## 4 Discussion  
Non-linear analyses must be carried out when designing a segmental tunnel due to the presence of joints, as the flexibility of the joints modifies the structural response of the lining. Structural designers need simple and reliable modeling methodology to analyze and design this type of construction project. There are many 3D models which include different types of nonlinearities in their elements and materials based on Finite Element Method (FEM) to analyze them; these are complex and incur a significant computational cost. In this paper, a practical model based on FEM composed of beam and spring elements is proposed in order to help engineers in practice to perform a \"realistic\" structural analysis and design segmental tunnel linings using a simplified numerical model (1D model), reducing the computational effort (in the case study, a reduction of 40 times). The difference of this model with other 1D models is the incorporation of circumferential joints to include the interaction between rings--a parameter generally ignored in simplified numerical models. If this parameter is not included, the structural load capacity of a segmental tunnel lining is underestimated, as was possible to observe by considering the stiffness of the axial springs of the circumferential joints with a value of zero, simulating that the rings work in an independent way (as single rings). In this case, the simplified models with single rings underestimated the structural load capacity, obtaining approximately 80% of the values obtained from the experimental and 3D models presented in this study. However, if the simplified models consider the interaction between rings (an interaction included in the proposed  
Figure 11: Additional numerical results of the 3D model: (**a**) strain of longitudinal reinforcing steel (Point A, scaling factor 30 for deformed configuration); (**b**) strain of reinforcing steel (Point B, scaling factor 30 for deformed configuration); (**c**) damage in the segments (Point A); (**d**) damage in the segments (Point B). practical model), the ultimate loading capacity obtained was approximately 95% of the values obtained from the experimental and 3D models, practically the same structural capacity. The proposed model presented some differences compared to experimental and 3D models. The first stiffness changes produced in the experimental and 3D models observed in the ovalization load-deformation curves occurred due to the joint apertures and the microcracking given in the segments, while in the proposed model, these changes were only shown for the mechanical behavior of joints due to the elastic behavior of the segments. However, from the engineering point of view, the global structural behavior is very similar in the three models (1D, 3D and experimental models). The deformed configuration and displacements given in the proposed model and 3D model were practically the same. These results were obtained at two points (the yielding load of the lining denoted as Point A, and the ultimate deformation of the tunnel, called Point B) of the ovalization load-deformation curve. In Point A, it was possible to observe from the numerical results of the 3D model that the damage in the segments is minor. Therefore, the structural behavior depends on the mechanical behavior of the joints. For this reason, the displacements of the proposed model were practically the same displacements as those in the 3D model. On the other hand, for Point B, the failure was shown in both models given by the joint apertures, including for the 3D model the damage in the segments. The configuration the displacements was very similar at this point because the collapse mechanism was governed by the mechanical behavior of the joints. It should be noted that the collapse mechanism of a segmental tunnel depends on the interaction between rings and therefore of the longitudinal axial load due to its construction process (TBM method).