# Research on Lateral Deformation Control Criteria of Metro Shield Tunnels with Excessive Ellipticity
## 3 Finite-Element Analysis
### Shield Tunnel Loading Results  
In Figure 10, the blue color signifies that the structure remains in the elastic stage, corresponding to a damage value of 0. In this region, the unit remains in the elastic stress stage, and the unit stress is below the elastic limit stress of the concrete plastic damage constitutive model. Regions other than blue suggest that the unit is in the plastic stage, with damage values ranging between 0 and 1. A higher value indicates more severe damage within that specific structural region.  
For varying foundation parameters, denoted as \(k\), even under approximate deformation conditions, the damage inside the tunnel segment for the erroneous seam tunnel consistently occurs in the same location. This consistency is due to the regularity observed in the structural loading damage results. This analysis focuses on the foundation spring coefficient set to 10 MPa/m. When the ellipticity ranges between 10% and 15%, the concrete in the tensile zone displays almost identical cracks. The tunnel's top is susceptible  
Figure 8: Full-scale model test of shield segment in the literature [24].  
Figure 9: Schematic diagram of the loading of the validation model.  
to cracks, while excessive deformation at the tunnel's waist can result in localized pressure collapse, manifesting as a block collapse phenomenon. The exterior of the tunnel's waist is also susceptible to damage. Although damage to the outside of the tunnel is not readily observable, it may still lead to leakage from the waist side of the tunnel and even corrosion of exposed reinforcement.  
Figures 11-13 show that a critical distinction between the algorithms is the overall yield point of the structure during loading. Analyzing the loading-deformation curve, a significant slope change in the curve indicates a decrease in the loaded stiffness, implying that the entire ring structure is in the yielding stage. For a \(k\) value of 5, 10, 20, and 30 MPa/m, the yielding begins when horizontal convergence reaches roughly 90, 80, 70, and 60 mm, respectively.  
Figure 11: Vertical load-horizontal convergence values with different ground resistance coefficients.  
Figure 10: Damage cloud (perspective view) of the tunnel segment with ellipticity development when the foundation spring coefficient is taken as 10 MPa/m.  
Figure 14 depicts the damage cloud diagram for each scenario at an ellipticity of 25%. With smaller \(k\) values, the red region in the arch waist is minimized, indicating lesser damage and the structure's ability to bear the normal load. Conversely, larger \(k\) values lead to pronounced damage in the arch waist. Comparing this with the loading curve reveals that the structure is almost in the yielding phase.  
In summary, under symmetrical loading conditions, an increase in vertical loading exposes the arch's inner side as the most vulnerable region due to concrete tension-induced cracks. However, the arch's base, linked to the roadbed, does not demand significant damage monitoring. Stratigraphic conditions considerably restrict the lateral deformation of the tunnel segments.  
Figure 12: Vertical load-vertical convergence values with different ground resistance coefficients.  
Figure 13: Vertical load-ellipticity curve.