## 4 Numerical Simulation Verification and Analysis
### Numerical Results and Analysis  
The maximum vertical displacement at the top of the span, the friction stress of the pipe-soil contact surface, and the maximum contact opening of the pipe-soil contact surface before the HDPE pipe enters the yield stage were analyzed under different working conditions. We considered the working condition C9 as an example. Figure 13 shows the plastic zone of the HDPE pipe, the friction stress of the contact surface, and the clearance distance of the pipe-soil contact surface before the point in the simulation when the calculation does not converge. Figure 13a shows that the HDPE pipe first starts to yield at the junction of the subsidence area and the nonsubsidese area and eventually yields at the midspan position. Thus, the junction of the subsidence and nonsubsidese areas can be taken as the section of the pipe body over which yielding should be controlled. The maximum vertical displacement (MVD) over the span and the maximum contact opening (MCP) between the pipe and the soil surface before the HDPE pipe starts to yield were used to analyze the degree of coordinated deformation between the pipe and the soil. Figure 13b shows that the contact friction stress reaches a maximum in the yield control section and then gradually decreases to the midspan region. Figure 13c shows that there is a large clearance distance only in the yield control section within a small range at the top and bottom of the pipeline. As the contact stress and friction stress near the yield control section are relatively large, the junction between the subsidence and nonsubsidese areas is more likely to have a larger displacement. The clearance between the pipe and soil is very small at most positions, indicating that most sections of the pipe are in good contact with the soil mass and that the pipe deforms effectively with the soil mass. Figure 14a shows the MVD and MCP before yielding under the working conditions C1-C9. The MVD decreases as the elastic modulus of the pipe increases. Thus, reducing the elastic modulus of the HDPE pipe can effectively increase the OFEB measurement range;  
Figure 13: Numerical simulation results obtained under the working condition C9: (**a**) equivalent plastic strain; (**b**) frictional shear force; and (**c**) contact opening. however, the increase in the MCP weakens the coupling between the pipe and the soil and causes the control section of the pipe to enter the yield stage earlier. Although the displacement could be measured before the yield stage, the plastic yield section near the yield control section of the pipe would gradually increase in size and ultimately affect the deformation measurement accuracy at each position. Figure 14b shows the MVD and MCP before yielding under the working conditions C7, C10, and C11. As the elastic modulus of the soil increases, the MVD decreases and the MCP increases. Thus, increasing the elastic modulus of the soil has a limited effect on improving the deformation between the pipe and the soil. The most suitable ratio of the elastic moduli (the pipe elastic modulus \(E_{\text{g}}\) normalized by the soil elastic modulus \(E_{0}\)) under the optimal code formation condition was determined. Figure 15 shows the MVD and MCP before yielding versus the normalized elastic modulus. At excessively small or large \(E_{\text{g}}\) values, the MVD exhibits an opposite trend in the normalized elastic modulus to the MCP. The coupling performance between the pipe and the soil is poor over a large measurement range. For moderate \(E_{\text{g}}\) values or small \(E_{0}\) values, the coupling between the pipe and the soil is good over a reasonable measurement range. The elastic modulus of a road bed after completion of construction is generally standardized controlled. As \(E_{\text{g}}\) is the main influence factor for the performance of the coupling between the pipe and the soil, it is recommended that the ratio between \(E_{\text{g}}\) and \(E_{0}\) be controlled to between 30 and 40.