## 2 Engineering Background
### Establishment of Numerical Model  
The finite element software PLAXIS 3D was used for numerical calculation, with a model size of 90 m \(\times\) 120 m \(\times\) 40 m (see Figure 7). The tunnel lining and bearing platform were simulated by solid elements. This paper mainly focuses on the responses between the piles and the tunnel. The simplified modelling of the lining was carried out by numerical simulation, and the influences of the segmented lining on the piles were not considered [26, 27]. The pile body was simulated by an Embedded Pile structural element, which is a friction end bearing pile with a length of 23 m and a diameter of 1.2 m. The interaction between the pile and soil layers was determined through the coupling layer between the pile and soil layers. The coupling layer embedded in the pile element defaults to the stiffness and friction parameters in PLAXIS 3D. In order to obtain more accurate results, the elements near the piles were densified to make the grid size no more than 1 m. The load on the pile foundation is divided into two parts, one is the dead weight of the bridge, the other is the lane load and crowd load. The dead load of the bridge acting on each pile top was considered to be 1 MN, and the movable load of vehicles was considered to be 200 kN. The load calculated according to relevant specifications was added to the foundation, and the load of each part of the model was determined in PLAXIS 3D [16]. To better simulate the interaction between the tunnel, soil layers and bridge piles, all soil layers were simulated by a small-strain Hardening-Soil constitutive model. The hardened soil model was developed by [PERSON] [28] and then extended to the small strain range by [PERSON] [29]. It is one of the most advanced soil constitutive models, and has been successfully used in deep foundation pits, large foundation rafts, diaphragm or sheet pile walls, and other applications, in the past decade [30, 31]. The small strain Hardening-Soil model considers the volumetric and shear hardening mechanism. The deformation characteristics of the soil in the initial stage of the constitutive model are not entirely linearly elasticity but elastic-plastic deformation with high modulus is affected by stress history and loading path. Fully considering the nonlinear characteristics of the soil under small strain, the constitutive model can better reflect the interaction between soil and structure under a working load [32]. In combination with the tunnel geological survey data [33] and relevant specifications [16], the soil layer parameters are shown in Table 2. The empirical relationship between \(E_{50}^{ref}\) and \(E_{ur}^{ref}\), refer to the Plaix 3D manual [34]. Poisson's ratio is \begin{table}
\begin{tabular}{c c c} \hline
**Control Items** & **Range of Control Items** & **References** \\ \hline Surface settlement & +10 mm\(\sim\)–30 mm & [PERSON] et al., [21] \\ Pier settlement & \(\leq\)15 mm & [PERSON] et al., [22]; [PERSON] et al., [14] \\ Pier settlement speed & \(\pm\)3 mm/day & \\ Differential settlement between & \(\leq\)2 mm & [PERSON] et al., [23]; [PERSON] et al., [24]; [PERSON] et al., [25] \\ longitudinal adjacent piers & \(\leq\)3 mm & \\ transverse adjacent piers & & \\ \hline \end{tabular}
\end{table} 
Table 1: Deformation control standard. determined by the triaxial drained loading and unloading test. The reference pressure of the power exponent \(m\) of the stiffness stress level correlation is 100 kPa. It is worth noting that the effect of grouting reinforcement is achieved by increasing the soil modulus of the corresponding reinforced stratum [35]. The shell element in PLAXIS 3D simulates the shell material of the shield machine, and the solid element simulates the segments. The linear elastic model parameters of the shield machine shell and segments are shown in Tables 3 and 4.