## Methodology
### Numerical Analysis  
#### 2.3.1 General Hypotheses  
In the numerical study performed, the following hypotheses have been adopted:  
* The rock mass is considered to be homogeneous and isotropic. This isotropy hypothesis is particularly important because the possible presence of singularities or discontinuities in the rock mass volume would lead to a drastic decrease in the values of the ultimate load that would be obtained.
* The finite difference method is used, with a perfect elastoplastic behaviour. The software employed is FLAC 3 Dv 5.01 (ITASCA, Minneapolis, MN, USA) [28]. It allows for the parameterisation of the variable dimensions of the model in a simple way. Additionally, this code can simulate the effect of the foundation by applying a very reduced speed movement (analysis in displacement control) to correct possible convergence problems when approaching plasticisation.
* The modelling domain considers the separation of the lateral boundaries to allow for the assumption that they have no influence on the results obtained. In relation to the lower boundary, it has always been placed in such a way that there is a thickness of rock mass under the upper cavity at the height thereof. These distances have been adjusted and verified in each model based on the geometry and the analysis considered.
* The geometry of the caverns modelled present an elliptical cross-section, with horizontal axis 2a, vertical axis 2b, and depth or cover thickness H. Different values for the dimensions a and b have been considered, as well as different slenderness ratios b/a. The cover is shown in relation to the vertical semi-axis: H/b.
* The surface load has been introduced by imposing a uniform vertical movement (simulation by displacement control), equivalent to a rigid foundation, with a width B, and eccentricity E, measured from the axis of the cavern. Different values for the dimensions B and E have been considered. The eccentricity is shown in relation to the horizontal semi-axis: E/a.
* The modelled ground, despite being three-dimensional, includes boundary conditions to simulate the hypotheses of a flat deformation (considered to have a thickness of 1 m). The boundaries in X and Y are established according to the ellipse parameters (a,b), to guarantee that their presence does not affect the results.
* The failure criterion adopted is the modified Hoek-Brown criterion, which is an evolution of the generalised Hoek-Brown criterion incorporated in the Flac3D software [29] and which allows for a more advanced control over the dilation angle and the plastic flow rule. Different values of the geotechnical parameters have been considered (RMR\({}_{\text{b}}\) or GSI, UCS and m\({}_{\text{i}}\)).  
With all these hypotheses, the ultimate bearing capacity (q\({}_{\text{u}}\)) is obtained for each case with the combination of the variables described, from a detailed analysis of the vertical load Q (kN/m\({}^{2}\)) and the vertical movement imposed(mm) curve of the middle point of the surface load.  
The obtention of the relative ultimate bearing capacity (q\({}_{\text{u}}\)/q\({}_{\text{u},\text{H}\text{H}\text{H}}\)) is of particular interest. Its value is the ratio of the ultimate bearing capacity (q\({}_{\text{u}}\)) for each case resulting from the combination of variables to the ultimate bearing capacity without the existence of the cavern (q\({}_{\text{u},\text{H}\text{H}\text{H}}\)).  
Through this methodology, it is possible to evaluate the influence of the presence of an elliptical cavern on the ultimate bearing capacity of a shallow foundation, based on its geometrical characteristics (dimensions and relation between the ellipse axes and vertical/horizontal disposition of the major axis), the cavity cover, and the geotechnical parameters of the rock mass, together with the size and eccentricity of the surface load (Figure 1).