Seismic Behavior of Pile Group Foundations in Soft Clay: Insights from Nonlinear Numerical Modeling

Abstract

Pile foundations are commonly used to support structures subjected to complex loading conditions. In seismic-prone regions, understanding the soil–pile interaction under cyclic loading is essential for ensuring the stability and safety of these foundations. Numerical modeling is an effective tool for predicting the nonlinear behavior of soil under seismic excitation, but selecting an appropriate constitutive model remains a significant challenge. This study investigates the seismic behavior of pile groups embedded in soft clay using advanced finite element analysis. The piles are modeled as aluminum with a linear elastic response and are analyzed within a soil domain characterized by two kinematic hardening constitutive models based on the Von Mises failure criterion. Model parameters are calibrated using a combination of experimental and numerical data. The study also examines the influence of pile spacing within the group on seismic response, revealing notable differences in the response patterns. The results show that the nonlinear kinematic hardening model provides a more accurate correlation with experimental centrifuge test results compared to the multilinear model. These findings contribute to enhancing the understanding of soil–pile interaction under seismic loading and improving the design of pile foundations.

1. Introduction

Pile foundations are critical in geotechnical engineering, particularly in seismic-prone regions where soil–pile interaction under cyclic loading plays a significant role in structural stability. Past earthquakes, such as the 1995 Kobe and 1964 Niigata events, have highlighted significant pile failures [1,2]. Despite extensive research, existing design codes primarily emphasize inertial effects, often overlooking kinematic interactions with the surrounding soil [3,4]. However, kinematic effects play a crucial role in accurately predicting pile behavior under seismic forces [5,6,7].

Soft clay significantly influences dynamic structural responses, making pile-supported structures in these environments particularly challenging [8]. Studies suggest that structures founded on clay-rich soils, such as piles, micro-piles, and tunnels, exhibit greater seismic vulnerability than those on sandy soils [9]. Despite advancements, seismic studies on piles embedded in soft soils remain limited, posing challenges for geotechnical engineers.

Soil behavior is inherently nonlinear, even at low strain levels induced by seismic ground motion. Various constitutive models have been developed to capture this complexity, yet selecting an appropriate model remains a challenge as it directly impacts the accuracy of seismic response predictions for structures such as moment-resisting frames and pile-supported wharves. Practical applications are difficult due to the intricate nature of soil behavior and the need for extensive parameter calibration. Several studies have sought to address these issues through numerical modeling. For instance, Maheshwari et al. [10] investigated the seismic response of single and group piles using the HISS model (Hyperbolic Incremental Stress–Strain), incorporating incremental stress–strain relationships under harmonic and transient motions with a peak ground acceleration (PGA) of 0.32 g. Their findings emphasized the distinct roles of pile caps in structural inertia and ground motion mitigation. Similarly, Tuladhar et al. [11] analyzed soil–pile behavior under monotonic and cyclic loads using the Ohsaki model, highlighting the impact of cap deformation on dynamic response and reduced cyclic loading capacity.

Further research on seismic soil–pile interaction has employed various constitutive models. Kang Ma et al. [12] utilized a hypoelastic model in Abaqus 2022 to study pile–raft foundations on clay, revealing increased resonance periods in clay-dominated sites. Castelli et al. [13] applied pseudo-static hyperbolic p-y curves (pressure–displacement) for the seismic analysis of single and group piles, stressing the necessity of methodological adjustments when scaling from single piles to groups. Maheshwari et al. [14] extended their work on soil–pile interaction using an advanced HISS model in the time domain, demonstrating that pile behavior at low frequencies is dominated by soil nonlinearity.

To address the limitations of existing approaches, this study employs nonlinear and multilinear kinematic constitutive models to predict pile behavior under seismic loading. These models effectively capture soil nonlinearity while reducing the need for extensive calibration, providing accurate estimations of deformations and bending moments in pile groups. Experimental and numerical methods, including centrifuge testing [15,16] and finite element simulations [17,18], have been widely used for seismic analysis. However, numerical approaches have gained prominence due to practical limitations associated with experimental testing.

For instance, physical model tests such as centrifuge experiments are often constrained by high costs, limited scalability, and challenges in simulating complex soil layering or large-scale boundary conditions. Additionally, the accurate measurement of internal stress and pore pressure within small-scale models remains technically demanding.

The multilinear and nonlinear combined kinematic hardening models, incorporating both isotropic and kinematic hardening effects, are particularly suited for modeling cyclic soil behavior [19]. This study applies these models in Abaqus to evaluate the seismic response of a pile group embedded in kaolin clay.

Notably, the piles are modeled as aluminum with a linear elastic response under seismic loading. This assumption is based on the fact that aluminum has a relatively high yield strength compared to the anticipated seismic forces (PGA = 0.843 g). Given the primary focus on soil–pile interaction, the linear elastic model is deemed appropriate for this study. Future research may consider nonlinear material models to explore the impact of potential pile yielding or pile–soil interface nonlinearities under more extreme conditions.

Unlike previous studies that often focus either on simplified constitutive models or single-pile configurations, this research uniquely integrates two advanced kinematic hardening models within a three-dimensional (Three-Dimensional) 3D finite element framework to simulate the complex behavior of group piles in soft clay. The novelty lies in the comparative assessment of these two constitutive approaches—nonlinear and multilinear kinematic hardening—under realistic seismic conditions, combined with a detailed evaluation of pile spacing effects. Furthermore, the calibration of model parameters using a hybrid dataset from laboratory tests and numerical simulations enhances the realism and applicability of the findings to geotechnical design practices. This dual-model implementation not only contributes to a more realistic and reliable representation of soil behavior but also offers valuable insights into the influence of constitutive model selection on seismic performance, particularly in the case of soft clay and pile group systems.

Additionally, key factors such as pile length, earthquake intensity, and pile spacing significantly influence seismic performance. This research aims to comprehensively assess these effects through three-dimensional finite element analysis using Abaqus software. The findings indicate that the nonlinear kinematic hardening model yields results closely aligning with centrifuge test observations [20], with slight deviations from the predictions of the multilinear model. By advancing the understanding of soil–pile interaction under seismic loading, this study contributes to the improved design of pile foundations in geotechnical engineering.

2. Constitutive Models and Calibration

2.1. Nonlinear Kinematic Hardening Constitutive Model

The nonlinear kinematic hardening model plays a crucial role in analyzing the dynamic response of geotechnical structures under seismic loading. This constitutive model consists of two key components: isotropic hardening, which is effective for materials with negligible creep effects, and kinematic hardening, which enhances the prediction of inelastic behavior in soils. Originally developed by Armstrong and Frederick [21], the nonlinear kinematic hardening model has been primarily applied to clayey soils, requiring modifications for sandy deposits. In this study, we employ an advanced nonlinear kinematic hardening model, integrated with the Von Mises failure criterion, to simulate the seismic response of saturated clay using 3D numerical modeling in Abaqus. The evolution of stress under cyclic loading conditions is characterized based on the formulations established by Lemaitre and Chaboche [22].

$$\mathsf{\sigma }={\mathsf{\sigma }}^{\mathrm{o}}+\mathsf{\alpha }$$

(1)

Here, ${\mathsf{\sigma }}^{\mathrm{o}}$ represents the instantaneous initial yield stress, and its unit is typically Pascal (Pa), and α denotes the backstress (Pa).

The yield surface function within the isotropic hardening model, which includes the Von Mises criterion, is given by the following equation (Chaboche [23]):

$$\mathrm{f}\left(\mathsf{\sigma },\mathrm{R}\right)={\mathrm{J}}_2\left(\mathsf{\sigma }\right)-\mathrm{R}-{\mathsf{\sigma }}_0$$

(2)

Here, R represents the isotropic hardening parameter (measured in Pa), and ${\mathrm{J}}_2\left(\mathsf{\sigma }\right)$ signifies the 2nd invariant of the deviatoric stress, and its unit is (Pa). This equation provides a crucial insight into the material’s response to external loading conditions and is pivotal in understanding the behavior of structures subjected to complex stress states.

