# Dynamic Lateral Response of the Partially-Embedded Single Piles in Layered Soil
## 5 Parameter Analysis
### Effect of Elastic Modulus of Soil  
This section analyzes the effect of the elastic modulus of the first layer of soil on the dynamic characteristics of the partially-embedded single pile. The first layer of soil around the pile is: mucky soil layer (\(E_{s1}=7\) MPa), loose sand layer (\(E_{s1}=10\) MPa), silty sand layer (\(E_{s1}=15\) MPa) and medium sand layer (\(E_{s1}=20\) MPa), successively. The second layer is the medium sand layer (\(E_{s2}=20\) MPa). The third layer is the tight sand layer (\(E_{s3}=50\) MPa). Table 4 shows the physical parameters of the single pile and layered soil field required for the calculation.  
Table 5 shows the variation of the first-order natural frequencies of the partially-embedded single pile and the response amplitudes of the pile top under varied scour levels with the elastic modulus of the first layer of soil. When the scour level is 10, the first layer of soil is completely cleared due to scour, so the dynamic characteristics of the single pile are the same. It can be seen from Table 5 that under each scour level, the response amplitudes of the pile top decrease significantly with the increase in the parameter \(E_{s1}\), and the first-order natural frequencies of a single pile increase with the increase in the parameter \(E_{s1}\). For example: if the elastic modulus of the soil increases by about 0.5 times, one time, and two times, the first-order natural frequencies of the single pile increase by about 10%, 20%, and 30%, respectively. The subgrade reaction coefficients of the elastic foundation increase with the increase in the elastic modulus of the soil, and accordingly, the lateral constraint of the soil field around the pile is strengthened.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Meaning, Symbols and Units** & **Numerical Value** \\ \hline Elastic modulus of the first layer of soil \(E_{s1}\) [MPa] & 7, 10, 15, 20 \\ Elastic modulus of the second and third layers of soil \(E_{s2}\), \(E_{s3}\) [MPa] & 20, 50 \\ Poisson’s ratio of elastic foundation \(v_{s}\) & 0.3 \\ Thickness of the first, second and third soil layers \(h_{i}\) [m] & \(h_{1}=h_{2}=h_{3}=2.19\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Physical parameters of the single pile and foundation in the parameter analysis.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of Single** & **Elastic Modulus** & **Scour Levels** & \multicolumn{2}{c}{**Dynamic Characteristics of Pile**} \\
**Pile** & **[MPa]** & & **Natural Frequency [Hz]** & **Amplitude of Pile Top [m]** \\ \hline \hline \multirow{8}{*}{Free-fixed pile} & \multirow{8}{*}{10} & 0 & 22.69 & 0.00311 \\  & & 5 & 14.49 & 0.00673 \\  & & 10 & 10.87 & **0.01343** \\  & & 0 & 24.75 & 0.00270 \\  & & 5 & 15.13 & 0.006  
Figure 8 shows the variation of the mode shapes of the partially-embedded single pile under varied scour levels with the change of the elastic modulus of the first layer of soil. Overall, the effect of parameter \(E_{\mathrm{s1}}\) on the mode shapes of the pile body is significant. With the increase in the parameter \(E_{\mathrm{s1}}\), the lateral displacement amplitudes of the pile top of the single pile under each scour level decrease, while the displacement amplitudes at the inverted point of the pile body increase referring to the detail drawn in Figure 8. Corresponding to the results shown in Table 5, with the increase in the elastic modulus of soil around the pile, the lateral constraint of soil around the pile on the pile foundation increases. The pile-soil interaction effect is enhanced and the lateral deformation of the partially-embedded single pile is reduced.