Numerical Study and Parametric Insights of Mechanized Shaft Excavation in Soft Clay

Abstract

The excavation of deep shafts using Vertical Shaft Sinking Machine (VSM) technology in stratified soft soils involves complex soil-structure interaction (SSI) mechanisms that are often oversimplified by conventional numerical approaches. This study develops a robust three-dimensional numerical framework to investigate ground deformation induced by VSM operations, explicitly incorporating the phased construction sequence, segmental lining installation, and site-specific stratigraphy. The model is calibrated and validated against high-resolution field monitoring data, employing a prediction envelope approach and statistical performance metrics (RMSE and $R^2$). The results suggest that ground response during VSM excavation is predominantly stiffness-controlled under the investigated conditions. Mobilized shear stresses remain significantly below the available soil capacity, indicating that deformation under serviceability conditions is driven by progressive strain accumulation. Horizontal displacement profiles suggest a relatively stable depth of influence, indicating that the excavation process amplifies deformations within a pre-established domain without significant deep-seated propagation. Sensitivity analyses indicate soil stiffness modules ($E_{50},E_{oed},E_{ur})$ and the SSI interface factor ($R_{inter}$) as the primary drivers of deformation magnitude. Furthermore, stratigraphic contrasts specifically clay-sand sequences, act as a mechanical filter, concentrating strains in soft layers while limiting vertical propagation through stiffer strata. The proposed framework provides a mechanically coherent basis for serviceability-oriented design, deformation prediction, and risk-mitigation strategies for mechanized shafts in saturated soft ground.

1. Introduction

The accelerating development of underground infrastructure in rapidly urbanizing regions has intensified the need for safe, efficient, and mechanized methods of shaft construction. The Vertical Shaft Sinking Machine (VSM) has emerged as a key technology, enabling the excavation of deep shafts under complex hydrogeological and spatial constraints with reduced environmental impact compared with traditional open-cut or caisson methods [1,2]. Recent field and numerical investigations have studied the growing importance of the VSM method in soft-ground environments, particularly for metro access, drainage, and utility structures [2]. Nevertheless, despite increasing implementation, the three-dimensional (3D) soil-structure interaction (SSI) mechanisms governing VSM excavation remain insufficiently understood.

Excavations in soft cohesive soils are inherently complex because of their pronounced stress-dependent stiffness, anisotropy, and time-dependent consolidation behavior [3,4]. Recent studies on constitutive degradation and deformation evolution in geomaterials have further emphasized the importance of accurately representing nonlinear stiffness response under complex loading conditions [5]. Stress relief during excavation triggers nonlinear stiffness degradation and pore-pressure redistribution that strongly influence deformation around the shaft lining [6,7]. Empirical approaches and two-dimensional (2D) formulations often fail to capture these mechanisms, leading to underestimation of displacements, especially in clustered-shaft layouts or stratified deposits [8].

Foundational works on ground movement and excavation behavior established empirical correlations between geometry and settlement [9,10,11], while subsequent Rankine lectures emphasized the role of small-strain stiffness and anisotropy in soil response [12]. Modern numerical analyses have expanded these insights, demonstrating that stress redistribution and progressive yielding in soft clays demand a 3D representation to correctly reproduce both ground settlement and lining internal forces [2,13,14]. Recent studies on nonlinear deformation evolution and progressive mechanical response in geomaterials subjected to complex loading paths have further emphasized the importance of accurately representing stress redistribution and stiffness-related behavior in numerical analyses [5,15]. Advancements in constitutive modeling such as the Hardening Soil (HS) and Hardening Soil with Small-Strain Stiffness (HSS) have markedly improved predictions of excavation-induced deformation [16,17]. These models account for stress-dependent stiffness and strain hardening, outperforming classical Mohr-Coulomb formulations in both serviceability and ultimate-limit simulations [18]. However, the number of validated 3D FE applications focusing specifically on VSM shafts are observed to remain limited.