The second invariant of the deviatoric stress, ${\mathrm{J}}_2\left(\mathsf{\sigma }\right)$, is computed as follows:

$${\mathrm{J}}_2\left(\mathsf{\sigma }\right)=\sqrt{{\displaystyle \frac{3}{2}}\acute{\mathsf{\sigma }}.\acute{\mathsf{\sigma }}}$$

(3)

Here, $\acute{\mathsf{\sigma }}$ represents the stress deviatoric tensors (measured in Pa), a key component in assessing the material’s response to loading conditions. As previously mentioned, the variation in the size of the yield surface in the stress space is quantified by the isotropic behavior model, expressed by the following equation:

$${\mathsf{\sigma }}^{\mathrm{o}}={\mathsf{\sigma }}_0+{\mathrm{Q}}_{\mathrm{\infty }}\left(1-{\mathrm{e}}^{-\mathrm{b}{\mathsf{\epsilon }}_{\mathrm{p}\mathrm{l}}}\right)$$

(4)

In this equation, ${\mathsf{\sigma }}_0$ represents the initial yield stress (in Pa), while ${\mathrm{Q}}_{\mathrm{\infty }}$ (in Pa) and $\mathrm{b}$ are essential isotropic hardening parameters. Specifically, ${\mathrm{Q}}_{\mathrm{\infty }}$ signifies the maximum change achievable in the yield surface, whereas $\mathrm{b}$ characterizes the rate at which this change unfolds. It is worth noting that when the alterations to the yield surface are zero (${\mathrm{Q}}_{\mathrm{\infty }}=0$), the isotropic/kinematic hardening model simplifies into the nonlinear kinematic hardening model. This transition is of paramount importance in understanding the model’s behavior under varying loading conditions, shedding light on the material’s response to complex stress states.

The nonlinear kinematic hardening model used in this study follows the standard implementation provided in the ABAQUS User Manual [24]. In this model, the yield surface is defined based on the backstress parameter α, and its evolution is governed by the Armstrong–Frederick rule [21], which introduces a recall term to relate backstress to the plastic strain rate. The evolution law is given as follows:

$$\mathrm{d}\mathsf{\alpha }=\mathrm{C}{\displaystyle \frac{1}{{\mathsf{\sigma }}_0}}\left(\mathsf{\sigma }-\mathsf{\alpha }\right)\left(\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{p}\mathrm{l}}\right)-\mathsf{\gamma }\mathsf{\alpha }\left(\mathrm{d}{\mathsf{\epsilon }}_{\mathrm{p}\mathrm{l}}\right)$$

(5)

In this equation, $\mathrm{C}$ (in Pa), $\mathsf{\gamma },$ and $\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{p}\mathrm{l}}$ represent the initial kinematic modulus, kinematic parameter, and equivalent plastic strain rate, respectively, while ${\mathsf{\sigma }}_0$ characterizes the initial size of the yield surface. Notably, when $\mathsf{\gamma }$ is equal to zero, the nonlinear kinematic model simplifies into a linear kinematic model.

The initial yield surface size is typically defined as a fraction $\mathsf{\lambda }$ of the ultimate yield stress ${\mathsf{\sigma }}_{\mathrm{y}}$:

$${\mathsf{\sigma }}_0=\mathsf{\lambda }{\mathsf{\sigma }}_{\mathrm{y}}$$

(6)

Here, $\mathsf{\lambda }$ is the material constant.

Further details, including derivations and graphical representations of the yield surface and backstress evolution, are well documented in the ABAQUS manual and thus are not repeated here. A conceptual illustration of the combined hardening mechanism is shown in Figure 1.

The determination of the maximum yield stress for clay involves utilizing Equation (7), which derives from the Von Mises failure criterion and accounts for data obtained from cyclic shear testing:

$${\mathsf{\sigma }}_{\mathrm{y}}={\sqrt{3}\mathrm{S}}_{\mathrm{u}}$$

(7)

where ${\mathrm{S}}_{\mathrm{u}}$ (in Pa) represents the undrained shear strength of the clay. This equation plays a critical role in characterizing the clay’s behavior and serves as a valuable reference point for assessing its response to varying conditions, particularly in geotechnical analyses.

It should be noted that, while some of the equations in this section (such as Equations (1)–(6)) are presented in simplified scalar or one-dimensional forms to facilitate conceptual understanding, the actual implementation of the nonlinear and multilinear kinematic hardening models in the finite element simulations was fully tensorial and consistent with 3D stress–strain behavior. To avoid unnecessary mathematical complexity in the main text, the governing equations were initially presented in a reduced form to emphasize the physical significance of the model parameters and to facilitate comprehension. However, a brief appendix has been included to summarize the key tensorial formulations employed in the implementation (see Appendix A).

2.2. Calibration of Kinematic Hardening Model Parameters

The calibration of constitutive model parameters is essential in geotechnical engineering to accurately represent soil behavior under various loading conditions, which directly impacts foundation stability and performance. Different calibration methods, typically involving experimental data and finite element simulations, are used depending on the constitutive model employed [25]. Proper calibration is critical for generating reliable numerical predictions. This study focuses on the parameter calibration of the kinematic hardening model applied to the seismic response of undrained kaolin clay.

The kinematic hardening constitutive model effectively simulates deformation behavior across multiple cycles and accurately represents stress history. To substantiate and validate this model, experimental results [26] and simulation outcomes on a soil sample under cyclic loading are depicted in Figure 2. Figure 2 illustrates a direct comparison between the experimental response reported in [26] and the numerical results obtained from the present model. The close agreement between the two confirms the adequacy of the selected constitutive framework in reproducing cyclic deformation behavior under undrained conditions. This comparison further validates the model’s capability to represent accurately the stress–strain relationship over multiple loading cycles, as previously described in the experimental setup and cyclic loading conditions. Initially, a consistent pressure of 200 kPa was uniformly applied to all outer surfaces of the specimen. Subsequently, a cyclic load (q = 70 kPa) was imposed on the specimen’s top. The graph illustrating deviatoric stress versus axial strain was then plotted. The next section provides a comprehensive explanation of the process for calibrating the parameters of the constitutive model.

2.2.1. Calibration of Parameters C and λ

In this paper, the shear modulus equation presented in reference [20] is employed to determine the parameter C for undrained kaolin clay. It is important to highlight that this equation was originally adapted from Viggiani and Atkinson’s work [27].

$${\mathrm{G}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=2060{\left({\acute{\mathrm{P}}}_0\right)}^{0.653}$$

(8)

where ${\acute{\mathrm{P}}}_0$ is the initial mean effective normal stress (in $\mathrm{k}\mathrm{P}\mathrm{a}$), derived from the average of the following equations.

$${\mathsf{\sigma }}_1=\acute{\mathsf{\gamma }}\mathrm{h}$$

(9)

$${\mathsf{\sigma }}_2={\mathsf{\sigma }}_3={\mathrm{K}}_0\acute{\mathsf{\gamma }}\mathrm{h}$$

(10)

Here, $\acute{\mathsf{\gamma }}$ and $\mathrm{h}$ represent the effective unit weight (in ${\displaystyle \raisebox{1ex}{$\mathrm{k}\mathrm{N}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^3$}\right.}$) and the depth of soil (in $\mathrm{m}$), respectively. ${\mathrm{K}}_0$ represents the lateral coefficient, and its calculation is determined by employing Equation (11) for clay soils.

$${\mathrm{K}}_0=(0.95-\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\varphi })$$

(11)

The parameter $\mathsf{\varphi }$ represents the internal friction angle, measured in degrees.

The undrained Young’s modulus (${\mathrm{E}}_{\mathrm{u}}$) is determined using the following equation:

$${\mathrm{E}}_{\mathrm{u}}=\mathrm{E}=2\mathrm{G}\left(1+\mathsf{\nu }\right)$$

(12)

where $\mathrm{G}$ (in Pa) represents the shear modulus, and $\mathsf{\nu }$ denotes the Poisson ratio.

C can be determined from equations $\frac{\partial \mathsf{\alpha }}{\partial {\mathsf{\epsilon }}^{\mathrm{p}\mathrm{l}}}{|}_{{\mathsf{\epsilon }}^{\mathrm{p}\mathrm{l}}\to 0}\mathrm{C}=\mathrm{E}$ and (12), where C is equal to both $\mathrm{E}$ and ${\mathrm{E}}_{\mathrm{u}}$ ($\mathrm{C}=\mathrm{E}={\mathrm{E}}_{\mathrm{u}})$ (in Pa).

In the absence of undrained cyclic triaxial test data, parameter λ can be calibrated using cyclic shear test results under strain control conditions through simulation. This process relies on Vucetic and Dobry’s experimental equations [28], which consider the soil plasticity index. Calibration requires a dataset representing the shear modulus–shear strain relationship ($\mathrm{G}$-$\mathsf{\gamma }$ curve). The first step involves generating hysteresis loops for cyclic shear tests at different strain amplitudes. Figure 3 shows these loops, highlighting how increased strain enhances damping due to soil plasticity, reducing shear modulus. For each strain amplitude, the shear modulus at the stabilized cycle (red points in Figure 4) is determined, forming the $\mathrm{G}$-$\mathsf{\gamma }$ data for calibrating λ. Figure 4 verifies the kinematic hardening model against the $\mathrm{G}$-$\mathsf{\gamma }$ curve from Vucetic and Dobry [28].

