## 4 Results and Discussion
### Comparison between Simulated and Measured Results of the Pipes with Pre-Existing Cracks
#### 4.3.1 Effect of Crack Length  
On the crown of the concrete pipe, 25 mm-deep cracks with a length of 400, 300, and 200 mm are prefabricated to study the influence of crack length on the mechanical response of the pipe, as displayed in Figure 14. The load is applied to the pipes 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 illustrated in Figure 15. It is clear that the stress distribution of the pipe obviously varies with the change in the crack length and is mainly concentrated at the crown position. The distribution area of compressive stress at the bottom of the crack and tensile stress at the inner wall of the crown increase with an increase in crack length, which will lead to a significant decrease in the ultimate bearing capacity of the concrete pipe. (1) Load-strain curve  
Because the prefabricated cracks with different lengths are located at the crown, and the concrete pipes are generally subjected to tensile damage, under the action of the external ultimate load, the first damaged parts of the pipe are the outer wall at the springline and the inner wall at the crown. Therefore, the points corresponding to the positions where the strain pages are affixed to the outer wall at the springline and to the inner wall at the crown are selected in Section 2.2 of the pipe model, and load-strain curves are drawn, as depicted in Figure 16. It is clear that under the action of the external load, the strain of the outer wall at the springline and that of the inner wall at the crown rise with an increase in the external step-by-step load, which can roughly be divided into three stages: 1) elastic stage: the strain at the top of the pipe changes linearly in this stage, indicating that the pipe is in the elastic stage; 2) elastic-plastic stage: when reaching a certain load value, the strain begins to follow a non-linear trend; thus, it can be considered that the pipe enters the plastic stage, and the process in this stage is extremely short; 3) plastic-failure stage: when the load reaches the limit of the failure load, the strain varies abruptly, indicating that the crown cracks and that the pipe loses its bearing capacity. Figure 14: Pipes containing cracks with different lengths: (**a**) 400 mm, (**b**) 300 mm, and (**c**) 200 mm. inner wall at the crown is too large, which implies that the crack is the place where the concrete pipe is extremely vulnerable. When the pipe loses its bearing capacity, the measured curve represents a maximum strain of 104.6 \(\mu\)\(\epsilon\) for a crack length of 300 mm. Also, according to the simulation curve, when the crack length is 400 mm, the maximum strain reaches 76.8 \(\mu\)\(\epsilon\). The analysis of the load-strain curve of the pipe with a 400 mm-long crack: According to Figure 16a, the simulated and measured curves of the outer wall at the springline respectively reveal that when the load reaches 11.4 kN/m and 11.2 kN/m, the strain changes abruptly and the outer wall cracks at the crown. In Figure 16b, the simulated and measured curves of the inner wall at the crown respectively show that when the load reaches 10 kN/m and 11 kN/m, the inner wall cracks at the crown. Therefore, it can be concluded that the failure load of the pipe with pre-existing cracks obtained by the simulation is about 10 kN/m, and the value calculated experimentally is about 11 kN/m. In fact, the existence of cracks causes the bearing capacity of the concrete at the crown to drop, while the inner wall is under tension at the crown. Therefore, under the action of an external load, the inner wall cracks at the crown before the outer wall breaks at the springline. Therefore, when the pipe loses its bearing capacity, the simulated failure load values of the pipes having cracks with a length of 400, 300, and 200 mm are respectively 10, 11.7, and 13.2 kN/m, and their ultimate bearing capacities are respectively reduced by 54, 46, and 39% compared to the failure load values of the intact pipes.