*4.1. The Establishment of Model*

As shown in Figure 4, the model size is set as 50 m (length) × 80 m (width) × 50 m (height) to avoid boundary effects [19], and the size of the foundation pit is 14 m (length) × 5.2 m (width) × 5.9 m (height). The dimensions of other components, such as old pile,

cap, underpinning beam, and new underpinning pile are also the same as those of the engineering case. The two new underpinning piles are 1.2 m in diameter and 27 m in length. The model unit is C3D8R, and the total number of units is 32,700.

**Figure 4.** 3D-FEM modelling configuration and boundary conditions.

The left and right boundaries of the model are constrained in the X direction, the front and rear boundaries are constrained in the Y direction, the bottom boundary is fixed, and the top boundary is free. The mechanical behavior of the soil obeys the Mohr–Coulomb criterion, and the underpinning beams, caps and piles are assumed to be homogeneous and linear elastic materials [19]. The corresponding model material parameters are listed in Table 2. A parametric study was conducted for the variability that the surrounding terrain could bring to the structural analysis, and the results improve that the model parameters adopted in the simulation can effectively avoid boundary effects. Considering that only the impact of the construction process on the substructure of the bridge can be analyzed in numerical simulation, continuous mechanical change cannot be monitored in the same way as the engineering. Therefore, in the modelling process, the "jack" was directly regarded as the concrete block after the pile sealing has been completed. The details of the underpinning are shown in Figure 5a, and the relationship between the underpinning pile and the tunnel is shown in Figure 5b.

**Table 2.** Material parameters used in numerical modelling.


**Figure 5.** Details of the 3D-FEM modelling: (**a**) underpinning diagram; (**b**) location of the underpinning pile and shield tunnel; (**c**) application of jacking force; and (**d**) old pile cutting and new pile sealing.

#### *4.2. Simulation Procedure*

There are 17 steps in the model analysis, and the specific settings are as follows:



**Table 3.** Equivalent jacking pressures applied on the pile platforms.
