# Graphical and Analytical Quantitative Comparison
## 5 Results and Discussion  
In this section, we compare the results obtained from TLA, MTLM and MEA. Restricting ourselves to the case of axis-symmetric structures (for which MTLM and TLA can be applied), a first consideration can be made of the similarities of the thrust lines, which cannot coincide. In fact, introducing hoop forces in the formulation modifies the thrust lines obtained with the TLA, even when starting with the same thrust value at the top of the vault.  
The similarities and differences in these methods can be evaluated by comparing the thrust at the base of the dome and the radial and circumferential stress distributions, in the case in which a similar thrust line profile is followed, and the optimized GSF in the case in which the objective of the analyses is optimization.  
Concerning the thrusts, with particular reference to graphical methods, in Table 1 we notice that, following a similar thrust line profile, for TLA, the horizontal thrust is the same along the meridian, and, of course, the hoop forces are vanishing; on the contrary, after adopting the MTLM, the radial thrust increases along the meridian due to the presence of the hoop forces.  
Concerning the stresses, MTLM and MEA can be compared after a suitable conversion of the static entities is made. In MTLM, the computation is made in terms of forces, whereas MEA is formulated in terms of generalized stress, and finally, using the Pucher stresses, which are the projection of the membrane-generalized stress \(T\) onto the domain of the membrane geometry.  
To compute the circumferential Pucher stress from the MTLM hoop forces, for each block, \(i\), we divide the force \(H_{i}\) by the lateral area of the block, and then we multiply it by the vertical average thickness of the voussoir itself.  
Concerning the radial stresses, we start from the contact force \(P_{i}\) along the meridian between two blocks: we consider the component along the \(x\)-axis and divide this by the contact area, and multiply again by the vertical thickness of the voussoir.  
A comparison between the results, in terms of Pucher stresses, is summarized in Table 2 and shown in Figure 16.  
Figure 15: Thrust line obtained using MEA in the safest case.  
From Figure 16 we notice that, for the same (or similar) thrust line, there is a slightly different stress distribution for the three methods, especially in the upper part of the dome (small values of the radius). In fact, when hoop stresses are admitted (MTLM and MEA), in the central part, high hoop stresses arise and the radial stress remains quite low. Some differences remains between the two methods, due to the distribution of the load of the lantern among the first voussoirs from the top (MTLM) and a ring around the lantern for MEA. In fact, MEA cannot handle concentrated loads; therefore, the load of the lantern must be spread over a finite area. TLA does not allow hoop stresses, and therefore, for similar thrust lines with respect to the other two methods, there are different (higher) values for the radial stresses near to the dome crown. At the impost, conversely, due to the absence of hoop forces in all the methods, the radial stresses are very similar.  
Since, with MTLM, it is possible to choose the hoop forces' distribution, this method allows the highest control on the thrust line shape, and, consequently, returns the highest GSF value. On the contrary, since the thrust path obtained through TLA is controlled by only two parameters, it is less flexible and, therefore, a lower GSF value is obtained.  
MEA, in general, is the most complete method for the evaluation of the stability of domes, but, for this kind of application, the results are inappropriate, since it cannot access the double curvature thrust surfaces that can be accessed using simpler methods, such as the MTLM.