# Load-Bearing Performance of a Reinforced Fill Structure with Pile Penetration
## 5 Analysis of Results
### The Distribution Law of Pile-Bending Moments
#### 5.2.4 Effects of Different Slope Rates  
Figure 12 illustrates the variation in the bending moment along the pile of the pile for slope ratios of 0.6, 0.4, and 0.2. Table 14 presents the values of the maximum bending moments for the structure at different slope ratios.  
In Figure 12a, it can be observed that when the slope ratio is 0.6, the bending moment along the pile decreases nonlinearly with depth within the range of 140-300 mm during the early and middle stages of loading. From a depth of 300 mm to 460 mm, the bending moment follows a parabolic trend. In the later stages of loading, as the load increases, there is a significant increase in the bending moment below a depth of 220 mm in the pile. This deformation pattern exhibits only positive bending moments, similar to the passive load-bearing mode of a cantilever structure. The maximum bending moment of 59.85 N\(\cdot\)m is observed at a depth of 380 mm, which closely resembles the distribution pattern of bending moments in a single vertical pile of a micro-pile reinforced earth retaining wall [32].  
According to Figure 12b, the bending moment distribution along the pile follows a parabolic trend with depth, with a slope ratio of 0.4. The maximum bending moment of 26.11 N\(\cdot\)m is observed at a depth of 300 mm.  
Based on Figure 12c, a slope ratio of 0.2 and loading in the middle and later stages result in a point of inflection at a depth of 340 mm along the pile, where the bending moment becomes zero. Above this point, a parabolic distribution of bending moments is observed on both sides. The bending moment in the pile is positive above the inflection point (bending in the same direction as soil sliding), while it is negative below the inflection point (bending in the opposite direction to soil sliding). This bending moment behavior resembles that of a micro-pile reinforced earth retaining wall [18; 19; 22]. With an increasing load, the positions of the maximum bending moment and the inflection point remain relatively constant, but the magnitude of the bending moment increases significantly, reaching a peak of 115.88 N\(\cdot\)m.  
The analysis of Figure 12 and Table 14 reveals that the slope ratio has a significant effect on the distribution of the bending moments during the middle and later stages of loading. Within a certain range, reducing the slope ratio can decrease the maximum bending moment, leading to a more reasonable distribution of bending moments in the pile and preventing localized stress concentrations in the pile. However, if the slope ratio is too small, the absolute magnitude of the bending moment increases.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Condition**} & **The Number of** & **Maximum Bending** & **Vertical** & **Pile Depth/mm** \\  & **Isofercement** & **Moment/(N\(\cdot\)m)** & **Load/kN** & **Pile Depth/mm** \\  & **Layer n** & & & \\ \hline
3 & 0 & 52.61 & 30.0 & 140 \\
7 & 1 & \(-\)61.64 & 30.0 & 140 \\
6 & 2 & 86.19 & 45.0 & 140 \\
5 & 3 & 26.11 & 67.5 & 300 \\ \hline \end{tabular}
\end{table}
Table 13: Maximum bending moment values with different reinforced layers.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Condition**} & **Ratio of Slope** &
\begin{tabular}{c} **Maximum Bending** \\ **Moment/(N\(\cdot\)m)** \\ \end{tabular} & **Vertical** & **Pile Depth/mm** \\  & **Pile Depth/mm** \\ \hline
9 & 0.6 & 59.85 & 45.0 & 380 \\
5 & 0.4 & 26.11 & 67.5 & 300 \\
8 & 0.2 & 115.88 & 52.5 & 220 \\ \hline \end{tabular}
\end{table}
Table 14: Maximum bending moment value of structures with different slope rates.