## 4 Numerical Analyses Result and Discussion  
### Pipe Diameter Effect  
To examine the effect of pipe diameter on stresses, HDPE pipes of 1.524 m in diameter --specified as F type--and 0.762 m in diameter--specified as C type-were considered and are shown in Tables 1 and 3. Stresses around the pipe are shown for the ITI model in Figure 7 and for the ETI model in Figure 8. In Figures 7 and 8, the left and right sides show the values for the pipe with a diameter of 1.524 m and 0.762 m, respectively. Table 7 shows the calculated stresses in the pipe crown, invert, and spring line under a 30 m fill. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Backfill Parameters** & **Cr. S** & **Sand** & **Clay** \\ \hline Natural unit volume weight, \(\gamma_{\text{n}}\) (kN/m\({}^{3}\)) & 22 & 17.7 & 20.4 \\ \hline Reference mean  sec ant module, \(E_{50}^{ref}\) (MPa) & 18 & 9.5 & 5.2 \\ \hline Reference unloading – reloading module, \(E_{ui}^{ref}\) (MPa) & 54 & 28.5 & 15.6 \\ \hline Cohesion, c (kN/m\({}^{2}\)) & - & - & 34.5 \\ \hline Shear strength angle, \(\varphi\) (\({}^{\circ}\)) & 40 & 37 & 15 \\ \hline Dilatancy angle, \(\psi\) (\({}^{\circ}\)) & 5 & 7 & - \\ \hline Exponential power for the stress–level dependency of stiffness, m (\(\cdot\)) & 0.5 & 0.5 & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: HS model parameters for backfill soils. Figure 6: (**a**) Axial stress–axial strain. (**b**) Volumetric strain–axial strain graphs for EPS15 geofoam material. According to the analyses (in Table 7, Figures 7 and 8), the effect of diameter is examined by considering HDPE as the pipe material. The calculated stresses in the 1.524 m diameter pipe were generally higher than the 0.762 m diameter pipe. In the ITI and ETI models, uniform vertical stress occurred in the pipe crown across the width of the EPS, but the stresses in the spring line and invert were larger in the ITI model than in the ETI model. Thus, the importance of the geometry of the compressible region around the pipe was revealed. In the ITI model, a compressible zone was only created on top of the pipe, which caused greater stresses on the pipe spring line and the invert in comparison to the ETI model. Figure 7: In the ITI model, stresses in the change of pipe diameter for the 30 m embankment. Figure 8: In the ETI model, stresses in the change of pipe diameter for the 30 m embankment. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**1.524 m (F) (HDPE)**} & \multicolumn{3}{c}{**0.762 m (C) (HDPE)**} \\ \cline{3-8}  & & **Crown** & **Spring Line** & **Invert** & **Crown** & **Spring Line** & **Invert** \\ \hline \multirow{3}{*}{Reference} & Cr. S & 267 & 331 & 248 & 267 & 235 & 209 \\ \cline{2-8}  & Sand & 326 & 348 & 287 & 316 & 254 & 256 \\ \cline{2-8}  & Clay & 397 & 366 & 370 & 379 & 297 & 338 \\ \hline \multirow{3}{*}{ITI model} & Cr. S & 195 & 197 & 216 & 156 & 154 & 178 \\ \cline{2-8}  & Sand & 207 & 224 & 246 & 176 & 185 & 224 \\ \cline{2-8}  & Clay & 248 & 280 & 334 & 237 & 257 & 301 \\ \hline \multirow{3}{*}{ETI model} & Cr. S & 163 & 151 & 134 & 138 & 97 & 121 \\ \cline{2-8}  & Sand & 171 & 167 & 147 & 153 & 113 & 136 \\ \cline{2-8}  & Clay & 209 & 222 & 205 & 206 & 193 & 207 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Stresses (in kPa) under the HDPE pipe diameter effect for 30 m embankment. It was determined that the ETI model reduced the stresses acting on the pipe and caused a more uniform stress distribution around it. When comparing the stresses acting on the 1.524 m and 0.762 m diameter pipe, the larger diameter pipe was approximately 20%, 15%, and 35% higher at the crown, invert, and spring lines, respectively. A similar study conducted by [58] that considered HDPE pipes with 0.2, 0.3, and 0.40 m small diameters stated that there was a 10% increase in vertical and horizontal stresses as the diameter increased. The type of backfill soil around the pipe also affected the stresses.