## 2 Influenced Zone of Deep Excavation on Adjacent Tunnel
### Numerical Simulation of the Effect of Deep Exavation on Adjacent Tunnel
#### 2.1.2 Constitutive Model and Parameters  
Some studies [15; 16; 17; 18; 19; 20; 21; 22] have demonstrated that the hardening soil model with small strain stiff (HSS) is an advanced constitutive model for soil in numerical simulations related to excavation issues, and the simulation results using it can obtain reasonable retaining structure and soil displacements. Therefore, the HSS model for soil was used in this study. In addition to the basic physical parameters such as the unit weight \(\gamma\), void ratio \(e\), the HSS model includes 13 constitutive model parameters: \(c^{\prime}\) is the effective cohesive force, \(\varphi^{\prime}\) is the effective internal friction angle, \(\psi\) is the dilatancy angle, \(E_{50}^{\text{ref}}\) is the secant stiffness in standard drained triaxial test, \(E_{\text{cod}}^{\text{ref}}\) is the tangent stiffness for endometer primary loading, \(E_{\text{ur}}^{\text{ref}}\) is the triaxial unloading-reloading stiffness, \(G_{0}^{\text{ref}}\) is the small stain shear stiffness modulus, \(\gamma_{0.7}\) is the shear strain amplitude at 0.7 \(G_{0}^{\text{ref}}\), \(m\) is the power for the stress-level dependency of stiffness, \(\
u_{\text{ur}}\) is the Poisson's ratio for unloading-reloading, \(P^{\text{ref}}\) is the reference stress for stiffness, \(R_{\text{f}}\) is the failure ratio, and \(K_{0}\) is the stress ratio of the horizontal effective stress to the vertical effective stress in a normally consolidated state. Figure 1: FEM numerical model used to study the effect of adjacent excavation on existing tunnel. [PERSON] et al. (2022) [23] collected 42 case histories in which the existing tunnel was affected by adjacent deep excavation and found that the cases that occurred in the soft soil represented by silty clay occupy a dominant position in actual engineering. Therefore, silty clay was chosen as the soil for calculation in the model, and it is assumed to be a single homogeneous layer to eliminate the influence of soil stratification. As this study is an extension of the study by [PERSON] et al. (2016, 2017) [11; 12], the parameters of the HSS model for the soil are the same, as listed in Table 1.  
The retaining wall, horizontal struts, and existing tunnel adopt the linear elastic constitutive model. The unit weight is \(\gamma\) = 24.5 kN/m\({}^{3}\), the elastic modulus is \(E\) = 30 GPa, and the Poisson's ratio is \(\
u\) = 0.2. In the model, the shield tunnel is regarded as a homogeneous ring without transverse and longitudinal joints but, actually, the tunnel segments are assembled by blots. Therefore, the effective stiffness ratio of tunnel \(\eta\) (\(\eta\leq 1\)) is introduced to reflect the weakening effect of segment joints on the tunnel stiffness, i.e., the equivalent stiffness of the tunnel is \(\eta EI\), where \(EI\) is the bending stiffness of the homogeneous tunnel before stiffness reduction. Some studies [24; 25; 26; 27; 28] demonstrated that the value of \(\eta\) is related to the soil conditions, the form of the segment joints, and the manner of segment assembly. In this study, the value of \(\eta\) = 0.75 is adopted, referring to the above studies. The contacts of the wall-soil and tunnel-soil adopt the interface model.