Discrete element study on mechanical properties of layered sand-cobble strata under unloading stress path of shield construction

Abstract

Aiming at the deformation problem of sand-cobble composite strata induced by tunnelling, the stress path of soil during tunnel excavation was analyzed by numerical simulation. Two stress path schemes (σ₁ reduction-σ₃ reduction, σ₁ reduction-σ₃ stability) were proposed. The triaxial test model of composite stratum is constructed. The DEM simulation results demonstrated significant differences in the macroscopic mechanical response of soil under different stress paths. The simultaneous unloading of axial pressure and confining pressure will accelerate soil softening, while the decrease of confining pressure unloading rate can improve the toughness of soil. With the increase of the thickness of cobble stratum, the bearing capacity and toughness of the soil show an increasing trend, and the increase of cobble particles enhances the overall stability of the soil. In addition, the increase of stress ratio also leads to the improvement of soil strength and stability, so that the soil can better resist the softening effect caused by stress release during unloading.

Introduction

With the continuous development of China’s economy and society, underground space has been progressively developed and utilized, facilitating the extensive growth of urban rail transit systems that primarily composed of subways. According to statistics from the China Association of Metros, according to the China Association of Metros, 59 cities in mainland China had operational urban rail transit systems by the end of 2023, with a total network length of 11,232.65 km, of which 76.1% (8547.76 km) was accounted for by subways. Composite strata with alternating soil layers are extensively distributed across China’s vast territory. However, the significant disparity in mechanical properties between these layers, particularly under varying stress paths, leads to less predictable soil deformation during shield tunneling. Yu et al.¹ established a correlation between the composition ratio and shield tunneling parameters in sand-clay composite strata, based on field excavation tests from the Bailiu section of Shijiazhuang Metro Line 1, thereby providing a basis for further investigation.

The stability of composite strata is critically governed by the stress state of the constituent soils and rocks. During shield tunnel excavation, the soil experiences a complex evolution of stress paths, including compression, shearing, and unloading, which presents significant challenges for stability analysis. The stress path, defined as the evolution of the stress state during loading, is a critical factor controlling the mechanical properties and failure mechanisms of geotechnical materials. Different stress paths lead to distinct variations in the soil’s stress-strain response, strength properties, and failure modes. The investigation of stress paths enables a more accurate characterization of soil mechanical behavior under realistic engineering conditions, thereby providing a sound theoretical basis for design and construction. For example, Zhu et al.² investigated the strength characteristics of soil under different stress paths via triaxial tests, demonstrating the significant influence of stress paths on soil strength. Zhao et al.³ systematically examined the mechanical properties and strength criteria of soils during triaxial unloading tests along various stress paths. Their work offers valuable insights into the mechanical behavior of soils under complex stress conditions. These studies establish that stress paths significantly influence the mechanical properties of geomaterials, especially during underground excavations where stress states are complex and variable, making the investigation of these effects of paramount importance. Although scholars worldwide have conducted extensive research on the mechanical properties and failure mechanisms of geomaterials under various stress paths through laboratory experiments and numerical simulations, their work has predominantly focused on single soil types. For example, conventional uniaxial and triaxial compression tests typically^4,5,6,7 employ a loading method with constant confining pressure and varying axial pressure^8,9,10. While these tests can determine important mechanical properties such as stress-strain curves, strength, and failure mechanisms of soil, a primary shortcoming is their inability to fully replicate the in-situ stress state of soil during underground excavation, especially for the unloading process that widely encountered in underground engineering owing to limitations in test conditions and variations in soil properties. Stress-strain behaviors, as well as the mechanical parameters of surrounding soil under unloading state, are different with that under loading state as simulated in traditional laboratory triaxial tests. In addition, current field monitoring or model test results in shield tunneling engineering mainly records macro-level parameters such as ground surface settlement, segment deformation, and pore water pressure^11,12,13,14, while direct and continuous measurements of the stress path evolution and the associated stress-strain relationship within the soil mass remain technically challenging and rarely available in practice. Some scholars suggest that stress paths have only a marginal influence on soil strength^15,16,17, which further highlights the limited applicability of existing research conclusions. More importantly, existing research is predominantly concerned with single soil types, while studies on composite strata (complex formations consisting of multiple geotechnical materials) remain comparatively limited. As show in present researches, Composite strata are prevalent in practical engineering, where they exhibit complex mechanical behavior due to interactions between different soil layers^18,19,20. It is necessary to conduct stress path studies that accurately simulate in-situ conditions, with a particular focus on composite strata to investigate their mechanical properties and failure mechanisms under different stress paths. Such research will provide a more reliable theoretical basis for engineering design and construction.

