## 1 Introduction  
Due to industrial development and increasing energy demand, structures are being constructed on a large scale, often in previously undeveloped areas. As the size of structures increases with advancements in construction and its technologies, pile foundations have become the most essential and widely used type of foundation [1]. Loads from the upper structure are transferred to the subsoils through piles and resisted through tip resistance and skin friction. Pile foundations govern the overall stability of the structure, resisting significant challenges from geological or meteorological hazards that have recently gained attention. In particular, open-ended piles (OEPs) are employed to facilitate drivability and to prevent pile destruction by allowing the intrusion of soil into the pile during pile installation [2]. They are frequently used not only in buildings but also in infrastructure projects such as wind turbines, oil platforms, and bridges, both in onshore and offshore environments, due to their efficient construction [3]. When an open-ended pile begins to penetrate, soil enters the central cavity of the OEP, and the soil-pile friction gradually corrects the internal soil as penetration continues. This phenomenon is referred to as the 'plugging effect', and the compacted soil inside the pile is called a'soil plug', which is a distinctive feature of OEPs. In this state, the friction between the soil plug and the interior of the OEP significantly contributes to the static resistance of the pile compared with conventional pile foundations. As driving continues, the degree of the soil plug transitions into a 'fully plugged' state, at which point no further soil inflow is allowed, causing the OEP to perform as a closed-ended pile [4; 5]. This phenomenon is governed by the interplay between two forces acting on the soil plug within the pile: (a) the drag-down friction from the internal skin of the pile and (b) the uplifting thrust at the bottom of the soil plug. Due to the significant influence of the 'plugging state' on pile behavior, several researchers have studied factors affecting plug formation [5; 6; 7; 8; 9; 10; 11]. It has been consistently reported that three major factors influence the formation of soil plugs: (1) pile geometry, (2) soil conditions, and (3) installation method. In summary, the optimal conditions for the longer soil plug development are (1) a large pile diameter [4; 6; 10], (2) dense ground conditions [7; 11], and (3) dynamic driving rather than static driving methods [8; 9; 12; 13]. According to the literature, the relation between the soil plug length and the pile penetration depth has been used to represent the plugging state [14; 15; 16; 17; 18; 19; 20] because the stress state measurement of the soil plug is highly limited. Two typical plug length indices are the Plug Length Ratio (\(\mathit{PLR}\)) and the Incremental Filling Ratio (\(\mathit{IFR}\)). The \(\mathit{PLR}\) is defined as the total plug length over the total penetration length at a specific driving stage, while the \(\mathit{IFR}\) is defined as the increment in plug length divided by the incremental penetration length for each driving impact. These indices can also be expressed using the following equations:  
\[\mathit{PLR}=\frac{L_{p}}{D_{p}} \tag{1}\]  
\[\mathit{IFR}=\frac{\Delta L_{p}}{\Delta D_{p}} \tag{2}\]  
Here, \(L_{p}\) and \(D_{p}\) present the soil plug length and the depth of pile penetration, respectively, with \(\Delta\) referring to the increment of the variable. An \(\mathit{IFR}\) is known to better represent the degree of plugging state than a \(\mathit{PLR}\) during open-ended pile installation [4; 7]. An \(\mathit{IFR}\) close to '1' indicates that the pile is installed in a fully cored state, and '0' means that a soil plug inside the pile exhibits a fully plugged mode, preventing the soil from below the pile tips from entering the open-ended pile. When the plugging effect occurs within open-ended piles, especially when it is impact-driven in sands, two major challenges occur.