## 1 Introduction  
The shield tunneling method is widely used in soft soils especially in urban areas because it is efficient for minimizing disturbances to surroundings. In this method, segmental lining is commonly used. The segmental lining is a connection of segments by longitudinal joints and circumferential joints, and its stability is ensured by the surrounding ground. There is interaction between the lining and the surrounding ground, constraining the lining in both normal and tangential directions. This mechanism results in reactions from the ground in both directions. A segmental lining design is carried out for (1) transient situations before segment installation, during tunnel boring-machine (TBM) tunneling, and during additional temporary works; (2) persistent situations due to the ground and groundwater, tolerance of construction, surrounding load conditions, and tunnel-use conditions; (3) accidental situations; and (4) seismic situations [1]. This study focuses on the final loading stage due to the ground and groundwater since it is a critical stage especially in soft soils and shallow depth conditions. In the case of the segmental lining design for the final loading stage, the following two methods are commonly used. The first approach involves analytical solutions [2; 3; 4; 5; 6; 7; 8], where the lining is modeled by a continuous beam without longitudinal joints and circumferential joints, and the ground is modeled by springs or a continuum medium. In these analytical solutions, initial earth pressure is applied directly on the lining, and then the ground reaction to the lining due to its deformation is assessed. This approach is fast and simple to use. However, it cannot take the complexity of a tunnel into account such as the aforementioned joints and effects of tunnel construction. In addition, linear elastic behaviors of lining and ground are assumed for the analytical solution. The second approach is to use numerical analysis. The segmental lining is usually modeled by the beam-spring model, as shown in Figure 1a, which can consider the effect of a staggered arrangement of segments under the following assumptions: (1) the plane strain conditions in the transverse section for the lining and ground, (2) the linear-elastic behavior of the lining, and (3) the non-tension linear-elastic characteristic of the ground. The two rings are connected by circumferential joints, which are modeled by shear springs in the normal and tangential directions. The segments are connected by a longitudinal joint, which is modeled by a rotational spring representing the relationship between the rotational angle and moments, and is tied at both ends of the segments. The ground is modeled by continuum elements or ground springs in the numerical analysis. The first ground model uses soil continuum elements and ground-lining interface elements to represent the surrounding ground and the ground-lining interaction, respectively [9; 10; 11; 12; 13; 14]. This model can analyze both lining and ground behavior, and offer the ability to model sophisticated properties of ground and lining structure, construction process, and complex scenarios; that is, material non-linearity, different geological strata, gap-filling process, and effects on adjacent structures. However, it is difficult to simulate their behavior in shallow tunnels in soft ground, needs a lot of computational effort, and is time-consuming. The second ground model represents the ground-lining interaction by the ground spring, as shown in Figure 1b. The initial earth pressures in the normal and tangential directions are loaded directly onto the lining. This model can only analyze lining behavior, but is commonly applied in tunnel lining design as recommended by most current tunnel lining design standards [1; 15; 16; 17; 18; 19; 20; 21; 22] and in research such as [23; 24; 25; 26; 27; 28].