In recent years, numerical simulation methods based on the discrete element method (DEM) have become increasingly important in shield tunnel research^21,22,23,24. Li et al.²⁵ utilized the discrete element method (DEM) to simulate the micromechanical properties and failure mechanisms of soil under various stress paths. Their research revealed the crack evolution process from a micromechanical perspective, offering a novel approach for investigating soil failure mechanisms. Research has demonstrated that different unloading paths result in distinct macroscopic strength characteristics of soil specimens. The discrete element method (DEM) simulates soil deformation and failure under complex stress states by representing geotechnical materials as assemblies of discrete particles. This approach provides insights into both macroscopic mechanical properties and the microscopic mechanism of crack development, thereby offering new avenues for in-depth investigation of soil failure mechanisms.

This study utilized the discrete element method to first analyze the evolution of stress paths in the soil mass ahead of the tunnel face during shield tunneling. Subsequently, numerical triaxial tests are conducted on composite specimens of sandy soil and cobble under various stress paths. The research investigates the differences in macroscopic strength and deformation characteristics of various composite soil formations under the influence of unloading effects during shield construction. The findings provide valuable insights for analyzing soil mass unloading failure during shield tunnel construction.

Analysis of soil stress paths during shield tunnel construction

The stress path concept was first introduced by Lambe²⁶. Subsequently, numerous researchers have conducted extensive studies focusing on stress path alterations in the surrounding soil induced by shield tunnel excavation, employing laboratory model tests²⁷, numerical simulations^28,29, and field monitoring^30,31. These investigations have established that soil stress paths are significantly modified during underground construction, and that soil at different locations experiences distinct stress paths.

Soil stress path analysis

To analyze the variation characteristics of soil stress paths at different locations during shield tunnel excavation, a discrete element numerical analysis model was constructed based on the centrifugal model tests of tunnel excavation face instability conducted by scholars such as Tang et al.³², as shown in Fig. 1. The area marked in Fig. 1 represented the boundary range selected for this numerical simulation. This study selected a tunnel with a burial depth ratio (C/D) of 1 as the control group. The model dimensions were 750 mm × 800 mm, with a tunnel crown burial depth (C) of 100 mm, a tunnel diameter (D) of 100 mm, and a tunnel penetration depth through the soil of 250 mm.

In the discrete element model, the rigid boundaries of the experimental apparatus were simulated using wall boundaries. The corresponding simulated soil was created via the gravity deposition method. A linear elastic contact model was used for the interactions between soil particles. The simulation parameters are summarized in Table 1. The discrete element model employs the same retraction method as used in field tests to simulate soil loss induced by tunnel excavation. A retraction velocity is applied to the excavation face, causing it to gradually retreat into the tunnel and thereby induce instability in the soil ahead of the excavation face. Three stress monitoring points were installed at three locations directly ahead of the tunnel face, from bottom to top, and designated as Point A, Point B, and Point C, as shown in Fig. 2.

Fig. 1

Model box size diagram³².

Table 1 The value of soil simulation parameters.

Fig. 2

Shield tunnel model and measuring point layout.

Analysis of stress path results​

The stress conditions at the three monitoring points following the simulated tunnel excavation are presented in Fig. 3. Following excavation, stress redistribution occurred within the surrounding rock mass. The vertical stresses decreased to varying degrees due to the excavation-induced unloading effect. In contrast, the horizontal stresses at different burial depths exhibited two distinct response patterns: a noticeable reduction or a state of relative stability. These observed stress variations are consistent with the changes identified in laboratory model tests. As shown in Fig. 3a, both the maximum and minimum principal stresses at monitoring point A decreased significantly with increasing excavation face displacement. Monitoring point B, shown in Fig. 3b, exhibits a stress variation trend similar to point A; however, the reduction in its minimum principal stress (the horizontal stress) is only half that of point A. For monitoring point C as shown in Fig. 3c, its shallow burial depth results in a smaller reduction in the maximum principal stress (vertical stress) during tunneling, while the horizontal stress (minimum principal stress) remains nearly constant.

