## 3 Simulation Results and Discussion
### Local Stress State of the Soil Plug  
The analysis subsequently focused on how the stress state within the soil plug changed with the progression of installation. It was observed that the stress distribution within the plug varied depending on its location. Therefore, the soil plug was divided into three equal parts (top, middle, and bottom) based on its length. The variation in the stress state due to penetration is illustrated in Figure 11.  
Figure 10: Horizontal and vertical stress distribution within the soil plug and the contact force chain at the penetration depth of (**a**) 292 mm, (**b**) 555 mm, and (**c**) 596 mm. In the top portion, neither vertical nor horizontal stresses were found to exceed their self-weight; in fact, lower stress states than those of self-weight were confirmed. In the middle part, at the initial stages of penetration, a considerable horizontal stress of 45 kPa was recorded, exceeding the average 18.8 kPa observed in the bottom part during the same phase. This led to clogging from the middle down to the bottom section, with a greater degree of plugging noted in the middle section. As penetration deepened to a depth of 300 mm, all additional stresses were alleviated. Furthermore, at the bottom section, at a penetration depth of 432 mm, corresponding to 60% of the final penetration depth, a maximum average horizontal stress of 349 kPa was observed. A significant observation is that, even during subsequent penetrations, this stress was not entirely relieved until reaching a 555 mm penetration depth, along with the persisting plugged state. For a detailed analysis of the soil state variations in this section, the overall stress state of the soil inside the open-ended pile at the penetration stages of 432 mm and 555 mm in terms of lateral earth pressure coefficient (\(K\)) is shown in Figure 12. It was plotted against the distance from the pile tip in order to track the relative movement of the soil plug and soil arching. Maximum earth pressure coefficients at each stage were recorded as 3.4 and 2.8, significantly higher than the external coefficients of 1 and 1.8, respectively. These results indicate that the horizontal stresses developed earlier were partially maintained and transferred upward along with the incoming soil. In the post-432 mm penetration stage, it was observed that horizontal stresses in the middle section were formed even without blocking from the top section. Moreover, an earth pressure coefficient of 1.6 in the middle section at the final stage suggests a partial alleviation from the initial maximum values. The distance between these two points was found to correspond with the depth penetrated during the sustained plugging state, from 432 mm to 555 mm. The maximum \(K\) of 3.4, approximately nine times higher than \(K_{0}\), was noted to be located at a distance of 1 \(D_{in}\) from the pile tip, as confirmed by the contact force chain analysis. Additionally, the beta index at the position that shows the maximum \(K_{0}\) inside the soil plug was calculated from both the \(PLR\) and the stress state, and they were finally compared to each other. A relationship between the \(PLR\) and the beta index studied by [PERSON] and  
Figure 11: Local stress state of the soil plug at the top, middle, and bottom sections. Salgado [7] was used to derive the beta index from the soil plug length (\(\beta_{PLR}\)). The equation is expressed as follows:  
\[\frac{q_{s,in}}{K_{0}\sigma_{v}^{\prime}tan\delta_{c}\alpha}=7.2-4.8\cdot PLR \tag{3}\]  
in which \(q_{s,in}\) = the internal skin friction (also equals \(\beta_{PLR}\cdot\sigma_{v}^{\prime}\)), \(\alpha\) = a function of the relative density of the sand bed (1.0 for dense sand), and \(\delta_{c}\) = the critical-state interface friction angle between the pile and the soil. Consequently, as the beta index relates to the skin friction and the stress state within soil after the pile penetration, \(\beta_{stress}\) was obtained from the observed internal stress state within the soil plug.