## 4 Verification of the Analytical Solution  
### Numerical Verification of the Analytical Solution  
The finite difference software FLAC3D was used to simulate the jacking construction of rectangular pipes. The stratum deformation caused by time effect was not considered during the construction of pipe jacking. Moreover, the deformation and deformation rate under initial gravity stress were not considered. In the actual construction process, each pipe section is rigidly connected by welding; hence, the shell element can be used to simulate the overall steel pipe jacking. The selection of the numerical simulation model should fully consider the influence range of the analytical solution. In this paper, the size of the numerical simulation model was 50 m in width and 30 m in depth, and the longitudinal excavation length was 36 m. The lateral distance between the pipe and the model boundary was greater than five times the pipe width, which could eliminate the influence of the model boundary on the calculation results. For the boundary of the numerical model, the upper face took \(z=-30\) m as the free boundary, and the bottom face was constrained in the z-direction at \(z=-30\) m. The left and right sides were constrained in the x-direction at the positions \(x=-27\) m and \(x=23\) m. The front and rear sides were constrained in the y-direction at the positions \(y=0\) m and \(y=36\) m.  
The isotropic elastic-plastic model was adopted for the soil, and the yield criterion was the Mohr-Coulomb criterion. Assuming that the soil layers were evenly and horizontally distributed. The thickness of the pipe wall was much smaller than the section size of the pipe; thus, the steel pipe can be regarded as a thin-walled tube. In the comparative calculation, the sizes of the pipe section were \(A=2\) m and \(B=2\) m. The length of the pipe jacking machine was \(s=2\) m. The distance between the axial of the pipe and the ground was \(H=5\) m. The axial distance between the pipes was \(L=3\) m. The additional thrust, friction, and grouting pressure were 20 kPa, 5 kPa, and 20 kPa, respectively. The calculation model is presented in Figure 9.  
Firstly, under the action of gravity, the initial stress field of the model was balanced, and then the displacement was cleared to obtain the stress field of the unexcavated state before construction. Secondly, the unbalanced force of the outer node of the excavation unit was obtained by excavating the first unit and running a calculation. The unbalanced force on these nodes was multiplied by a coefficient less than 1 and reversely applied to the corresponding nodes. In this paper, the stress release rate was 15% to simulate the stress release in the construction process. Then, a uniform load was applied on the excavation face to simulate the additional horizontal thrust of the pipe jacking machine. The shell structure unit was added to simulate the corresponding pipe, and then the calculation process was started to balance the stress in the model. Lastly, 3.6 m was taken as the cycle of excavation step, and excavation continued until the pipe jacking construction was complete. (1) Comparison of transverse surface settlement  
The working condition where the left pipes 01 and 02 were completed while the right pipe was jacked for 18 m was taken as an example (i.e., the excavation face of jacking machine 03 was located at the position of \(x=0\)). The lateral settlement curves of the ground surface obtained by analytical calculation and numerical simulation are shown in Figure 10.  
The analytical calculation results of the transverse surface settlement curve are consistent with the overall trend of numerical simulation results. With the monitoring points away from the axis of the pipe, the ground surface settlement gradually decreased and tended to be stable, and the influence range was about five times the width of the pipe.