# Numerical Analysis of the Settlement Behavior of Soft Soil Improved with Stone Columns
## 2 Numerical Analysis of Field Trials at Bothkennar Site
### Field Trials at Bothkennar Site  
The Bothkennar test site was selected by [36] to install stone columns using the dry bottom feed technique. The dry bottom feed technique is considered an appropriate method to construct stone columns in weak soft soils with high ground water table and undrained shear strength in the range of 15 kPa to 30 kPa. Floating stone columns beneath narrow footings were installed in an attempt to better understand the general behavior of stone columns and their application in the ground improvement context. A design treatment depth of 5.5 m was calculated below the foundation level [36]. Stone column lengths of 3.7 m, 5.7 m, and 7.7 m with a stone column diameter, d, of 750 mm were adopted in the field trials to investigate the effect of column length on the performance of the foundation. Table 1 and Figure 2 summarize the stone column and footing arrangements in the field trials investigated by [36]. Figure 1: Soil stratigraphy and key soil properties for the Bothkennar site after [38]. Figure 2: Trial footing arrangements at Bothkennar site after [36]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Trial Footing**} & \multicolumn{1}{c}{\begin{tabular}{c} **Footing** \\ **Dimensions** \\ **(L \(\times\) B) (m \(\times\) m)** \\ \end{tabular} } & \begin{tabular}{c} **Number of** \\ **Columns** \\ **Columns** \\ \end{tabular} & \begin{tabular}{c} **Column Spacing** \\ **(m)** \\ \end{tabular} & \
\begin{tabular}{c} **Founding Depth** \\ **(m)** \\ \end{tabular} &
\begin{tabular}{c} **Column Length** \\ **Below Footing** \\ **(m)** \\ \end{tabular} \\ \hline
1 & \(6\times 0.75\) & 4 & 1.5 & 0.5 & 5.7 \\
2 & \(6\times 0.75\) & 3 & 2.0 & 0.5 & 5.7 \\
3 & \(3\times 0.75\) & 2 & 1.5 & 0.5 & 3.7 \\
4 & \(3\times 0.75\) & 2 & 1.5 & 0.5 & 5.7 \\
5 & \(3\times 0.75\) & 2 & 1.5 & 0.5 & 7.7 \\
6 & \(3\times 0.75\) & 2 & 1.5 & 1.2 & 5.7 \\
7 & \(1.5\times 1.5\) & 2 & 1.2 & 0.5 & 5.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bothkennar trial footing arrangements. The stone columns are generally one order of magnitude stiffer than the surrounding weak soil and will attract a greater proportion of the load compared to the surrounding soft soil as defined by [12]:  
\[\frac{P_{C}}{P_{S}}=\frac{1+2\Big{(}\frac{E_{S}}{E_{C}}\Big{)}K_{S}\ln\Big{(} \frac{a}{r_{o}}\Big{)}}{2\Big{(}\frac{E_{S}}{E_{C}}\Big{)}K_{C}\ln\Big{(}\frac{a }{r_{o}}\Big{)}} \tag{1}\]  
\[P_{o}A_{o}=P_{C}A_{C}+P_{S}A_{S} \tag{2}\]  
\[a=\sqrt{\frac{A_{o}}{\pi}} \tag{3}\]  
\[\frac{A_{o}}{A_{C}}=\frac{B\ L}{N\frac{\pi d^{2}}{4}} \tag{4}\]  
where \(P_{C}\) is the stress on stone column (kPa), \(P_{S}\) is the stress on soil (kPa), \(P_{o}\) is the imposed pressure from foundation (kPa), \(E_{S}\) is the modulus of deformation for soil (kPa), \(E_{C}\) is the modulus of deformation for stone column (kPa), \(K_{S}\) is the earth pressure coefficient for soil, \(K_{C}\) is the earth pressure coefficient for stone column, \(A_{S}\) is the cross-sectional area of treated soil (m\({}^{2}\)), \(A_{C}\) is the cross-sectional area of stone column (m\({}^{2}\)), \(A_{o}\) is the unit area per stone column (m\({}^{2}\)), \(r_{o}\) is the stone column radius (m), \(A_{o}/A_{C}\) is the area replacement ratio, \(B\) is the footing width, \(L\) is the footing length, \(d\) is the stone column diameter, and \(N\) is the number of stone columns. Two load increments with average bearing pressures of 33 kPa and 70 kPa were applied on top of the trial footings, as shown in Table 2, to calculate the settlement performance. Two further load increments with average pressures of 107 kPa and 125 kPa were applied to footings 4 and 6 (Table 2) to determine the ultimate bearing capacity. A typical working pressure of 50 kPa was adopted by [29] following the trials at the Bothkennar site by [41]. In this work, only the first two load increments (Table 2) were used to assess the stone column settlement behavior.