The interaction between adjacent shafts or nearby deep excavations can significantly amplify deformation due to overlapping plastic zones, stress redistribution, and pore-pressure coupling. Experimental and numerical studies have shown that when excavation spacing is small relative to the shaft diameter, the ground settlement and lining internal forces increase nonlinearly [19,20]. Recent calibrated FE simulations in soft clays suggest that spatial interaction effects and construction sequence govern both horizontal displacements and lining internal forces [21]. Parametric investigations further indicate that stiffness and interface parameters dominate model sensitivity, underscoring the importance of robust constitutive calibration for reliable predictions of mechanized shaft performance [2,22].

Despite the increasing adoption of Vertical Shaft Sinking Machine (VSM) technology in urban environments, most existing studies have focused on the mechanical performance of the machine itself or on simplified 2D analytical solutions. There remains a significant knowledge gap regarding the three-dimensional soil-structure interaction mechanisms in complex stratified soft clays, where the interplay between hydrostatic pressure during submerged excavation and the installation of segmented linings dictates the displacement field. Furthermore, the sensitivity of the ground response to specific soil stiffness parameters under the unique stress-path of mechanized vertical sinking is not yet fully understood. This study addresses these gaps by developing a high-fidelity 3D finite element framework that incorporates the actual construction sequence and performs a systematic parametric evaluation to identify the governing geotechnical variables in stratified alluvial deposits.

In this context, the present study employs and evaluates a proposed framework to investigate shaft excavation using Vertical Shaft Sinking Machine (VSM) technology in stratified soft cohesive soils. The analysis focuses on the serviceability response of the surrounding ground, where excavation-induced deformation governs system behavior while mobilized stresses remain below soil strength limits. Unlike many previous numerical studies that rely on simplified geometries or idealized construction sequences, the proposed model incorporates realistic excavation stages, segmental lining installation, soil-structure interface behavior, and stratigraphic stiffness contrasts within a validated 3D framework.

The present work constitutes the calibrated numerical framework stage of a broader research effort aimed at understanding the three-dimensional propagation of excavation-induced deformation in dense urban environments where multiple shafts may interact and affect nearby infrastructure. The modelling framework is verified through detailed comparison with field monitoring data from a full-scale VSM shaft excavation, complemented by sensitivity and robustness analyses. This integrated approach enables a physically systematic interpretation of settlement development and deformation propagation around large-diameter mechanized shafts in soft ground.

2. Project Overview

The case study considered in this research is the Caisson-Type Underground Intelligent Parking Garage in the Jianye District of Nanjing, China, which is characterized by dense urban infrastructure and deep, water-saturated alluvial deposits. The project employs the Vertical Shaft Sinking Machine (VSM) to construct two circular, vertical-lift parking shafts, approximately 69 m deep and 12.0 m in internal diameter, formed using precast reinforced-concrete segmental linings that provide a continuous watertight structural shell capable of resisting hydrostatic and lateral pressures.

Figure 1 presents the geometric layout of the twin shafts, whose center-to-center spacing varies between 12.0 m and 14.40 m depending on construction stage and spatial constraints. The outer modeling boundary delineates the deformation influence zone adopted in the finite element simulations to capture soil-structure interaction between the two shafts. Each shaft includes a top ring foundation, foundation beam, and a raft slab at the base, ensuring structural rigidity and uniform load transfer, a configuration representative of contemporary mechanized shaft systems increasingly used in China’s underground infrastructure.

The site is underlain by a thick sequence of soft to medium silty clay interbedded with thin silty sand layers, transitioning to stiff clay below 35–40 m. Groundwater lies near the surface, and elevated pore pressures critically influence stability and deformation. These subsurface conditions pose significant challenges for conventional open-cut or pneumatic caisson excavation, which can trigger large settlements, basal instability, and excessive groundwater inflow. The VSM method provides a more controlled alternative, enabling submerged excavation with continuous segment installation and limited disturbance to the surrounding ground.

