## 5 Numerical Modeling of the Vertical Section of the Cava Madre Cavern
The following figure (Figure 8) reproduces the future excavation and support phases considered in the numerical modeling of the cross section of the entrance area:  
* Completion of the excavation in the right area at the current level (bench 0) (Figure 8a);
* Underpinning of the support structures of the concrete strut beams on the right side of the cavern (Figure 8b);
* Removal of the central concrete partition temporarily supporting the existing strut beams (Figure 8b);
* Completion of the excavation in the left area at the current level (bench 0) (Figure 8c);
* Construction of a row of future strut-beams at a lower level (Figure 8d);
* Creation of bench 1 to deepen the foot of the cavern, following the same procedure as for the previous bench 0 (Figure 8e-i). The developed numerical model has very fine detail by adopting quadrilateral elements measuring 0.25 m \(\times\) 0.25 m. The total number of numerical elements used in the model is approximately 1 million seven hundred thousand. The model has a total size of 325 m \(\times\) 325 m. The boundary conditions were entered by applying adequate pressure on the lateral edges in order to recreate the previously defined lateral thrust coefficient (k\({}_{0}\) = 1.4): at the base of the model, a constraint on vertical movements was imposed, while on the two lateral borders, a horizontal pressure variable with depth (triangular trend of the load diagram) and defined on the basis of the coefficient k\({}_{0}\) (horizontal to vertical in situ stresses ratio) was applied; no constraints or even pressures were applied to the upper border of the model, as it coincides with the ground surface. The model mesh was chosen thanks to an in-depth analysis, which made it possible to identify the size of the elements capable of achieving high calculation precision: the size of the elements was varied, starting from larger values towards smaller ones, until some representative results of the calculation (displacements and stresses at the center of the roof and at half the height of the side walls) reached a precision deemed acceptable. The mechanical properties of the rock mass were represented in the numerical model based on the geomechanical characterization developed and considering the actual geometry of the lithotypes present, also for petrographical issues as done in similar exploitation [23]. For the materials of the support and reinforcing elements (concrete and steel), reference was made to the material tests carried out: steel elastic modulus 210 GPa, steel tensile strength: 540 MPa, concrete elastic modulus 32 GPa and concrete compressive strength 30 MPa. The section of the cavern that was represented in the two-dimensional numerical model is the one that emerged from the topographic surveys carried out in the entrance zone, in particular in correspondence with the central beam strut among the three created. Figure 8: Representation of cross sections of the future excavation and support phases planned for the Cava Madre of Candoglia to allow the continuation of mining activities in the lower area of its entrance. The solution adopted by the used calculation program (Flac 2D ver. 8.1) is of the finite difference type, capable of simulating the excavation phases in soils and rocks and evaluating the stress and strain even when the limits of the elastic behavior have been reached and a plastic behavior occurs. The potential of the calculation program consists of being able to develop analyses in the presence of structural elements introduced to simulate support and rock reinforcement interventions, in the dynamic field [24; 25], and in the presence of groundwater. The most interesting results to analyze in order to evaluate the static behavior of the cavern and of the surrounding rock mass are:  
* induced stresses both in the rock mass and in the structural elements (in particular inside the new concrete strut beams);
* the movements of the cavern walls.