Based on the aforementioned results, two stress paths are identified for the soil during shield tunnel excavation: one with a decrease in both σ₁ and σ₃, and the other with a decrease in σ₁ while σ₃ remains stable. For the first path, soils at different burial depths exhibit different unloading rates. Conventional triaxial loading tests, which typically maintain a constant confining pressure while increasing axial pressure, often fail to realistically replicate the stress paths experienced by the surrounding soil during the tunnel excavation process. Based on the simulation results, this study considers two main factors when determining the stress paths for subsequent triaxial unloading test simulations: Firstly, for the first stress path, the simulation is conducted by simultaneously reducing σ₁ and σ₃ in the triaxial test. Taking into account the influence of unloading, the reduction magnitude and rate of σ₁ and σ₃ vary at different burial depths. Therefore, different unloading rates can be set for σ₁ and σ₃ in the simulation. Secondly, for the second stress path, since σ₃ changes minimally, the simulation maintains σ₃ constant via servo control and only reduces σ₁ to simulate the unloading process.

Fig. 3

Stress changes of measuring points during shield tunnel excavation: (a) Monitoring point A; (b) Monitoring point B; (c) Monitoring point C.

Discrete element triaxial model construction

Parameter calibration

The particle-particle contact simulations in this study encompass sand–sand, sand–cobble, and cobble–cobble interactions. With reference to^29,33, A linear contact model is applied to all three types. A more sophisticated model such as Hertz–Mindlin contact model and Rolling Resistance Linear contact model would provide a more ‘realistic’ value. However, it will not affect the main behavior of stress-strain curve of sand-cobble composite sample with different combination of sand particles and cobble particles. To calibrate the micromechanical parameters for the three contact types, discrete element simulations of triaxial tests were conducted on sand samples with 0% stone content and cobble samples with 100% stone content. These simulations referenced triaxial test results for sand-cobble soils from the existing literature³⁴.

The particle size distribution of soil samples for the discrete element triaxial calibration model was determined according to the distribution from the large-scale triaxial tests referenced in the literature. Owing to the fine particle size of natural sandy soil, the size of the sand particles was scaled up by a factor of 10 in this study to ensure computational efficiency^35,36,37. However, since cobble particles are relatively large, their particle size distribution was simulated using a gradation identical to that of the actual soil. The particle gradation curves for the two soil discrete element models are presented in Fig. 4. The sand particles have a median particle size (d₅₀) of 6.95 mm and a coefficient of uniformity (C_u) of 1.21. The cobble particles have a median particle size (d₅₀) of 63.2 mm and a coefficient of uniformity (C_u) of 1.03. The particle size of cobble particles is represented by spherical particles of the same volume. Sand particles are simulated using spherical particles, while cobble particles are simulated using clustered particles comprising two spherical particles. Here, r_s denotes the radius of the spherical sand particles, r_c represents the diameter of the spherical particles within a cobble particle cluster, and s indicates the center-to-center distance between the two spherical particles in the cluster. A ratio of s/2r_c = 0.7 is adopted in this discrete element model. The specific configuration is illustrated in Fig. 5.

Fig. 4

Particle gradation diagram of sand cobble numerical sample.

Fig. 5

Sand (right) and cobble (left) discrete element numerical model.

Table 2 The value of soil simulation parameters.

Figure 6 presents a comparative analysis of the stress-strain curves derived from numerical simulations under confining pressures of 0.5 MPa, 1.5 MPa, and 2.5 MPa alongside laboratory test data. The values of soil simulation parameters are provided in Table 2. The results indicate that the discrete element triaxial simulations, using the selected parameters, show close agreement with well with the laboratory large-scale triaxial tests. These parameters effectively represent the stress-strain relationship of both sandy soil and cobble specimens. For the contact between sand and cobble particles, since the simulated specimen in this study consists of layered sand and cobble particles, sand-cobble contacts only occur at the interface between the two soil layers. These contact parameters have a minor influence on the overall stress-strain behavior of the layered specimen under triaxial unloading stress paths. Therefore, to simplify the analysis, the relevant contact parameters were set to the midpoint value between those of sand–sand contacts and cobble–cobble contacts.

Fig. 6

Comparison of indoor test and model test curve: (a) Stress–strain curve variation for sandy soil; (b) Stress–strain curve variation for cobble.

