## 3 Numerical Model for the Simulation of Jacking Reinforced Concrete Circular Pipes
### Geometric Model Establishment and Material Properties  
The numerical model was based on the aforementioned pipe jacking engineering, which passes underneath the existing cultural relics. Figure 2 shows the numerical model of pipe jacking construction, in which the model size is nearly four times that of the soil mass around the pipe jacking engineering, which is 100 m long (y direction), 50 m wide (x direction), and 30 m high (z direction). Some assumptions of boundary conditions were adopted in the numerical model. The four lateral sides of the model were constrained horizontally, the bottom was constrained horizontally and vertically, and the upper bound  
Figure 1: Section diagram of the reinforced concrete circular pipes in pipe jacking engineering.  
ary was a free boundary. In the numerical model, there were three reinforced concrete circular pipes with inner diameters of 4.5 m, 3.5 m, and 3 m, and the corresponding outer diameters of 5.4 m, 4.2 m, and 3.6 m, which were labelled as 1#, 2# and 3#, respectively. The jacking lengths of the three reinforced concrete circular pipes were all 100 m, the stand length of a single prefabricated reinforced concrete pipe section was 2.5 m, and the spacings between the outer walls of adjacent reinforced concrete circular pipes were 4.2 m and 3.6 m, respectively, which are the same as those in the field.  
As demonstrated in Figure 2, the grid divisions of both the jacking pipes and soil mass were suitable to ensure the accuracy and running speed of the model calculation, and the tetrahedral grid was adopted for partitioning the numerical model. There were 17,041 nodes and 89,970 elements in the numerical model, and the 1.0 m size was used for the pipe gallery and 2.0 m size was employed in the soil mass.  
The soil layer was divided into four layers according to the engineering survey, and the Mohr-Coulomb model was used to simulate the soil mass. The specific parameters of the material properties are shown in Table 1. The circumferential gap was generated between the soil mass and reinforced concrete circular pipe during the pipe jacking process, and the drag-reducing thixotropic slurry was injected around the reinforced concrete circular pipe to reduce the frictional resistance force thereafter. In the numerical model, the soil loss area after grouting the aforementioned drag-reducing thixotropic slurry was equivalently replaced by the reduced elastic modulus within a certain range of an equivalent layer with 20 mm thickness and set around the reinforced concrete circular pipe as demonstrated in Figure 3, in which the reduced elastic modulus of the equivalent layer with drag-reducing thixotropic slurry grouting was 2500 MPa, and the Poisson's ratio and density of the equivalent layer with the drag-reducing thixotropic slurry grouting were 0.2 and 1200 kg/m\({}^{3}\), respectively. Moreover, the concrete adopted in the reinforced concrete circular pipes had an elastic modulus of 34.5 GPa, a Poisson's ratio of 0.3, and density of 2500 kg/m\({}^{3}\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Soil Layer**} & **Depth** & **Density** & **Elastic Modulus** & **Poisson’s Ratio** & **Cohesive Force** & **Friction Angle** \\ \cline{2-7}  & **m** & **kg/m\({}^{3}\)** & **MPa** & & **kPa** & \({}^{\circ}\) \\ \hline miscellaneous filled soil & 2 & 1850 & 4.5 & 0.15 & 8 & 12 \\ plain filling soil & 3 & 2000 & 4.5 & 0.3 & 8 & 10 \\ silty clay & 3.5 & 1900 & 6 & 0.3 & 33 & 30 \\ strongly weathered sandstone & 21.5 & 2100 & 20 & 0.26 & 20 & 15 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of soil layers.  
Figure 2: Numerical model of pipe jacking construction.