2.2.2. Calibration of Parameters σ₀ and γ

To calibrate these parameters, the initial step involves obtaining data from monotonic undrained triaxial tests to determine the undrained shear strength of the soil (${\mathrm{S}}_{\mathrm{u}}$). Monotonic triaxial tests are a well-established practice in geotechnical engineering for assessing the undrained shear strength of soils. Undrained shear strength represents a soil’s ability to withstand shearing forces without undergoing significant pore water drainage. Consequently, monotonic triaxial tests offer essential insights into the undrained shear strength and stress–strain characteristics of soils.

Generally, the value of ${\mathrm{S}}_{\mathrm{u}}$ (in Pa) can be determined by conducting monotonic undrained triaxial tests (deviatoric stress vs. axial strain ($\mathrm{q}-\mathsf{\epsilon }$)) using the following equation:

$${\mathrm{S}}_{\mathrm{u}}={\displaystyle \frac{{({\mathsf{\sigma }}_1-{\mathsf{\sigma }}_3)}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}={\displaystyle \frac{{\mathrm{q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}$$

(13)

The research in article [20] provides data on soil undrained shear strength obtained through conducted tests. It is observed that soil undrained shear strength exhibits an approximately linear relationship with depth (${\mathrm{S}}_{\mathrm{u}}$ = 1.43 d, where ‘d’ represents the depth of the soil (unit: meters (m))), as illustrated in Figure 5. The readers are advised to refer to [20] for additional details regarding the value of undrained shear strength.

As the undrained shear strength (${\mathrm{S}}_{\mathrm{u}}$) values increased with depth, corresponding ${\mathrm{S}}_{\mathrm{u}}$ values were assigned to various depths within the models, as depicted in Figure 5, taking into account the specific soil properties (see

Table 1. Properties of soft Kaolin clay and sand.

Depth (m)	Bulk Density (kN/m2)	Elasticity Modulus (kN/m2)	Poisson Ratio, ν	Cohesion, C (kN/m2)	Internal Friction Angle, ϕ (°)	Dilatancy Angle, Ψ (°)	Coefficient of Lateral Pressure, (K0)
0–15.5	18	Var.	0.49	Var.	15	0.1	0.527
15.5–16	19	150,000	0.30	5	40	10	0.357

and

Table 2. Properties of the hardening parameters of soft Kaolin clay.

Soil Deposit	Depth (m)	PI (%)	C (KPa)	γ	σ0 (KPa)	Su (KPa)	α	β
Layer 1	0–1.5	33	12,942	9289	0.46	1.1	8.15821
Layer 2	1.5–3.5	33	28,407.4	6117	1.54	3.6	9.065172	9.19 × 10−4
Layer 3	3.5–5.5	33	41,699	4988	2.88	6.4	10.98304	8.27 × 10−4
Layer 4	5.5–7.5	33	53,016.3	4390.7	4.02	9.3	12.38411	6.83 × 10−4
Layer 5	7.5–9.5	33	63,166.6	4000	5.26	12.2	13.51774	6.06 × 10−4
Layer 6	9.5–11.5	33	72,512.65	3717.6	6.5	15	14.48328	5.55 × 10−4
Layer 7	11.5–13.5	33	81,256.7	3500	7.74	17.87	15.33168	5.18 × 10−4
Layer 8	13.5–15.5	33	89,526.3	3323.7	8.97	20.7	16.09294	4.89 × 10−4

).

It is noted that the linear relationship used (${\mathrm{S}}_{\mathrm{u}}$ = 1.43 d) does not imply a zero shear strength at the ground surface, which may only be realistic for seabed or soft marine deposits. In the context of this study, however, the uppermost soil layer has a finite initial undrained shear strength (e.g., ${\mathrm{S}}_{\mathrm{u}}$ = 1.1 kPa at depth 0–1.5 m). Therefore, a nonzero offset exists.

To better represent this condition, the general form of the undrained shear strength profile can be written as ${\mathrm{S}}_{\mathrm{u}}$ = 1.43 d + c. Here, c is a constant representing the initial undrained shear strength at the ground surface (i.e., the intercept at d = 0). This formulation allows for a more realistic representation of soil conditions where the shear strength does not start from zero, as is the case in this study. For example, based on the measured data, c may be taken as approximately 1.1 kPa to reflect the near-surface strength more accurately. The original linear relationship ${\mathrm{S}}_{\mathrm{u}}$ = 1.43 d is adopted from article [20], and in this study, it has been extended by introducing a constant offset to better fit the measured soil conditions.

Furthermore, while a more detailed piecewise interpretation of the raw ${\mathrm{S}}_{\mathrm{u}}$ data could potentially improve the local accuracy of the model inputs, the chosen linear approximation is deemed adequate for this study. It enables the simplified calibration of constitutive parameters without significantly affecting the relative comparisons and conclusions derived from the simulations.

After determining the value of ${\mathrm{S}}_{\mathrm{u}}$, the maximum yield stress (${\mathsf{\sigma }}_{\mathrm{y}}$) is calculated using Equation (11) (${\mathsf{\sigma }}_{\mathrm{y}}={\sqrt{3}\mathrm{S}}_{\mathrm{u}}$). Parameter ${\mathsf{\sigma }}_0$ (unit: Pa) represents a fraction ($\mathsf{\lambda }$) of (${\mathsf{\sigma }}_{\mathrm{y}}$) at great plastic strain, as determined by the following equation:

$${\mathsf{\sigma }}_0=\mathsf{\lambda }{\mathsf{\sigma }}_{\mathrm{y}}$$

(14)

The maximum yield stress value (in Pa) is calculated using the following equation:

$${\mathsf{\sigma }}_{\mathrm{y}}={\displaystyle \frac{\mathrm{C}}{\mathsf{\gamma }}}+{\mathsf{\sigma }}_0$$

(15)

After determining all the aforementioned parameters and considering Equations (7) and (15), the parameter ($\mathsf{\gamma }$), which signifies the rate of decrease in the kinematic hardening modulus, is derived from the following Equation (16).

$$\mathsf{\gamma }={\displaystyle \frac{\mathrm{C}}{{\mathsf{\sigma }}_{\mathrm{y}}-{\mathsf{\sigma }}_0}}={\displaystyle \frac{\mathrm{C}}{{\sqrt{3}\mathrm{S}}_{\mathrm{u}}-{\mathsf{\sigma }}_0}}$$

(16)

2.3. Multilinear Kinematic Hardening Constitutive Model

The multilinear kinematic hardening model in this study defines uniaxial yield stress based on uniaxial plastic strain, with the number of stress–plastic strain pairs corresponding to model subvolumes. Using the soil backbone curve (Figure 6), the stress–plastic strain curve is derived (Figure 7) for this investigation. This model evaluates soil response to seismic loading by inputting stress–plastic strain data into Abaqus. While parameters for multilinear and nonlinear kinematic hardening models align, the multilinear model requires an optimized backbone curve, derived from hysteresis loops at various strain levels (Figure 3). This backbone curve forms the basis for the stress–plastic strain curve used in the model (Figure 7). For further details, refer to Meijuan et al. [29].

The multilinear kinematic hardening model used in this study follows the standard formulation available in the ABAQUS User Manual [24] and is widely employed to simulate nonlinear cyclic plasticity in geomaterials. While the full derivation can be found in the manual, here we present two key equations that are essential to understanding how stress is distributed among subvolumes.

The plastic strain is decomposed as the sum of weighted contributions from each subvolume:

The multilinear kinematic hardening constitutive model, employed in this study, is defined by Equation (17), which outlines the flow rule governing its behavior:

$$\mathrm{d}{\mathsf{\epsilon }}^{\mathrm{p}\mathrm{l}}=\sum _{\mathrm{k}=1}^{\mathrm{N}}{\mathsf{\omega }}_{\mathrm{k}}\mathrm{d}{\mathsf{\epsilon }}_{\mathrm{p}\mathrm{l},\mathrm{k}}{\displaystyle \frac{\partial {\mathrm{F}}_{\mathrm{k}}}{\partial {\mathsf{\sigma }}_{\mathrm{k}}}}$$

(17)

The total stress is similarly obtained as a weighted sum of subvolume stresses:

$$\mathsf{\sigma }=\sum _{\mathrm{k}=1}^{\mathrm{N}}{\mathsf{\omega }}_{\mathrm{k}}{\mathsf{\sigma }}_{\mathrm{k}}$$

(18)

In these equations, $\mathrm{N}$ represents the total number of subvolumes, while ${\mathsf{\omega }}_{\mathrm{k}}$, ${\mathrm{F}}_{\mathrm{k}}$, $\mathrm{d}{\mathsf{\epsilon }}_{\mathrm{p}\mathrm{l},\mathrm{k}}$, and ${\mathsf{\sigma }}_{\mathrm{k}}$ denote the weights, Mises yield surface, equivalent plastic strain rate, and stress of the kth subvolume (in Pa), respectively.

