## 2 Improved Penzien Model  
### Model Details  
The Penzien model is composed of the superstructure, pile, and equivalent soil, as shown in Figure 1[51]. The soil is divided into different layers according to its properties, and the masses of the soil layers are concentrated at the interfaces of each soil layer. The model simplifies the superstructure and piles into a series-wound multi-mass system. All piles are combined into one pile, and an equivalent rotational spring is added to the pile head to replace the anti-rotational stiffness of the original pile foundation. The pile foundation is divided into several elements according to the soil layers. The mass of the pile is concentrated at the interface of each horizontal soil layer. The effect of additional soil mass is simulated by the additional mass directly connected to the pile. The interaction between the soil and pile is represented by the horizontal springs and dampers of the soil-pile interaction. The horizontal ground motion displacement of each layer of the free field is input at the other end of the horizontal spring and damper. In the improved Penzien model, the additional soil mass is given a more specific physical meaning, as shown in Figure 2. The soil around the pile in a certain range is considered attached to the pile, and they move together. [PERSON]'s [58] results showed that the response of the structure in taking one time the area of the pile cap as the additional soil area is almost the same as that in taking 10 times the area of the pile cap. Thus, in the improved model, the range of the soil around the pile is taken as one times the area of the pile cap. The soil around the pile is called the equivalent soil, to which shear springs and dampers are added. The free field model is added to change the multi-point input problem of the original Penzien model into a dynamic response problem in the relative coordinate system input only by the bedrock, in which the coordinate origin of the coordinate system is at the bottom of the pile. The relative coordinate system makes the calculation easier. Based on the Penzien model, the following improvements are presented in this paper. The finite element software ANSYS is used to realize the dynamic nonlinearity simulation of the soil, which is applied to the free field mass system [59; 60]. The improved Penzien model is composed of two parts: the structure system, including the superstructure, the pile, and the equivalent soil; and the free-field system. The free field is represented by the soil column per unit area, and the free-field system is simplified into a shear mass system. The site soil is divided into different horizontal soil layers according to their properties, and the soil masses are concentrated at the interface of each soil layer. The superstructure, the pile, and the interactions between the soil and pile are the same as in the previous Penzien model. The improved Penzien model is used as the analytical model for the soil-pile-structure interaction based on the following assumptions:  
1. Only the horizontal translational degrees of the foundation soil and the free field are considered. 2. Vertical and torsion effects of the structure are not considered. 3. The pile foundation under the bearing pile cap maintains synchronous movement and can be combined. 4. The interactions between pile groups are ignored. Figure 1: Penzien model. The motion equation of the improved Penzien model is as follows:  
\[M\ddot{U}+C\dot{U}+KU=-M\ddot{U}_{g}+C_{h}\dot{U}^{G}+K_{h}U^{G} \tag{1}\]  
where \(U\), \(\dot{U}\), and \(\ddot{U}\) denote the displacement, velocity, and acceleration vectors of the structure system, respectively. \(U^{G}\) and \(\dot{U}^{G}\) are the displacement and velocity vectors of the free-field system, and they are obtained by the ANSYS finite element model. \(M\), \(C\), and \(K\) are the mass, damping, and stiffness matrixes of the structure system, respectively. \(K_{h}\) and \(C_{h}\) refer to the stiffness and damping matrixes of soil and pile interaction, respectively.