## 3 Analysis and Interpretation of Measurement Results
### Numerical Modelling  
In order to investigate the influence of arch forces, assumed to have caused the initial tilting of the towers, a 3D numerical model of the Cathedral structure was created with discontinuities at the locations of major existing cracks of the facade wall, the joints between the tower and the naive, internal arches, and vaults. Figure 11 depicts the general shape of the model, while Figure 12 depicts the FE mesh of the Cathedral and the surrounding soil that was taken into account for the calculations. The FE analysis was performed with software Ansys(r) Academic Teaching Mechanical, Release 17. The masonry was simulated with the SOLID65, 3D element type from the Ansys Element Library, while for the soil, the SOLID185 element was used. Special attention was devoted to finding an adequate FE mesh, in order to optimize the accuracy of the results, convergence and hardware limitations. As a result, the masonry was modelled with 45,115 elements, and the soil with 192,070 elements, 237,185 elements in total. Timber elements of the Cathedral roof structure were introduced as loads applied on the locations that represent support points for the real structure. Compared to the rest of the structure, these structures have relatively small effects. The bell section of each tower has a weight of 37.5 tons compared to the total weight of each tower, which is 3200 tons. The weight of the roof over the naive is 225 tons, compared to the total weight of the naive of approximately 7000 tons. Since the numerical model is relatively large, material nonlinearities were utilized only for the soil. The masonry was modelled with the linear material model, without cracking. The effect of the existing cracks was introduced in the form of voids that  
Figure 11: 3D model of the Cathedral—general shape of the model. Figure 12: FE mesh of the Cathedral numerical model with the surrounding soil. extend throughout the walls. This simplification was adopted after the series of simulations with material nonlinear models both for the masonry and for the soil proved to be too demanding for hardware that was available at the time. On the other hand, the results obtained from the simplified model have a good agreement with the damages observed on the structure itself. The soil was modelled with the Mohr-Coulomb material model. The numerical values used for the definition of this material model were obtained from numerous in situ soil testing, laboratory testing of the soil samples from the site and for the modulus of elasticity, as shown in Section 3.1 of this paper. The boundary conditions on the outer surfaces of the soil, Figure 12, were defined to have no displacements. Overall calculations related to this structure consisted of both static and dynamic analysis, but for the purpose of this paper, the results showed here are related to the effects of self-weight, which govern the damaging long-term settlements that led to such extreme cracks observed on the Cathedral. Figures 13-17 therefore all show the effects of the permanent gravity loads on the structure. Figure 14: Characteristic points for contact stresses below the foundations of the Cathedral. Figure 13: Vertical contact stresses in the soil at the idealized (flat) bottom surface (N/m\({}^{2}\)). Figure 16: Horizontal displacement of the structure \"north-south\" direction (m). Figure 17: Modal analysis—first modes in two orthogonal directions. Figure 15: Vertical stresses in the masonry (N/m\({}^{2}\)). Calculations have shown (Figures 13 and 14) that the maximum edge pressure is below the southeast and northeast corners of the tower, and is about 816 kPa, which is much greater than the centric stress. The divergence of the towers is about 6.0 mm/m.