## 5 Conclusions  
In this work, through digital simulation and the mutual verification of the field tests, concrete pipes containing various cracks (different crack lengths crack depths, and crack locations) are studied and analyzed. The main conclusions drawn are as follows:  
(1) Through ABAQUS finite element software, the concrete damage plastic model is selected to carry out a numerical simulation analysis on the concrete pipes with pre-existing cracks. Also, using the actual field tests, the simulation results are analyzed and verified. The obtained test results are in good agreement with the simulation data, which indicates that the developed model can well predict the test outcomes. (2) With the gradual increase of the external load, the circumferential strain on the concrete pipe also rises. The change in strain versus load can be divided into three stages: (i) the elastic stage (in which the load-strain curve shows a linear relationship); (ii) the elastic-plastic stage (in which the load-strain curve displays a nonlinear relationship); (iii) the plastic-failure stage (in which the load-strain curve varies suddenly). Moreover, the elastic-plastic phase occurs for a short time and is not transparently significant, which is due to the fact that concrete is a brittle material, that its compressive strength is far greater than its tensile strength, and that there is no significant plastic deformation stage. (3) The length, depth, and location of the cracks greatly affect the bearing capacity of the concrete pipes. The greater the length and the depth of the cracks are, the smaller the bearing capacity of the concrete pipe becomes; in fact, the measured failure load values of the pipes containing cracks with a length of 400, 300, and 200 mm are 11, 12.5, and 14 kN/m respectively, and their ultimate bearing capacities are reduced by 52, 49, and 43% respectively. The failure load values of the pipes containing cracks with depths of 25, 20, and 15 mm are 12.5, 13, and 14 kN/m respectively, and their ultimate bearing capacities are reduced by 46, 43, and 39% respectively. When the crack is located either at the crown or at the invert, its influence on the bearing capacity of the pipe is basically the same, and it \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Contrast**} & \multirow{3}{*}{**Working Condition**} & \multirow{3}{*}{**Intact**} & \multicolumn{8}{c}{**Crack Length (mm)**} & \multicolumn{2}{c}{**Crack Depth (mm)**} & \multicolumn{2}{c}{**Crack Location**} \\ \cline{3-11}  & & & **400** & **300** & **200** & **25** & **20** & **15** & **Crown** & **Spring-Line** & **Invert** \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Numerical simulation \\ \end{tabular} } & Average failure load & 21.7 & 10 & 11.7 & 13.2 & 11.7 & 13 & 13.4 & 11.7 & 13.1 & 12 \\ \cline{2-11}  & Reduction rate (\%) & - & 54 & 46 & 39 & 46 & 40 & 38 & 46 & 40 & 45 \\ \hline \multirow{3}{*}{Test} & Average failure load & \multirow{3}{*}{23} & \multirow{3}{*}{11} & \multirow{3}{*}{12.5} & \multirow{3}{*}{14} & \multirow{3}{*}{12.5} & \multirow{3}{*}{13} & \multirow{3}{*}{14} & \multirow{3}{*}{12.5} & \multirow{3}{*}{13.6} & \multirow{3}{*}{12.3} \\  & (kN/m) & & & & & & & & & & \\ \cline{1-1} \cline{2-11}  & Reduction rate (\%) & - & 52 & 49 & 43 & 46 & 43 & 39 & 46 & 41 & 47 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Failure loads of the concrete pipesis greater than when the crack is located at the springline; indeed, when the crack is located at the crown, springline, and invert, the measured failure load values of the pipe are 12.5, 13.6, and 12.3 kN/m respectively, and their ultimate bearing capacities are reduced by 46, 41, and 47% respectively. (4) Stress concentration occurs at both ends of the crack on the pipe with the tip stress far exceeding the apparent stress, causing the local stress to be greatly higher than the allowable stress value of the material. Therefore, in the tests, when the concrete pipes are damaged by the ultimate load, they all break at the crack positions.