## 4 Numerical Simulation and Analysis  
### Numerical Model Establishment and Validation  
A finite element model was established using Plaxis3D to analyze the soil deformation during the VSM sinking process. The model employed the hardening soil model (HSS), which accounts for small-strain stiffness behavior, to simulate the stratigraphy. The HSS model is highly applicable in the numerical analysis of excavations in soft soils, because it not only considers both the shear hardening and compressive hardening of the soil, but also accounts for the attenuation of the shear modulus in the small-strain range. This capability makes the HSS model more accurate and reliable in capturing the complex mechanical behavior of the soil during excavation processes, particularly in soft soil areas [31]. Figure 10 shows a schematic of the finite element model. The overall water level in the model was -3 m. To eliminate boundary effects, the dimensions of the soil model were 150 m \(\times\) 150 m \(\times\) 120 m [32]. The shaft lining segments, ring beam, and riverbank protection on both sides of the river are simulated using plate elements. The elastic moduli for the lining segments, ring beam, and riverbank protection are 26 GPa, 28 GPa, and 35 GPa, respectively, with a Poisson's ratio of 0.2 and a unit weight of 26 kN/m3. To account for the effects of staggered joints, the stiffness efficiency of the lining segments is 0.75 [33]. The ring beam is subjected  
Figure 10: Schematic of the finite element model. to a uniformly distributed load to simulate the load from the equipment and the self-weight of the shaft. The load increases incrementally as the lining segments are generated. The pressure grouting piles beneath the ring beam are modeled using solid elements, with an elastic modulus of 30 GPa. The 12 bored piles are modeled using embedded pile elements. The shear strain \(\gamma_{0.7}\), which corresponds to the reduction in the shear modulus to 70% of the initial shear modulus, is 2.5 \(\times\) 10\({}^{-4}\). The reference stress value (\(p^{\text{ref}}\)) is 100 kPa, and the power index m related to the modulus stress level is 0.5 for sandy soil and 0.8 for clayey soil. The soil model parameters were derived from the geotechnical investigation report provided by the monitoring unit and were further refined on the basis of empirical parameters for the hardening soil small (HSS) model, as explored by [PERSON] et al. [34]. The stratification and corresponding parameters are detailed in Table 1.  
The simulation process uses the \"element birth-death\" feature of the Plaxis\({}^{\text{3D}}\) 24.1 software to model the shaft sinking process. After simplification, the river width is 18 m and the depth is 4 m, with a distance of 4 m from the ring beam. In the preliminary steps, riverbanks are generated on both sides, and the river is \"frozen\" to simulate dewatering measures before excavation in the actual project. After the retaining structure is generated, the soil is excavated layer by layer, and a 10 mm slurry is generated around the soil. After the excavation is complete, the lining segments are generated. The construction of each segment is followed by the excavation of the next layer of soil and the generation of slurry around the newly excavated soil. The slurry is modeled using a linear elastic model, with a unit weight of 11 kN/m\({}^{3}\), a Poisson's ratio of 0.38, and an elastic modulus of 7 MPa [35]. The load applied to the ring beam increases layer by layer during the sinking process until excavation and sinking are completed. Unloading is performed after the construction stage at the bottom of the shaft is finished. Figure 11 shows a comparison between the measured data and the numerical simulation results, with the deformation trends of horizontal displacement and surface settlement closely matching the monitoring results.