## 2 Three-Dimensional Finite Element Modelling and Validation
### Finite Element Model  
This analysis presents a finite element model of staggered shield tunnel linings, incorporating elements such as precast concrete segments, bent bolts, bolt washers, bolt sleeves, and reinforcement. The reinforcement's quantity and distribution mirror the actual setup of the precast concrete segments. The finite element model, showcasing the staggered arrangement in the shield tunnel lining, is illustrated in Figure 2. Figure 2a presents a schematic of the finite element model for the full-ring structure. The numerical model in this study was established based on a selection derived from full-scale model tests, consisting of two semi-cut lining rings (each with a width of 750 mm) and one complete lining ring (with a width of 1500 mm), with the central ring being the complete one. The primary focus of the full-scale tests is on the transverse deformation characteristics of the lining, with a special emphasis on the deformation properties of the full-ring lining structure [12]. The inclusion  
Figure 1: Schematic diagram of lining blocking and joint construction. (**a**) Ring 1; (**b**) Ring 2; (**c**) Ring 3; (**d**) joint construction.  
of two semi-cut lining rings is designed to consider the boundary conditions of misaligned assembly in the complete lining ring. Since this study primarily investigates the impact of ground surcharge on the transverse deformation of the lining ring and also takes into account the complexity of a refined finite element model of the lining structure, the effects of using a complete lining ring in place of two semi-cut lining rings for the misaligned assembly boundary conditions are fundamentally consistent. To avoid the need for remodeling semi-cut lining rings, the finite element model also utilizes complete lining rings for the semi-cut lining rings on both sides. Figure 2b illustrates the finite element model of the segmental structure, which includes circumferential and longitudinal bolt access holes for the positioning of circumferential and longitudinal bolts, respectively. To achieve precise deformation simulations, our model employs reduced integration eight-node linear elements (C3D8R), optimizing the accuracy and computational efficiency. These elements are crucial for representing concrete segments, bolts, washers, and sleeves, as they allow for the detailed analysis of stress distribution and deformation under load. Similarly, the use of three-node truss elements (T3D3) for simulating reinforcement is vital for accurately capturing the behavior of structural steel within concrete, reflecting its contribution to the overall structural integrity. For instance, in a simulated load test, C3D8R elements enabled us to observe the segments, while T3D3 elements illustrated how reinforcement bars absorbed tension forces.  
The structural and finite element models developed in this study are based on a three-ring staggered assembly of tunnel segments. This design aims to mimic the boundary constraints of staggered assemblies, although some differences from actual conditions may persist. In practical engineering, shield tunnels consist of multiple rings of segments that are more adept at managing deformation than models with fewer rings, potentially resulting in a symmetrical deformation pattern. Despite this, models with a single ring or three rings serve as vital representations of localized sections within shield tunnels, capturing essential aspects of their behavior. These models are particularly valuable for understanding the deformation and stress responses of segment structures, offering high reliability in these aspects.