## Validation  
### Validation 1  
In this section, a numerical model is built in FLAC-3D. The numerical model aims to better understand the performance of the pile-supported embankment with a conical head and verify the theoretical solution. The foundation soil consists of three layers. The first layer is an artificial working platform with a thickness of 2 m, which is the pre-treatment layer for carrying the machinery load during construction. This is equivalent to the uppermost clayey crust with relatively high plasticity. The second layer is a soft soil layer with a thickness of 10 m. The third layer is a hard soil layer with a thickness of 6 m. The pile penetrates into the soft soil layer until the toe reaches the hard soil layer. As presented in Figure 6a, the numerical analysis (FLAC-3D) is conducted under an axisymmetric model, which adopts the unit-cell concept applied in the theoretical approach. The axisymmetric boundary conditions are adopted. The vertical boundary only allows for vertical displacement and the bottom boundary is fixed in any direction. Figures 6b and 6c are the profile of the pile with conical head and the equal section pile, respectively. The pile is simulated as a linear elastic material. Meanwhile, the soft soil and embankment fills are simulated as linearly elastic-perfectly plastic material with the Mohr-Coulomb failure criterion. The interface parameters between the pile and soil include normal stiffness \(k_{n}\), shear stiffness \(k_{s}\), cohesion \(c\), and internal friction angle \(\varphi_{s}\). \(k_{n}\) and \(k_{s}\) can be taken as ten times the stiffness of the hardest soil layer in the adjacent contact surface [35]. The value of \(c\) and \(\varphi_{s}\) are respectively equal to 0.8 times the cohesion and internal friction angle of the hardest soil layer [35], as shown in Equation (39). The material parameters and geometric parameters used in the model are listed in Table 1.  
\[k_{n}=k_{s}=10\text{max}[\frac{K+\frac{4}{3}G}{\Delta z_{\text{min}}}] \tag{39}\]where \(\Delta z_{\rm min}\) is the least grid size in this model, and \(K\) and \(G\) represent the bulk modulus and shear modulus of the soil respectively. Figure 7 shows the variation of vertical stress within the inner cylinder and the outer hollow cylinder versus the height of the embankment. The curves of the proposed solution are much closer to the numerical simulation. The vertical stresses between the inner cylinder and the outer hollow cylinder are equal when \(z/h\) is less than 0.625. It means that there is a plane of equal settlement at \(z/h=0.625\). Below the plane of equal settlement, the vertical stress gradually deviates from the gravity while the vertical stress of the outer hollow cylinder gradually decreases and that of the inner cylinder increases. The height of the equal settlement plane is 1.41 \(s_{n}\). The result agrees with that obtained from the experiment of [PERSON] et al. 2007 [PERSON] et al. (2007). Thus, the proposed solution is feasible for predicting the soil arching effect for a pile-supported embankment with a conical head. Figure 8 indicates the effect of \(D\) on the pile-soil stress ratio (\(n\)) and the load sharing ratio (\(n_{e}\)). The pile-soil stress ratio decreases (Figure 8a) while the load sharing ratio is increased (Figure 8b) with the increase in \(D\). The comparison indicates that the predicted result is highly close to the numerical result, where the relative error of the load sharing ratio floats within the range of 1.6%-9.6%. The relative error of the two curves increases with the value of \(D\). It is because the pile-soil stress ratio is the ratio of the average vertical stress of the pile to that of the soft soil at the plane of the pile top. The larger \(D\) is, the greater the deviation between the calculated value of the average stress and the actual one will be.