## 4 Results and Discussion
### Results of the Parametric Study
#### 4.6.3 Effect of the Stiffness of the Pipe and Backfill  
The analyses were repeated while keeping the backfill modulus of elasticity (Es) constant and considering a range of pipe modulus of elasticity (Ep) from 450 MPa to 2800 MPa. The increment in Ep significantly reduces the u\({}_{\rm pvv}\) component in S\({}_{\rm d}\), while the increment in stresses acting on the central soil prism resulted in a limited increase in the u\({}_{\rm sys}\) component. In particular, the reduction in u\({}_{\rm pvv}\) prevents the formation of a heart formed deformation. The stresses applied to the buried pipe system are distributed based on the relative stiffness of the pipe and the surrounding backfill. An increase in pipe stiffness amplifies the stresses carried by the pipe [19; 28]. Additionally, a reduction in Pv decreases the stresses transmitted to the adjacent soil prism. The decrease in K implies a reduction in the horizontal effect of vertical stresses in the central soil prism. Figure 18 illustrates the fluctuation in stresses derived from the analyses and Equation (2) for various stiffness ratios. Increasing the Ep leads to a significant reduction in the horizontal and vertical pipe deflection. This reduction in Ph results in a substantial decrease in \(\sigma_{\rm h1}\). These stresses become negligible at a distance of D/4. Between D/4 and 1.5D, the horizontal component of the stresses applied from the surface remains effective, but its effectiveness decreases significantly beyond this distance. The gradual decline in \(\sigma_{\rm v1}\) corresponds to the disappearance of the arching. The decrease in horizontal stress is more pronounced close to the tank wall than the decrease in vertical stress. Moreover, increasing the modulus of elasticity (Ep) not only restrains pipe deflection but also facilitates stress transmission from the crown to the invert. Figure 17: Variation of K at SP-1 with regard to h (**a**) K for different stresses (**b**) mean of K values. The rvs and rvd values were computed for various Ep/Es ratios, using the stress and deflection or displacement obtained with an Ep value of 450 MPa as a reference (Figure 19). The term stiffness ratio (rs) is defined to express these different Ep/Es ratios. The heart-shaped deformation pattern was gradually disappeared as Ep increased, resulting in a more uniform distribution of deflections at the top of the pipe. This led to a greater decrease in Pv than in Ph, and the d\({}_{r}\) gradually approached 1. For rs values of 16, 27, 54, and 100, the d\({}_{r}\) values were determined to be 0.79, 0.85, 0.91, and 0.94, respectively. The coefficient c\({}_{1}\) for K in Equation (7) was computed as 0.90, 0.85, 0.78, and 0.73 based on the rs values. The change in K for different stress levels and rs values was derived from the analyses and Equation (7) (Figure 20). Figure 19: Stress and deformation variation ratios for different rs (**a**) rvd (**b**) rvs. Figure 18: Variation of \(\sigma_{\text{crown}}\) values. The gradual increase in the stiffness of the backfill from 28 MPa to 115 MPa, while keeping the Ep constant, leads to a decrease in the \(\mathrm{u_{ys}}\). Similarly, the reduction in the stresses carried by the pipe causes to a reduction in the \(\mathrm{u_{ypv}}\) component. The higher stiffness of the backfill surrounding the pipe enhances the soil support provided to the pipe by the backfill [28; 57]. The soil support enables a significant reduction in pipe deflections. Generally, the changes in the system resulting from the increase in pipe stiffness for arching are reversed by the increase in backfill stiffness. The \(\mathrm{r_{vs}}\) and \(\mathrm{r_{vd}}\) values were computed for various \(\mathrm{r_{s}}\) with reference to the Es value of 28 MPa (Figure 21). Increasing the stiffness of the backfill enhances \(\mathrm{Fv}\)[14; 58], amplifying the horizontal stress component acting on the central soil prism.