# Analysis of Shield Tunnels Undercrossing an Existing Building and Tunnel Reinforcement Measures
## 3 Mathematical Model  
The mathematical model was established to simulate the construction process of the double shield tunnels by the Midas GTS NX software. The dimensions of the model for length, width, and height were 100 m, 50 m, and 50 m, respectively. The flow chart of the modeling steps of the model is shown in Figure 3.  
The modeling steps of the model are as follows. 1. [label=(0)]
2. Plain fill, silty clay, completely weathered slate, and strongly weathered slate were simulated by 3D solid units. The constitutive models were the isotropic Mohr-Coulomb model [26], abbreviated as the MC model, an ideal elastic-plastic model that combines [PERSON]'s law and Coulomb's failure criterion widely used to simulate most geotechnical materials. The MC model used the Coulomb formula as the shear strength formula, and the failure criterion was based on whether the shear stress reached the shear strength. As shown in Equation (1), when the shear stress at a point on any plane in the soil was equal to the shear strength of the soil, that point was in a critical state of near failure, which was called the \"ultimate equilibrium state\". The relationship between various stresses in this state was called the \"ultimate equilibrium condition\". In the MC model of Midas GTS NX, elastic modulus and Poisson's ratio controlled elastic behavior, while cohesion, internal friction angle, and shear expansion angle controlled plastic behavior. \[\tau_{f}=f(\sigma)\] (1) where \[\tau_{f}\] is the shear strength of soil, and \[\sigma\] is the soil stress. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Soil (Rock) Layers** & **Severe/** & **Cohesive Force/** & **Friction Angle/** & **Modulus of Elasticity/** & **Poisson’s** \\
**and Materials** & **(kN\(\cdot\)m\({}^{-3}\))** & **(kN\(\cdot\)m\({}^{-2}\))** & **(\(\lx@paragraphsign\))** & **(kN\(\cdot\)m\({}^{-2}\))** & **Ratio** \\ \hline Plain filled soil & 19 & 16 & 11 & 4000 & 0.28 \\ Silty clay & 19 & 15 & 8 & 6000 & 0.31 \\ Fully weathered slate & 20 & 15 & 35 & 8500 & 0.23 \\ Strongly weathered slate & 23.3 & 50 & 29 & 120,000 & 0.2 \\ segment & 25 & — & — & 3.45 \(\times\) 107 & 0.2 \\ Grouting layer & 21 & — & — & 1000 (soft) & 0.2 \\ Pile & 25 & & & 2 \(\times\) 106 & 0.3 \\ Shield shell & 78.5 & & & 2.1 \(\times\) 1010 & 0.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Calculation parameters of soil (rock) layer and material. Figure 3: Flow chart of modeling steps of the model. 2. Bearing piles, retaining piles, basement columns, and building columns were simulated by beam element, while the basement floors and building floors were simulated by 2D plate element with the constitutive model of the isotropic elastic model. Suppose the property of a material did not change with direction, which was isotropic. The isotropic elastic material had the same elastic modulus, Poisson's ratio, thermal expansion coefficient, and thermal conductivity in all directions. The expression for isotropic elasticity is shown in Equation (2). \[\sigma_{ij}=\frac{E}{1+v}\epsilon_{ij}+\frac{vE}{(1+v)(1-2v)}\epsilon_{kk} \sigma_{ij}\] (2) where \(E\) is elastic modulus, \(v\) is Poisson's ratio, and \(\sigma\) and \(\epsilon\) are the stress and strain of the material. (3) The segment and shield shell were simulated by a 2D plate element, and the constitutive model was the isotropic elastic model. The grouting body was simulated by a 3D solid element, and the constitutive model was an isotropic MC model. Step 2: Assign material properties. Assign various parameters and attributes to the corresponding geometric model. Step 3: Create mesh. In order to highlight the research focus, the pile and shield tunnel structures were divided into mesh with a spacing of 2 m, and the soil was divided into mesh with a spacing of 4 m. Because of the higher accuracy of the hexahedron element, the hexahedron hybrid mesh generator was used to generate soil and shield tunnel structures. Step 4: Apply boundary conditions.