# Numerical Modeling for Tunnel Lining Optimization
## 5 Comparison with Analytical Methods  
Very often, simplified methods such as those based on the development of ground softening pressure on the tunnel vault [13; 15] may lead to costly designs and unsafe scenarios  
Figure 31: Measured and predicted convergences at the monitoring section in the secondary lining. Figure 29: Measured and predicted ground movements at (**a**) tunnel crown and (**b**) sidewalls, respectively. during tunneling execution. Those design approaches ignore the actual soil-tunnel lining interaction mechanism, and therefore, they should only be used in preliminary tunnel design. Other design approaches often used to estimate lining pressures approximately, which attempt to account for soil-tunnel lining interaction and deformations, are the so-called Convergence-Confinement method (CCM) [44]. This methodology is based on the development of the ground reaction curve (GRC) and the support reaction curve (SRC), which allows the calculation of the stress relief due to soil unconfinement during excavation, and the estimation of deformations around the tunnel in the sequence of lining installation, as the excavation continues gradually [19,45]. The CCM allows us to compute the radial pressure acting on the primary lining, which is defined as the intersection between the GRC and the SRC. In this way, only the pressure that has not yet been redistributed in the ground is considered as active pressure on the lining. To obtain the corresponding displacement to that stress redistribution, it is necessary to calculate another curve, conventionally called the longitudinal displacement profile (LDP), which provides the displacement expected around the tunnel as the excavation face approaches the control section and the displacement as the excavation face moves away from the control section. Although this is a more reasonable way to obtain the lining pressures, most of the existing equations have been derived from very idealized conditions, (e.g., isotropic stress field K0 = 1, circular tunnel cross-sections, one excavation stage, plane-strain conditions). Figure 32b shows the GRC and the SRC, calculated with the equations proposed by several authors, as well as the calculated with calibrated models in FLAC\({}^{3\text{D}}\) accounting for the support installation, on the crown and on the sidewalls. The average GRC includes the computed curves from [PERSON] et al. [46], [PERSON] [47], [PERSON] [45], and [PERSON] [19], which led to similar results because they consider initial properties; on the other hand, the curve from [PERSON] [48] predicted larger displacement values because it accounts for residual strength parameters in the plastic region around the tunnel. Figure 32a shows the LDP curve, as well as the pre-convergence displacement, u0/umax. As can be seen, all the analytical models considered underestimate both ground displacement and pressure distribution acting on the primary lining. This is a major source of ground settlement during tunnel construction that can easily exceed the state limits of service of nearby structures in urban areas. Furthermore, differential settlements  
Figure 32: (**a**) Ground reaction and support reaction curves, calculated with analytical methods and with the calibrated numerical model, and (**b**) longitudinal displacement profiles, LDPs, calculated with analytical methods and with the calibrated numerical model [25,44,47,48,49]. associated with this fact can potentially lead to structural damage to adjacent structures. In addition, in Figure 32a, the fitted curve from the calibrated numerical model exhibits a different behavior than the other proposed equations derived from several measured data [49], elastic analysis [44], and numerical models [25]. This behavior can be attributed to the idealized conditions considered to derive the equations to computed GRC and LDP (i.e., circular tunnel cross-section, and the assumption of \(\text{K}_{\text{o}}=1\)).