The subvolume-specific hardening modulus ${\mathrm{H}}_{\mathrm{k}}$ is computed from input stress–strain data using the following equation:

$${\mathrm{H}}_{\mathrm{k}}={\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{k}+1}-{\mathsf{\sigma }}_{\mathrm{k}}}{{\mathsf{\epsilon }}_{\mathrm{k}+1}^{\mathrm{p}\mathrm{l}}-{\mathsf{\epsilon }}_{\mathrm{k}}^{\mathrm{p}\mathrm{l}}}}$$

(19)

This model allows the stress–strain curve to be approximated using piecewise linear segments, making it especially suitable for cyclic loading analysis.

3. Model Descriptions

3.1. Finite Element Analysis

In this study, we used 3D Abaqus finite element software (2022 version) to analyze the response of clay–pile models under seismic loads. The numerical results from the KHM (kinematic hardening model) and NKH (nonlinear kinematic hardening) models were compared with centrifuge model findings from [20], showing a good agreement.

In static analyses of soil–pile interaction, soil boundaries can be assumed fixed, but this assumption fails in dynamic analyses due to wave propagation effects. In numerical modeling, only part of the foundation is included within limited geometric boundaries. Neglecting proper boundary conditions may cause incidents and reflected earthquake waves to interfere, generating additional waves and leading to significant errors. Thus, boundary conditions play a crucial role in dynamic analysis, critically affecting the seismic behavior of the model.

Various approaches have been proposed to simulate boundary conditions in dynamic soil–structure interaction analyses, including infinite element methods [30,31] and Tie Degree of Freedom techniques [32,33,34]. In this study, the infinite element method is employed to effectively reduce the reflection of seismic waves at the model boundaries, thereby improving the reliability of the dynamic response.

In pile–soil interaction, damping effects are present, but damping between the pile and soil should be minimized [35]. The Rayleigh damping mechanism, commonly used in dynamic analyses, plays a key role. However, its effective use requires a preliminary frequency analysis of the soil–pile model, as performed in this study, to determine the Rayleigh damping coefficients using the equations provided.

The fundamental equation governing Rayleigh damping is as follows:

$$\left[\mathrm{C}\right]=\mathsf{\alpha }\left[\mathrm{M}\right]+\mathsf{\beta }\left[\mathrm{K}\right]$$

(20)

The mass matrix is denoted by $\mathrm{M}$ (unit: kg), and the stiffness matrix is represented as $\mathrm{K}$ (unit: N/m). The damping ratios, $\mathsf{\alpha }$ (unit: 1/s) and $\mathsf{\beta }$ (unit: s), are proportional to mass and stiffness, respectively, and are obtained from the following equations:

$$\mathsf{\alpha }=\mathsf{\xi }{\displaystyle \frac{2{\mathsf{\omega }}_1{\mathsf{\omega }}_2}{{\mathsf{\omega }}_1+{\mathsf{\omega }}_2}}$$

(21)

$$\mathsf{\beta }=\mathsf{\xi }{\displaystyle \frac{2}{{\mathsf{\omega }}_1+{\mathsf{\omega }}_2}}$$

(22)

where ${\mathsf{\omega }}_1$ and ${\mathsf{\omega }}_2$ represent the angular frequencies of the 1st and 2nd modes (in rad/s), respectively. Material damping ($\mathsf{\xi }$) is assumed to be 10% for soft soils, as indicated in reference [36].

$${\mathsf{\omega }}_1=2\mathsf{\pi }{\mathrm{f}}_1 \mathrm{and} {\mathsf{\omega }}_2=2\mathsf{\pi }{\mathrm{f}}_2$$

(23)

where ${\mathrm{f}}_1$ and ${\mathrm{f}}_2$ denote the first and second frequencies (in Hz), respectively.

$${\mathrm{f}}_1={\displaystyle \frac{{\mathrm{V}}_{\mathrm{s}}}{4\mathrm{H}}} \mathrm{and} {\mathrm{f}}_2={\displaystyle \frac{3{\mathrm{V}}_{\mathrm{s}}}{4\mathrm{H}}}$$

(24)

where ${\mathrm{V}}_{\mathrm{s}}$ represents the shear wave velocity (in m/s), and $\mathrm{H}$ is the depth of the soil (in m). Upon completion of all the aforementioned procedures, the Rayleigh coefficients (α and β) were computed, as depicted in Table 2.

In this study, a finite element model was developed to simulate the dynamic behavior of the soil–pile–caps system under seismic loading, using solid elements for the soil ((Continuum 3D 8-node reduced integration element) C3D8R) and beam elements for the pile group or single pile and cap ((2-node linear beam element in 3D space) B31). The mesh size was carefully chosen to ensure accuracy, particularly refining the mesh near the pile and cap structures. This finer mesh improves the representation of the dynamic behavior and enhances the analysis’s overall accuracy. To further enhance the reader’s understanding of the reduced integration technique, we have referred to the study by the authors of [37], who provide a clear and detailed explanation of this method and its applications within advanced numerical modeling.

Hence, the mesh size, denoted as the maximum mesh length (${∆\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$), is a critical parameter that significantly influences the simulation’s accuracy. To ensure reliable results, the maximum mesh length (${∆\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$) used in the analysis must satisfy the following criterion (in m), as prescribed by reference [34].

$${∆\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}~{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}\right){\mathsf{\lambda }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$$

(25)

Here, ${\mathsf{\lambda }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ represents the wavelength of the shear wave with the lowest velocity (in m).

A mesh convergence study was conducted to further ensure mesh adequacy. The refinement level adopted in this analysis was based on convergence results that satisfied the criterion ${∆\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}~{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}\right){\mathsf{\lambda }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$, as prescribed in reference [34]. This approach confirmed that further refinement did not significantly change the output responses, particularly in stress and displacement. With 12,179 elements for the soil domain and 166 elements for the structural components, the selected mesh density was found to be sufficient for accurately capturing the dynamic interactions within the system. To further ensure the reliability and accuracy of the numerical results, a comprehensive mesh sensitivity analysis was conducted before finalizing the mesh configuration. Multiple levels of mesh refinement were examined, particularly around the pile and cap regions where stress concentration and soil–structure interaction are more pronounced.

Following established dynamic analysis practices, the maximum mesh size was determined based on the minimum shear wave wavelength, satisfying the criterion ${∆\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\le \left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}~{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}\right){\mathsf{\lambda }}_{\mathrm{m}\mathrm{i}\mathrm{n}}$. The refinement process continued until further reduction in element size caused negligible changes in the key output parameters such as bending moment and acceleration. This convergence-based approach confirmed the adequacy of the selected mesh and ensured that discretization errors had a minimal effect on the dynamic response. Consequently, the adopted mesh represents a balanced solution between computational efficiency and numerical precision.

Here, a meshing approach was used, resulting in 12,179 elements for the soil domain and 166 elements for the structural components (pile group and cap). This approach was designed to capture the dynamic interactions between the soil and structure. Figure 8 shows the 3D soil model with the pile group and cap, illustrating the meshing configuration. To compare the behavior of piles in a pile group under seismic loading to that of a single pile, two primary soil–pile systems were used: a soil–single pile system and a soil–pile group with two piles, both meshed using the same technique.

3.2. Properties of Materials

The soil medium comprises two distinct components: saturated kaolin clay and saturated sand soil. The overall dimensions of the analytical model encompass a length of 25 m, a width of 14.25 m, and a height of 16 m. The model’s total depth is 16 m, with a 15.5-m-thick layer of clay situated atop a 0.5-m-thick layer of sandy soil. It is important to note that as the depth of the soil increases, its stiffness also increases. Consequently, the soil’s stiffness varies at different depths, as illustrated in Figure 9. The specific soil properties utilized across all models are detailed in Table 1 [20].

