Seismic response of pile-supported structures considering the coupling of inertial and kinematic interactions in different soil sites

Abstract

Dynamic soil−pile−superstructure interaction is crucial for understanding pile behavior in earthquake-prone ground. Evaluating the safety of piles requires determining the seismic bending moment caused by combined inertial and kinematic interactions, which is challenging. This paper addresses this problem through numerical simulations of piles in different soil sites, considering soil nonlinearity. Results reveal that the period of the soil site significantly affects the interaction among soil, piles, and structures. Bending moments in soft and hard soil sites exceed those in medium soil sites by more than twice. Deformation modes of piles exhibit distinct characteristics between hard and soft soil sites. Soft soil sites exhibit a singular inflection point, while hard soil sites show two inflection points. In soft soil sites, pile-soil kinematic interaction gradually increases bending moment from tip to head, with minor influence from superstructure’s inertial interaction. In hard soil sites, significant inertial effects from soil, even surpassing pile-soil kinematic effects near the tip, lead to reversed superposition bending moment. Superstructure’s inertial interaction notably impacts pile head in hard soil sites. A simplified coupling method is proposed using correlation coefficient to represent inertial and kinematic interactions. These findings provide insights into complex seismic interactions among soil, piles, and structures.

Graphical abstract

Keywords

pile seismic response / soft soil / inertial interaction / kinematic interaction

1 Introduction

Seismic pile response has gained significant importance, especially in the aftermath of notable seismic events including the Niigata earthquake in 1964, Kobe earthquake in 1995, Chi-Chi earthquake in 1999, and the Turkey earthquake in 2023, which highlighted the potential for damage to pile foundations in soil sites consisting of weak soils like clay, sand and silt during strong earthquakes [1–4]. Dynamic behavior of piles in earthquakes is crucial for ensuring structural safety, particularly in soil sites associated with the Seismic-Soil-Pile-Superstructure Interaction (SSPSI) [5–8]. The SSPSI refers to the seismic interaction between the soil, superstructure, and piles of the supporting structure in seismic events, impacting significantly on the overall structural seismic response.

Since the 1970s, pioneering researchers have embarked upon the investigation of soil−pile kinematic interaction, on which the stiffness of piles considering dynamic effect was studied based [9–11]. The “p−y” method was introduced to capture the relationship between force and displacement in soil-pile seismic interplay during earthquakes [12,13]. It incorporated the nonlinear characteristics of soil, pile and their dynamic interaction by using a series of “p−y” springs with varying stiffness and damping properties [14], which are often calibrated based on laboratory testing or in situ measurements. The method also includes gap elements to account for the separation between the pile and soil as the load varies cyclically. However, the “p−y” method has limitations in accurately capturing the dynamic evolution of soil−pile interaction and incorporating the influence of superstructure on their interaction. The inertial and kinematic interactions add a level of complexity to the pile seismic analysis [15]. Inertial effects refer to the pile dynamic response to the earthquake-induced ground motion. They are related to the mass and acceleration of both the pile and the supporting superstructure. In general, the inertial effects are more pronounced for stiffer piles and harder soil conditions. The inertial effects can cause significant bending moment, shear force in the pile, resulting in the pile failure or damage. Kinematic effects refer to the displacement of the soil around the pile induced by earthquake motions. This deformation can lead to lateral soil-pile interaction forces affecting the pile behavior [16,17]. Aldimashki et al. [18] summarized that the inertia force of the structure during earthquakes may be one of the primary reasons for pile damage. During earthquakes, the inertial interplay between the pile and structure exerts a substantial influence on the dynamic response of piles in soft soil sites and can be more dangerous than the kinematic interaction before ground failure. The kinematic interaction mainly relates to the relative movement of the soil to pile, which is significant when the site soil undergoes significant deformation. During an earthquake, liquefaction can cause significant soil deformation and lateral flow, leading to pile damage between the liquefied and non-liquefied soil layers [19].