This project offers a field-consistent framework due to the availability of detailed stratigraphy, a well-documented construction sequence, and comprehensive field monitoring records. These datasets were used to develop a three-dimensional PLAXIS 3D model replicating excavation stages, segment installation, and coupled soil-structure interaction under realistic hydraulic and boundary conditions. This case study therefore establishes the foundation for the model calibration and validation presented in the subsequent sections.

3. Numerical Model

The Hardening Soil (HS) model was selected as the constitutive framework for all simulations due to its ability to represent stress-dependent stiffness, plastic hardening, and nonlinear deformation behavior in soft clays. The model distinguishes between primary loading stiffness ($E_{50}$), oedometer stiffness ($E_{oed}$), and unloading-reloading stiffness ($E_{ur}$), enabling realistic simulation of stress-path effects during excavation and staged construction. All parameters were derived from triaxial, oedometer, in situ tests and published studies, and reference stiffness values were assigned at 100 kPa confining pressure. The HSS was used only in sensitivity analyses to evaluate small-strain stiffness effects, while the baseline model exclusively employed the HS formulation [23,24].

3.1. Selection and Capabilities of Simulation Software

A fully three-dimensional finite element framework was employed because the excavation of segmental-lined VSM shafts inherently induces spatially complex stress redistribution, sequential ring installation effects, and potential multi-shaft interaction phenomena that are difficult to represent within plane strain or axisymmetric idealizations. Although axisymmetric or symmetry-reduced formulations may provide computationally efficient approximations for isolated shaft analyses, the present study intentionally adopts a full three-dimensional framework in order to preserve the spatial interaction effects between adjacent shafts, staged excavation sequences, and non-uniform stress redistribution mechanisms associated with the investigated urban excavation configuration. These aspects are difficult to represent consistently using simplified axisymmetric idealizations. PLAXIS 3D provides appropriate capabilities for this level of fidelity, including staged construction procedures, explicit soil-structure interface elements, coupled groundwater flow-deformation processes, and rigorous K₀-based in situ stress initialization, all of which are important for reproducing observed ground responses in recent 3D back-analyses of deep excavations and intersecting underground structures [25,26]. The subsurface behavior follows the HS formulation previously described.

3.2. General Configuration and Boundary Conditions of the Numerical Model

The 3D FE domain was defined as a large rectangular volume with horizontal boundaries placed several excavation depths away from the shaft, a fully fixed base, roller side boundaries, and a free ground surface, following recommended practice for deep-excavation modeling of CIRIA C760 and excavation monographs [27,28,29]. Initial stresses and pore-water pressures were established through a K₀ procedure calibrated to stress history (K₀ & OCR relations), followed by initialization of pore-water pressures under hydrostatic conditions [30,31,32,33]. Local mesh refinement was applied around the shaft lining, interfaces, and raft, with convergence checks on deformations and internal forces [28]. Soil-structure interaction was modeled through interface elements as described in Section 3.2.4.

In the numerical model, the interaction between the concrete shaft lining and the surrounding soil is represented using the Hardening Soil model. In PLAXIS, the interface shear strength is defined using a reduced Mohr-Coulomb criterion, where the soil strength parameters are reduced by the interface reduction factor $R_{inter}=\mathrm{t}\mathrm{a}\mathrm{n} \delta /\mathrm{t}\mathrm{a}\mathrm{n} \varphi$ [24]. Experimental studies indicate that the interface friction angle typically ranges between 0.67φ and 0.9φ depending on surface roughness [34,35], corresponding to $R_{inter}$ values between approximately 0.6 and 0.85 for concrete-soil interfaces. Based on these considerations and calibration against field displacement profiles obtained at different excavation stages (R1–14, R15–28, and R29–41), an interface reduction factor of $R_{inter}=0.7$ was adopted as the baseline value in the numerical model.

