## 3 Finite Element Method (FEM)  
### Finite Element Model and Verification  
Because of the limitations of test conditions, cost, and time, the mechanical property information of the soil mass obtained in the SP\(-\)PSC pile group tests was not enough. Therefore, in order to obtain more detailed results of soil mechanical properties under different conditions, a series of numerical studies on the SP\(-\)PSC pile group were carried out using ABAQUS software. The finite element analysis models are shown in Figure 5. Because the stiffness of the reamed steel pipes pile was far greater than that of the soil, the reamed steel pipe piles were assumed to be a rigid body in the numerical models. The Mohr-Coulomb model was adopted for the soil mass, and the material parameters were obtained through the soil tests (see Section 2.1). The surface-to-surface contact model was selected for the contact between the reamed steel pipe piles and the soil mass. In the contact normal direction, the hard contact mode was adopted. In the contact tangential direction, the penalty function method was selected. The friction coefficient was taken as 0.3, which could be calculated by the tangent value of the internal friction angle \(\varphi\) of soils (taken as the average value of different soil samples in Table 2, \(\varphi=17.96\)\({}^{\circ}\)) [20]. The meshes of the numerical models were divided using a hexahedron by a three-dimensional eight-node reduced integral cell (C3 DSR). In the process of numerical analysis, the penetration of the reamed steel pipe piles was controlled by displacement loading. To avoid the boundary effects, the boundary of the semi-infinite soil mass was set at 20\(D\). The distance between the pile hole boundary and the soil mass boundary was greater than 6\(D\). Since the length of the piles was 7 m, the depth of the semi-infinite soil mass was set as 20 m. Thus, the boundary extent and size were enough. As described in the literature [21,22], the horizontal displacement constraint was applied only to the vertical boundaries of the model, while both horizontal and vertical constraints were applied simultaneously to the bottom boundary. Additionally, no constraints were applied to the model's top boundary.  
Figure 5: Numerical models of the SP\(-\)PSC pile group construction process. (**a**) Total model. (**b**) Pile penetration FEM 1 (Pile 1). (**c**) Pile penetration FEM 2 (Pile 2). (**d**) Pile penetration FEM 3 (Pile 3). (**e**) Formed pile holes.  
The compaction efficiency of the soil within the SP\(-\)PSC pile group in the process of penetration of the reamed steel pipe pile was much higher than that in the process of tamping the plain loess layer by layer, which was proved by the results of the SP\(-\)PSC pile group tests. For the above reasons, the results of finite element analysis were obtained from the state of the reamed steel pipe pile penetrating to the design depth. During the subsequent tamping of the plain loess, it was considered that only the collapsibility of the backfilling loess in the hole was eliminated, and the influence on the soil within the SP\(-\)PSC pile group was ignored.  
The measurements of the stress and compaction coefficient of the soil within the SP\(-\)PSC pile group based on the finite element analysis are shown in Tables 3 and 4. It can be observed that the numerical results are in good agreement with the test results.