# Evaluation on Improvement Zone of Foundation after Dynamic Compaction
## 3 Parametric Studies
### Effect of Drop Number
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Formula Definition** & **Physical Meaning** \\ \hline \(S_{\rm i}^{\prime}\) (\(\rm i=1,2\)) & \(S_{\rm i}^{\prime}=\frac{\int_{\rm r}^{\rm h}f(I_{\rm r},\,\rm depth)\,\, \rm d(depth)}{(\rm b-a)}\) (\(\rm i=1,2\)) where & S\({}_{\rm i}^{\prime}\) and S\({}_{\rm 2}^{\prime}\) are the mean value theorem of integral \\ \(S_{\rm i}^{\prime}\) (\(\rm i=1,2\)) & \(a=0\) m means the ground surface; \(b=18\) m means & computation results between \(I_{\rm r}\) and depth after tamping \\  & the depth at 18 m from the ground surface. & of the 1 st and 2 nd impact points. \\ \(Z_{\rm i}^{\prime}\) (\(\rm i=1,2\)) & - & \(Z_{\rm i}^{\prime}\) and \(Z_{\rm i}^{\prime}\) are the improvement depth corresponding to \\ \(A_{\rm i}^{\prime}\) (\(\rm i=1,2\)) & - & \(A_{\rm i}^{\prime}\) and \(A_{\rm i}^{\prime}\) are the improvement area corresponding to \\ \(S_{\rm R}^{\prime},Z^{\prime},R\) and \(A_{\rm R}^{\prime}\) & - & impact points. \\ \(\delta_{\rm U,i}\) (\(\rm i=1,2\)) & \(\delta_{\rm U,i}=\frac{S_{\rm i}^{\prime}}{\rm N}(\rm i=1,2)\) & \(R=1.2\) m after the completion of the 1 st impact points. \\ \(\delta_{\rm z,i}\) (\(\rm i=1,2\)) & \(\delta_{\rm z,i}=\frac{Z_{\rm i}^{\prime}}{\rm Z_{\rm R}^{\prime}}(\rm i=1,2)\) & \(\delta_{\rm z,1}\) and \(\delta_{\rm z,2}\) are the normalized improvement depth \\ \(\delta_{\rm A,i}\) (\(\rm i=1,2\)) & \(\delta_{\rm A,i}=\frac{A_{\rm i}}{\rm N_{\rm R}}(\rm i=1,2)\) & \(\delta_{\rm A,1}\) and \(\delta_{\rm A,2}\) are the normalized improvement area after \\  & & damping of the 1 st and 2 nd impact points. \\ \(\Delta\delta_{\rm U,i}\) (\(\rm i=1,2\)) & \(\Delta\delta_{\rm U,1}=\delta_{\rm U,1},\Delta\delta_{\rm U,2}=\delta_{\rm U,2}- \delta_{\rm U,1}\) & \(\Delta\delta_{\rm U,1}\) and \(\Delta\delta_{\rm U,2}\) are used to investigate the soil \\  & & improvement below point 3 between impact points) \\  & & during impact at points 1 and 2, respectively. \\ \(\Delta\delta_{\rm z,i}\) (\(\rm i=1,2\)) & \(\Delta\delta_{\rm z,1}=\delta_{\rm z,1},\Delta\delta_{\rm z,2}=\delta_{\rm z,2}- \delta_{\rm z,1}\) & \(\Delta\delta_{\rm z,1}\) and \(\Delta\delta_{\rm z,2}\) are used to investigate improvement \\  & & depth during impact at points 1 and 2, respectively. \\ \(\Delta\delta_{\rm A,i}\) (\(\rm i=1,2\)) & \(\Delta\delta_{\rm A,1}=\delta_{\rm A,1},\Delta\delta_{\rm A,2}=\delta_{\rm A,2}- \delta_{\rm A,1}\) & \(\Delta\delta_{\rm A,1}\) and \(\Delta\delta_{\rm A,2}\) are used to investigate the \\  & & improvement area corresponding to the zone of \\  & & \(I_{\rm r}=7.7\%\) during impact at points 1 and 2, respectively. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The meaning of normalized indicators. consistent with the results of [PERSON] et al. (2019), [PERSON] et al. (2019), and [PERSON] (2020). Moreover, the value of \(I_{\rm r}\) along the depth decays rapidly for single-point tamping, while for multipoint tamping, the value of \(I_{\rm r}\) increases along with the depth within around 3 m where the maximum value of \(I_{\rm r}\) occurs. Beyond the depth of 3 m, \(I_{\rm r}\) decreases with depth. This may be due to the effect of the stress wave (surface wave) at the bottom of the tamper diffusing along a certain angle and lateral confinement effect, resulting in a weak zone in the upper soil layer ([PERSON] et al., 2019), where it requires a further impact at the intermediate point between the impact points of the first pass in DC. Figure 9 shows that both \(\triangle\delta_{\rm z,1}\) and \(\triangle\delta_{\rm U,1}\) generally increase with the drop numbers, indicating that the soil improvement between impact points can be improved as DC continues. However, the increasing trend gradually slows down after the eighth blow, which is consistent with the results of Figures 7 and 8. Moreover, for single-point tamping, the first six blows can account for 80% of the improvement effect.