## 3 Simulation Results and Discussion
### Effect of Pile Diameter  
#### 3.6.1 Influence on Plug Length  
Figure 13 represents the model overview at the final penetration depth of 670 mm for four different diameters of piles, where the length of the soil within the pile was measured throughout the entire penetration process. Initially, the length of the soil follows the fully coring line, but as penetration progresses, the trend shows that the smaller the diameter, the further it deviates from this line. The soil plug lengths vary with pile diameter: D90 resulted in a 320 mm long soil plug, D100 in 407 mm, D110 in 526 mm, and D130 in 584 mm. In addition, the plug length indices were subsequently obtained and are shown in Figure 13c,d. They indicate that with an increase in pile diameter, the values of the \(PLR\) at the final penetration depth increased from 0.48 to 0.87, a trend consistent with existing research. Contrary to the \(PLR\), the \(IFR\) shows more significant variations with penetration depth; however, the magnitude of reduction from 1 diminishes with the increase in pile diameter. This also aligns with existing studies, suggesting that smaller pile diameters tend to lead to a higher tendency for the plugging effect. Furthermore, due to the high variability in the \(IFR\) and the difficulty of measuring it during pile installation ([PERSON] and [PERSON] [7]), a correlation was derived to predict the \(IFR\) directly from the \(PLR\) values:  
\[IFR(\%)=144\cdot PLR-61 \tag{4}\]  
As shown in Figure 14, this relationship demonstrates relatively high accuracy for \(PLR\) values above 0.85, but the accuracy is relatively low for \(PLR\) values below 0.8.  
Figure 12: Distribution of the lateral earth pressure coefficient inside the soil plug at the 432 mm and 555 mm penetration stages.  
3.6.2. Impact on the Internal Stress State of the Soil Plug  
Since the bottom portion of the plug reacts most sensitively to the plugging state, the horizontal stress of this portion is depicted for each pile diameter throughout the penetration process, as shown in Figure 15.  
Figure 14. Correlation between the _IFR_ and the _PLR_[7; 45].  
Figure 13. Results of the numerical analysis of (**a**) soil plug length, (**b**) configuration, (**c**) plug length ratio, and (**d**) incremental filling ratio of each model at the final penetration depth.  
The greatest incidence of plugging occurs at a penetration depth corresponding to 60-65% of the final depth. It is noted that apart from the 90 mm diameter pile, plugging occurs intermittently and is relieved periodically. At this depth, the soil plug has probably experienced substantial compaction and shear, which increases the likelihood of particle interlocking and frictional resistance. This intermediate depth is a crucial transition zone, where the soil particles are not fully stabilized but are compacted enough to create conditions favorable for a fully plugged state. Additionally, the stress distribution and load transfer within the soil column may peak at this penetration range, facilitating the formation of the soil plug. The maximum horizontal stress within the soil plug is not significantly affected by the pile diameter. However, in the case of 110 D, an unusually low-stress pattern was observed compared with other cases due to the uncertainty of the plugging phenomenon.  
Furthermore, based on the maximum stress states observed within the soil plug, the determination of the \(\beta\) coefficient from the beta method, which calculates the frictional force on the internal surface of open-ended piles, yielded a range from 1 to 3.6. This suggests that the methodology presented by the Canadian Geotechnical Society (CGS) [21] better reflects the internal stress state of the soil plug in the bearing capacity than the method proposed by the American Petroleum Institute (API) [22].