Development of a triaxial test model for composite soil layers​

In order to investigate the stress-strain behaviours of the sand-cobble composite strata in the triaxial test, an upper-lower layered specimen as shown in Fig. 7a is adopted in this study. It should be noted that in more common state, these two kinds of particles are randomly mixed in natural sand-pebble strata. The specimen in this study serve as an simplified conceptual model to isolate and study the fundamental mechanical interactions and load-transfer mechanisms between two clearly defined material units. The model was generated according to the following steps:

• (1) Sand particles and cobble particles are randomly generated with an initial porosity of 0.40^33,38,39,40 in the up region and bottom region within a box with dimension of 0.45 m×0.45 m×1.125 m (as shown in Fig. 7), respectively. The region of sand particles and cobble particles is separated by a rigid wall.

• (2) The generated model is cycled to equilibrium with the soil simulation parameters listed in Table 2.

• (3) The rigid wall separated the two kinds of particles is deleted and the whole sample is consolidated to the designated stress state under servo control.

• (4) After reaching the designed initial state, the sand-cobble composite sample can be unloaded according to the designed stress path. The generated sample is shown in Fig. 7b.

Fig. 7

Numerical model diagram: (a) Plan view of the model; (b) Three-dimensional view of the model.

Design of the loading test program for various stress paths​

Based on established research and prior numerical simulation results, three stress path simulation methodologies have been developed to capture the two distinct stress path behaviors observed in the surrounding soil during shield tunneling.

Considering the composition ratio of the composite strata (m_i), the stress ratio between the maximum and minimum principal stresses (k_i), and different combinations of stress paths, twelve sets of simulations were designed. The composition ratio represents the height ratio of sand to cobble in the triaxial test.

$${\text{mi = }}\frac{{{\text{h1 }}}}{{{\text{h2 }}}}$$

(1)

where h₁ denotes the height of the sand layer, and h₂ denotes the height of the cobble layer in the model.

$${\text{ki = }}\frac{{{\sigma _z}}}{{{\sigma _{x/y}}}}$$

(2)

where σ_z denotes the axial stress, and σ_x/y denotes the confining pressure.

Based on existing true triaxial test results, unloading tests under different stress paths were conducted, taking the critical state point from the loading test under a confining pressure of 0.5 MPa as the unloading point. The unloading process uses stress control, with three unloading schemes designed:

Scheme 1: the unloading rates for both σ_z and σ_x/y are 0.01 MPa/s.

Scheme 2: the unloading rate for σ_z is 0.01 MPa/s, whereas that for σ_x/y is 0.005 MPa/s.

Scheme 3: the unloading rate for σ_z is 0.01 MPa/s, while σ_x/y is held constant.

The specific simulation schemes are presented in Table 3.

Table 3 Numerical simulation test groups.

Analysis of simulation results

Analysis of results under different stress paths

Fig. 8

Simulation results of composite ratio under different stress paths (k_i = 1.0): (a) m_i = 0.5; (b) m_i = 1.0; (c) m_i = 2.0.

Fig. 9

Simulation results of different composite ratios under the same stress paths (k_i = 1.0).

Figures 8 and 9 show the simulation results for three stress paths under different composition ratios. The results demonstrate significant variations in the macro-mechanical response of the composite strata across these stress paths.

In comparison to Scheme 1, Scheme 2 exhibits a slower stress release process in the specimen, resulting from the decreased unloading rate of the confining pressure. Consequently, the stress-strain curve maintains relative stability during the initial unloading stage before undergoing a gradual decline, indicating the toughness of the soil as it progressively adapts to the stress alterations. The time to failure, which is characterized by the deviatoric stress dropping to zero, is also markedly delayed. Under stress path scheme 3, the confining pressure is held constant while the axial stress is reduced. The specimen demonstrates a more stable stress-strain response, showing a smoother unloading curve. This indicates that soil deformation is constrained under constant confining pressure, resulting in high overall stability.

Under different stress paths, the internal stress state and deformation mechanisms of soil exhibit distinct characteristics. The simultaneous unloading of both axial and confining pressures induces rapid stress release within the soil, weakening interparticle interactions and accelerating the softening process. In contrast, when the confining pressure unloading rate is reduced, the soil gains sufficient time to adjust its internal stress state, thereby preserving interparticle contacts and enhancing soil toughness. This results in a delayed stress release process. Under constant confining pressure conditions, the soil maintains a relatively stable internal stress state with strong interparticle forces, which effectively restricts the soil’s deformation capacity.

