4.1 Base Dome: Models 1 and 2 of the dome with different densities
The first two models analyzed are solely of the dome. Those models are used to validate the values obtained with the numerical method by comparing them with the results showed by Mark and Hutchinson (Mark & Hutchinson 1986). Subsequently, they are analyzed by the graphical method and an assessment of the results is obtained by comparing them to the data previously recorded.
Two models were used. Model 1 consisted ofa hemisphere with an interior radius of 21.81 m and a thickness of 1.5 m. According to Mark, 2,200 kg/m3 were taken as the unit weight. It was not common among Romans to use 2,200 kg/m3 for vaults, but we use that unit weight to be able to compare the results obtained in this article with their results. Model 2 included a hemisphere with a unit weight of 1,350 kg/m3 in the dome. Both have been solved with graphical and numerical analyses.
It is assumed that the base of the dome is supported only against vertical loadings. There are no horizontal buttresses. That simplification allows us to analyze the dome using the shell theory. The results obtained make it possible to know the maximum value of circumferential tensile stress.
4.1.1 Study carried out with Abaqus
The values obtained through the graphical analysis have been taken at the center of gravity of the volume of the each voussoir. (Figure 2)
4.1.2 Study carried out with graphical analysis Figure 3 shows the section of the dome used to obtain results in the gravity center of each voussoir (volume) and the force triangle corresponding to the centroids.
Figure 2. Model 2: maximum tensile stress (kg/cm2).
Table 1. Internal hoop and meridional stress, at the base of the dome (kg/cm2).
Figure 2. Model 2: maximum tensile stress (kg/cm2).
Figure 3. Force triangle.
Model 1 Heavy density
Model 2 Light density
The vertical line is the sum of each weight. The horizontal line is an arbitrary one; the meridional forces are parallel to segments that link two adjoining centroids. The hoop forces are perpendicular at the meridians and pass by the final points of the meridional forces where they intersect with the horizontal line.
The value of the magnitude of the meridional and hoop forces are measured and divided by the contact area, yielding data that will be compared with previous results.
Table 1 shows the results obtained at the base of the dome with both methods and the values presented by Mark and Hutchinson. Mark does not give data about
Hoop Meridional Hoop Meridional forces forces forces forces
*these values were not provided by Mark & Hutchinson.
the mesh used, which could also affect the accuracy of the results and the values of the meridional forces.
Additional information that is important to know is to identify the point at which the internal hoop stress changes from compression to tensile. The graphical analysis shows that in voussoir number nine, where the center of gravity is placed at 52.5° from the axis of revolution, the hoop stress changes. The numerical analysis coincides at 50°. These results correspond approximately to the values expected for a thin hemispherical dome [51° 49'].
The vertical reactions in the base of the dome vary from 8.5T - in the exterior of the dome- to 6.5T in the interior.
4.2 Models 3-6: Influence of the uppermost level and rings
The next four models deal with determining the influence in the structural behavior of different construction levels. First, the lower and upper level are added to the dome, secondly the uppermost and finally the stepped-rings.
In order to evaluate the influence of the uppermost level and the stepped-rings, four models were created. In Model 3, the pier has been added to Model 2. The pier has been reduced to 5.5 m and without taking the chambers into account. In Model 4, the uppermost level has been added to Model 3. Both are assumed to use lightweight concrete (Figure 2). Model 5 is a full model, with stepped rings added. The density of heavy concrete was taken at 2,200 kg/m3 for unit weight. Model 6 is similar to Model 5 with different densities, 1,350, 1,600 and 1,750 kg/m3, according to figure 2. Models 5 and 6 deal with how to determine the influence of the stepped-rings built over the dome.
4.2.1 Study carried out with Abaqus Figure 4 shows the models studied with Abaqus. Model 3 assumes that the dome is embedded at the base. That supposition modifies the values obtained. The traction zone changes, the tensile stress reaches a value of 1.21 kg/cm2 and is placed approximately 72° from
Figure 4. Models 3-6 analyzed with Abaqus.
the axis (Figure 9a). The base of the dome is compressed and the values of the hoop forces vary from -0.1 kg/cm2 to -0.83 kg/cm2 at the exterior of the cupola. The maximum value -0.83 kg/cm2 is taken outside of the cupola and coincides with the line of discontinuity.
In Model 4, the lower zone of the dome is embedded in the uppermost level of the wall. The values of the table are taken at the base and at three points: where the rings start, where they end and at an intermediate point on the interior surface of the dome. The traction zone changes and is placed on the exterior of the uppermost level, reaching a value of 0.6kg/cm2. In Figure 5b we notice a line of discontinuity where the dome tries to open itself. At that point the value reached is -0.1 kg/cm2.
Models 5 and 6 show the influence of the rings over the dome. The values obtained at the base of the dome in model 6 hardly present differences from model 4. The hoop forces are about 0.25 kg/cm2 and the meridional forces about -2.7 kg/cm2.
4.2.2 Study carried out with graphical analysis Models 4 and 6 were analyzed with the graphical analysis method. The profile of the dome has changed. In Model 4 the dome begins where the uppermost level ends (Figure 5a). The embrace angle is approximately 70°, taken from axis.
In Model 6 the weight of each stepped ring has been considered and the profile of the dome is taken, as shown in Figure 5b.
The values obtained with the Graphical Analysis method were reached with the steps indicated in 4.2 and are taken at the center of gravity of each voussoir.
4.3 Models 7-9: cracked model
The models with cracks completely change the behavior of the dome. The internal hoop forces disappear. The dome behaves like an array of arches.
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