The lateral coefficient, denoted as ${\mathrm{K}}_0$, is determined separately for clay and sand using Equations (11) and (26), respectively.

$${\mathrm{K}}_0=(1-\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\varphi })$$

(26)

To account for the depth-dependent variability of soil hardening parameters, the clay layer was discretized into eight layers, with the first layer having a thickness of 1.5 m and the subsequent layers having a thickness of 2 m each. This discretization scheme was chosen to accurately capture the nonlinear behavior near the surface while maintaining a balance between computational efficiency and the need to represent the gradual variation in stiffness and strength properties with depth. This division was essential in conducting a comprehensive investigation of the seismic response of both the soil–pile model and the soil–pile group system. Consequently, via the methodology outlined in Section 2.2, the kinematic hardening parameters specific to Kaolin soft clay were meticulously calibrated, and the resulting values, presented in Table 2, were directly obtained from this calibration process.

In the numerical analysis conducted in this study, the pile group configuration includes two rows, each containing one pile. These piles have dimensions of 12 m and 14 m in height, with a diameter of 1 m, and are constructed from aluminum material. Additionally, a cap with dimensions of 12.5 m in length, 3.75 m in width, and 1.2 m in thickness, also made of aluminum, is used. The piles are installed vertically with 9 m spacing (9D). The material properties of the piles and cap are shown in

Table 3. Material properties of the piles and cap.

Unit Weight (ρ)-($\raisebox{1ex}{$\mathbf{Kg}$}\!\left/ \!\raisebox{-1ex}{${\mathbf{m}}^{\mathbf{3}}$}\right.$)	Young’s Modulus (E)-(GPa)	Poisson’s Ratio (ν)
2700	70	0.33

. Figure 10 [20] illustrates the model arrangement, including the soil, pile group, and cap used in the analysis. Notably, the piles are modeled as aluminum with a linear elastic response under seismic loading. This assumption is based on the fact that aluminum has a relatively high yield strength compared to the anticipated seismic forces (PGA = 0.843 g). Given the primary focus on soil–pile interaction, the linear elastic model is deemed appropriate for this study. Future research may consider nonlinear material models to explore the impact of potential pile-yielding or pile–soil interface nonlinearities under more extreme conditions.

The use of aluminum for the piles and cap is not intended to replicate materials used in infrastructure projects but rather to represent the physical properties of the piles used in the referenced centrifuge tests [20], which serve as benchmarks for this study. This choice allows for the study to focus on isolating the nonlinear response of the surrounding soil under seismic excitation rather than on structural yielding. The high yield strength of aluminum (approximately 200–250 MPa) ensures that the piles remain within the elastic regime under the seismic loading (PGA = 0.843 g), making the linear elastic model appropriate for this analysis. This approach aligns with the validation dataset and provides a reliable basis for investigating soil–structure interaction mechanisms.

The selection of the 14-m pile length was made to match the configuration used in the referenced centrifuge test [20], which provides a validated experimental dataset for model benchmarking and comparative analysis. To investigate the influence of pile length on seismic performance under otherwise identical conditions, a second model with 12-m-long piles was developed. This alternative length was chosen to represent a moderately shorter pile embedded within the same soft clay layer, ensuring full embedment in both cases.

From a geotechnical perspective, moderate changes in pile length within the embedded soil layer influence lateral stiffness and the mobilization of passive resistance, especially in soft clay where confinement tends to improve with depth. The chosen lengths (12 m and 14 m) represent a practical and relevant range based on the referenced experimental setup and typical design scenarios, ensuring full embedment and avoiding complexities associated with excessively long piles that might introduce different failure mechanisms or modeling uncertainties.

This controlled variation enables the focused assessment of the effect of pile length on seismic response, particularly lateral displacements and bending moment distribution, while maintaining comparability between models.

In the final section, the effect of pile spacing on the dynamic behavior of pile groups under 0.845 g seismic loading is investigated, with pile distances ranging from 3D to 9D. These results are compared with those from soil–single pile models, where all conditions are identical to the pile-group models, allowing for the comparison of seismic behavior between pile groups and individual piles.

Figure 11 presents the time history of the applied acceleration to the model. Initially, to validate all models created in Abaqus, the simulated results of one of the investigated models were compared with those from the experimentation model [20] (Figure 12). For this purpose, the soil–pile group model (piles of 14 m in length) was subjected to an earthquake with an intensity of 0.06 g. This loading condition, represented by a peak ground acceleration (PGA) of 0.06 g [20] and a dominant frequency below 1 Hz, is vital for evaluating the system’s dynamic response under realistic seismic conditions. Subsequently, to examine the influence of earthquake intensity on the seismic behavior of the soil–pile model, three earthquakes with varying intensities were used.

4. Results and Discussion

This research employs a three-dimensional finite element method to investigate the effects of soil–pile–raft interaction in geotechnical engineering applications. Saturated clay was chosen as the soil medium for analysis, motivated by the relative scarcity of studies focusing on the seismic behavior of soft soils compared to coarser soils like sand and the increasing need for foundation systems on soft soil substrates. The study delves into the influence of several key parameters, including variations in earthquake intensity, the pile–soil stiffness ratio achieved through alterations in pile length, and pile spacing within a pile group. The primary objective is to evaluate these parameters’ influence on the seismic behavior of the pile in geotechnical applications where earthquake loading plays a crucial role.

Upon a rigorous analysis of the numerical results in comparison to data obtained from centrifuge model experiments, it becomes evident that both the multilinear and nonlinear kinematic hardening models are suitable constitutive models for studying the dynamic behavior of soft soils under earthquake loading. Notably, the nonlinear kinematic hardening model exhibits a closer alignment with the centrifuge model data than the multilinear kinematic hardening model, as demonstrated in Figure 12. Consequently, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 are exclusively dedicated to the nonlinear model.

The subsequent sections provide a comprehensive elaboration of the obtained results, offering insights into the seismic resilience of pile foundations in geotechnical engineering projects.

4.1. Model Verification

In this study, after analyzing the behavior of a pile group comprising two piles with a single cap in saturated kaolin clay under seismic loading, it was found that the bending moment distribution along the pile group increases from the bottom of the piles to the top, reaching its peak value at the interface between the piles and the cap. Figure 12 shows the bending moment distribution along the pile subjected to a seismic load with a PGA of 0.06 g. It is evident from this figure that the outcomes obtained through both the multilinear and nonlinear kinematic hardening models exhibit a rational and acceptable correspondence with the experimental data. Particularly noteworthy is the proximity of the results obtained from the nonlinear model to those derived from the centrifuge tests commonly employed in geotechnical engineering. It is worth noting that the presented results pertain specifically to the pile on the right, as illustrated in Figure 10. These findings contribute to a better understanding of soil–pile interaction mechanisms, where cyclic loading due to seismic and environmental forces significantly influences foundation stability.

At the base of the piles, where they are embedded in the soil, the transfer of loads from the surrounding soil to the piles is relatively uniform. As you move up the length of the piles, the soil–pile interaction becomes increasingly pronounced, leading to a greater concentration of load transfer. This results in higher bending moments as the seismic forces are more effectively transmitted to the piles. Also, piles typically exhibit a stiffness gradient along their length, with higher stiffness at the base and decreasing stiffness towards the top. The lower part of the piles resists deformations more effectively and, as a result, experiences lower bending moments. Conversely, the upper part of the piles, being more flexible, undergoes greater deformations and, consequently, higher bending moments. In dynamic environments, where loading from waves and currents compounds seismic effects, such variations in stiffness and load transfer become even more critical for foundation stability.

Seismic forces induce lateral shear forces in the soil, which are transmitted to the piles. Near the base of the piles, these shear forces may primarily cause translational movement without significant bending. However, as one moves upward along the pile length, the shear forces lead to both translational and rotational movement, resulting in increased bending moments. In geotechnical engineering, understanding these interactions is essential for ensuring the resilience of pile-supported structures, such as wind turbines, platforms, and coastal infrastructure.

Soft, cohesive soils like saturated kaolin clay exhibit non-linear stress–strain behavior. As seismic waves pass through the soil, they induce deformations, causing variations in stress and strain along the pile length. These variations contribute to a differential load distribution on the piles, influencing the bending moment distribution. The behavior of clay under seismic loading is particularly important due to its tendency to undergo excess pore pressure buildup, which may lead to strength degradation and affect the overall foundation performance.

The presence of the cap connecting the two piles in the group introduces additional stiffness and load-sharing characteristics. The cap redistributes loads within the group, concentrating forces towards the pile–cap interface. This redistribution maximizes the bending moment at the interface due to the enhanced transfer of forces and moments between the piles and the cap.

