## 2 Numerical Analysis
### Result of Numerical Analysis  
This chapter presents an in-depth analysis of the pile axial stress distribution when tunnel excavation is carried out directly beneath the pile (OH = 0.0 D, OV = 0.5 D). The results are compared with the findings of [PERSON] et al. [11] based on field measurements and numerical analysis to verify the accuracy of the current study. Figure 3 illustrates the axial stress distribution of a pile at the pre-loading stage of tunnel excavation and at the end of tunnel excavation (TF = 0) in dense soil.  
As demonstrated in Figure 3, the axial stress experienced by the pile decreases uniformly along its entire length during tunnel excavation from the pre-excavation stage in dense sand. The degree of reduction in the axial stress is found to be more pronounced in the central pile, as evident from Figure 3a, as compared with the left and right piles in Figure 3b,c. Furthermore, as depicted in Figure 4, the reduction in the axial stress is observed along the entire length of the pile in loose sand. However, in contrast with dense sand, the reduction in the axial stress is significantly greater in the middle and toe sections of the pile than in the upper portion. Additionally, while the central pile exhibits a greater reduction in axial force than the left and right piles in dense sand, the reduction in the axial force is observed across all piles in loose sand.  
Figure 1: Layout of numerical analysis.  
Figure 2: Modeling of numerical analysis: (**a**) mesh generation; (**b**) piled raft and tunnel excavation.  
Figure 3: _Cont._  
Figure 4: _Cont._  
Figure 3: File axial stress due to tunneling in dense sand: (**a**) center pile; (**b**) left pile; (**c**) right pile.  
The changes in the pile axial stress are presented in Figures 5 and 6, depicting the results for dense and loose sand, respectively. As demonstrated in Figure 5, the rate of change in the pile axial stress varies depending on the length of the pile in the center pile. Nevertheless, the axial stress in the center pile and at the top of the pile decreases overall during tunnel excavation. When the length of the pile is shorter, the middle and toe of the pile exhibit a more significant decrease compared with the top. However, for longer piles, the reduction in the pile axial stress is more pronounced.  
Figure 4: Pile axial stress due to tunneling in loose sand: (**a**) center pile; (**b**) left pile; (**c**) right pile.  
Figure 5: Increase in pile axial stress due to tunneling in dense sand: (a) center pile; (b) left pile; (c) right pile.  
Figure 6: _Cont_.  
Figure 6: Increase in pile axial stress due to tunneling in loose sand: (**a**) center pile; (**b**) left pile; (**c**) right pile.  
In the scenario where the tunnel excavation was carried out directly underneath the pile, the maximum pile axial stress was observed in the center pile, while the left and right piles exhibited a slight decrease or increase, in agreement with the findings of [PERSON] et al. [11]. Moreover, it can be inferred that the highest pile axial stress occurs at the top during reloading, implying that the results of the numerical analysis conducted in this study conform to the prevailing literature.