Various pseudo-static analysis methods assessing the seismic behavior of piles treat the inertial and kinematic actions on piles as decoupled for simplification. Boulanger et al. [20] and Tabesh and Poulos [21] obtained the pile response by adding the inertial action calculated by the maximum response spectrum to the kinematic action. However, Liyanapthirana and Poulos [22] found that adopting the maximum acceleration of the response spectrum overestimated the structure inertial action and suggested calculating this inertial action based on the peak acceleration of the ground and superposing the maximum displacement of the ground to assess the stress and deformation of the piles. Some of these methods have been employed by design guidelines. For example, in the Chinese Technical Code for Building Pile Foundations [23], the inertia force of the superstructure acting on the pile is directly superimposed on the internal force of the pile caused by soil deformation. However, the inertial and kinematic effects are coupled and depend on the dynamic properties of the structure and soil, the stiffness of the piles, and the characteristics of the ground motions, among other factors. Shaking table tests were executed by Xu et al. [24] to study the pile−soil−structure seismic interactions in sand with different water conditions. Hussein and El Naggar [25] performed a parametric study by FEA to assess influence of ground motion levels and piles characteristics on the kinematic and inertial effects. The study revealed that piles with larger cross-section exhibit a substantial increase in bending moment caused by kinematic interaction. Furthermore, the influence of kinematic interaction was more pronounced in piles with higher elastic modulus. Tokimatsu et al. [15] conducted a shaking table test on a soil−pile−superstructure (SPS) system to study the phase relation of inertial and kinematic interactions under different superstructure natural periods. A simplified analysis method for the internal force of the pile during an earthquake was proposed according the test results. When the structure period was less than the site period, the inertial and kinematic interactions were in phase, and the maximum bending moment of the pile was the sum of the two actions. However, when the structure period was greater than the site period, there was a 90° phase offset between the two interactions, and the maximum bending moment of the pile was the root sum square (RSS) of inertial and kinematic actions. It was pointed out that the coupling of the two interactions is not simply a linear superposition, and the phase difference between the two interactions can significantly affect the pile dynamic response during earthquakes. Wang et al. [26,27] analyzed the SSPSI using a finite element model and simplified the coupling mode of kinematic interaction and inertial interaction by assuming that the structure period had no effect on the kinematic interaction. They found that the coupling mechanism of the two interactions of the pile with and without a cap was different. Ullah et al. [28] proposed a simplified analytical formulation for the superposition of kinematic and the inertial interaction to analysis the response of the SPS system and found that soil damping and the relative stiffness of the soil to pile affected the amplification ratio and frequency relationship. Accurately predicting the pile seismic response in soil sites requires more advanced numerical methods involving material model, modeling methodology, etc., that incorporate the interactions in SPS system with the consideration of their coupling. The seismic motions induced by medium to extreme earthquakes often cause the soil deposit and pile to exhibit nonlinear elastic-plastic behavior. Pile nonlinearity trends to decrease the kinematic bending moment experienced along the pile shaft by a reduction in the contrast of stiffness between the pile and soil, which had been already demonstrated in the parametric study by Mucciacciaro and Sica [29]. In numerical simulations the utilization of the initial shear modulus of soil for the system is limited to capture the site response under small strains. Paying attention to the nonlinear characteristics of the site soil is necessary for obtaining reliable seismic response of the SPS system [29]. Pressure Dependent Multi Yield (PDMY) model [30] and Pressure Independent Multi Yield (PIMY) model [31] are two elastoplastic constitutive models that are respectively applicable to clay and sandy soils. They effectively capture the nonlinear behavior of soils under seismic loading conditions and consider the influence of groundwater, allowing for the simulation of liquefaction phenomena in sandy soils [32–35]. The PDMY material model can simulate the pressure-sensitive characteristics of sandy soil. It is an elastic−plastic model employing the multi-yielding surface to formulate plasticity and a non-associative flow rule, which can simulate the dilatancy effect of the sandy soil. Compared to sand, clay’s shear behavior under dynamic loading is less sensitive to confinement changes. Characteristics of the clay can be captured by the elastic−plastic material model PIMY model, where the volumetric stress−strain and the deviatoric stress−strain that produce plasticity are independent.

In this paper, a series of finite element numerical simulations were conducted to analyze the dynamic response of SSPSI and study the evolution of inertial and kinematic interactions and their effects on the pile. A finite element model of a SSPSI system was established, taking into account the heterogeneity of the actual site and the nonlinear constitutive behavior of the soil. The site fundamental period is a critical factor affecting the system response, but has been rarely studied in previous research. Herein, we compared the dynamic responses of systems with different site fundamental periods with consideration of soil nonlinearity, including pile bending moments, soil displacement, and structural acceleration, and analyzed their correlations. A correlation coefficient was introduced to consider the time series of the inertial and kinematic interactions. The correlation of the two interactions, represented by variations in pile head moment, were calculated for different soil sites. The coupling mechanism was studied for SPS interaction systems with different natural periods of the soil sites.

