Figure 8. The macro-elements calculation in the analysis of the masonry dome structures.
emerged during the "in situ" investigation. The solid continuum mechanics, especially the finite element method, offers the most suitable and practical models for skeletal structures macro-modelling.
Then, a 3D model of the dome has been built, by using a simplified solid elaboration. A precise geometrical survey restitution is very important for the construction of the model; however the correct simplification of the numerous data of the 3D survey (as the laser-scan model) it is also of fundamental value. In order to obtain a quite precise mesh of the structure and of the output, the finite element used for the FEM (Abaqus) elaboration is a tetrahedron solid with 4 nodes. Based on the simple structure of the dome, we built up a regular solid, on hexagonal base, symmetric, which grows in height and follows the circular profiles of the real dome edges, as measured in the survey. The used 3D model is constituted by a single shell, with a variable thickness, starting from 30 cm at the abutment up to 20 cm at the top.
Finally, the main problem is to choose the constitutive law for masonry. As observed by Di Pasquale (1984), it can be considered that traditional masonry structures have a very low tensile strength, which decreases with time. The masonry can also be considered as an ideal no-tension material, especially in the case of domes, in which the prevalence of compressive stresses is obvious.
Then, we have attributed to this 3D model the mechanical characteristics of an elastic, homogeneous,
isotropic solid. Precisely, the following values have been applied:
Young modulus E = 10.000 kg/cm2
Poisson ratio n = 0.2
Mass density g = 1800 kg/m3.
The previous conditions have been applied both to the intact and the cracked structure. The material is assumed to be linear-elastic; however, the non-linear geometry has been introduced by inserting the cracks in the undamaged structure. In this way, the non-linearity of the material has been concentrated in the discontinuities inserted in the cracks, which have been actually surveyed in the real dome.
In this phase, it's important to compare the obtained deformation and strengthens results with the measured ones. The comparison with the empiric methods is recommended for the control, based on the uncertainty of the mechanical characteristics of the material "masonry".
The utilized finite elements have defined the structure through a mesh which has fitted the volume, by considering its singularities. Gradually proceeding from the more simplified model of the structure, up to simulate the real critical load and restraint conditions on the dome, different load cases and dome configurations have been considered:
Case 1. Intact dome subject to the only dead weight. The model isn't realistic but it's useful to simulate the previous damage. The results pointed out the higher tensile stresses in the zones which are actually cracked (Fig. 9).
Case 2. Cracked dome subject to the only dead weight. In this model the real cracks on the edges have been inserted, with their actual measured thickness and height. Also in this case, the model has verified the
presence of tensile stresses in the central zone of each panel, in which we can observe the other major cracks.
Case 3. Dome, symmetrically cracked, both on the edges and on the centre of the six panels, subject to the dead weight (Fig. 10).
Case 4. Cracked dome, as before, subject to dead weight and to the pointed weight of the roof.
Case 5. Insertion of an encircling steel tie on the cracked dome. The absence of tensile stresses incompatible with the masonry properties, has confirmed the validity of the hypothesis which have been assumed.
3.4 Comparison on findings with finite-element analysis and empiric-experimental method
The results of the previously described ideal model of the dome, geometrically regularized, verified the outcomes of the first empiric-experimental phase. Moreover, these findings have confirmed the collapse mechanism hypothesis, based on the observation of the cracks.
The numerical results of the first case (Case 1) stresses that the radial deformation on the vertical section, under the only dead weight, is so low to justify the use of a linear model for the whole structure. In fact, in the undamaged configuration, the horizontal tension stresses are higher in the edges (in which the main cracks actually are. Fig. 9). The founded tensile pattern matches with the actual cracks, which really represent the main continuous monitoring system of the structure.
The analysis on the cracked dome (Case 2) underlines tensile stresses on the central part of each panel. If we image to cut the obtained model about 1/3 of the whole height of the dome, we can observe very high tensile stresses on the limits of the cracks. Now, these values are controlled by a recent installed precision monitoring system, which would give information on the trend of the cracks, also considering the seasonal variations.
Obviously, the presence of the cracks has gradually modified the structural behaviour of the dome. The structure has changed from a substantial axial-symmetric functioning (in which each parallel slice gives a contribution to the meridian ones) to that of 6 different parts (6 panels) which are linked at the top and which are supported by the common abutment.
The cracked model (Case 3) simulates the real behaviour of the structure, even if it uses a linear-elastic constitutive law for the finite elements. In fact, the insertion of the real cracks of the dome in the model allows to not consider the masonry extremely low resistance to tension and shear strain: in this way the non-linearity is concentrated in the cracks.
It's interesting to note that the load due to the pointed weight of the roof, applied to the cracked model of the dome (case 4) doesn't provoke significant variations in the horizontal global thrust. This result further supports the collapse mechanism hypothesis, which pointed the global thrust of the dome, due to its self weight, as the main responsible of its damage.
After these analysis, we can say that the elastic linear model, with the empiric insertion of the cracks, could represent a good approximation of the reality, of easy application and control. The presence, in some models, of tensile stresses, has evidenced the existing discrepancy, but it has also allowed to identify the fracture zones.
In fact, on valuable historical buildings, it's useful to know, not only the limit collapse last resistance, but even the rise of the actual damage state level. Indeed, a linear model permits to simply simulate the behaviour of the structural organism and then to clearly identify the risk zones.
4 THE EXPERIMENTAL INTERVENTION 4.1 The steel encircling
As stated before, the historical debate about the Vatican dome cracks (during the XVII and XVIII centuries) has for good clarified that a dome exercises a radial thrust force which has to be nullified.
Thanks to the symmetry of the structure, the solution of this constitutive problem can be solved by the insertion of a constraint element, which would neutralize this force: a tie.
Then, the thrust force value, which has been calculated both by empiric-experimental method and by finite element analysis, has been used for the dimensioning of the steel encircling tie, which has to be put in work on the dome.
The area of the tie section could be easily calculated by the equation At = H/o adm.
(where A is the area of the tie, H is the thrust force of the dome and o adm is the still admissible tension).
The section of the tie has to be the lowest, to reduce the reaction of the steel to temperature variations. Therefore, in order to determine the better solution for the consolidation of the structure, the experimentation on this type of intervention suggests to compare the performances of different materials.
In keeping whit this, we hypothesize to use a tie constituted by an high resistance steel, e.g. type DYWIDAG (which has a o adm = 6500kg/cm2). At the end of the calculation we have found that the use of this type of steel allows a significant reduction of the tie diameter: from 2,6 cm to 2,15 cm large. This could assure a lower dilatation during the seasonal variations.
Then, in the last case analyzed by finite element method (Case 5) we have determined the effects of an encircling tie on the dome. In order to gain the better results for consolidation intervention, different type of materials have been simulated, with different diameter dimensions.
Precisely, an encircling intervention has been simulated, through the insertion of a tie on the line of the windows in the dome. The numerical results which we have obtained stress the different reactions to the temperature variations and the different restraint conditions.
It is in course of study an hypothesis of using fiber reinforced material, instead of the projected steel tie, for the encircling of the dome. Some model have been produced which are waiting the comparison with the experimental results.
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