*Hoop stress does not exist in this part of the dome.

HF = Hoop Forces, MF = Meridional Forces.

*Hoop stress does not exist in this part of the dome.

HF = Hoop Forces, MF = Meridional Forces.

The results are shown in table 3. The values are taken in three points (interior surface of the dome) coinciding with the rings and at the base of the dome.

4.3.2 Study carried out with graphical analysis The stability or equilibrium approach is the most important concept to assess the safety of these structures. Equilibrium can be visualized using a line of thrust, the theoretical line or inverted catenary, which represents the path of the resultants of the compressive forces throughout the structure. The forces' line passes through the pier up to the ground-line. At this point the Stability Factor (SF), or Rankine factor, is obtained (SF = b/2c, b being the pier weight). This factor was determined by W. J. Rankine in the mid-nineteenth century. If that factor takes a value 1, it indicates the collapse point of the pier: a value 3 is considered safe. That study was conducted with a Graphical Analysis method.

Various suppositions have been made. Firstly, two thrust lines were found for both models.

Figures 7 shows the two thrust lines found for each supposition of the dome profile of the Pantheon. The weight of the uppermost level is 13.10 N. Stability

Figure 7. Thrust line for profile Model 9 and Model 8. A comparison.

factors of SF8 = 2.9 and SF9 = 3.9 were obtained for model 8 and Model 9, respectively.

In order to know the influence of different densities used in the construction of the Pantheon, three variables were taken for Model 9 in function of the unit weight: a) the dome at 1,350 and 1,600 kg/m3, b) the dome at 1,350 kg/m3 (lighter) and c) the dome at 1,600 kg/m3 (heavier). The stability factors obtained are: SF9 a = 3.9, SF9 b = 3.1 and SF9 c = 2.9. Those values are compared with the values obtained by Lancaster and are reflected in table 4.

4.3.3 Thrust line in a geometrical context We can propose another question. What is the maximum height allowed for the Rotunda? What is the relation between geometry and the thrust line? Figure 8 shows different stability values for model 9 in function of the height of the wall. That factor is taken at the base of the pier and takes a value 1, indicating the collapse point of the pier. The actual height is a diameter of one sphere (73.75 Roman Feet) inscribed in its interior. If the high value increases 82.35 Roman Feet the SF is 1 and the thrust line passes out of the pier. If the high increases 27 Roman Feet the SF value is 2, where stability starts to be critical.

The thrust line is tangent to the extrados at point A. That point coincides approximately with the vertex of the so-called Diophantine triangle. Point A is

Table 4. Comparative graph between Lancaster and Morer & Goni
Figure 9. Model 10, main tensile stress.

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