2 Simulation of the pile response

2.1 Finite element simulation setup

A two-dimensional (2D) finite element numerical simulation was employed to investigate the seismic response of a two-story frame structure with a pile foundation in soil below the groundwater level. The simulation was finished by means of the software OpenSees. The size of the soft soil model was 70 m × 20 m, including a clay layer, two sand layers, with a thickness of 2, 10, and 8 m, respectively, as shown in Fig.1. In the 2D numerical model, the soil was modeled by four-node quad u−p elements, where each node possesses displacement and pore pressure degrees of freedom. They are plane-strain elements using bilinear isoparametric formulation capable of simulating deformation and flow coupling in soil below the groundwater level. The clay layer adopted the PIMY model, while the two sand layers adopted the PDMY model according to the manual [36,37]. The soil parameters used are shown in Tab.1.

A two-story reinforced concrete space frame with piles embedded in clay and sand soil was used in the simulation, as shown in Fig.1. The model has been previously employed in literature to study pile-supported structures [38]. Each story had a height of 4.5 m and a width of 10 m, with cross-sectional areas of 0.4, 0.5, and 0.6 m² for columns, girders, and piles, respectively. The flexural stiffness of the piles E_PI_P is 8.6 × 10⁸ N·m². The frame structure has a fundamental period of 0.1 s. The length of the piles was 10 m. Beam−column elements were utilized to model the columns, girders, and piles, with Concrete01 Material and Reinforcing Steel Material used to represent the constitutive behaviors of concrete and reinforcing steel, respectively. The two models are uniaxial material models with degraded linear unloading/reloading stiffness. The appropriate constraints were applied on the lateral boundaries of the model, in which the horizontal translational degree of freedom of nodes on one side of the model was coupled with the corresponding node on the opposite side. This boundary condition has been demonstrated to produce accurate results [39,40]. The seismic motion is applied at the base of the model by inputting the acceleration time history from the bottom boundary. Beam Contact 2D elements were employed to model frictional and contact interactions at the interface between the piles and soil. The recorded ground motion in the Parkfield (2004) earthquake with a peak ground acceleration (PGA) of 0.24g from the PEER Strong Motion Database was used as the seismic excitation, as it has been previously used for centrifuge testing [26]. Fig.2 and Fig.3 depicts the seismic excitations and their elastic response spectra. The Vs₃₀ of the recorded motion site is 510.92 m/s, assuming it to be close to the motion at the bedrock outcrop and ignoring the differences between it and the motion at a certain depth in the soil site. No formal deconvolution analysis was employed. It should be noted that 2D finite element models used for obtaining some insights on SSPSIs herein due to computational efficiency and relative simplicity, but also with inherent limitations. The 2D model is a simplification that may not capture all the geometric features of a structure and interactions between structural components may lead to inaccuracies in stress distribution and deformation patterns.

2.2 Soil sites in simulation

The fundamental site period (FSP), T_g, is a key parameter helping to quantify the effects of local soil conditions on shaking intensities, determining the soil classes and estimating ground peak acceleration. An estimation of the site period can be derived from the shear wave velocity profile via the equation of 4 × h/V_s, where h is the overburden thickness, and V_s is the average shear wave velocity at this depth. In a soil site, the fundamental period approximates the period of vibration at which the greatest soil amplification is anticipated [41]. During an earthquake, the motions of a structure can be amplified significantly when the structure natural period approaches that of the underlying ground. Thus the FSP reflecting the site stiffness significantly affect the inertial and kinematic effects of a SSPSI system. Vibrations and loads transferring between soil, pile and structure as shear wave dissemination from bedrock to the ground level, needs to be quantified for structural and geotechnical design. Chinese Code for Seismic design of Buildings (GB 50011-2010) [42] provides a subsoil classification system, which categorizes sites into four classes, I to IV. The boundary FSP of these four kinds of sites is 0.045, 0.374, 0.966, and 1.281 s, respectively. Classes I and II usually refer to rock, dense gravel, hard clay, whereas classes III and IV, usually representing soil site including sandy soil, silt soil, clay soil. Soil sites are prone to complex interactions among piles, soil, and structures. The motivation behind this research stems from the challenges faced by practitioners and researchers in predicting pile seismic responses for SPS interaction systems in soil sites. Specifically, this study considered FSPs of 0.39, 0.70, and 1.48 s, which were achieved by modifying the shear modulus of the sand 1 layer in the model, listed in Tab.2. Notably, the fundamental periods of these three soil sites exceed the structural period 0.1 s.

