# Graphical and Analytical Quantitative Comparison
## 1 Introduction  
Proper modelling of the mechanical behavior of masonry domes is a crucial task for the correct evaluation of their structural safety, and consequently, to assure their preservation. Masonry structures do not behave as elastic continua, and, due to the intrinsic anisotropic and non-linear response, most of the difficulties in masonry modelling come from its mechanical properties, its heterogeneity (as it is made of blocks usually bonded with mortar) and, of course, from the geometric complexity of the artefact.  
Although non-linear analysis remains an essential field of structural investigation in the current scientific context, many studies are looking at on limit analysis as a reliable tool to evaluate the safety factors of masonry constructions.  
The current article provides a further deepening of some methods used to examine the state of unreinforced masonry structures, based on the equilibrium approach to limit analysis under [PERSON]'s assumptions of rigid no-tension materials [1].  
The theoretical investigation leans on the case study of San Francesco di Paola in Naples, a large dome in masonry conceived of as an assemblage of rigid blocks and subject to the self-weight action. In previous works, this dome has been investigated, with a particular focus on graphical analysis of the dome [2, 3], combined with kinematic analysis [4], to define a range of possible displacements, assessing the relative stability of the structure. The dome safety assessment has been evaluated under horizontal actions in some of these works, by performing a thrust membrane analysis through a tilting test [5, 6, 7].  
These works represent the background and the starting point of the present study. As is well known, the fundamental aspect of the graphic statics in the modelling of masonry domes is the analysis of a three-dimensional structural behavior in a bi-dimensional space. Despite being a powerful tool to explain and examine the stability of two-dimensional structures, Thrust Line Analysis (TLA) does not consider the hoop forces, thus producing overly conservative results when applied to three-dimensional cases [8]. On this basis, the present paper aims to explore the role of variable internal meridional and hoop forces by introducing the Modified Thrust Line Method (MTLM) [9] to investigate the equilibrium of the dome. The results obtained are compared with the Membrane Equilibrium Analysis [10], an analytical and general method that can be applied to three-dimensional cases, disregarding the axis-symmetry of the geometry and of the loads. The methods are not further developed with respect to the current literature. The main contribution of this research is to investigate the differences and the points of agreement of the different methods, highlighting the strengths and weaknesses of each of them, also with reference to the presented case study.  
Out of the authors' knowledge, there are few works comparing different methodologies applied to masonry structures. TLA and MEA have been used in [5] to assess the seismic vulnerability of San Francesco di Paola; in [11], some differences between TLA and MEA have been highlighted, and both methods are discussed together with the Finite Element Method, following the idea proposed in [12].