## 4 Numerical Analysis and Parameter Determination
### Parameter Analysis  
In this section, parameter analysis is conducted to analyze the influence of each parameter on the computation of the ultimate toe capacity. In order to facilitate comparison, pile length = 20 m, pile diameter = 0.5 m, and the unit weight of soil = 19 kN/m\({}^{3}\), while groundwater is not considered. Figure 9 shows the ZEL net for an illustrative example. The influence of parameters on the computed ultimate toe capacity is shown in Figures 10-12. Figure 10 presents the influence of the inclination angle of the failure zone boundary on the computed toe capacity. Figure 10 shows that as the inclination angle of the failure zone boundary increases, the computed toe capacity rapidly decreases and trends to a stable value. This is because assuming the equivalent frictional strength is fully mobilized, the inclination angle of the failure zone boundary determines the size of the Mohr stress circle of the point at the failure zone boundary. When the inclination angle of the failure zone boundary decreases, the mean stress and the radius of the Mohr stress circle of the computing nodes at the failure zone boundary increases sharply, resulting in a sharp increase in the calculated vertical force acting on the side of the rigid cone beneath the pile tip. Figure 9: The zero-extension line (ZEL) net for an illustrative example: \(\phi=30^{\circ},v=0.1\phi,\alpha_{SA1}=40^{\circ}\). There should be a specific failure zone at the boundary of which the equivalent frictional strength is fully mobilized at the ultimate state. Any inclination angle greater or less than the angle of this specific failure zone will result in significant errors in the predicted ultimate toe capacity. Therefore, the parameter inclination angle of the failure zone boundary should be further determined by fitting the measured ultimate toe capacity of engineering practice pile cases. However, Figure 10 shows that a larger inclination angle than the actual value is more acceptable because a smaller one will cause a considerable upward deviation of the computed toe capacity from the actual ultimate toe capacity. In the next section, an equation is proposed for calculating the inclination angle of the failure zone boundary and determining the range of the failure zone, in order to achieve good prediction of an engineering practice case. For simplicity, the dilation angle is taken as a constant in the proposed method, although the dilation angle varies in actual situations, such as soil compression processes and shear processes. Figure 11 presents the variation of the computed toe capacity when the dilation angle \(v\) is taken as 0.1\(\phi\), 0.2\(\phi\), 0.3\(\phi\), 0.4\(\phi\), 0.5\(\phi\), 0.6\(\phi\), and 0.7\(\phi\), respectively. Figure 11 shows that the dilation angle has an impact on the computed toe capacity. As the dilation angle increases, the computed toe capacity increases to a certain extent. This is because in the proposed method, the dilation angle affects the size of the failure zone. As the dilation angle increases, the range of soil with strength fully mobilized increases, leading to a certain degree of increase in the computed toe capacity. Figure 11: The influence of dilation angle on the computed toe capacity: \(\phi=30^{\circ}\), \(\alpha_{SA1}=37.5^{\circ}\). Figure 10: The influence of the inclination angle of the failure zone boundary on the computed toe capacity: \(\phi=30^{\circ}\), \(v=0.1\phi\). Figure 12 presents the influence of friction angle on the computed toe capacity. It is shown in Figure 12 that as the friction angle increases, the computed toe capacity increases, and the rate of increase becomes faster. In the case of the parameter \(\alpha_{SA1}=37.5^{\circ}\), \(v=0.1\phi\), when the friction angle increases from \(18^{\circ}\) to \(20^{\circ}\), the computed toe capacity increases by about 1.3 times (from 517 kN to 653 kN), and when the friction angle increases from \(34^{\circ}\) to \(36^{\circ}\), the computed toe capacity increases by about 2.6 times (from 7016 kN to 18,085 kN).