2.3 Dynamic responses of the soil−pile−superstructure system

The response of a pile is primarily attributed to the kinematic deformation of the soil and the inertial acceleration of the structure. In this section, the time histories of the pile bending moment at different depths, superstructure acceleration, and ground surface displacement are presented. The curves of three different soil sites with FSP T_g values of 0.39, 0.70, and 1.48 s are compared to illustrate the variation in response under different soil conditions.

Fig.4 depicts the moment histories of piles at depths of 0, 4, and 8 m, respectively, obtained from numerical simulations. The results demonstrate the maximum bending moment occurs at the pile head, which is a critical consideration in pile design. This is consistent with the results that the bending strains reach their maximum magnitude either at the pile head or at the interface where a significant contrast in stiffness between soil layers. obtained using simplified BNWF (Beam on Nonlinear Winkler Foundation) and finite element analysis [43–45], as well as experimental investigations [46,47]. The simplified solutions for assessing the kinematic bending moment at the pile-head were provided for both fixed- and free-head piles in Di Laora’s study [48]. Herein, the dynamic response of the SPS system in different soil sites is discussed under the coupling of kinematic interaction and inertial interaction among the three components.

Fig.4 shows the bending moment at three positions along the pile shaft was found to be maximum at T_g of 1.48 s, followed by T_g of 0.39 s, and minimum at T_g of 0.7 s, indicating that significant bending moments occur in both soft and hard soil sites, exceeding the bending moment of piles in medium soil sites by more than twice. Fig.5 displays the ground surface displacement time histories for three different soil sites. The ground surface displacement was found to be maximum at T_g of 1.48 s, followed by T_g of 0.7 s, and minimum at T_g of 0.39 s. For the soft soil site with T_g of 1.48 s, cycles with large ground surface displacements were generally accompanied by substantial bending moments exerted on the pile, except for the final stages of the time history in which the soil has accumulated significant plastic deformation and stiffness degradation. It suggested that the kinematic effects of soil deformation played a significant role in bending moment in the pile in the soft soil site. In the soil site with T_g of 0.7 s, the cycles with large ground surface displacement corresponded to the large cycles of bending moment at depths of 4 and 8 m in the pile, but not to the bending at the pile head. It was found that the kinematic effects of soil deformation played a role in the bending moment of the lower part of the pile. In the soil site with T_g of 0.39 s, there was no clear correlation between the variation of ground surface displacement and the bending moment exerted on the pile. Fig.6 presented the structural acceleration time histories for three different soil sites. The structural acceleration on the soil site with T_g of 0.39 s was significantly higher than the other two soil sites. The cycles of large structural acceleration on the soil site with T_g of 0.39 s were generally consistent with the cycles of large bending moment in the pile, particularly with the trend of the bending moment at the pile head. This indicated that the inertial effects of the structure played a significant role in the bending moment of the pile, especially at pile head. The displacement response at the ground surface had a low frequency and narrow bandwidth. In the response of the bending moment in the lower part of the pile, in addition to the low-frequency components, there were also some high-frequency components in the range of 3–5 Hz. The pile head bending moment was mainly composed of low-frequency components. The piles exert filtering effect [49,50]. The high frequency components were filtered out at the pile head, as demonstrated in Fig.4(a). This observation is consistent with the phenomenon in which high-frequency components of seismic waves are attenuated, while low-frequency components are preserved during the propagation process in soil sites. The filtering effect was significant in soft soil site with T_g of 1.48 s, while play a less role in hard soil site with T_g of 0.39 s. The dominant frequencies of pile head bending moments are consistent with the dominant frequencies of the soft soil and hard soil sites respectively, indicating the response of pile bending moments is coordinated with the response of the soil. However, in the medium soil site with T_g of 0.70 s, the influencing factors on pile shaft bending moments are more complex.

3 Soil−structure interaction on pile bending in different soil sites

3.1 Pile bending moment

Pile moment is correlated with soil displacement and structure acceleration as discussed in Section 2. However, the inversion of these correlations due to variations in the FSP highlights the need to explore the influence of soil sites on pile bending induced by soil−structure interactions, including the inertial interaction and kinematic effect. Fig.7 illustrates the variations in ground surface displacement, structure acceleration, and pile head moment are mutually dependent in different soil sits, reflecting the influence of kinematic effects (a, c, and e) and inertia effects (b, d, and f ), respectively.

