# Dynamic Lateral Response of the Partially-Embedded Single Piles in Layered Soil
## 2 Solving Equations of Motion  
### Dynamic Model  
Figure 1 shows the analysis diagram of the partially-embedded single pile in the layered soil with the pile top subjected to the lateral dynamic force \(P_{0}(t)\) and the bending moment \(M_{0}(t)\). The physical parameters of the pile are: length \(L\), radius \(R\), elastic modulus \(E_{P}\), the moment of inertia \(I_{P}\), which \(l_{1}\) is the length of the non-embedded section pile. As shown in Figure 1a, taking the three-layer soil field as an example, the analysis area is divided into five layers according to the embedded depth: the first layer is the non-embedded section; the second, third, and fourth layers are the embedded section; the 5 th layer is the soil section below the pile end; the distance from the bottom of each layer to the pile top is \(H_{i}\) (\(i=1,2,3,4,5\)), and \(H_{0}=0\). Lame constants of the elastic foundation are \(\lambda_{s}=\frac{E_{p}l_{i}}{(1+\kappa)(1-2\kappa)}\) and \(G_{s}=\frac{E_{p}}{2(1+\kappa)}\). Taking the undeformed pile top as the origin \(O\), the cylindrical coordinate system \(O-r\theta z\) is established. In Figure 1a, \(u_{r}\), \(u_{\theta}\), \(w_{z}\) are the radial, tangential, and vertical displacement components of the pile body displacement along with the cylindrical coordinate system with the central axis of the pile as the \(z\)-axis. \(\theta\) is the angle between any point and the \(r\) axis. The adopted Vlasov model simplifies the basic equation of the isotropic linear elastic continuum by introducing the constraints of displacement and stress and reduces the complexity of the half-space model. For simplicity, modeling and analysis are based on the following assumptions: the pile is slender, vertical and linear elastic; the soil field is a uniform, isotropic and of linear elastic material; there is no slippage or detachment at the pile-soil interface; the displacement of the pile body outside the direction of external excitation is ignored. The displacement components of the soil field around the partially-embedded single pile can ignore the vertical displacement. They can be written as:  
\[u_{r}(r,\theta,z,t)=u(z,t)\phi(r)\cos\theta \tag{1}\]  
\[u_{\theta}(r,\theta,z,t)=-u(z,t)\phi(r)\sin\theta \tag{2}\]  
\[w_{z}(r,\theta,z,t)=0 \tag{3}\]  
where \(u(z,t)\) is the lateral displacement of the pile, \(w_{z}\) is the vertical displacement components of the pile body displacement, and \(\phi(r)\) is the dimensionless function representing the variation of the soil displacement along the \(r\) direction. Based on the assumption of displacements given in Equations (1)-(3) and the stress-strain relations, the potential energy of the pile and soil system is given by:  
Figure 1: Schematic diagram of the partially-embedded single pile in layered soil: (**a**) Laterally-loaded single pile; (**b**) Analytical model of pile-soil system. \[U=\frac{1}{2}\int_{R}^{\infty}\int_{0}^{\infty}\left[\pi(\lambda_{s}+3G_{s})u^{2}( \frac{\mathrm{d}\phi}{\mathrm{d}r})^{2}+2\pi G_{s}(\frac{\mathrm{d}u}{\mathrm{d }z})^{2}\phi^{2}\right]r\ \mathrm{d}r\ \mathrm{d}z+\frac{1}{2}\int_{0}^{L}E_{P}I_{P}(\frac{ \mathrm{d}^{2}u}{\mathrm{d}z})^{2}\mathrm{d}z+\frac{1}{2}\int_{L}^{\infty}\pi R ^{2}G_{s}(\frac{\mathrm{d}u}{\mathrm{d}z})^{2}\mathrm{d}z \tag{4}\]  
Correspondingly, the kinetic energy of the pile and soil system can be written as:  
\[T=\frac{1}{2}\int_{0}^{L}m_{P}(\frac{\mathrm{d}u}{\mathrm{d}t})^{2}\mathrm{d}z +\frac{1}{2}\int_{L}^{\infty}m_{s}(\frac{\mathrm{d}u}{\mathrm{d}t})^{2}\mathrm{ d}z+\pi\int_{0}^{\infty}\int_{R}^{\infty}\rho_{s}(\frac{\mathrm{d}u}{\mathrm{d}t})^{2} \theta^{2}r\ \mathrm{d}r\ \mathrm{d}z \tag{5}\]  
where \(m_{s}\) and \(m_{P}\) are the mass of the soil under the pile and the mass per linear meter of the pile, respectively, \(\rho_{s}\) is the density of the soil around the pile.