## 4 Results and Discussion
### Comparison between Simulated and Measured Results of the Pipes with Pre-Existing Cracks
#### 4.3.3 Effect of Crack Location  
In order to study the influence of crack location on the mechanical response of the concrete pipes, a pipe model with prefabricated cracks with a length of 300 mm and a depth of 25 mm at the crown, springline, and invert locations is selected as displayed in Figure 22. An external load is applied to the pipe with pre-existing cracks step by step. The stress nephogram, when the maximum principal stress of the pipe reaches the ultimate tensile strength (2.66 MPa), is shown in Figure 23. It is inferred from the maximum principal stress distribution in Figure 23 that when the cracks are located either at the crown or at the invert, the largest tensile stress occurs on the inner wall of the crown and invert. However, when the crack is located at the springline, the stress on the bottom of the crack is small. Similarly, stress concentration occurs at both ends of the cracks at these three positions. (1) Load-strain curve  
At Section 2.2 of the pipe with pre-existing cracks, we draw the load-strain curves of the inner and outer walls at different crack positions as can be seen in Figure 24. The analysis of the simulated and measured load-strain curves in Figure 24 clearly proves that the load-strain curve has three stages of changes while the external load increases. At first, the strain curve of the inner wall varies abruptly, which implies that the damage first occurs on the inner wall of the pipe at the crack position. Moreover, the strain curves of the crown and invert positions basically coincide, which confirms that when the cracks are located either at the crown and or at the invert their influence on the pipe is basically the same. According to Figure 24a,b, when the crack is located either at the crown and or at the invert, the measured and/or simulated strain values are basically equal, but both are larger than the strain value when the crack is located at the springline. In fact, when the crack is located either at the crown and or at the invert, it greatly impacts on the bearing capacity of the concrete pipe, and the pipe is more easily damaged. Figure 22: Pipes containing cracks at different positions: (**a**) crown, (**b**) springline, and (**c**) invert. Furthermore, the simulation data (Figure 24b) reveal that when the crack is located at the crown, springline, and invert, the failure load values of the pipe are 11.7, 13.1, and 12 kN/m respectively, and their ultimate bearing capacities are reduced by 46, 40, and 45% respectively. However, the test data show that when the crack is located at the crown, springline, and invert, the failure load values are 12.5, 13.6, and 12.3 kN/m respectively, which are 46, 41, and 47% lower than the actual failure load values of the intact pipes. (2) Circumferential strain  
Figure 23: Nephogram of the maximum principal stress of the pipes containing cracks at different positions: (**a**) crown, (**b**) springline, and (**c**) invert. Figure 24: Load–strain curves of the pipes containing cracks at different positions: (**a**) outer wall and (**b**) inner wall. The circumferential strains on the inner and outer walls of the pipe are extracted under the destructive load at eight positions, and the circumferential strains on the pipe at three different crack positions are analyzed. The strain curves are presented in Figure 25.  
It is clear that the circumferential strain of the inner and outer walls of the pipe is still distributed symmetrically in the center; the circumferential strain on the outer wall of the pipe is distributed as 'concave' from top to bottom and 'convex' from left to right; nevertheless, the circumferential strain on the inner wall of the pipe is distributed as 'convex' from top to bottom and 'concave' from left to right. These findings are consistent with the stress distribution law of concrete pipes under an external load.