# Numerical Analysis of the Settlement Behavior of Soft Soil Improved with Stone Columns
## 2 Numerical Analysis of Field Trials at Bothkennar Site
### Boundaries, Meshing and Interface  
A sensitivity analysis was conducted to investigate the influence of the lateral and bottom boundaries upon the accuracy of the numerical modeling results. The lateral boundaries of the model allow vertical displacement and restrict lateral movement. A pad footing of 3 m length (L) and 0.75 m width (B) was investigated to position the boundaries at a sufficient distance to avoid influencing the results. Three points, A at 0 m, B at 1 m, and C at 2 m below the center of the footing, were selected to measure the vertical displacement, \(u_{z}\), and mean effective stress, \(p^{\prime}\), as shown in Figure 4. The distance from the footing center to the boundary ranged from four to ten times the footing length. The effect of the boundary position is evaluated by comparing the normalized vertical displacements and mean effective stresses at points A, B, and C: \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{\begin{tabular}{c} **Trial** \\ **Footing** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **1 st Load** \\ **Increment** \\ **(kPa)** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Measured** \\ **Settlement** \\ **(mm)** \\ \end{tabular} } & \multicolumn{3}{c}{**PLAXIS 3D**} & \multirow{3}{*}{\
\begin{tabular}{c} **2 nd Load** \\ **Settlement** \\ **(mm)** \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} **Measured** \\ **Settlement** \\ **(kPa)** \\ \end{tabular} } & \multicolumn{3}{c}{**PLAXIS 3D**} \\ \cline{1-1}  & & & & **Settlement** & & **Settlement** & **Settlement** & **Settlement** & **Settlement** \\  & & & **Drained** & **Undrained** & **(kPa)** & **(mm)** & **Drained** & **Undrained** \\ \hline
1 & 35.5 & 27.5 & 16.5 & 15.0 & 72.0 & 50.0 & 55.0 & 56.0 \\
2 & 32.9 & 25.0 & 18.0 & 16.0 & 67.1 & 42.5 & 57.0 & 57.0 \\
3 & 33.1 & 17.5 & 14.0 & 13.0 & 67.8 & 37.5 & 37.5 & 39.0 \\
4 & 34.9 & 20.0 & 12.0 & 11.5 & 71.1 & 42.5 & 42.5 & 41.0 \\
5 & 32.1 & 22.0 & 9.5 & 8.0 & 67.0 & 40.0 & 35.0 & 31.0 \\
6 & 34.2 & 24.0 & 11.0 & 11.0 & 69.6 & 45.0 & 40.0 & 36.0 \\
7 & 32.7 & 20.0 & 12.0 & 10.5 & 67.0 & 42.5 & 43.5 & 39.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Measured [36] and predicted settlement for the field trials at the Bothkennar site. Figure 3: Validation of soil profile and parameters for a pad footing on untreated soft clay. \[\text{Normalized error}\;\;for\;\text{vertical displacement}\;(u_{z})(\%)=\left(\frac{u_{z,1}-u_{z,2}}{u_{z,1}}\right)\times 100 \tag{5}\]  
\[\text{Normalized error}\;\;for\;\text{mean}\;effective\;\text{stress}\;\left(p^{ \prime}\right)(\%)=\left(\frac{p^{\prime}_{z,1}-p^{\prime}_{z,2}}{p^{\prime} _{z,1}}\right)\times 100 \tag{6}\]  
where \(u_{z,1}-u_{z,2}\) and \(p^{\prime}_{z,1}-p^{\prime}_{z,2}\) are the differences between the vertical displacements and mean effective stresses, respectively, for the boundary placed at different distances from the footing center, for example 4 L and 6 L. The convergence of the settlement and stress with increasing boundary distance indicated that locating the lateral boundaries at a distance between eight and ten times the footing length from the center of the pad footing is sufficient to avoid influencing the results (Tables 5 and 6). It was found that lowering the bottom boundary of the model has a minimal effect on settlement and mean effective stress. Therefore, the boundaries were positioned conservatively at distances of 10 L from the center of the pad for all subsequent analyses. Medium mesh is used for the model with refinement to very fine mesh around the stone columns to increase the level of accuracy. The in-situ coefficient of earth pressure at rest, K\({}_{\text{o}}\), at the Bothkennar site was varied from 1.5 in the crust layer to 0.75 at depth [39]. Similar K\({}_{\text{o}}\) values have been used in this research to capture column installation effects. Rigid interface is assumed between the stone column aggregate and the surrounding soils.