## 4 Results and Discussion
### Pipe Deflection  
The stress distribution and vertical deflection of the HDPE pipe under localised loading conditions without pavement on the topsoil surface are shown in Figure 10.  
The analysis indicates that the vertical deflections of the pipe are non-uniform and concentrated at the crown, depending on the stress distribution. The heart-shaped deflection pattern emerges as a consequence of the crown descending more than other parts of the top of the pipe. This distinct deformation is a characteristic outcome of the specific deflection behaviour observed in the system (Figure 10). In addition to the vertical deflection of the flexible pipe, the applied stresses also cause horizontal deflection in the pipe. This has a significant effect on the behaviour of the backfill at the sidewalls of the pipe.  
Ph reaches a maximum value approximately at the springline. Considering the deformation form of the pipe, the vertical deflection of the pipe (Pv) is greater than that of Ph. In addition, the ratio of the horizontal deflection of the pipe to the vertical deflection (Ph/Pv) is expressed as the deformation ratio (d\({}_{r}\)). The deformation ratio ranges from 0.76 to 0.87, with a mean of 0.81 for the test, and from 0.76 to 0.80, with a mean of 0.79 for the analysis (Figure 11). Although the d\({}_{r}\) for the analysis is consistent, the d\({}_{r}\) obtained for the test data fluctuates at high stress levels.  
Ph is the primary parameter determining the horizontal stresses in the springline. Similarly, Pv increases the vertical stresses transmitted to the sides of the pipe by enhancing the arching effect. Therefore, a relationship exists between the pipe deflections and the stresses determined at the sides of the pipe. To clarify this relationship further, the measurement points where L9 and L10 are located are denoted as SP-1. The variations of horizontal stresses (\(\sigma_{\mathrm{h1}}\)) at SP-1 as a function of Ph and the variations of vertical stresses (\(\sigma_{\mathrm{v1}}\)) at SP-1 as a function of Pv are illustrated in Figure 12.  
Figure 10: Distribution of stress and deflection of pipe subjected to localised loading.  
The relationship between Ph and \(\sigma_{\rm h1}\) at SP-1 is defined by Equation (3) for both test and analysis, while Equation (4) describes the relationship between Pv and \(\sigma_{\rm v1}\). However, the preferred backfill and pipe properties for buried pipe systems vary widely depending on the application. Therefore, the relationships presented in this study are valid for the loading conditions, backfill, and pipe properties.  
\[\sigma_{\rm h1t} =22.22\times\rm{Ph_{t}} \tag{3a}\] \[\sigma_{\rm h1a} =23.98\times\rm{Ph_{a}}\] (3b) \[\sigma_{\rm v1t} =26.67\times\rm{Pv_{t}}\] (4a) \[\sigma_{\rm v1a} =26.51\times\rm{Pv_{a}} \tag{4b}\]  
In Equations (3) and (4), \(\sigma_{\rm h1t}\), \(\sigma_{\rm v1t}\), \(\rm{Ph_{t}}\) and \(\rm{Pv_{t}}\) signify the measured values in the test, while \(\sigma_{\rm h1a}\), \(\sigma_{\rm v1a}\), \(\rm{Ph_{a}}\) and \(\rm{Pv_{a}}\) signify the values computed from the analysis.  
Figure 11: Pipe deflection versus stress levels (**a**) Pv and Ph variation (**b**) variation of \(\rm{d_{r}}\).  
Figure 12: Variation of stresses at the SP-1 with respect to pipe deflection (**a**) Ph and \(\sigma_{\rm h1}\) (**b**) Pv and \(\sigma_{\rm v1}\).