## 2 Computation Model  
The focus of this paper is on laterally loaded long piles in layered foundations. As shown in Figure 1, \(P_{0}\) and \(M_{0}\) represent the transverse load and bending moment, \(L\) is the pile length, \(R\) is the pile radius, and \(E_{p}\) and \(I_{p}\) are the elastic modulus and moment of inertia of pile body, respectively. The pile body is embedded in the n-layer elastic foundation, where \(H_{i}\) denotes the depth of the bottom surface of the \(i\)-th layer foundation, and \(\lambda_{\text{si}}\) and \(G_{\text{si}}\) are the Lame constants of the \(i\)-th layer foundation. The cylindrical coordinate system is established with the central axis of the pile as the z-axis. Any point in the coordinate system can be represented by the radial coordinate \(r\), the circumferential coordinate \(\theta\), and the vertical coordinate \(z\). Figure 1: Laterally loaded long pile in layered foundation. As shown in Figure 2, \(u_{\theta}(z,r)\) and \(u_{r}(z,r)\) represent the lateral displacement of the soil field around the pile, and \(u_{z}(z,r)\) represents the axial displacement. To facilitate the analysis and highlight the research focus of this paper, the following assumptions are made: (1) Both the foundation and the pile are considered as linearly elastic isotropic bodies. (2) The pile-soil contact surface friction is ignored. (3) If the pile-soil contact surface of a stratum is detached, the pile-soil interaction of the stratum is completely ignored. (4) The deformation of the pile is assumed to be in an elastic state, and there is a nonlinear effect of the pile-soil interaction. The axial displacement of the pile foundation is ignored, and only the lateral displacement \(u_{\theta}(z,r)\) and \(u_{r}(z,r)\) of the elastic foundation is considered as follows [43; 44]:  
\[\left. \begin{array}{l}u_{r}(z,r)=w(z)\phi(r)\cos\theta\\ u_{\theta}(z,r)=-w(z)\phi(r)\sin\theta\end{array} \right\} \tag{1}\]  
where \(w(z)\) is the lateral displacement of the pile foundation; \(\phi(r)\) is the attenuation function of foundation displacement along the radial direction \(r\). From the geometric relationships in cylindrical coordinates, we can derive the following Equation (2) based on Equation (1):  
\[\left[ \begin{array}{c}\varepsilon_{rr}\\ \varepsilon_{\theta\theta}\\ \varepsilon_{zz}\\ \gamma_{rz}\\ \gamma_{\theta z}\end{array} \right]=\left[ \begin{array}{c}-\frac{\partial u _{r}}{\partial r}\\ -\frac{u_{r}}{r}-\frac{\partial u_{\theta}}{\partial\theta}\\ -\frac{\partial u_{z}}{\partial z}\\ -\frac{1}{r}\frac{\partial u_{r}}{\partial\theta}-\frac{\partial u_{\theta}}{ \partial r}+\frac{u_{\theta}}{r}\\ -\frac{\partial u_{z}}{\partial r}-\frac{\partial u_{r}}{\partial z}\\ -\frac{1}{r}\frac{\partial u_{z}}{\partial\theta}-\frac{\partial u_{\theta}}{ \partial z}\end{array} \right]=\left[ \begin{array}{c}-w(z)\frac{\mathrm{d} \phi(r)}{\mathrm{d}r}\cos\theta\\ 0\\ 0\\ w(z)\frac{\mathrm{d}\phi(r)}{\mathrm{d}r}\sin\theta\\ -\frac{\mathrm{d}w(z)}{\mathrm{d}z}\phi(r)\cos\theta\\ \frac{\mathrm{d}w(z)}{\mathrm{d}z}\phi(r)\sin\theta\end{array} \right] \tag{2}\]  
Consequently, the elastic foundation physical equation in cylindrical coordinates is as follows:  
\[\left[ \begin{array}{c}\sigma_{rr}\\ \sigma_{\theta\theta}\\ \sigma_{zz}\\ \tau_{r\theta}\\ \tau_{\theta z}\\ \tau_{\theta z}\\ \tau_{\theta z}\end{array} \right]=\left[ \begin{array}{cccccc}\lambda_{s}+2G_ {s}&\lambda_{s}&\lambda_{s}&0&0&0\\ \lambda_{s}&\lambda_{s}+2G_{s}&\lambda_{s}&0&0&0\\ \lambda_{s}&\lambda_{s}&\lambda_{s}+2G_{s}&0&0&0\\ 0&0&0&G_{s}&0&0\\ 0&0&0&0&G_{s}&0\\ 0&0&0&0&G_{s}\\ \end{array} \right]\left[ \begin{array}{c}\varepsilon_{rr}\\ \epsilon_{\theta\theta}\\ \tau_{r\theta}\\ \gamma_{r\theta}\\ \gamma_{rz}\\ \gamma_{\theta z}\end{array} \right] \tag{3}\]  
According to the principle of minimum potential energy [44], the total potential energy of the pile-soil system is as follows:  
\[