative Comparison
## 3 Problem Statement
### Materials and Methods
#### 3.3.3 Membrane Equilibrium Analysis  
The Membrane Equilibrium Analysis (MEA) is a complete analytical tool to assess the stability of a vault and find a possible state of stress, under the restrictions of Rigid  
Figure 4: (**a**) Bottom, plan view of hemispherical dome and a lune. Top, elevation view of a lune divided into blocks. (**b**) three-dimensional view of a lune and the forces acting on it. The hoop forces are highlighted in green; in blue, the masses of blocks, and in red, the thrusts. No-Tension (RNT) materials. The driving criterion is again the Safe Theorem of Limit analysis, particularized to three-dimensional structures. Differently from the previously considered methods, MEA is a more general numerical tool, which does not need any preliminar consideration of the symmetry of the geometry and of the loads, and is, therefore, also adopted in complicated vaults, such as cloister vaults or cross vaults. In MEA, the equilibrium of a vault is assessed through the search for a membrane surface, comprised inbetween the intrados and the extrados of a vault, capable to equilibrate the assigned load with a purely compressive membrane stress field [34; 40; 41]. The theoretical framework adopted here has been proposed in previous works [10; 40; 42], and consists of solving the equilibrium equation  
\[div\mathbf{T}+\mathbf{b}=\mathbf{0} \tag{3}\]  
for membranes, where \(\mathbf{T}\) is the membrane generalized stress--that is, the stress for unit thickness of the membrane--and, therefore, is measured as a force per unit length, and \(\mathbf{b}\) is the applied load. The membrane behavior allows for the assumption that the stress can be expressed in a membrane reference system as  
\[\mathbf{T}=T^{\alpha\beta}\mathbf{a}_{\alpha}\otimes\mathbf{a}_{\alpha} \tag{4}\]  
where the assumption of summation over repeated indices is implicit; moreover, \(\mathbf{a}_{\alpha}\) is a curvilinear covariant basis vector, tangent to the membrane geometry, and \(T^{\alpha\beta}\) is a contravariant component of the generalized stress. After some algebra, which is out of the scopes of this work, but is reported in [41; 43], we reach the transversal equilibrium equation of the membrane  
\[F_{,22}\,f_{,11}+F_{,11}\,f_{,22}-2F_{,12}\,f_{,12}+p=0 \tag{5}\]  
where \(f=f(x_{1},x_{2})\) is the geometrical description of the membrane, \(x_{1}\) and \(x_{2}\) are the Cartesian coordinates of a horizontal plane; \(F\) is a stress potential, defined such that \(F_{,11}=S^{22}\), \(F_{,22}=S^{11}\), and \(F_{,12}=-S^{12}\), \(S^{\alpha\beta}\) being the components of the Pucher stresses--that is, the projection of the generalized stress \(\mathbf{T}\) onto the planform of the membrane--and, as such, are measured again as a force per unit length, that is, in kN/m. In particular, \(S^{\alpha\beta}=J\,T^{\alpha\beta}\), where \(J\) is the ratio between an infinitesimal area onto the membrane and its projection onto the horizontal plane; moreover, a subscript after a comma denotes partial derivative; finally, \(p\) is the distributed vertical load. Equation (5) is a second-order partial differential equation, where the stress potential and the shape are both unknown. It can, however, be solved if at least one of the two functions are assigned. This can be done by assigning a shape (comprised between the geometrical boundary of the vault), and assuming the stress potential as unknown: in this case, the compressive state of the resulting stress has to be checked, and in case, a new membrane shape has to be assigned [41]; in this case, to numerically solve Equation (5), the ellipticity of the Equation that corresponds to the concavity of the assigned surface should be assured. On the contrary, Equation (5) can be solved for the membrane shape, given an assigned compressive stress state. In this case, the Equation is elliptic by definition, since the assigned stress state is compressive under the Safe Theorem, and, therefore, the stress potential is concave.