## 3 Results and Discussion
### Underpinned Foundation Behavior
#### 3.2.1 Load Settlement Behavior of Underpinned Foundations  
Figure 6 shows the load settlement behavior of foundations underpinned with four types of micropile during the initial and additional loading stages. The response curves clearly changed at the point that the micropile is connected. The load settlement behavior in the initial loading stage is mainly attributed to the behavior of four PCP. As mentioned before, the micropile was installed before initial loading, as soil disturbance and experimental error may influence the load settlement behavior of the foundations before underpinning. As shown in Figure 6, the stiffness of the foundation after the connection of a micropile was larger than that in the initial loading stage without underpinning. The settlement in the cases underpinned with WMP1, CMP1, WMP2, and CMP2 at the final loading stage (1825 N) was 0.88, 1.02, 1.03, and 1.43 mm, respectively. Underpinning in each case reduced settlement by 63%, 54%, 54%, and 35%, respectively, in comparison with the case without underpinning. These results clearly demonstrate that the settlement reduction increased with the increasing axial stiffness of the micropile.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multicolumn{3}{c}{**Underpinning Pile (UP)**} & \multicolumn{3}{c}{**Existing Pile (EP)**} \\  & **WMP1** & **CMP1** & **WMP2** & **CMP2** & **PCP** \\ \hline \(Q_{alt}\) (N) & \(>1000\) & 998 & 612 & 141 & 460 \\ \(Q_{all}\) (N) & \(>333\) & 332 & 204 & 47 & 153 \\ \(K_{slope}\) (N/mm) & 1500 & 888 & 814 & 200 & 552 \\ \(K_{eq}\) & 1277 & 766 & 798 & 478 & 647 \\ (N/mm) & 2.70 & 1.60 & 1.47 & 0.36 & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Bearing capacity and axial stiffness of model piles.  
Figure 5: Load settlement behavior of single piles (\(s_{1}\) is 0.1D of micropile; \(s_{2}\) is 0.1D of PCP).  
[PERSON] and [PERSON] proposed a simplified method for estimating the ratio \(\chi\)of the stiffness of the existing foundation to that of the underpinned foundation [20]. This ratio provides an overview of the effectiveness of the pile installation for underpinning the shallow foundation. It is calculated as follows:  
\[S_{TFP}=S_{ti}+\chi(S_{TF}-S_{ti}) \tag{4}\]  
\[U_{s}=\frac{S_{ti}}{S_{TF}} \tag{5}\]  
\[RSR=U_{s}+\chi(1-U_{s}) \tag{6}\]  
where \(RSR\) is the ratio of settlement reduction; \(S_{TFP}\) is the total final settlement of the foundation underpinned by piles; \(S_{ti}\) is total settlement of the existing foundation at the time \(t_{i}\) of underpinning; \(S_{TF}\) is the final settlement of the existing foundation without underpinning; and \(U_{s}\) is the degree of total settlement.  
Figure 7 shows the reduction in final settlement (RSR) observed under maximum loading with respect to the stiffness ratio of the existing foundation and the underpinned foundation. The theoretical curve from [PERSON] and [PERSON] is also plotted for comparison. The contribution of \(\chi\) to the reduction of the foundation's final settlement is significant. The reduction of the final settlement of foundations increased linearly with the increase of the underpinned foundation's stiffness, which is in agreement with the findings of [PERSON] and [PERSON] [20].  
Figure 6: Comparison of load settlement behavior on the raft during initial loading and additional loading on the underpinned foundation.  
Figure 7: Relation between the reduction of foundation settlement by the underpinned foundation and the normalized stiffness of the existing foundation.