For the soft soil site, where T_g of 1.48 s, the highest positive moment occurred when the soil displacement was close to its maximum positive value, coupled with a maximum negative acceleration of the structure. The phase of the pile moment was almost reversed with the structure acceleration but coincided with the soil displacement, consistent with the findings of Wang’s study on single piles without caps [27]. For the medium soil site, where T_g of 0.7 s, the peak pile moment decreased significantly, while soil displacement and structure acceleration varied little compared to the soft soil site. In this case, the peak negative moment occurred at almost the maximum negative soil displacement and the maximum negative structure acceleration. The phase of the pile moment was almost reversed with the two other responses. For the hard soil site, where T_g equals 0.39 s, the peak structure acceleration increased significantly, while soil displacement decreased, and the pile moment changed little compared to the soft soil site. In this case, the peak positive moment occurred at almost maximum negative soil displacement and positive structure acceleration. The phase of the pile moment was almost coincident with the structure acceleration but reversed with the soil displacement. The result demonstrates the complexity of SPS interactions and demonstrates the vital role of accurately modeling the soil behavior in predicting pile seismic response in various soil sites.

3.2 Pile deformation

Pile moment and deformation are two fundamental aspects of pile mechanical behavior and are closely related. Fig.8 illustrates the distribution of pile horizontal displacement when the pile moment reaches its maximum in different soil sites. The deformation distributed along the pile decreased as the FSP T_g decreased. Under the same superstructure, the pile exhibited different deformation modes in different soil sites. In the soft soil site, the displacement increased from the pile tip to below the pile head and decreased approaching the ground surface. The structure restricted the bending of the pile head, causing the deformation depicted in Fig.8(a). The inflection point was located at a depth of 1.9 m, with the maximum displacement of 0.068 m. In the medium soil site, the restriction to the pile head by the structure strengthened the opposite pile deformation near the pile head compared to the soft soil, as shown that the deeper inflection point at a depth of 2.3 m. In the hard soil site in Fig.8(c), the restriction effect on the pile head was further enhanced and spread to the depth of the pile. The maximum displacement located at a depth of 2.5 m. The opposite effect on the pile tip deformation resulted in the second inflection point at a depth of 7.6 m. This finding highlights the importance of accurately modeling soil behavior in predicting the deformation and moment behavior of piles in different soil sites, which exhibit varying levels of restriction to pile bending due to the presence of structures.

Deformation modes are primarily attributed to the inertial and kinematic interactions in different soil sites, which require further investigation. The loading mechanism of pile in soft and hard soil sites respectively in Fig.9, illustrating how inertial and kinematic interactions created the specific deformation mode. M_i and M_k represent bending moments due to inertial and kinematic interactions. The large soil deformation produced in the soft soil site resulted in a bending moment generated by the kinematic interaction from the pile tip to the head that was always larger than that generated by the inertial interaction. However, the bending by the inertial interaction from the structure superposed near the pile head as it approached the ground surface, exceeding the bending caused by the kinematic interaction. In the hard soil site, a large inertial acceleration occurred, especially in the deeper regions of the soil site. The bending moment by the soil inertial effect was larger than that by the soil kinematic effect near the pile tip. The soil kinematic effect increased as it left the pile tip, with increasing soil deformation. The bending moment by the kinematic effect gradually increased along the pile, and at a certain position, it was equal to that by the soil inertia effect, forming the second inflection point. Then, the bending moment by the soil inertial effect was less than that by the soil kinematic effect in the midsection of the pile. The moment of bend due to inertial interaction from the superstructure also acted on the pile head in addition to that caused by soil interaction.

3.3 Coupling mode

Coupling mode of inertial and kinematic interactions is one of important aspects to understanding what drives the seismic pile response essentially. The interlation and phase offset between inertial and kinematic interactions were quantified by means of cross-correlation analysis. A function which can be used to calculate the correlation coefficient and hysteresis order between two discrete time series of data f[t] and g[t] is introduced, which used in signal processing [51].

(1)(f\times g)[\mathrm{\Delta }{]}^{\mathrm{d}\mathrm{e}\mathrm{f}}=\frac{{\displaystyle {\sum }_{t=1}^{n}f[t]g[t+\mathrm{\Delta }]}}{\sqrt{\left({\displaystyle {\sum }_{t=1}^{n}f[t]f[t]}\right)\left({\displaystyle {\sum }_{t=1}^{n}g[t]g[t]}\right)}},

where t represents the position of a specific data point within the sequence, with values extending from 1 to n, and Δ denotes the delay or time lag. The correlation coefficient is between −1 and +1, with −1 signifies a perfect inverse correlation, while + 1 indicates a perfect direct correlation.

