# Graphical and Analytical Quantitative Comparison
## 2 Mechanical Behavior of Masonry Domes
### Evolution of Classical Methods for Assessing the Stability of Vaults and Dome  
The most critical studies on domes have begun from the 18 th Century onward, even though the construction of masonry domes dates back to the 17 th century. [PERSON]'s centuries-old law of inversion for which \"_as hangs a flexible cable, so inverted, stand the touching pieces of an arch_\" remains a milestone of structural theory. [PERSON] had an intuition that led him to state that an arch in compression has the same, reversed shape of a hanging chain under its own weight [18]. It is possible to analyze complicated vaults and domes with reference to the theory of masonry arches. Though complex, a dome can always be considered as a set of arches and obtained by slicing the dome with meridional planes, forming several arches in the meridional direction [19]. The thrust transmitted by the two ideal half-arches connecting at the keystone must be counteracted at the base to make the arch-system stable. The slicing technique, which analyses two-dimensional sections to assess three-dimensional structures, dates back to [PERSON] (1748). In his structural analysis of the cracked basilica dome of St. Peter, in Rome [20], [PERSON] considered the cupola as a series of lunes with internal forces in only meridional directions, namely as wedge-shaped arches not requiring hoop forces to be stable. Inspired by [PERSON]'s law, the method proposed by [PERSON] can be considered a first statement of the static theorem of limit analysis of masonry structures [20]. Many architects have followed [PERSON]'s principle, among which, [PERSON], at the beginning  
Figure 3: Stresses on the meridians for the hemispherical dome. (**a**) In the case of a single surface of revolution, the stresses on the meridian planes are tensions and compressions with zero stress at \(51^{\circ}49^{\prime}\) from the vertical. (**b**) In the dome with an oculus, the stresses on the meridian planes are similarly tensions and compressions with a point of zero stress with a greater angle than \(51^{\circ}49^{\prime}\), which varies with the opening amount. (**c**) In the dome surmounted by a lantern, the extent of the tensile stress is increased, and the point of zero stress rises [16]. Figure 2: Crack pattern on hemispherical domes [1]. of the 20 th century, applied the same principle of the catenary in 3D hanging models to design the Church of Colonia Guell (near Barcelona) and additional complex structures. A different approach to the analyses of domes was introduced by [PERSON] in 1866 [17], who developed the basic equilibrium concept of the membrane theory, which was subsequently refined by [PERSON] (1904) in [21]. In 1878, [PERSON] published a graphical method to analyze masonry domes by determining the hoop forces, in terms of both tension and compression, for the membrane stress condition of the vault [9]. While recognizing that hoop forces can mutate within the surface, [PERSON] considered the dome to be a radial series of arches and, to maintain compression everywhere in the section, he forced the solutions to stay within the middle-third of the section. In the 1920s, [PERSON] published a graphical procedure for masonry domes [22] that can be seen as a graphics-based version of the membrane theory introduced by [PERSON]. Like the membrane theory, this method fixed internal hoop forces to the values needed to equilibrate the meridional forces. [PERSON]'s approach led to the subsequent development of the Modified Thrust Line Method (MTLM) [23]. During the 19 th century, the thrust line analysis and graphic statics marked a further development in the theory of structures by providing a practical method based on the catenary principle. Out of all the modern theories to support classical approaches developed in the first half of the 20 th Century, [PERSON]'s Limit Analysis [1] appears to be the most appropriate approach to understand and analyze masonry structures.