## 3 Analysis of Deformation Characteristics of Underground Structure  
### Stress Analysis of Existing Structures on the Pasternak Foundation Model  
The [PERSON] model [22], built upon the Winkler material model, assumes the existence of shear action between foundation units. This shear action is achieved through \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Cases** & **New Tunnel** & **Geological Condition** & **Construction Method** & \(H\) **(m)** & \(R\) **(m)** & \(S_{\text{max}}\) **(mm)** & \(V\) **(\%)** \\ \hline
1 & Urumqi Metro Line 1 [44] & Silty clay, muddy silstone & Bench method & 14.83 & 3.34 & \(-\)21.4 & 1.13 \\
2 & Xishipo Tunnel [45] & Gravel soil, sandstone, strong & Bench method & 28.23 & 3.23 & \(-\)11.33 & 1.22 \\
3 & Trial tunnel [46] & London clay, River Thames & & & & & \\
4 & Beijing Subway Line 4 [47] & Round gravel pellets, pebbles, sil clay & Bench method & 12.80 & 3.30 & \(-\)19.03 & 0.887 \\ \hline \hline \end{tabular} Where \(H\) is the buried depth of the tunnel axis, \(R\) is the virtual radius of the new tunnel, \(S_{\text{max}}\) is the maximum settlement, and \(V\) is the formation loss rate.  
\end{table} 
Table 2: Mining method construction case. Figure 3: Surface settlement: (**a**) Case 1, (**b**) Case 2, (**c**) Case 3, and (**d**) Case 4 [21, 48]. a shear layer that can only produce lateral shear deformation and is incompressible, as shown in Figure 4. The governing equation for the impact of additional soil deformation on the existing structure can be derived as follows:  
\[EI\frac{d^{4}\omega}{dx^{4}}-G_{\mathrm{P}}b\frac{d^{2}\omega}{dx^{2}}+kb\omega= bq(x), \tag{8}\]  
where \(EI\) represents the flexural stiffness of the extant structure, \(E\) is the elastic modulus of the existing structure, \(I\) represents the moment of inertia of the section of the existing structure, \(q(x)\) denotes the additional load applied to the beam, \(G_{\mathrm{P}}\) indicates the foundation shear stiffness, \(k\) is the foundation spring stiffness, \(\omega\) is the deflection of the beam, and \(b\) is the beam cross-section width. Regarding stiffness equivalence of existing structures, the structure is often assumed to be a rectangular beam for calculation, as shown in Figure 5. Equating them to a rectangular beam using Equation (14) gives the following:  
\[I=\frac{1}{12}\Big{(}bh^{3}-b_{1}h_{1}^{3}\Big{)}, \tag{9}\]  
where \(b\) is the width of the existing structure, \(h\) is the existing structural height, and \(b-b_{1}\) is the thickness of the existing structure. [PERSON] [49] and [PERSON] [50] introduced formulas for the foundation shear stiffness \(G_{\mathrm{P}}\) and the foundation spring stiffness \(k\), respectively. \[k=\frac{0.65}{b}\bigg{(}\frac{E_{\mathrm{s}}b^{4}}{EI}\bigg{)}^{\frac{1}{12}} \frac{E_{\mathrm{s}}}{1-\
u_{\mathrm{s}}^{2}}, \tag{10}\]  
\[G_{\mathrm{P}}=\frac{E_{\mathrm{s}}t}{6(1+\
u_{\mathrm{s}})}, \tag{11}\]  
where \(E_{\mathrm{s}}\) and \(\
u_{\mathrm{s}}\) are the elastic modulus and Poisson ratio of soils, respectively, and \(t\) is the thickness of the shear layer. Regarding Equation (8), [PERSON] [25] utilized the finite difference method to derive the Pasternak foundation model for tunnel-soil-beam interaction. Figure 4: Pasternak foundation beam-soil interaction model. Figure 5: Equivalent schematic diagram of the foundation beam of the existing structure. Based on the assumption that the finite long beam is divided into \(n\) equal segments, the ordinary differential equation can be transformed into a difference equation by applying the finite difference method:  
Figure 3:0  
where \(i\) = 0, 1, 2, 3,\(\ldots\), \(n\); and coefficients \(\lambda\), \(\mu\), and \(\xi\) follow the following relationship:  
\[ \begin{bmatrix}\lambda\\ \mu\\ \xi\end{bmatrix} = \begin{bmatrix}1&0&0\\ -4&-1&0\\ 6&2&1\end{bmatrix}