Tab.3 shows the correlation coefficients (c₁) between the structure acceleration and the pile head moment in soil sites with different FSP T_g, as well as the correlation coefficients (c₂) between the surface displacement and the pile head moment in different soil sites. Herein the negative correlation coefficients reflect resulting in pile moments in opposite direction. When the FSP T_g is less than 0.2 s, the both correlation coefficient are less than 0.2, representing contribute little to pile head moment in very stiff site. The correlation coefficients between the structural acceleration, representing the structure inertial effect, and the bending moment at the pile head, as well as the correlation coefficients between the surface displacement of the site, representing the soil kinematic effect, and the bending moment at the pile head, change continuously as the fundamental period of the site.

Fig.10 shows the variation of (a) peak structural acceleration, (b) peak surface displacement and (c) peak pile head moment with the FSP T_g. The maximum structural acceleration occurred at T_g equal to 0.39 s in variation curve of peak structural acceleration as T_g. The influence of this peak ranged from 0.2 to 0.7 s. The peak surface displacement increased as T_g when T_g less than 1.0 s. As for the pile head moment, the first peak occurred at T_g equal to 0.39 s and moment was great when T_g more than 1.0 s, where the maximum structural peak acceleration and the maximum surface peak displacement occurred. As T_g increased the contributions of interactions to the pile head bending moment were constantly changing, which were correlated with the variation of the structural peak acceleration and surface peak displacement.

The dynamic pile head moment can be decomposed into the moment caused by kinematic interaction multiplied by its correlation coefficient and the moment caused by inertial interaction multiplied by its corresponding correlation coefficient, which can be expressed as follows:

(2)M=c_1{M}_i+c_2{M}_k.

Fig.11 depicts the coupling model of the kinematic and inertial interactions for the pile in different soil sites. The total dynamic interaction-induced moment of the structure is approximated as the sum of the two interaction-induced bending moments multiplied by their respective correlation coefficients. The composite diagram illustrates four typical points, with the lengths of the arrowhead segments reflecting their respective acting amplitudes, and the phase difference of the bending moments due to the two interactions reflected as the angle between the segments. When the FSP was small (point ①), little pile moment was induced by both interactions. At point ②, the inertial interaction reached its maximum value, and the bending due to the two interactions were reversed. At point ③, the bending moment caused by the two interactions were in almost the same phase, and the kinematic interaction remained dominant. As shown in point ④, when the FSP was large, the bending resulting from the two interactions were reversed, with the bending by the kinematic interaction being much larger than that caused by the inertial interaction.

4 Conclusions

This paper investigates the seismic response of the piles supporting structure in different soil sites, with a focus on the deformation and moment behavior of piles by means of numerical simulation. The conclusions can be obtained as follows.

1) The seismic dynamic response of the pile supporting structure varies significantly in different soil sites, with surface soil displacement, structure acceleration, and bending at the pile head exhibiting distinct patterns. Soft soil sites tend to produce large soil displacement, while hard soil sites tend to produce small soil displacement but large inertial acceleration. The bending generated by the pile-soil kinematic interaction from the pile tip to head in soft soil sites is always larger than that generated by the inertial interaction, but the bending caused by the inertial interaction from the structure can also significantly contribute to the bending moment. Accurately modeling soil behavior is crucial for assessing the deformation and moment behavior of piles in various soil sites, which exhibit varying levels of restriction to pile bending due to the presence of structures.

2) The loading mode and deformation pattern of pile foundation in layered soil depends on the role of two interactions in different soil sites. In soft soil sites, the bending generated by kinematic interaction is larger than that generated by the inertial interaction, while in hard soil sites, the bending due to the inertial interaction is larger than that generated by the kinematic interaction near the pile tip. However, the bending caused by the latter interaction increases as leaving the pile tip, with increasing soil deformation, and at a certain position, it is equal to that caused by the inertial interaction, forming the second inflection point of pile in hard soil sites.

3) The contributions of the two interactions to the pile head bending constantly change as the FSP varies, which is correlated with the variation of the structural peak acceleration and surface peak displacement. The dynamic pile head moment can be decomposed into the moment caused by kinematic and inertial effects.

This research paper employed a numerical simulation that was limited to a single seismic motion. The main focus was to analyze the complex seismic interactions among soil, piles, and structures in different soil sites considering soil nonlinearity. However, investigating the effects of seismic motions with varying intensity levels and different spectral characteristics on seismic interactions will be addressed as another systematic study in future research, and the coupling mode of kinematic and inertia interactions proposed in Fig.11 may need significant revisions following further investigations.

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