In summary, the increase in bending moment distribution along the pile group from the bottom of the piles towards the top, reaching its maximum value at the interface between the piles and the cap, results from the interplay of factors such as load transfer mechanisms, stiffness gradients, shear forces, pile–soil interaction, and the load redistribution effects introduced by the cap. These findings provide valuable insights into the design of pile foundations in geotechnical applications, contributing to improved engineering solutions for foundations subjected to seismic loading. Understanding these mechanisms is crucial for the design and analysis of pile foundations in seismic-prone regions.

Figure 13 illustrates the stress–strain behavior of soft clay as simulated using the multilinear kinematic hardening (KHM) model and the nonlinear kinematic hardening (NKH) model. In both cases, the material initially exhibits a hardening response, characterized by an increase in deviatoric stress with increasing axial strain. The KHM model approximates the stress–strain relationship through a series of discrete linear segments, resulting in abrupt reductions in the tangent stiffness at pre-defined strain thresholds. Conversely, the NKH model produces a smooth, continuous stress–strain curve, capturing the gradual degradation of stiffness with increasing strain more realistically. The continuous reduction in the secant modulus observed in the NKH model better represents the progressive yielding and strain accumulation typically encountered in soft cohesive soils under monotonic or cyclic loading.

A comparative assessment of the two models reveals that the NKH model offers a superior representation of post-yield softening, avoiding artificial stiffness transitions inherent to piecewise linear approximations. This continuous depiction of material degradation is particularly critical in dynamic analyses, where energy dissipation and strain accumulation mechanisms govern the seismic performance of soil–structure systems. While the KHM model provides computational simplicity and may suffice for preliminary analyses, its inability to fully capture smooth nonlinear softening could lead to an underestimation or overestimation of critical dynamic responses, particularly in simulations involving significant cyclic degradation. Therefore, the use of the NKH model is recommended when accurate simulation of soil behavior under seismic or cyclic loads is required to ensure reliable predictions of foundation performance.

Figure 13. Stress–strain response of soft clay.

Figure 13. Stress–strain response of soft clay.

4.2. Influence of the Ground Motion Effect on the System Response

The primary objective of this paper is to provide a comprehensive exploration of the inelastic seismic behavior exhibited by pile–raft foundations, with a particular focus on the interplay between soil–pile interactions under the influence of ground motions recorded during significant seismic events. The assumptions underpinning our analysis are presented below to foster a comprehensive understanding of our approach. The anticipated findings not only promise to enhance our physical understanding of the inelastic seismic behavior within such systems but may also hold implications for design considerations.

Within the realm of seismic response analysis for soil–pile systems, it is crucial to recognize that earthquake characteristics play a significant role in influencing system behavior. Hence, this study rigorously investigates the seismic response of piles constructed in soft clay under the effects of three distinct earthquakes, each characterized by varying levels of intensity. This multipronged investigation aims to shed light on the implications of earthquake-induced loading on the dynamic response of structural models, particularly in geotechnical applications where pile foundations serve as critical elements.

Figure 14a juxtaposes the bending moments experienced along the length of the pile when subjected to earthquake loading, characterized by peak accelerations of 0.6 g, 0.4 g, and 0.843 g (as detailed in Figure 15). The objective here is to discern the influence of earthquake intensity on the seismic behavior of the soil–pile system. Notably, the length of the piles remains consistent at 12 m across all three seismic loads, facilitating direct comparisons.

Figure 14b extends this analysis by investigating the implications of varying earthquake intensities (0.4 g, 0.6 g, and 0.843 g) on the bending moment distribution along a 14-m-long pile. A noteworthy observation is that, irrespective of the earthquake intensity, the maximum bending moment is consistently situated above the pile. However, the magnitude of bending moments is markedly amplified in the presence of the 0.843 g earthquake.

The relatively consistent behavior of the 12-m piles under varying earthquake intensities (Figure 14a) can be attributed to their lower slenderness and higher stiffness, which result in a stiffer soil–pile system with limited relative displacement and reduced curvature under seismic loading. In contrast, the 14-m piles, due to their increased length and flexibility, exhibit greater bending moment amplification. The longer pile length leads to a longer fundamental natural period, which increases the likelihood of resonance when the dominant frequency content of the ground motion aligns with it. This dynamic interaction amplifies lateral demands and bending moments as earthquake intensity increases.

This observation underscores a critical relationship between earthquake intensity and soil–pile seismic behavior. As earthquake intensity escalates, several interconnected factors come into play. Soil strength diminishes, and soil displacement amplitudes increase, culminating in heightened forces exerted upon the pile. Furthermore, in the context of saturated soils, especially under more severe seismic events, an increase in pore water pressure within the soft clay seabed layer leads to a reduction in soil stiffness, which, in turn, exacerbates displacement amplitudes among soil particles and precipitates a concomitant decline in soil strength. These multifaceted interactions highlight the need for a nuanced understanding of earthquake characteristics when evaluating the seismic response of piles in geotechnical applications and also raise concerns about potential soil liquefaction under certain conditions.

Figure 14. Comparison of maximum pile bending moment under the different earthquakes: (a) pile length = 12 m; (b) pile length = 14.

Figure 14. Comparison of maximum pile bending moment under the different earthquakes: (a) pile length = 12 m; (b) pile length = 14.

Figure 15. Adopted earthquake records: 1995 Kobe earthquake.

Figure 15. Adopted earthquake records: 1995 Kobe earthquake.

As shown in Figure 14, the magnitude of bending moments induced in the piles increases with rising earthquake intensity. Higher seismic acceleration results in greater inertial forces acting on the piles. These inertial forces lead to larger lateral displacements and deformations of the soil–pile system. As the seismic excitation intensifies, the piles must undergo larger rotations and displacements to accommodate these forces, necessitating an increase in the bending moment experienced by the piles. The behavior of saturated kaolin clay, like many cohesive soils, is characterized by non-linear stress–strain relationships. Under stronger seismic loading, the hydro-mechanical interactions within the clayey soil become more pronounced. This nonlinearity affects the distribution of forces along the piles, causing variations in the transfer of loads from the soil to the piles and influencing the bending moment distribution.

Variations in earthquake intensity have a notable impact on the acceleration response in different soil regions, warranting further discussion in the context of the seismic stability of foundations. This phenomenon is elucidated by investigating Figure 15, which illustrates the acceleration response at various soil depths within models subjected to 0.4 g and 0.843 g seismic events. Generally, across these models, earthquake-induced acceleration tends to decrease from greater depths toward the soil surface. Notably, in most depth regions, the acceleration response is higher in the model exposed to the most severe earthquake load (0.843 g). This behavior can be attributed to the soft kaolin clay present in the models, which has a dissipative effect on seismic wave propagation in the soil, contributing to energy attenuation across the soil domain.

As previously mentioned, it is evident that the earthquake acceleration response amplifies with greater depth from the soil surface to the bedrock, as illustrated in Figure 16. The increasing acceleration with depth can be attributed to the inherent stiffness gradient in the soil. Typically, soils become stiffer and more resistant to deformation with increasing depth, resulting in the greater transmission of seismic forces to deeper layers and heightened acceleration responses. In geotechnical engineering, such depth-dependent acceleration patterns are critical for evaluating the seismic resilience of pile foundations. As seismic waves propagate through the soil, they experience changes in velocity and impedance, which can lead to wave reflections and refractions. These wave interactions can amplify ground motion at certain depths, contributing to the observed increase in acceleration. Additionally, in clay deposits, excess pore water pressure generation under cyclic loading may further alter the seismic response, impacting both acceleration distribution and soil–structure interaction. Furthermore, the pile group itself can impact the acceleration distribution within the soil. Interactions between the pile, soil, and pile cap can influence the transmission of seismic forces and lead to variations in the acceleration response at different depths. Such interactions are particularly significant in dynamic engineering applications, where pile groups support critical infrastructure exposed to complex loading conditions. In summary, the observed increase in acceleration response with depth in the soil models subjected to seismic excitation can be attributed to a combination of soil properties, wave propagation effects, and the presence of the pile group.

Figure 16. Acceleration response in soil depth under the different earthquake intensities.

Figure 16. Acceleration response in soil depth under the different earthquake intensities.

The increase in acceleration with depth is primarily influenced by soil stiffness, wave propagation effects, and pile–soil interactions. As seismic waves travel deeper, the soil becomes stiffer, leading to amplified acceleration. Additionally, pile–soil interaction and wave reflections contribute to this depth-dependent acceleration pattern, especially in soft soil conditions.

