## 3 Results and Discussion
### Underpinned Foundation Behavior
#### 3.2.3 Comparison of Underpinning Piles' Load Sharing Behavior  
Underpinning by micropiles aims to share partial loads caused by vertical construction in order to ensure the stability of existing foundations. Therefore, understanding the load sharing capacity of each pile is key to effective underpinning design. The load sharing ratio (LSR) is used to evaluate a pile's load sharing capacity, and is defined as follows:  
\[LSR_{EP}=\frac{Q_{EP}}{Q_{a}} \tag{7}\]  
\[LSR_{UP}=\frac{Q_{UP}}{Q_{a}} \tag{8}\]  
where \(Q_{EP}\) is the carried load of an existing pile under additional loading; \(Q_{UP}\) is the carried load of an underpinning pile under additional loading; \(Q_{a}\) is the additional applied load to the underpinned foundation; \(LSR_{EP}\) is the load sharing ratio of an existing pile under additional loading; and \(LSR_{UP}\) is the loading sharing ratio of an underpinning pile under additional loading. Variables such as soil strength and pile type and stiffness affect the load sharing capacity of the pile. [PERSON] et al. (2018, 2019) proposed a simplified model for reliability analysis of uncertainties influencing the design of slopes ([PERSON] et al., 2018; [PERSON] et al., 2018). The present study considered only the effects of axial stiffness on the load sharing behavior of each pile. Figure 11 presents the load sharing by each micropile under additional loads applied to the underpinned foundation. The load sharing ratio of the underpinning piles increased with the applied load. The load sharing ratio for CMP2 increased the most slowly and converged to around 12.5% as the final load of 235 N exceeded its ultimate bearing capacity of 141 N. In contrast, the other pile models showed continuous increases of the load sharing ratio. At the final loading step, the load sharing ratios of WMP1, CMP1, and WMP2 were 34%, 24%, and 22% of the additional load, respectively. Figure 12 shows the normalized load sharing ratio of the underpinning piles to an existing pile with an increase in the additional applied load. The results show that load sharing ratios of WMP1, CMP1, and WMP2 increased to over 5, 2.5, and 2 times that of the existing pile, respectively, while CMP2 appeared to share the load almost equally with the existing pile. This clearly demonstrates that the load sharing capacity of a micropile increased with its increasing axial stiffness. Figure 11: Load sharing ratio (LSR) of underpinning pile with respect to additional loading level. The load sharing of piles are generally determined by structural analysis software such as Midas. Such software that are for the design of foundation underpinning consider piles as springs with an input parameter of axial stiffness. Therefore, a pile's stiffness is an important factor influencing its load sharing capacity. To evaluate quantitatively the effect of the stiffness on the load sharing capacity of each pile, the relationship between the normalized LSR of an underpinning pile to the existing pile, \(\alpha\), and the normalized stiffness of the underpinning pile to the existing pile, \(k\), is illustrated in Figure 13 and Table 2. Figure 13 shows WMP1 to have the highest \(\alpha\) and \(k\). The normalized LSR of CMP1 and WMP2 were similar because of their similar axial stiffness. The plot shows \(\alpha\) to increase linearly with \(k\) with a gradient about 1.8. This implies that underpinning by waveform micropiles or increasing the underpinning piles' stiffness is an effective way to enhance the load sharing capacities of underpinning piles, which can contribute to reducing the number of underpinning piles required, and thus save construction cost. This finding could be used as a reference when designing underpinning pile schemes. Figure 12: Load sharing ratio of the underpinning pile (UP) normalized by that of the existing pile (EP) plotted with respect to the additional loading level.