# Study on the Time Domain Semi Analytical Method for Horizontal Vibration of Pile in Saturated Clay
## 2 Mechanical Model
### Horizontal Coupled Vibration Model of Pile Soil  
As shown in Figure 2, end-bearing piles are buried in \(a\) saturated clay foundation using a cylindrical coordinate system (\(r\)-\(\theta\)-\(z\)) to build a pile-soil coupling system model. Figure 1: Schematic diagram of the fractional Kelvin model. The bottom of the pile is a fixed end, with a pile length of H and a pile diameter of \(r_{0}\). The pile top is subjected to a sinusoidal horizontal vibration load \(P(t)\), which is defined as:  
\[P(t)=P_{0}\sin(\omega t) \tag{6}\]  
In the formula, \(P_{0}\) is the load amplitude, \(\omega\) is the load frequency. Introducing the Laplace transform of time t into Equation (6), the expression of cyclic load in the Laplace transform domain is obtained as:  
\[\tilde{P}(s)=P_{0}\frac{\omega}{s^{2}+\omega^{2}} \tag{7}\]  
where \(\widetilde{P}(s)\) is the Laplace transformation of \(P(t)\), and \(s\) is the transformation parameter. For the problem of horizontal vibration, ignoring the vertical displacement of the soil, the motion equation of saturated clay in the cylindrical coordinate system is:  
\[\
abla\cdot\sigma_{a}=\rho\ddot{\mathbf{u}}+\rho_{f}\ddot{\mathbf{w}} \tag{8}\]  
\[-\
abla p=\rho_{f}\ddot{\mathbf{u}}+\frac{\rho_{f}}{n}\ddot{\mathbf{w}}+\frac{\gamma_{ w}}{k_{f}}\dot{\mathbf{w}} \tag{9}\]  
where \(\mathbf{u}\) is the displacement tensor of the soil skeleton, \(\mathbf{w}\) is the relative displacement between the pore water and the soil skeleton, \(\rho\) denotes the bulk density of the porous material, and \(\rho=(1-n)\rho_{s}+n\rho_{f}\). \(n\) is the porosity, and \(\rho_{s}\) and \(\rho_{f}\) are the density of the soil skeleton and pore water, respectively. In addition, \(k_{f}\) is the permeability coefficient of the soil. \(\gamma_{w}=\rho_{f}g\) is the gravity density of the pore water, where g denotes the gravitational acceleration, \(\bigtriangledown\) represents the Hamiltonian operator, and \(\
abla=\frac{\partial}{\partial\tau}+\frac{1}{r}\frac{\partial}{\partial\tau} +\frac{\partial}{\partial\varepsilon}\) in the cylindrical coordinate system. \(\sigma_{\text{a}}\) is the total stress tensor on the porous material unit, and \(p\) is the pore water pressure. Taking compressive stress as the positive direction, the principle of effective stress is expressed as  
\[\sigma_{a}=\sigma^{\prime}-\alpha_{s}p\mathbf{I} \tag{10}\]  
where \(\sigma^{\prime}\) is the effective stress tensor of the soil skeleton, and \(\alpha_{s}\) is the Biot-Willis coefficient. The seepage continuity equation is expressed as  
\[-\dot{p}=\alpha_{s}M\
abla\cdot\dot{\mathbf{u}}+M\
abla\cdot\dot{\mathbf{w}} \tag{11}\]  
where \(M\) is a Biot coefficient to describe the compressibility of the pore water; the superscript \".\" represents the first-order differential of time. Figure 2: Horizontal coupled vibration model of pile–soil system. The fractional order Kelvin model is introduced into the soil skeleton; that is, the effective stress \(\sigma^{\prime}\) satisfies Equation (4). Substituting Equations (4) and (10) into Equation (8) and then performing Laplace transform, the following equation can be derived:  
\[\mu\
abla^{2}\hat{\mathbf{u}}+(\lambda+\mu)\
abla\hat{\theta}-\Big{(}\alpha_{s} \rho_{f}+\rho\Big{)}s^{2}\hat{\mathbf{u}}-\Big{(}\alpha_{s}K_{d}+\rho_{f}s^{2} \Big{)}\hat{\mathbf{w}}=0 \tag{12}\]  
where \(K_{d}=\rho_{f}s^{2}/n+\gamma_{w}s/k_{f}\). Due to the study of horizontal vibrations of pile foundations in this article, the pile top pressure and constraint stiffness can be ignored.