These findings contribute to a deeper understanding of seismic wave transmission in soft soil, providing insights for the design and stability assessment of pile foundations.

4.3. Influence of the Pile Length on the Bending Moment

Another highly influential parameter governing the seismic response of soil–pile systems is the ratio of the pile’s elasticity modulus to that of the surrounding soil. In this study, we investigate the impact of varying pile length on the bending moment distribution along the pile, considering that an increase in pile length not only enhances the pile’s stiffness relative to the surrounding soft clay but also influences its bending behavior. Generally, investigating the impact of pile length in seismic models, such as assessing pile length within a pile group situated in saturated clay soil under seismic loading, holds paramount importance for several reasons.

Firstly, pile length directly influences the distribution of seismic forces among the piles, thereby affecting the load-sharing mechanism within the group. Secondly, the stability of the pile group during seismic events is significantly influenced by pile length, as longer piles provide enhanced resistance against lateral deformation and potential group instability. Furthermore, the choice of pile length is closely linked to cost considerations, as longer piles may incur higher project costs but provide improved seismic performance and structural safety for foundation systems. Lastly, in dynamic structural analyses under seismic loading, pile length serves as a critical parameter, potentially exerting a direct influence on the analysis outcomes in geotechnical engineering. Thus, the careful selection of pile length in seismic contexts, especially within pile groups in soft, saturated soils, is imperative due to its multifaceted impact on seismic force distribution, structural stability, project economics, and dynamic analysis results.

Figure 17a illustrates the distribution of bending moments generated within 12-m and 14-m-long piles under the influence of a seismic load with a PGA of 0.4 g. Additionally, Figure 17b and Figure 17c depict the bending moment distribution along the same piles when subjected to seismic excitations with PGAs of 0.6 g and 0.843 g, respectively. The findings from Figure 17 elucidate a notable trend: the magnitude of the bending moment induced in the piles demonstrates an upward trajectory as their length increases. This phenomenon is intimately tied to the structural stiffness relative to the surrounding soil conditions, where an augmentation in structural stiffness results in a corresponding escalation in the bending moment experienced by the piles. In simpler terms, this behavior can be attributed to the enhanced stiffness of longer piles, which enables them to better resist the lateral forces induced by seismic events, leading to greater bending moments.

In summary, the relationship between pile length and bending moment distribution, as depicted in Figure 17, underscores the critical interplay between structural stiffness, soil–pile interaction, and seismic design considerations. These insights contribute to a more comprehensive understanding of foundation behavior under seismic conditions and inform the design process to enhance structural resilience in earthquake-prone areas.

Figure 17. Comparison of maximum pile bending moment under the different earthquakes and piles: (a) PGA = 0.4 g; (b) PGA = 0.6 g; (c) PGA = 0.843 g.

Figure 17. Comparison of maximum pile bending moment under the different earthquakes and piles: (a) PGA = 0.4 g; (b) PGA = 0.6 g; (c) PGA = 0.843 g.

Figure 17 clearly illustrates that longer piles (14 m) consistently experience higher bending moments compared to shorter piles (12 m) under seismic loading. This increasing trend in bending moment magnitude with pile length is attributed to the enhanced structural stiffness relative to the surrounding soft clay, enabling deeper soil resistance mobilization and increased moment demands along the pile. Such behavior becomes especially evident at higher PGA levels, highlighting the role of pile stiffness in controlling dynamic response.

4.4. Analysis of Pile Group Effects on Settlement

In geotechnical engineering, the behavior of pile groups is of greater significance compared to that of single piles due to the complex soil–structure interaction and load distribution mechanisms. The collective response of piles as a group plays a pivotal role in ensuring the stability, load-bearing capacity, and overall performance of the structure under complex hydrodynamic and seismic conditions. Pile groups contribute to a more uniform allocation of loads within the soil. The collaborative interaction among the piles guarantees the equitable distribution of applied loads among the group, averting the formation of concentrated stress points and mitigating the risk of uneven settlement in soft soils. When piles collaborate within a group, they demonstrate an increased capacity for bearing loads compared to individual piles. The combined effect of adjacent piles empowers them to collectively bear greater loads, thereby enhancing the overall strength and stability of the foundation in dynamic environments.

In this section, two pile groups were subjected to seismic loading with an intensity of 0.843 g. The piles in one group have a length of 12 m, while in the other group, they are 14 m long. The center-to-center distance between the piles in both groups was varied, ranging from 3 times the pile diameter (3D) to 10 times the pile diameter (10D)—specifically 3D, 5D, 7D, 9D, and 10D—to investigate the maximum distance at which their behavior remains that of a pile group.

Commencing the analysis, the settlement values induced in individual piles, measuring 12 m and 14 m in length under a seismic acceleration of 0.843 g, were determined. Subsequently, these settlement values were compared with the settlements observed in the respective pile groups. The settlement in a single 12-m pile was contrasted with that in the pile group featuring 12-m piles (Figure 18), and similarly, the settlement in a single 14-m pile was evaluated against the settlement in the pile group with 14-m piles (Figure 19). This comparative analysis aimed to assess the collective behavior of the piles in response to seismic loading, considering variations in pile length and inter-pile distances within the groups to improve the understanding of soil–pile interaction in clay deposits.

Figure 18. Pile group settlement variation with 12-m piles under seismic loading at 0.843 g.

Figure 18. Pile group settlement variation with 12-m piles under seismic loading at 0.843 g.

Figure 19. Pile group settlement variation with 14-m piles under seismic loading at 0.843 g.

Figure 19. Pile group settlement variation with 14-m piles under seismic loading at 0.843 g.

The settlement results of both pile groups (Figure 18 and Figure 19) were compared, revealing a general trend: settlements in pile groups tend to increase as the distance between piles within the group increases (from 3D to 10D). Moreover, the settlement in a single pile exceeds that in pile groups with various inter-pile distances. Additionally, the settlement in the 12-m pile group exceeds that in the group with 14-m piles. For example, with a pile-to-pile distance of 5D, the settlement in the 12-m pile group is 34 mm, while in the 14-m pile group, it is 21 mm. Consequently, it can be inferred that the length of the piles significantly influences settlement behavior. This observation is particularly relevant in geotechnical applications, where pile foundations are subjected to complex interactions with soft soil layers.

Another notable finding is that in the 12-m pile group, the maximum distance between piles should be 9D for them to function as a pile group. As evident from the settlement results graph, when the distance between piles is 10D, the settlement (79 mm) is nearly equal to that of a single pile (82 mm). Therefore, if the distance between piles exceeds 9D, their behavior resembles that of individual piles. This phenomenon underscores the role of soil–pile interaction in seismic response prediction, as excessive spacing between piles can reduce the beneficial effects of group behavior, particularly in clay with low shear strength. In contrast, for the 14-m pile group, even with a distance between piles of 10D, their behavior continues to resemble that of a pile group. These results indicate that the length of piles has a significant impact on their performance as a group.

This difference in critical spacing between the 12-m and 14-m pile groups can primarily be attributed to pile length and the resulting soil–pile interaction zones. The longer 14-m piles extend into deeper, stiffer soil layers, generating larger and more overlapping zones of soil influence. This overlap preserves the group behavior even at wider spacings such as 10D. In contrast, the shorter 12-m piles interact mainly with shallower, softer soils, which produce smaller influence zones that become effectively isolated beyond a spacing of 9D. As a result, their behavior transitions to that of individual piles at larger spacings.

In Figure 20 and Figure 21, the percentage reduction in settlement for the pile groups, in comparison to that of a single pile, is presented for lengths of 14 and 16 m, respectively, as the inter-pile distance varies within the groups. As the results indicate, in pile groups, the percentage reduction in settlement increases with a decrease in the inter-pile distance up to the point where stress interference is avoided. For instance, in the 14-m pile group (Figure 21), when the distance between piles is 9D, there is a 90% reduction in settlement compared to the individual pile. However, reducing the spacing to 7D leads to a 137.5% reduction in settlement. This trend highlights the critical role of optimized pile spacing in enhancing foundation performance and structural resilience under seismic conditions. This trend and these results are similarly observed for the 12-m pile group (Figure 20). These findings contribute to the understanding of seismic-induced deformation in pile groups, providing valuable insights for improving foundation design in soft soil environments. The graphs illustrate that as the distance between piles decreases, the pile group’s ability to collectively reduce settlement improves.

Figure 20. Percentage reduction in settlement for the 12-m pile group compared to that of a single pile at 0.843 g.

