## 1 Introduction  
The widespread adoption of pile foundations is due to their great bearing capacity, high quality, and low cost. In addition to serving as the foundation for sustaining superstructures in urban areas, piles are also widely employed in anti-floating [1] and landslide prevention [2, 3]. So far, many studies have contributed to the field of piles, such as the prediction of pile settlement performance under different loads [4] and the response of piles to different types of loads [5, 6]. However, the squeezing effect and the ground-borne vibrations induced by pile driving result in permanent damage to adjacent structures. To reduce the squeezing effect and penetration resistance, the pre-cast open-ended pipe piles are used. During the installation of an open-ended pipe pile, a soil plug will be formed with soil being squeezed into the piles [7, 8]. Via model tests, [PERSON] et al. [9] found that the behavior of soil plugging during pile driving can be divided into three stages. Initially, the entire interior space of piles driven into the ground was completely filled with soil. As the pile descended, the ratio of the length of the soil plug to the penetration depth decreased. Finally, the ratio between the length of the soil plug and the penetration depth stabilized. Model tests conducted by [PERSON] and [PERSON] [10] showed that the soil-plugging effect in open-ended pipe piles was dependent on the installation technique, dimension of the pile, and characteristics of the soil. In general, the soil-plugging effect was more remarkable in the press-in pile than in the striking pile [11]. Using centrifuge tests, [PERSON] and [PERSON] [12] indicated that, when the striking method was employed, the length of soil plug increased with increasing sand compactness. [PERSON] and [PERSON] [13] found that,for open steel piles, the closure of soil plug was usually associated with the pile diameter: (1) fully plugged with D \(\leq\) 0.5 m; (2) full coring with D \(\geq\) 1.5 m; (3) partially plugged with 0.5 m < D < 1.5 m. Some scholars quantified the closure degree of soil plug by using the index of increment filling ratio (IFR) and plug length ratio (PLR). The fully plugged and fully coring modes corresponding to IFR were 0 and 100%, respectively. A value of IFR between 0 and 100% means that the pile was partially plugged. [PERSON] and [PERSON] [14] proposed a diameter criterion to give early insight into the plugging condition of a pile before driving. Via Discrete Element Method (DEM) simulation under 100 g gravity, [PERSON] et al. [15] found that the pile installation process involved a continual interchange between the instantly fully plugged mode and the partially plugged mode. [PERSON] et al. [16] integrated the inertial force and dynamic resistance factor into the displacement analytical model and obtained the analytical solution for plug displacement under both drained and undrained conditions by developing the ultimate equilibrium differential equations of the soil plug subjected to striking load. To protect the adjacent structures from strike-induced vibration, the dynamic response of open-ended pipe piles during the striking process needs to be explored. Many scholars have performed extensive research on the dynamic response of piles during the driving procedure. Based on the plane strain assumption, [PERSON] and [PERSON] [17] proposed an elastic-plastic solution regarding the use of the circular cavity expansion method to simulate the soil consolidation around the pile. The simulated absorption boundary method provided some insights to solve the reflective wave problem in artificial boundaries when using FEM to simulate the striking pile process [18]. However, the presence of soil plugs has a significant impact on the ultimate static capacity and dynamic behavior of the pipe pile [19]. By assuming the soil plug acts as a spring, [PERSON] and [PERSON] [20] concluded that the formation of soil plugging resistance was relevant to the deformation of soil around the pile. [PERSON] et al.