## 4 Results and Discussion
### Comparison between Numerical and Experimental Results of the Intact Pipe  
Firstly, the stress state of the control group, which is the intact concrete pipes under an external step-by-step load, is tested. The model of the intact concrete pipe is shown in Figure 11. We increase the external pressure load step by step until the maximum principal stress of the pipe reaches the ultimate tensile strength, i.e. 2.66 MPa. As shown in Figure 12, the nephogram of the maximum principal stress of the intact pipe in the failure state shows that the largest tensile stress is on the inner wall at the crown and invert locations, followed by that of the outer wall at the springline.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Group** & **Crack Length (mm)** & **Crack Depth (mm)** & **Crack Location** \\ \hline
1 & 400 & & \\ \hline
2 & 300 & 25 & Crown \\ \hline
3 & 200 & & \\ \hline
4 & 300 & 20 & \\ \hline
5 & 300 & 10 & Crown \\ \hline
6 & & & Springline \\ \hline
7 & 300 & 25 & Invert \\ \hline \hline \end{tabular}
\end{table}
Table 5: Design condition  
Figure 10: (**a**) Intact concrete pipe and (**b**) cracked concrete pipe.  
We extract the strains at the crown, shoulder, springline, haunch, and invert locations of the intact pipe in Section 2.2 and plot the load-strain curves respectively, as presented in Figure 13 (\(\varepsilon\) is the strain at the vertical axes of the diagrams and \(\mu\varepsilon\) express microstrain, where \(\varepsilon\)=\(\Delta\)L/L and \(\mu\varepsilon\)=\(\varepsilon\)\(\cdot\)10\({}^{-6}\)). It is clear that the distribution law and numerical value of the simulated and measured load-strain curves are basically consistent, which confirms that the test results are reliable. In addition, as can be inferred from Figure 13, in the elastic stage of the pipe, with the gradual increase of the external load, the strain at each measuring point increases linearly, and the duration of this stage is longer. With a further increase in the external load, the pipe transits from the elastic stage to the plastic stage. When the external load exceeds the ultimate bearing capacity of the pipe, the load-strain curve changes abruptly and the pipe is damaged. The transition from the elastic stage to the plastic stage and then to final failure is very short, which fully demonstrates the brittleness of the concrete materials.  
Figure 11: Intact pipe.  
Figure 12: Nephogram of the maximum principal stress.  
Figure 13: Comparison of load–strain curves of the intact pipe: (**a**) outer wall and (**b**) inner wall.  
In Figure 13a, the simulated load-strain curve of the outer wall of the pipe states that when the load reaches 22 kN/m, the strain value of the outer wall at the springline first changes abruptly. However, the simulated load-strain curve of the inner wall of the pipe (Figure 13b) shows that when the load reaches 21.7 kN/m, the strain at the crown and invert first changes suddenly. It demonstrates that, under an external load, the intact pipe is first damaged on the inner wall at the crown and invert. The measured load curve also confirms that the failure load of the intact pipe is about 23 kN/m, and the error value is only 5.7%, which further verifies the reliability of the simulation results.