Results for various composition ratios

Fig. 10

Simulation results of stress path under different composite ratios (Scheme 1, k_i = 1.0).

Fig. 11

Simulation results of stress path under different composite ratios (Scheme 2, k_i = 1.0).

Fig. 12

Simulation results of stress path under different composite ratios (Scheme 3, k_i = 1.0).

As shown in Figs. 10, 11 and 12, under stress path scheme 1, the deviatoric stress–strain curves of the specimens show clear variations as the composition ratio (m_i) increases from 0.5 to 1.0 and then to 2.0. When m_i = 0.5, the curve reaches the peak deviatoric stress at a relatively low axial strain, followed by a rapid decrease, indicating that failure occurs at low strain levels. When m_i = 1.0, the peak deviatoric stress rises, and the post-peak descending segment becomes relatively gentler, reflecting an improvement in pre-failure bearing capacity and a more gradual failure process. When m_i = 2.0, the peak deviatoric stress increases further, and the post-peak descending segment becomes even more gradual, demonstrating that the soil exhibits higher bearing capacity and enhanced toughness at larger composition ratios.

When the axial strain (ε_z) is zero, the specimen remains in an undeformed state. At this point, specimens under different stress ratios show differences in their initial stress states due to variations in confining pressure and axial stress. Under stress path scheme 2, as the composition ratio increases, the deviatoric stress-axial strain curves of the specimen also exhibit significant differences. With increasing composition ratio, the peak stress gradually decreases, and the rate at which the curve reaches peak stress slows down, leading to accelerated specimen failure.

Under stress path scheme 3, the deviatoric stress–axial strain curves of the specimen follow similar trends as the composition ratio increases, although specific values differ. At all composition ratios, the curves maintain relatively stable deviatoric stress levels during unloading, with a relatively gentle declining segment. When m_i reaches 2.0, the curve exhibits the highest peak deviatoric stress and the most gentle declining segment, indicating that the soil exhibits the strongest stability at higher composition ratios. In Scheme 3, where the confining pressure remains constant and only the axial pressure is reduced, the internal stress state of the soil is maintained in a relatively stable condition.

Under different stress paths, the mechanical response of soils with different formation composition ratios also varies. Overall, however, both the bearing capacity and toughness of the soil generally increase with higher composition ratios. Variations in the composition ratio influence the arrangement and distribution of particles within the soil, consequently modifying its mechanical properties. Sand and cobble exhibit different mechanical characteristics: sand particles, being smaller in size, are more susceptible to particle rearrangement and mutual displacement, whereas cobble particles, due to their larger dimensions, help enhance the overall stability of the soil. As the composition ratio decreases, the proportion of cobble particles in the soil increases, thereby improving the global bearing capacity and deformation resistance of the soil. The addition of cobbles enhances the stabilizing effect of cobble particles on the soil mass, enabling the soil to maintain better overall stability during unloading. Although the mechanisms of interaction and stress transfer between particles within the soil remain similar under different composition ratios, the specific values differ. These differences lead to variations in mechanical response under identical stress paths. Changes in the composition ratio affect the overall mechanical properties and deformation resistance of the soil. Compared to pure sand and pure cobble samples, the strength of composite soil specimens was significantly lower than that of single-type soil samples. The formation composition ratios are a key factor influencing the mechanical behavior of soils. Adjusting this ratio improves the soil’s bearing capacity and toughness, thereby enhancing engineering stability and safety. In fields such as underground engineering and geological exploration, the influence of stratum composition ratios must be thoroughly considered. Construction plans and parameters should be rationally designed to ensure proper project implementation and long-term stability.

Fig. 13

Evolution of average particle contact force during simulations (k_i = 1.0, Scheme 1) (a) m_i = 0.5; (b) m_i = 1.0; (c) m_i = 2.0.

Figure 13 presents the evolution of average contact force between sand-sand particles, cobble-cobble particles, sand-cobble particles, and the average contact force of the whole specimen during the simulations. As shown in these figures, the contact force of cobble-cobble contact is obviously large than the contact force of sand-sand contact and sand-pebble contact and therefore can provide more bearing capacity and toughness for the whole sample. As m_i increase, the average contact force between sand-sand particles, sand-cobble particles, and cobble-cobble particles don’t show significant variation, but the average contact force of the whole sample decreases because of the decrease of cobble particle contents. As a result, the bearing capacity of the sample is decreased as shown in Figs. 10, 11 and 12.

