## 4 Numerical Analyses Result and Discussion
### Horizontal and Vertical Arching  
Horizontal and vertical arching factors are generally used to calculate soil stresses on buried pipes. These factors determine the stresses acting on the pipe due to the pipe-soil interaction. While the arching factors used for rigid pipes are the ratio of the total earth pressure to the prism load, they are calculated with different parameters in thermoplastic pipes. The reason for this is that deformations occurring in thermoplastic pipes significantly affect the stresses in the soil. The equations proposed by [85] and presented in Equations (2) and (3) were used in calculating these factors. Therefore, it was essential to determine the arching factors to predict the stresses in the soil. In these equations, the arching factors of a buried pipe under load are calculated by considering the ring stiffness. Elastic solutions were made by [71] to perform these calculations, and equations were developed.  
Through the numerical analysis made within the scope of this article, horizontal and vertical stresses occurring around the pipe are known. Thus, the values for the vertical arching factor (VAF) and horizontal arching factor (HAF) were determined as the ratio of backfill load to prism load. The equations used in these calculations are given below:  
\[\text{VAF }=\frac{W_{e}}{PL}=\frac{2N_{sp}}{PL} \tag{2}\]  
\[\text{HAF }=\frac{W_{h}}{PL}=\frac{N_{c}+N_{i}}{PL} \tag{3}\]  
The expressions in the equations above are as follows:  
\(W_{e}\) = total vertical load, \(W_{h}\) = total horizontal load, \(PL\) = Prism load, \(N_{sp}\) = soil stress on the pipe in the horizontal axis, \(N_{c}\) = total soil stress on the pipe crown, and \(N_{i}\) = total soil stress at the bottom of the pipe.  
Figures 13 and 14 show VAF and HAF ratios depending on embankment height (H/D ratio) for 0.762 m PVC and 1.524 m diameter HDPE pipes, respectively. While VAF and HAF were high in the reference--the situation where EPS was not used--arching values decreased in the ITI and ETI model using EPS. The highest VAF and HAF value in clay backfill was calculated in both models and compared to the others.  
In Figure 13, for the models using EPS with 0.762 m diameter PVC pipe, VAF values were very close to crushed stone and sand backfills. In the ITI and ETI models, VAF decreased to 0.3 and 0.2, respectively, and to 0.4 and 0.34 in clay backfill. HAF decreased to 0.3 and 0.25 in the ITI and ETI model for crushed stone and sand backfills and to 0.36 and 0.30 in clay backfill, respectively. In Figure 14, the VAF values for crushed stone and sand backfills were very close to each other in the 1.524 m diameter HDPE pipe. In the ITI and ETI models, VAF decreased to 0.3 and 0.25, respectively, and to 0.4 and 0.34 for clay backfill. HAF decreased to 0.3 and 0.25 in the ITI and ETI models for crushed stone and sand backfills and to 0.45 and 0.36 in clay backfills, respectively. As the H/D ratio increased,the VAF and HAF values decreased, so as the embankment height increased, the vertical and horizontal stresses acting on the pipe decreased. VAF and HAF curves are similar regardless of backfill type and start from high values decreasing with the increase in the H/D ratio and becoming relatively horizontal. The largest pipe diameter considered in the numerical analysis was 1.524 m. The VAF and HAF values for large-diameter pipes are higher than for small-diameter pipes. According to the analysis results, the highest and lowest VAF and HAF values were obtained for clay and crushed stone backfill, respectively. With the increase in the H/D ratio in the ETI model, VAF and HAF values are the lowest for all three backfill types.