# Numerical Modeling for Tunnel Lining Optimization
## 4 Numerical Modelling  
Three-dimensional finite difference numerical models were developed using the program FLAC3D v. 5.0. [43], as depicted in Figure 25, to assess the load transfer mechanism in the primary and secondary lining. To simulate soil behavior, it was deemed appropriate to consider an elasto-plastic Mohr-Coulomb model because of the expected low levels of soil non-linearities during the construction process, and the brittle stress-strain behavior exhibited by these cemented soils. The initial parameters considered are those shown in Table 1, and the soil profile was idealized horizontally according to the geotechnical layering prevailing at the instrumented tunnel section (i.e., chainage 29+350 and 29+250). Figure 23: Recorded radial pressures. Figure 24: Recorded axial pressures. The model consists of 319,000 three-dimensional solid elements representing the soil mass through which the tunnel is excavated. Within the domain of the model, further refinement of the mesh was adopted on the periphery of the tunnel and in the established control section, where the elements have a dimension of 0.25 m on each side on average. Figure 25a shows the global model domain. Initially, the geo-static state of stress was determined in the soil mass by the application of a body force equal to the acceleration of gravity. As boundary conditions, only the vertical displacements were permitted on the lateral faces, and the base of the model was fixed in all directions. The modeling process consisted of the simulation of each excavation step, in which a null-type material [43] was used to represent the elements that were removed from the model. In the same way, the placement of the linings according to the construction process described previously was included in the modeling, for which SHELL elements were used to simulate the PL and the SL with solid elements. It is warranted to mention that a relaxation of the soil around the tunnel was allowed prior to placing the SHELL by solving the system of equilibrium in every excavation step to effectively simulate each advancement of the construction procedure. To simulate the PL behavior, a non-linear stress-strain relationship considering the evolution of shotcrete properties with time was used (see Figure 26). This consideration permits the relaxation of the soil (solid elements) in a more realistic way. Thus, the primary lining is flexible in the early stages and becomes stiffer as the excavation moves forward until reaching its design strength at 7 days, avoiding both the generation of spurious stress concentration in the lining and underestimation of the displacements. In the case of the SL, its behavior was simulated with a linear elastic model. Figure 25b shows the discretization of the mesh according to the excavation steps (upper-middle section, central bench, and lateral benches). Figure 25: Three-dimensional mesh of finite differences: (**a**) global domain of the model; (**b**) discretization of construction stages. A blind model prediction and back-analyses were conducted to calibrate the numerical models of the analyzed section. Initially, the properties shown in Table 1 were used to assess the potential foreseen capability of the model, comparing the data gathered from the instrumentation (i.e., topographic references, tunnel cross-section converges and divergences, and radial and axial pressure cells) with those obtained in the control points placed in the numerical model, whose locations correspond to every instrument (see Figure 25a). In a second step, the soil properties were adjusted slightly to better match the measured response. If there was a discrepancy, Young's modulus, cohesion, and friction angle were adjusted accordingly to account for changes associated with the excavation process at the soil adjacent to the excavation seeking to minimize the relative error. This step was repeated until the relative error between the computed and measured values (i.e., surface and tunnel crown settlements, radial and axial stresses) were minimized.