## 4 Parametric Studies
### Effect of the Maximum Diameter of the Conical Head (\(D\))  
Figure 10 shows that the vertical displacement of the embankment surface changes as \(D\) increases from 0.6 m to 1.4 m in the case \(d\) = 0.4m (Minimum pile diameter) and \(r_{e}\) = 1.2 m. As is anticipated, the settlement of the embankment surface decreases with an increase in \(D\), but when the diameter exceeds 1 m, the curve of settlement for the equal section pile flattens out gradually. The settlement of the embankment reinforced by the pile with conical head is less than that of the equal section pile when the maximum diameter of the conical head is equal to the diameter of the equal section pile. The difference of two curves become increasingly obvious as \(D\) increases, which further testifies to the superiority of the pile with a conical head as \(D\) grows. Table 3 shows the material consumption of two types of piles. The comparative result indicates that a pile with a conical head consumes less material and results in less settlement. It can be observed that the piles with conical heads have better working performance and practicability. Figure 11 shows the variation of the axial force of pile with \(D/d\). The axial force of the equal section pile increases with depth on the whole, and the maximum value is 196 kN at the neutral point (11 m below the ground surface). However, the neutral point of the pile with a conical head is located at 9 m below the ground surface with a maximum axial force of 116 kN. Below the conical head, the variation tendency of the axial force is similar and the difference is small although \(D/d\) differs. In the design of pile supported embankments, the pile diameter is determined by the maximum axial force and the construction machinery. The area replacement rate is a constant for the equal section pile, which results in a high area placement rate except for the neutral point. Compared with the equal section pile, the axial force below the conical head varies within a relatively small range, which is consistent with the principle that the greater the axial force is, the larger the cross section of the pile. As shown in Figure 11, the axial force at the pile top increases with the value of \(D/d\). While the axial force gradually decreases as \(z^{**}\) goes deeper within the conical head zone, and the bigger the \(D/d\) is, the faster the axial force decreases. Obviously, the axial force variation trend of the two types of pile stands opposite within the conical head zone. The conical head converts a portion of the vertical load into a horizontal load and transmits it into the uppermost crust (with relatively high plasticity, a water content slightly lower than the plastic limit, and a relatively high over consolidation ratio (OCR)) or artificial working platform. This process contributes to improving the bearing capacity of pile supported embankments. When \(D/d\) = 1.5, the maximum diameter of the conical head and the diameter of the equal section pile both are 0.6 m. The axial force at the top of the two types of piles is 76 kN. Within the conical head zone, the axial force of pile with a conical head remains unchanged while that of the equal section pile climbs to 110 kN. It can be concluded that the vertical interaction force between the pile and soil is zero within the conical head zone when \(D/d\) = 1.5. The reason will be unfolded below. Figure. 12 shows the vertical interaction stress (\(F(z^{**})\)) versus \(z^{**}\). For the pile with a conical head, as \(D/d\) increases, the vertical interaction stress changes from negative to positive within the conical head zone. This does not mean that the conical head shifts downward relative to the soil. As shown inFigure 5, the vertical stress of the soil to the pile is composed of two parts: the vertical component of friction and the vertical component of the normal force. Figure 13 illustrates the axial force at the top and bottom of the conical head.