# Dynamic Lateral Response of the Partially-Embedded Single Piles in Layered Soil
## 4 Numerical Calculations
### Validity of the Present Solutions  
In order to verify the validity of the finite difference method, the calculation model of [PERSON] et al. and the model and calculation method of this paper were used to solve the variation of the first-order natural frequencies of the steel pipe pile with a length of 8.76 m under scouring conditions. The results are compared with the results measured on-site [16]. As shown in Figure 3, when the pile (free-free pile or fixed-free pile) is analyzed under the scour conditions, the scour effect is considered by completely removing the entire soil layer to a certain depth. After scouring, part of the pile body is exposed to the soil surface, and the pile is treated as partially-embedded. In this paper, the pile is divided into 40 pile units. There is no pile-soil interaction above the scour depth so they can be regarded as beam elements. Moreover, they can be regarded as spring-beam elements because of the pile-soil interaction below the scour depth. In the process of numerical calculation, the initial pile's length of the non-embedded section is 2.19 m (scour level is 0), which can be represented as 10 beam elements and 30 spring-beam elements. The scouring process is simulated by removing spring supports from top to bottom. Figure 2: Flow chart of calculation. The physical parameters of the pile and the foundation are shown in Table 1. The elastic modulus of the foundation is [18]:  
\[E_{s}=\frac{\rho_{s}v_{C}^{2}(1+v_{s})(1-2v_{s})}{1-v_{s}} \tag{96}\]  
where \(\rho_{s}\) and \(v_{s}\) are the soil's density and Poisson's ratio, respectively, and \(v_{C}\) is the velocity of the compression wave (213 m/s for sand [18]). The subgrade reaction coefficient of the model of [PERSON] et al. is [24]:  
\[k_{0}=\frac{1.0E_{s}}{1-v_{s}2}[\frac{E_{s}D_{p}^{4}}{E_{p}I_{p}}] \tag{97}\]  
where \(D_{p}\) is the outer diameter of the pile, and \(E_{p}I_{p}\) is the flexural rigidity of the pile. Figure 4 shows the comparison of the first-order natural frequencies of the partially-embedded single pile under scouring obtained by the model of [PERSON] et al., the method in this paper, and the actual measured dates on-site. Referring to Equation (97), the subgrade reaction force in the calculation model of [PERSON] et al. has nothing to do \begin{table}
\begin{tabular}{c c} \hline \hline
**Physical Meaning, Symbols and Units** & **Numerical Value** \\ \hline Length of the pile \(L\) [m] & 8.76 \\ Outer diameter of the pile \(R\) [m] & 0.17 \\ Inter diameter of the pile \(r\) [m] & 0.157 \\ Density of the pile \(\rho\) [kg/m\({}^{3}\)] & \(7.8\times 10^{3}\) \\ Elastic modulus of the pile \(E_{p}\) [MPa] & \(2.0\times 10^{5}\) \\ Poisson’s ratio of the pile \(\
u\) & 0.3 \\ Poisson’s ratio of elastic foundation \(v_{s}\) & 0.3 \\ Density of elastic foundation \(\rho_{s}\) [kg/m\({}^{3}\)] & \(2\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the single pile and foundation during the verification. Figure 3: Numerical model of the finite difference method and setting of scour levels. with the pile diameter. The first-order natural frequencies of the single pile obtained by [PERSON] et al. 's calculation model have the same trend as the actual measured dates on-site. The first-order natural frequencies of the single pile calculated by the method in this paper are closer to the measured dates on-site, and they are highly consistent with the measured dates on-site within the range of common scour depth (1-3 m). The results of the calculation model in this paper can more effectively predict the dynamic characteristics of partially-embedded single piles under the scouring action. The displacement curve of the single pile with pile diameter \(d=0.3\) m, slenderness ratio \(\lambda=10\), dimensionless frequency \(a_{0}=0.3\) is calculated by using the proposed model and the model presented by the literature [25], which assumes that the pile has a fixed end and a pile top with a constrained angle.