## 3 Numerical Simulation Modeling
### Material Parameters  
When applying PLAXIS 3D analysis, the parameters required for the Hardening Soil mode are as follows:  
* \(c\): Cohesion [kN/m\({}^{2}\)];
* \(\varphi\): Angle of friction [\({}^{\circ}\)];
* \(\psi\): Angle of dilatancy [\({}^{\circ}\)], the default value is \(\psi=\varphi-30^{\circ}\), when \(\varphi>30^{\circ}\); the default value is \(\psi=0\), when \(\varphi<30^{\circ}\);
* \(E_{50}^{ref}\): Secant stiffness in standard drained triaxial test [kN/m\({}^{2}\)];  
Figure 10: Excavation support profile: (**a**) diaphragm wall; (**b**) sheet pile. Figure 9: Sectional profile of the relevant structures. * \(E_{\text{\tiny{odd}}}^{ref}\): Tangent stiffness for primary oedometer loading [khN/m2], the default value is \(E_{\text{\tiny{odd}}}^{ref}=0.7E_{50}^{ref}\), in this study, \(E_{\text{\tiny{odd}}}^{ref}=0.7E_{50}^{ref}\);
* \(E_{\text{\tiny{ur}}}^{ref}\): Unloading/reloading stiffness, the default value is \(E_{\text{\tiny{ur}}}^{ref}=3E_{50}^{ref}\);
* \(m\): Power for stress-level dependency of stiffness, its value is between 0.5 and 1.0;
* \(\
u_{\text{\tiny{ur}}}\): [PERSON]'s ratio for unloading-reloading, the default value is 0.2;
* \(p^{ref}\): Reference stress for stiffnesses, the default value is 100 kN/m2;
* \(K_{0}^{nc}\): K\({}_{0}\) value for normal consolidation, the default value is \(K_{0}^{nc}=1-\sin\varphi\);
* \(R_{f}\): Failure ratio, the default value is 0.9;
* \(\sigma_{\text{\tiny{tension}}}\): The allowable tensile strength, the default value is 0 kPa. The drilling depth of the drilling report for this research case was 30 m. After referring to the drilling reports of surrounding cases, the depth of this model was extended to 50 m below the ground to provide the response depth required for program analysis. Therefore, the soil layer of the model was divided into four layers, which included the backfill layer (SF)--its depth was from the ground level to 1 m underground; the silty clay layer (CL1)--its depth was 1 m to 3 m underground; the soil clay layer (CL2)--its depth was 3 m to 35 m underground; and the rock layer (RL)--its depth was 35 m to 50 m underground. The parameters of the Hardening Soil model are shown in Table 3.  
As mentioned above, the pipe roof was arranged in such a way that the steel pipes were connected with one another using mortise and tenon joints and embedded in the soil horizontally. The backfill grouting was carried out after the construction of each pipe roof. Then, each row of pipes was like a large impermeable plate structure. Both ends of the steel pipe were closely connected to the wall of the concrete and the wall of the steel sheet pile. In order to make the simulation of the software closer to the real project, this study used the tunnel designer-plate element to simulate the steel pipe and filled concrete of the pipe-roof structure. In addition, interfacial properties were added around plate elements to simulate water permeability properties. In the simulation analysis, the pipe-roof structure was regarded as a composite material composed of steel pipe and concrete. Assuming that the section of the pipe roof was a square section, the calculated cross-sectional area of a single pipe roof was 0.519 m2. After the equivalent transformation of the sectional area, the plate thickness of the pipe-roof structure was set to 0.72 m for simulation analysis. The elastic modulus \(E_{\text{\tiny{Pr}}}\) and unit weight \(\gamma_{\text{\tiny{Pr}}}\) of the composite material composed of steel pipe and concrete were calculated by using the transformed section in material mechanics, as shown in Equations (1) and (2), respectively. \[E_{\text{\tiny{Pr}}}=\frac{E_{s}A_{s}+E_{c}A_{c}}{A_{total}}, \tag{1}\]  
\[\gamma_{\text{\tiny{Pr}}}=\frac{\gamma_{s}v_{s}+\gamma_{c}v_{c}}{v_{total}}, \tag{2}\]  
where \(E_{s}=\) elastic modulus of steel pipe; \(E_{c}=\) modulus of elasticity of concrete; \(A_{s}=\) cross-sectional area occupied by the steel pipe in a single composite material; \(A_{c}=\) cross-sectional