Figure 20. Percentage reduction in settlement for the 12-m pile group compared to that of a single pile at 0.843 g.

Figure 21. Percentage reduction in settlement for the 14-m pile group compared to that of a single pile at 0.843 g.

Figure 21. Percentage reduction in settlement for the 14-m pile group compared to that of a single pile at 0.843 g.

As previously discussed, both pile length and spacing within a pile group play a critical role in influencing settlement behavior and overall performance, particularly under dynamic loading conditions. Larger pile lengths in pile groups exhibit a more substantial impact on reducing settlement and improving performance when compared to groups with shorter pile lengths. This behavior is particularly crucial for geotechnical infrastructure, where pile foundations must endure complex loading conditions, including seismic forces. Figure 22 illustrates the percentage reduction in settlement with a decreasing inter-pile distance for pile groups with lengths of 12 and 14 m.

In the case of the 12-m pile group, a reduction in the inter-pile distance from 7D to 5D results in a 20.59% reduction in settlement. Conversely, for the 14-m pile group, a similar reduction in inter-pile distance yields a more significant reduction of 52.38% in settlement. Furthermore, decreasing the inter-pile distance from 5D to 3D leads to a 36% reduction in settlement for the 12-m pile group and a substantial 110% reduction for the 14-m pile group. These findings emphasize the crucial role of pile spacing in optimizing the seismic resilience of foundation systems, particularly in soft soils, which exhibit nonlinear deformation characteristics under cyclic loading. Consequently, it can be concluded that an increase in pile length correlates with a decrease in pile group settlement. In other words, under comparable conditions, pile groups with longer piles exhibit lower settlement, and a reduction in inter-pile distance leads to a higher percentage reduction in settlement, indicating an overall improved performance of the pile group in dynamic soil environments.

4.5. Horizontal Displacement Behavior of Single and Group Piles Under Seismic Loading

In addition to vertical settlement, horizontal displacement plays a critical role in assessing the seismic performance of pile foundations. Excessive lateral displacement can compromise the structural stability of superstructures, particularly under seismic excitation. Figure 23 presents the horizontal displacement of single piles with lengths of 12 m and 14 m, as well as pile groups composed of 12 m and 14 m piles, all subjected to seismic loading with a peak ground acceleration of 0.843 g. The comparison reveals that group piles, irrespective of length, exhibit significantly lower horizontal displacements than their corresponding single-pile counterparts. This trend aligns with the earlier settlement results and emphasizes the beneficial effect of group interaction in mitigating seismic-induced lateral deformations. In this analysis, we included a pile group with an inter-pile spacing equal to three times the pile diameter (3D), as identified in Section 4.4, since this configuration demonstrated the lowest vertical settlement among all tested pile group arrangements. Specifically, the 14-m pile group demonstrates the minimum horizontal displacement among all configurations, followed by the 12-m pile group, the single 14-m pile, and finally the single 12-m pile with the highest lateral deformation. These results suggest that not only does the grouping of piles enhance vertical stability through load-sharing mechanisms, but it also reduces horizontal movement under dynamic conditions. The superior performance of group piles can be attributed to their collective resistance against lateral soil movement and increased system stiffness, which reduces the pile–cap displacement. Furthermore, longer piles contribute to improved lateral resistance due to their deeper embedment and greater soil–pile interaction along their length, especially in soft clay where the confining pressure increases with depth. The observed reduction in horizontal displacement with increased pile length and group arrangement underscores two important design implications. The first is the length effect: Piles with a length of 14 m consistently outperform 12-m piles in reducing lateral movement, both in single and group configurations. This indicates that longer piles engage a greater volume of confining soil and mobilize more passive resistance during seismic shaking. The second implication is the group effect: The horizontal displacements in pile groups are considerably lower than in single piles, even when subjected to identical seismic conditions. This implies that group pile foundations are more effective in dissipating seismic energy through distributed soil–pile interaction, which minimizes the risk of excessive lateral deformation and potential structural damage.

5. Conclusions

This study involved a comprehensive investigation of the seismic behavior of soil–pile–cap systems subjected to various earthquake loads, employing 3D Abaqus finite element software for analysis. Additionally, we conducted an in-depth investigation into how various parameters, such as earthquake intensity, pile length, and the spacing between piles within a pile group, affect the behavior of the soil–pile–cap model. This thorough investigation provided valuable insights into the dynamic response of such systems under diverse seismic conditions. The structural components, including the piles and cap, were modeled using aluminum material properties by a linear elastic model. The surrounding soil in our models consisted of saturated soft kaolin clay and was characterized using both multilinear and nonlinear kinematic hardening constitutive models. Through a meticulous calibration process for model parameters and rigorous analysis of the soil–pile systems, we achieved excellent congruence between our simulation results and those obtained from a centrifuge model, indicating the accuracy and reliability of our computational approach. It is noteworthy that the nonlinear kinematic hardening model demonstrated superior performance in capturing the cyclic behavior of soft clay, exhibiting a closer agreement with experimental data compared to the multilinear kinematic hardening model.

While the study provides valuable insights into the interaction between soil and pile foundations under seismic loading, it is important to acknowledge certain modeling simplifications. The piles were modeled with linear elastic material behavior, which is considered adequate for the range of seismic loading investigated (up to a peak ground acceleration of 0.843 g) but may not capture potential yielding or nonlinearity at the pile–soil interface under more extreme conditions. Additionally, the 3D numerical model assumes homogenous soil properties and does not account for spatial variability or heterogeneities typically found in clay deposits. These simplifications were made to maintain focus on the core objective of understanding soil–pile interaction under typical seismic scenarios, and the approach was validated using centrifuge test data. Nevertheless, future research may benefit from incorporating nonlinear material models for both soil and piles, along with more realistic boundary conditions and heterogeneous soil profiles, to enhance the model’s predictive capability in complex or high-intensity seismic environments.

The findings from this study are presented below:

• Selecting the appropriate and accurate constitutive model is the most crucial aspect of numerical analysis. While advanced constitutive models aim to reduce reliance on complex, time-consuming, and costly experiments by improving predictive capabilities, it is acknowledged that physical experimental data remain essential for the initial calibration and validation of these models. Thus, the use of advanced multilinear and nonlinear kinematic hardening models facilitates more efficient parameter tuning and can reduce—but not eliminate—the need for extensive experimental campaigns. Advanced multilinear and nonlinear kinematic hardening models exhibit these characteristics. Additionally, they effectively simulate hysteresis loops under cyclic loads and account for the degradation of soil stiffness under seismic conditions. Advanced multilinear and nonlinear kinematic hardening models exhibit these characteristics. Additionally, they effectively simulate hysteresis loops under cyclic loads and account for the degradation of soil stiffness under seismic conditions.
• The seismic response of soil-embedded piles is notably affected by earthquake characteristics, particularly intensity. Earthquake intensity exerts a substantial influence on the behavior of piles installed in saturated clay soils. As earthquake intensity increases, there is a consequential reduction in soil stiffness, leading to an increase in the bending moment experienced along the pile. The highest bending moment is typically concentrated at the interface between the pile and the pile cap. This phenomenon is critical in foundation systems, where seismic activity can exacerbate the effects of soil degradation.
• The bending moment experienced by piles can be influenced by various critical parameters, among which is the pile–soil stiffness ratio, achieved through modifications in pile length. The response of a pile to seismic loads varies with its length. As the pile length increases and the stiffness of the structure relative to the soil medium escalates, the magnitude of the bending moment generated also increases, as clearly observed in the results presented in Figure 17. This emphasizes the importance of incorporating stiffness-related effects in early seismic design evaluations. These insights are essential for optimizing pile foundation design to withstand dynamic forces and improve overall structural stability.
• The bending moment distribution within a pile group comprising two piles connected by a single cap and embedded in saturated kaolin clay under seismic loading exhibits an upward trend, with its peak occurring at the interface between the piles and the cap. This behavior highlights complex pile–soil interaction mechanisms that are crucial for foundation systems exposed to dynamic and cyclic loading conditions.
• In pile groups with longer piles, reducing the inter-pile spacing results in a significantly higher reduction in pile group settlement compared to groups with shorter piles. For example, in a 12-m pile group, reducing the spacing from 7D to 5D led to a 20.59% decrease in settlement. Conversely, in a 14-m pile group, a similar reduction in spacing caused a 52.38% reduction in pile group settlement. These findings underscore the importance of pile spacing optimization in geotechnical engineering applications, where reducing settlement is critical for the stability of foundation systems.