Analysis of results under different stress ratios

Fig. 14

Simulation results of stress path under different stress ratios (Scheme 1, m_i = 1.0).

Fig. 15

Simulation results of stress path under different stress ratios (Scheme 2, m_i = 1.0).

Fig. 16

Simulation results of stress path under different stress ratios (Scheme 3, m_i = 1.0).

As shown in Figs. 14, 15 and 16, under stress path scheme 1, the deviatoric stress-strain curves of the specimens also show significant differences with increasing stress ratio. When the stress ratio k_i = 1.0, the curve fluctuates around 0 during the whole simulation. When the stress ratio k_i = 1.25, as the deviatoric stress is not equal to 0 initially, the curve show continuously increase trend and tend to 0 finally. When the stress ratio k_i = 1.5 and 2.0, the stress strain curve shows similar tendency with that of k_i = 1.25. Through the combined effect of reducing the confining pressure unloading rate and increasing the stress ratio in Scheme 2, the soil maintains better overall stability during unloading. Higher stress ratios enhance interparticle interactions within the soil, enabling it to better resist softening caused by stress release during unloading. Under stress path scheme 3, the deviatoric stress-axial strain curves of the specimen demonstrate consistent trends with increasing stress ratio. At each stress ratio, the curves maintain relatively stable deviatoric stress levels during the unloading process, showing a relatively gentle decline segment. When k_i = 2.0, the curve exhibits the highest peak deviatoric stress and the most gradual decline segment. This behavior occurs because in Scheme 3, the confining pressure is maintained constant while only the axial stress is reduced, leading to a relatively stable internal stress state in the soil. The increased stress ratio enhances interparticle interactions within the soil, thereby improving its ability to maintain overall stability during unloading.

Conclusion

In this paper, triaxial unloading test of layered sand-cobble composite strata specimen is simulated by the discrete element method, and the macro stress-strain behavior of the sample under different unloading stress path that encountered in underground tunnelling is analyzed. The following conclusions are obtained:

• (1) Under different stress paths induced by shield tunneling disturbances, the macro-mechanical response of soil demonstrates significant variations. At the excavation face, simultaneous horizontal and vertical unloading caused by excavation rapidly releases internal soil stresses and accelerates the softening process. Above the excavation face, under constant confining pressure, the internal stress state remains relatively stable, which constrains the soil deformation capacity and leads to higher overall stability compared to the excavation face.

• (2) Interfacial weakness is identified in composite soil layers. As the composition ratio increases, both the bearing capacity and toughness of the soil demonstrate a trend of improvement. Variations in the composition ratio influence the arrangement and distribution of particles within the soil, consequently modifying its mechanical properties. The incorporation of cobble enhances the overall bearing capacity and deformation resistance of the soil, allowing it to maintain enhanced stability during the unloading process.

• (3) In strata with different stress ratios, an increase in the stress ratio induces particle rearrangement, reduces porosity, and enhances interparticle contacts. This improvement in interparticle interactions strengthens the soil’s mechanical properties, thereby increasing its overall stability. During unloading, a higher stress ratio enables the soil to better resist the softening effects induced by stress release, maintaining its structural integrity.

The findings can offer practical guidance for the design and construction of shield tunneling in layered sand-cobble composite strata. The distinct mechanical responses under different stress paths can assist engineers in predicting ground deformation and stress redistribution more accurately during excavation. For instance, recognizing the rapid stress release and softening at the excavation face supports the optimization of support systems and excavation rates to mitigate potential instability. However, while the study examines the effect of composition ratio and stress ratio, other factors like particle size distribution, interface roughness, and cyclic unloading-reloading conditions—common in real tunneling processes—were not extensively explored. In addition, the conclusions are derived from numerical simulations, while the layered specimen in this study can capture the main behavior of stress-strain curve of sand-cobble composite sample with different combination of sand particles and cobble particles, it may also overestimate localized bearing capacity and homogenizing interface effects. Validation through physical model tests or field monitoring data in the future work would strengthen their reliability and applicability.