## 5 Parameter Analysis through Numerical Analysis  
### Numerical Modeling  
In this study, parametric analysis was performed using FLAC\({}^{\text{3D}}\) for various tamper sizes, height of tamper drop, and number of drops in order to analyze the effect of compaction parameters on ground subsidence, range of ground improvement, and degree of ground improvement. The maximum impact load for dynamic analysis was estimated using Equation (2), derived from the momentum conservation law and the ground deformation according to the dynamic load.  
\[F_{max}=\sqrt{\frac{32 WHGrr_{0}}{\pi^{2}(1-v)}} \tag{2}\]  
where \(F_{max}\) = peak dynamic force; \(H\) = height of tamper drop; \(G_{d}\) = dynamic shear modulus; \(r_{0}\) = radius of tamper; and \(v\) = [PERSON]'s ratio.  
The geological strata for the numerical model were simplified to two layers, and the average values from lab and in-situ tests for each geotechnical property were used, as given in Table 6. Dense to very dense sand layer (SP-SM) was encountered at around GL-15-GL-25 m in all 31 boreholes. Only two boreholes were drilled down to 30 m below ground surface, and brown, hard clay layer was found at  
Figure 6: Shape of wave propagation generated by dynamic compaction [9].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Soil Category (Figure 3)** & **Soil Type** & **Ground Subsidence (\% of \(D\))** \\ \hline \multirow{2}{*}{Zone 1} & natural sands & 3–10 \\ \cline{2-3}  & granular fills & 5–15 \\ \hline Zone 2 & semi pervious soils & N/A \\ \hline \multirow{2}{*}{Zone 3} & clay fills & 3–5 \\ \cline{2-3}  & natural clay & 1–3 \\ \hline Other & refuse and peat & 7–20 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Surface subsidence for different deposits (Solocombe, 1993).  
around GL-25.3-GL-28.7 m. Dynamic elastic modulus and dynamic shear modulus were estimated using t Equations (3) and (4), which were proposed by [PERSON] [11].  
\[G_{d}=\rho V_{s}^{2}=\left(\frac{\gamma_{t}}{g}\right)V_{s}^{2} \tag{3}\]  
\[E_{d}=2(1+v)G_{d} \tag{4}\]  
\[V_{s}=a\times N^{b} \tag{5}\]  
where \(V_{s}\) = shear wave velocity (m/s); \(G_{d}\) = dynamic shear modulus; \(E_{d}\) = dynamic elastic modulus; \(\rho\) = density of soil; and \(a\) = 81, \(b\) = 0.33 (for alluvial soil) or \(a\) = 91, \(b\) = 0.32 (for dilluvial soil).  
Figure 7 shows the numerical model and boundary conditions for the parameter study. The model size was set to 50 m \(\times\) 50 m \(\times\) 50 m, and both quiet boundary and free-field boundary conditions were applied in order to minimize the wave reflections at model boundaries when dynamic loading was applied. The Mohr-Coulomb model was used for soil, and the large strain mode was applied to appropriately simulate the large ground deformation caused by dynamic compaction. In this study, the effect of pore pressure change with time after application of dynamic compaction was not considered.  
Figure 8, described by [PERSON] et al. (1994), shows the curve of deceleration response measured during dynamic compaction tests. In this study, the load of F\({}_{\max}\) was set to be applied on the ground surface for only 0.05 seconds and then removed to simu  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \multicolumn{2}{c}{**Depth (m)**} & \multicolumn{2}{c}{**SPT N**} & \multicolumn{1}{c}{**V\({}_{\text{s}}\)**} & \multicolumn{1}{c}{**Unit Weight**} & \multicolumn{1}{c}{**Dynamic Shear**} & \multirow{2}{*}{**V**} & \multicolumn{1}{c}{**Dynamic Elastic**} \\  & **from** & **to** & **Value** & **(m/sec)** & **(kN/m3)** & **Modulus, G\({}_{\text{d}}\) (kPa)** & & **Modulus, E\({}_{\text{d}}\) (kPa)** \\ \hline S1 & 0 & 10 & 10 & 173 & 16 & 49,000 & 0.35 & 130,000 \\ \hline S2 & 10 & 20 & 30 & 249 & 18 & 114,000 & 0.35 & 310,000 \\ \hline \end{tabular}
\end{table}
Table 6: Geotechnical properties.  
Figure 7: Numerical model and boundary conditions: (**a**) numerical model; (**b**) boundary conditions.