## 4 ABAQUS Simulation  
### Modeling  
The behavior of the driven PHC nodular pile cannot be analyzed in depth as no stress or strain transducers were equipped along the pile shaft. The finite element software ABAQUS (6.14) was then adopted to build a 3D finite element model of the PHC nodular pile to further analyze and study the load transfer mechanism of the PHC nodular pile and influencing factors of pile capacity [21; 22; 23]. The pile shaft of test piles was not damaged in the loading process, and the PHC nodular pile shaft was defined as linear elastic material. The 1:1 PHC nodular pile modeling was built according to the 450 (500) mm PHC nodular pile. Moreover, the 500 mm PHC pipe pile and 450 mm PHC pipe pile models were also built for comparison. The lengths of PHC nodular pile and PHC pipe pile were all set as 40 m in the modeling. According to Table 1, the driven PHC nodular pile was driven into the soft soil and clay layers, and the modified Cambridge model was adopted for the soft soil and clay [24]. Considering the symmetry of the model in the vertical loading process, the symmetric half model was built for the ABAQUS simulation. The diameter of soil around PHC pile was set to be 40 m in the plane, and the height of the soil layer was 80 m, which were both larger than 20 times the pile diameter. The soil parameters, logarithmic hardening modulus \(\lambda\), isotropic expansion index \(\kappa\) and damage constant \(M\), could be calculated by the following equations:  
\[\lambda =C_{\mathrm{c}}/2.3 \tag{1}\] \[\kappa =C_{\mathrm{s}}/2.3\] (2) \[M =6\sin\varphi/(3-\sin\varphi) \tag{3}\]  
in which \(C_{\mathrm{c}}\) is the soil compression index, and \(C_{\mathrm{c}}\) = 0.011(\(\omega\) \(-\) 7.95) according to the empirical relationship for soft clay; \(\omega\) is the water content; \(C_{\mathrm{s}}\) is the soil resilience index; and \(C_{\mathrm{s}}\) = \(C_{\mathrm{c}}/6.9\). The soil profiles and properties adopted in the ABAQUS modeling are presented in Table 2.  
Three pairs of pile-soil interfaces of the pile shaft and one pair of pile-soil interface of the pile base were established according to the soil profiles around the PHC nodular pile and at the pile base. The contact surface was defined as hard contact in the normal  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Soil Layer**} & \(H\) & \(\gamma\) & \multirow{2}{*}{\(e\)} & \(E_{\mathrm{s}}\) & \(\varphi\) & \multicolumn{3}{c}{**Parameters for Modified Cambridge Model**} \\  & **(m)** & **(kN/m\({}^{3}\))** & & **(MPa)** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **\(\lambda\)** & \(\kappa\)** & \(\
u\)** & \(M\)** \\ \hline Soft soil 1 & 21.3 & 17.6 & 1.12 & 24.6 & 9.6 & 0.154 & 0.022 & 0.36 & 0.353 \\ Soft soil 2 & 9.0 & 17.2 direction, and the Coulomb shear model was adopted in the tangential direction, in which the frictional coefficient could be estimated by the following equations [25]:  
\[\psi=\tan^{-1}(\frac{\sin\varphi\cdot\cos\varphi}{1+\sin^{2}\varphi}) \tag{4}\]  
\[\mu=\tan\psi \tag{5}\]  
in which \(\psi\) is the frictional angle of pile-soil interface.  
To avoid the influence of initial ground stress on the simulation results, the odb import method was adopted to balance the initial ground stress. Moreover, the model grid cell was set as C3D8R type, and the vertical load was applied by the displacement control method. The 3D finite element model of the PHC nodular pile and surrounding soil are presented in Figure 5.