## 3 Matrix of the Improved Penzien Model
### Mass and Stiffness Matrices of Pile Group
#### 3.1.2 Stiffness Matrix of Pile Group  
Starting from the basic case of two symmetrical piles in a pile group, the equivalent element stiffness matrix in this case is obtained by superimposing the lateral displacement and the angle of rotation at both ends of the element. This is then extended to the case of a pile group with n symmetrically arranged piles [70]. The relationship between the force and displacement of the pile group in section \(i\) can be obtained as follows. \[\left\{ \begin{array}{l}V_{i}\\ M_{i}\\ V_{j}\\ M_{j}\end{array} \right\}=\left[ \begin{array}{cccc}12a_{i}&6a_{i}l_{i}&-12a_{ i}&6a_{i}l_{i}\\ 6a_{i}l_{i}&b_{i}&-6a_{i}l_{i}&c_{i}\\ -12a_{i}&-6a_{i}l_{i}&12a_{i}&-6a_{i}l_{i}\\ 6a_{i}l_{i}&c_{i}&-6a_{i}l_{i}&b_{i}\end{array} \right]\left\{ \begin{array}{l }u_{i}\\ \theta_{i}\\ u_{j}\\ \theta_{j}\end{array} \right\} \tag{28}\]  
\[a_{i}=\frac{nE_{p}I_{p}}{(1+\phi_{i})l_{i}^{3}} \tag{29}\]  
\[b_{i}=\frac{(4+\phi_{i})nE_{p}I_{p}}{(1+\phi_{i})l_{i}}+\frac{E_{p}I_{p}}{l_{i}} \tag{30}\]  
\[c_{i}=\frac{(2-\phi_{i})nE_{p}I_{p}}{(1+\phi_{i})l_{i}}-\frac{E_{p}I_{p}}{l_{i}} \tag{31}\]  
\[J_{p}=\sum_{k=1}^{n}A_{p}r_{k}^{2} \tag{32}\]  
\[\phi_{i}=\frac{12E_{p}I_{p}\eta}{G_{p}A_{p}l_{i}^{2}} \tag{33}\]  
where \(V_{i}\) and \(V_{j}\) are the shear forces at both ends of the equivalent pile element, respectively. \(M_{i}\) and \(M_{j}\) denote the bending moments at both ends of the equivalent pile element, respectively. \(u_{i}\) and \(u_{j}\) represent the displacements at both ends of the equivalent pile element, respectively, and \(\theta_{i}\) and \(\theta_{j}\) are the angles of rotation at both ends of the equivalent pile element, respectively. \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are the replacement parameters in order to simplify the matrix form. \(l_{i}\) is the length of the equivalent pile element in section \(i\). \(n\), \(E_{p}\), \(I_{p}\), \(A_{p}\), \(r_{k}\), and \(G_{p}\) denote the total number of piles, the elastic modulus of the equivalent pile element, the inertia moment of a single pile, the area of a single pile, the distance from the section core of the pile \(k\) to the neutral axis of the pile cap, and the shear modulus of the equivalent pile element, respectively. \(\eta\) is the coefficient of uneven distribution of shear stress at the pile section. Ignoring the shear strain, \(\eta\) is equal to 0. \(\eta\)_is_ 10/9 and 1.2, when the pile section is circular and rectangular, respectively. Assuming that the pile cap is rigid, the equivalent single pile integrated by the pile group is divided into N elements according to the soil layer. The equivalent element stiffness matrix of the equivalent single pile is formed for each element, and the total stiffness matrix of the equivalent single pile is then formed after integrating the element  
\[K_{p22}= \begin{bmatrix}b_{1}&c_{1}\\ c_{1}&b_{1}+b_{2}&c_{2}\\...&...&...\\ &c_{i-1}&b_{i-1}+b_{i}&c_{i}\\ &&c_{i}&b_{i}+b_{i+1}&c_{i+1}\\ &&&&...&...&...\\ &&&&c_{N-1}&b_{N-1}+b_{N}&c_{N}\\ &&&&c_{N}&b_{N}\end{bmatrix}  \tag{42}\]  
Considering only the horizontal vibration under horizontal earthquake excitation, the whole stiffness matrix needs to be condensed. The specific steps are as follows. The equation of motion of the equivalent single pile is shown as follows:  
\[ \begin{bmatrix}M_{p11}&M_{p12}\\ M_{p21}&M_{p22}\end{bmatrix}\begin{pmatrix}\ddot{u}\\ \ddot{\theta}\end{pmatrix} +\begin{bmatrix}C_{p11}&C_{p12}\\ C_{p21}&C_{p22}\end{pmatrix} \begin{pmatrix}\dot{u}\\ \dot{\theta}\end{pmatrix} +\begin{bmatrix}K_{p11}&K_{p12}\\ K_{p21}&K_{p22}\end{pmatrix} \begin{pmatrix}u\\ \theta\end{pmatrix} =- \begin{bmatrix}M_{p11}&M_{p12}\\ M_{p21}&M_{p22}\end{bmatrix}\begin{pmatrix}\ddot{u}_{g}\\ 0\end{pmatrix}