# Evaluation on Improvement Zone of Foundation after Dynamic Compaction
## 4 Proposition and Application of Prediction Formula  
### Degree of Soil Improvement with Depth  
[PERSON] et al. [19] concluded that the better and the least soil improvement zones are below the impact point and at the middle of impact points, respectively. Thus, these two points (points 1 and 3 as shown in Figure 23) are used as the ideal quality control points for DC treatment in this section. For point 1 by single-point tamping, the variation of the value of \(I_{r,1}\) with the depth can be roughly represented by an exponential formula, and the fitting formula is:  
\[I_{r,1}=I_{r,\max 1}\exp\biggl{(}-\frac{h-h_{\max 1}}{h_{\max 1}}\biggr{)} \tag{8}\]  
For point 3 by multipoint tamping, the relationship between the depth and the value of \(I_{r3}\) can be fitted using the following piecewise function, and the formulas are:  
\[I_{r,3}=\left\{ \begin{array}{ll}I_{r,\max 3}\Bigl{[}\left(\frac{h-h_{\max 3}}{h_{\max 3}}\right)+1\Bigr{]},&if\ h\leq h_{\max 3}\\ I_{r,\max 3}\exp\biggl{(}-\frac{h-h_{\max 3}}{h_{\max 3}}\biggr{)},&if\ h>h_{\max 3} \end{array} \right. \tag{9}\]  
Figure 22: The sensitivity analysis for the multipoint tamping: (a) \(\delta_{x,2}\); (b) \(\delta_{\cup 2}\); (c) \(\delta_{\Lambda,2}\). where \(I_{\rm r,max1}\), \(I_{\rm r,max3}\), \(h_{\rm max1}\), and \(h_{\rm max3}\) are the maximum value of \(I_{\rm r}\), and the corresponding depth in the cases of single-point or multipoint tamping, as shown in Figure 23.  
Figures 24 and 25 show that the correlation coefficient \(R\)-value of the normalized \(I_{\rm r,1}/I_{\rm r,max1}\) and \(I_{\rm r,3}/I_{\rm r,max3}\) between the numerical results and those derived by Equations (8) and (9) are relatively higher than 90%. However, it appears to be overestimated for different construction parameters in the zone ranging from 2 to 3 of normalized depth. This may be attributed to the exponential function that is used in the analysis. The differences bring relatively lesser impact on this estimation method, which has been proved by Figures 26 and 27. Thus, we can assume that the effect of the influence factor on the zone of improvement can be approximatively \"eliminated\" by using Equations (8) and (9). According to [PERSON] et al [30], the coefficient \(I_{\rm r,max1}\), \(h_{\rm max1}\), \(I_{\rm r,max3}\), and \(h_{\rm max3}\) can be determined by using the following expressions:  
\[I_{\rm r,max1}=k_{1}N^{k_{2}}(MH)^{  
Figure 24: Normalized \(I_{\rm r1}/I_{\rm r,max1}\) (%) and \((h\)-\(h_{\rm max1})/h_{\rm max1}\) for construction parameter: (**a**) drop number; (**b**) drop energy; (**c**) tamper radius; (**d**) drop momentum. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Equation** & \multicolumn{4}{c}{**Fitting Coefficients**} & \multicolumn{2}{c}{**R-Square**} \\ \hline Equation (10) & \(k_{1}\) & \(k_{2}\) & \(k_{3}\) & \(k_{4}\) & 0.90 \\  & 0.32 & 0.34 & 0.072 & \(-\)0.44 & 0.90 \\ Equation (11) & \(l_{1}\) & \(l_{2}\) & \(l_{3}\) & \(l_{4}\) & \\  & 0.21 & 0.22 & 0.35 & \(-\)0.50 & 0.93 \\ Equation (12) & \(e_{1}\) & \(e_{2}\) & \(e_{3}\) & \(e_{4}\) & \\  & 0.9 & 0.23 & 0.54 & 0.20 & 0.94 \\ Equation (13) & \(f_{1}\) & \(f_{2}\) & \(f_{3}\) & \(f_{4}\) & \\  & 0.26 & 0.26 & 0.28 & 0.29 & 0.91 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Curve-fitting coefficients of Equations (10)–(13). Figure 25: Normalized \(I_{\rm t,3}/I_{\rm t,max3}\) (%) and \((h\)-\(I_{\rm max3})/h_{\rm max3}\) for construction parameter: (**a**) drop number; (**b**) drop energy; (**c**) tamping distance; (**d**) tamper radius; (**e**) drop momentum. The instruction of the estimation method for the improvement effect of the sandy soil is shown in Figure 28. The flow chart provides step-by-step instructions on how civil engineers can use the proposed formula to estimate soil improvement after DC. Figure 27: Measured relative density and estimated relative density: (**a**) below point 1; (**b**) in the center of the square of tamping. Figure 26: Measured relative density and estimated relative density in the center of the square of tamping.