## 4 Numerical Calculation of the Stress Change in the Shaft Tower Structure during the DHS Process  
### Simulation Scheme  
The solid mechanics module in the COMSOL software was used to carry out the numerical calculation of the stress in the shaft tower, and the geometric model was established according to the on-site conditions. Moreover, the boundary conditions were determined according to the elevation change of the shaft tower foundation, and the material properties of each part of the shaft tower were determined based on the existing measurement results. (1) Geometric model  
According to the structure and size of the shaft tower, a geometric model was established. Inside the pillars of the shaft tower, the longitudinal reinforcing bars were six steel bars of 28 mm in diameter, the stirrups were steel bars of 10 mm in diameter, and the spacing between the stirrups was 120 mm. The section size of the concrete main beam was 440 \(\times\) 1500 mm; the reinforcing bars were six steel bars of 32 mm in diameter, the stirrups were steel bars of 12 mm in diameter, and the spacing between the stirrups was 140 mm. In order to protect the concrete main beams of the shaft tower, steel plates with a thickness of 8 mm were wrapped on its surface. Since the objects in the established model varied greatly in size, the sub-regional meshing method was adopted. The reinforcing bars, steel plates, epilasters, and main beams used ultra-fine meshes, and the tower walls adopted fine meshes. The model generated a total of 0.73 million free-tetrahedron meshes, and the geometry and mesh diagrams of the model are shown in Figure 6. The model is calculated by a steady-state solver, and the number of degrees of freedom to calculate reaches 3,490,206. (2) Constitutive equations  
Since the shaft tower deformation is in a small range, it can be assumed that the shaft tower structure is linear elastic, and its deformation follows [PERSON]'s law:  
\[\sigma_{ij}=\lambda\delta_{ij}\epsilon_{y}+2\epsilon e_{ij} \tag{1}\]  
Figure 6: Geometry and mesh diagrams of the model. where \(\sigma_{ij}\) is the total stress tensor; \(\lambda\) is the Lame constant; \(\delta_{ij}\) is the Kronecker function; \(G\) is the shear modulus; and \(\epsilon_{ij}\) is the strain tensor. (3) Geometric deformation equation  
The deformation of the shaft tower is within a small range, and its geometric equation can be written as:  
\[\epsilon_{ij}=\frac{1}{2}(u_{ij}+u_{ji}) \tag{2}\]  
where: \(u_{ij}\) and \(u_{ji}\) are the displacement components. (4) Boundary conditions  
Since the bottom frame is in contact with the shaft tower foundation, this paper simulates the different inclination states of the shaft tower by changing the displacement on the bottom frame. Since the model is established according to the actual situation of the shaft tower, the remaining boundaries can be set as free boundaries, so that the shaft tower can tilt in different directions. The bottom of the model includes the frames that are shown in Figure 7, whose boundary conditions are set to predefined displacement boundaries. The other boundaries of the model are set to free boundaries. Since only the elevations of monitoring points J1, J2, J3, and J4 were observed but the tower foundation was quite rigid, it was assumed that the predefined displacements on the bottom frames were distributed linearly along the frames.