## 4 Substructure Pseudo-Dynamic Test of Soil-Pile-Structure Interaction
### Numerical Substructure
#### 4.2.2 Details of the Numerical Substructure Model  
The numerical substructure, considering the effect of SPSI, included the superstructure with five floors, the pile group, and the soil. The section sizes and material parameters of the superstructure were the same as the experimental substructure. The height of each floor was 2 m. The pile group was a 2 by 2 circular concrete pile with a diameter of 40 cm and a length of 16 m. The material was C30 concrete with an elastic modulus of 2.6 \(\times\) 104 MPa and a specific mass of 2500 kg/m\({}^{3}\). The numerical substructure of the test without considering the influence of SPSI only included the superstructure with five floors, and the parameters of the superstructure were the same as the numerical substructure of the test considering the effect of SPSI. The improved Penzien model was used to calculate the numerical substructure, and the calculation program was compiled using MATLAB. The integral method of the substructure pseudo-dynamic test was the PC-Newmark method with explicit and implicit combinations. The motion equation form of the PC-Newmark method is as follows:  
\[Ma_{i+1}+\overline{C}^{l}v_{i+1}+\overline{C}^{E}v_{i+1}+\overline{K}^{l}d_{i +1}+\overline{r}_{i+1}=f_{i+1} \tag{67}\]  
where \(\overline{C}^{l}\) and \(\overline{C}^{E}\) denote the damping matrixes of the numerical substructure and the experimental substructure, respectively. The damping matrix of the SPSI system is \(C=\overline{C}^{l}+\overline{C}^{E}\), and \(\overline{K}^{l}\) is the stiffness matrix of the numerical substructure. \(\overline{r}_{i+1}\) represents the restoring force measured from the experimental substructure after applying the predicted displacement. The displacement and velocity calculation formulas of the PC-Newmark method are shown in Equations (68)-(71). \[\overline{d}_{i+1}=d_{i}+\Delta tv_{i}+\Delta t^{2}\bigg{(}\frac{1}{2}-\beta \bigg{)}a_{i} \tag{68}\]  
\[\overline{v}_{i+1}=v_{i}+\Delta t(1-\gamma)a_{i} \tag{69}\]  
\[d_{i+1}=\overline{d}_{i+1}+\Delta t^{2}\beta a_{i+1} \tag{70}\]  
\[v_{i+1}=\overline{v}_{i+1}+\Delta t\gamma a_{i+1} \tag{71}\]  
where \(\overline{d}_{i+1}\) and \(\overline{v}_{i+1}\) are the predicted displacement and the predicted velocity, respectively. \(d_{i+1}\) and \(v_{i+1}\) denote the calibrated displacement and the calibrated velocity, respectively. Figure 8: Fitted \(G/G_{max}\)\(\sim\)\(\gamma_{d}\) and \(D\)\(\sim\)\(\gamma_{d}\) curves for silty. Combined with the improved Penzien model, Equation (67) is modified to Equation (72):  
\[Ma_{i+1}+\overline{C}^{l}v_{i+1}+\overline{C}^{\overline{c}}v_{i+1}+\overline{ K}^{l}d_{i+1}+\overline{r}_{i+1}=-Ma_{g,i+1}+C_{h}v_{i+1}^{G}+K_{h}d_{i+1}^{G} \tag{72}\]  
The substructure pseudo-dynamic test steps are as follows: first, substitute the result of the previous step into Equations (68) and (69), and calculate the prediction displacement and the prediction velocity of step \(i+1\). The corresponding displacement of the experimental substructure in the prediction displacement is then loaded on the test specimen, and the restoring force on the specimen is measured at the same time. Third, the acceleration, the calibrated displacement, and the calibrated velocity in step \(i+1\) are calculated according to Equations (72), (70), and (71), respectively. Calculate the predicted value in the next step and repeat the above operation until the end of the experiment. Because the parameters of the improved Penzien model and the derivation of the dynamic coefficient matrix are described in detail in Section 3, this section only describes the boundary treatment of the unmentioned numerical substructure stiffness matrix and the restoring force vector. According to the number of each element shown in Figure 9, assemble the element stiffness matrix into the overall stiffness matrix. When element 6 is used as the test substructure, the corresponding stiffness matrix of the numerical substructure is \(\overline{K}^{l}\), which is shown in Equation (73).