## 3 Results and Discussion
### Void Effects  
Voids are typically formed in the lower haunch area based on the difficulty of compaction during pipe installation. Voids may also form at other locations based on either exfiltration (leaking) or the infiltration of water into the pipe, which causes soil erosion. Regardless of the causes of void formation, the effect of voids on the buried concrete pipes' behavior must be understood to assess the structural stability of pipe-soil systems. To consider the effects of void size and location on pipe behavior, three-dimensional analysis is required because plane-strain analysis is only valid when voids are continuous throughout the entire buried concrete pipe. In this study, analyses were conducted separately for voids at two different locations: at the invert and the haunch. We considered one void at the invert and a separate void on each side of the haunch. One void at the haunch is located from 130\({}^{\circ}\) to 150\({}^{\circ}\) measured clockwise from the crown and the other was in the same location in the counterclockwise direction. The voids have a width of 12 cm and a length of 90 cm in the longitudinal direction of the pipe. These void dimensions were selected arbitrarily. Under the longitudinal loading condition, the effects of a void at the invert on the normal soil pressure distribution are presented in Figure 9 for gravelly sand, silty sand, and silty clay backfills. One can see that the presence of a void at the invert reduces the normal pressures at the invert, as expected. There is no apparent influence on the soil pressures at either the crown or springline. Based on the presence of the void, there is no soil pressure immediately below the invert. It appears that the reduced soil pressure around the invert area is transferred to the haunch area, resulting in greater soil pressure around the haunch area, as illustrated in Figure 9. Although Figure 9a-c reveals similar patterns for the normal soil pressure distributions of all three trench materials, there is a significant difference in the pressure distribution at the haunch between the silty clay trench material and the other two materials. This can be attributed to the relatively high compressibility of silty clay (Figure 9c) compared to silty sand (Figure 9b) and gravelly sand (Figure 9a). This higher compressibility leads to greater compression of the soil around the invert. As a result, the soil pressure at the haunch is significantly greater in silty clay than in either gravelly sand or silty sand. Figure 8: Crack depth through the pipe wall under moist, saturated, and submerged conditions with longitudinal loading. The effects of a void at the haunch on the normal soil pressure distribution are presented in Figure 9d for the gravelly sand backfill. As expected, the reduced soil pressure at the void is transferred to the adjacent soil on both sides of the void, resulting in greater soil pressure around the haunch area. There is little influence on the soil pressures at either the crown or springline. However, the soil pressure at the invert increases slightly based on the presence of a void at the haunch. This increased soil pressure at the invert can be attributed to the soil's increased compression at the invert based on a loss of soil support at the haunch. Comparisons of the crack depths in the pipe wall with no voids and with a void at the invert are presented under different longitudinal surface loading intensities in Figure 10.  
One can see that at both the crown and springline, the presence of a void at the invert has very little effect on crack propagation through the wall. Both with and without a void at the invert, the crack that initiates in the intrados of the crown propagates outward to approximately 0.38 cm under 200 kPa of surface loading and approximately 1.90 cm under 500 kPa of surface loading. At the springline, the crack that initiates in the extrados reaches a depth of approximately 0.76 cm under 300 kPa of surface loading and propagates inward to approximately 1.54 under 500 kPa of surface loading.