# Graphical and Analytical Quantitative Comparison
## 3 Problem Statement  
### A Few Initial Remarks  
In this work, we consider three different methodologies (the Thrust Line Analysis, the Modified Line Thrust Method, and the Membrane Equilibrium Analysis) in the framework of limit analysis and apply them to the case study of the dome of San Francesco di Paola, in Naples.  
The Thrust Line Analysis (TLA) is based on the slicing technique to reduce the initial three-dimensional problem of the vault in a two-dimensional analysis. Looking at a dome, it consists of dividing it into lunes defined by vertical cutting planes passing through the revolution-axis. In this approach, compressive hoop forces are totally neglected. This initial assumption is not arbitrary, but based on a peculiar physical behavior that can often be seen in masonry domes: the supports spread outwardly a little, and the dome shows typical meridional cracks that are often necessary to accommodate physiological settlements of the supports. In this light, the slicing technique can correctly reproduce this physical behavior by not running into over-conservative solutions. This phenomenon is due to the low tensile resistance of the material. As a result, part of the dome is cut into wedges, and its lower part is subjected to uniaxial compressive stresses in the direction of the meridional cracks.  
This aspect does not compromise the structural behavior of domes. The difference between use of the slicing technique and biaxial methods as the membrane theory lies mainly in the upper part of the dome, subjected to a biaxial stress state. Nonetheless, the value of the compressive hoop force is low and does not lead to high differences in terms of results. Although TLA allows for the direct evaluation of such effects, representing a widely accepted assessment strategy, in some cases, it could lead to over-conservative solutions [8].  
However, Membrane Equilibrium Analysis (MEA) is a more general method, which naturally considers the presence of biaxial stress regimes; the main drawback is that it requires a more sophisticated mathematical background, losing the characteristic simplicity of other graphical methods. The advantage is that it can be applied to a large variety of vaults, disregarding the symmetries of geometries and loads, such as cloister vaults, cross vaults, and others. Some comparative studies between TLA and MEA were recently developed in [11].  
The Modified Thrust Line Method (MTLM) conjugates the simplicity of TLA with the possibility of accounting for the presence of hoop forces. The advantage of MTLM is its simplicity, which allows the method to be included in the set of graphical techniques. Hence, the analyses executed through this method can be compared with the TLA analyses to show the influence of hoop forces and those obtained with the membrane solutions.  
The aim of this work is to highlight the analogies and differences between the three methods. Based on the case study of San Francesco di Paola, we highlight the differences, strengths and weaknesses of each method, outlining how the parameters used in each method may affect the results. The idea is to provide some hints about simple methods for the study of masonry domes. A simple but complete knowledge of the behavior of masonry domes, possibly with simple instruments, can help to assess the safety of existing masonry buildings, avoiding inappropriate interventions on historic buildings.