The “water drainage” stage considered in the construction sequence refers only to the removal of water from inside the shaft after the installation of the complete lining system and raft slab. At this stage, the shaft structure acts as a hydraulic barrier, preventing hydraulic interaction between the water inside the shaft and the surrounding soil mass. Consequently, the drainage process occurs primarily within the shaft, while the completed lining system substantially limits direct hydraulic interaction with the surrounding soil mass. Nevertheless, localized pore-pressure redistribution and minor time-dependent consolidation effects may still occur due to delayed hydro-mechanical response and imperfect hydraulic isolation, particularly during the post-drainage stage.

In the numerical model, the soil mass was therefore assumed to be fully saturated, and the groundwater table was conservatively placed at the ground surface to represent critical hydrogeological conditions typical of soft clay deposits such as those encountered in Nanjing, China.

3.2.1. Geometrical Model and Shaft Configuration

The 3D computational domain (150 m × 150 m) was extended to full stratigraphic depth, with boundaries placed sufficiently far from the shafts to avoid boundary-induced deformation effects. Two shaft center-to-center spacings were modeled to assess their influence on ground deformation and load transfer. Each shaft consisted of segmental precast concrete rings (12.0 m ID, 12.8 m OD, 0.4 m thickness), a top beam and foundation beam (12.8–15.8 m diameter, 3 m width), and 12 m-diameter, 3 m-thick raft slabs. All structural components were modeled as linear-elastic elements using stiffness parameters derived from design and laboratory data. Operational loads (931.95 kN total, 13.83 kN/m²) were applied on the top slab to replicate construction conditions (

Table 1. Mechanical properties of shaft structural components.

Component	Bulk Modulus K (MPa)	Shear Modulus G (MPa)	Elastic Modulus E(c) (MPa)	Density ρ (kg/m3)	Unit Weight γ (kN/m3)	Cross-Section Area A (m2)	Weight per Unit Length W (kN/m)	Volume V (m3)	Moment of Inertia I2 (m4)	Moment of Inertia I3 (m4)	Axial Stiffness EA (kN/m)
Rings	19,200	14,400	34,560	2500	24.525	15.5823	573.2338	23.3734	-	-	5.385 × 1011
Top Beam	18,000	13,500	32,400	2500	24.525	67.3872	4958.0148	202.1615	6.75	6.75	2.183 × 1012
Foundation	18,000	13,500	30,000	2300	22.563	67.3872	16,725.022	741.2588	332.75	24.75	2.021 × 1012
Raft Slab	17,500	14,400	33,901.35	2350	23.0535	113.0973	7821.8683	339.292	-	-	3.834 × 1012

).

Internal forces and deformations of all structural components, including lining rings, foundation beams, top beams, and raft slabs were monitored throughout the staged construction sequence. All elements remained within elastic response bounds throughout the excavation phases.

The lateral boundaries were located sufficiently far from the excavation to minimize boundary effects and approximate semi-infinite ground conditions, which is a standard practice in numerical modelling of deep excavations.

The full three-dimensional domain was retained in preference to axisymmetric or half-model simplifications in order to preserve the spatial interaction effects between the two adjacent shafts, whose asymmetric stress redistribution fields cannot be consistently represented using reduced geometric idealizations.

3.2.2. Stratigraphy and Soil Properties

The numerical model incorporated the complete site-specific stratigraphy to a depth beyond the excavation influence zone, comprising six layers: fill/organic clay (0–4 m), soft-medium silty clay (4–12 m), medium-stiff silty clay (12–28 m), silty sand (28–36 m), stiff silty clay (36–51 m), and very stiff silty clay to the model base, as shown in

Table 2. Basic geotechnical parameters of the soil layers.

Layer No.	Layer Description	Depth (m)	Thickness (m)	γ (kN/m3)	γd (kN/m3)	K0	c′ (kPa)	ϕ′ (°)
1	Fill and Soft Organic Clay	0–4	4	19.1	14.8	0.7	26	18.4
2	Soft to Medium Silty Clay	4–12	8	18.0	13.2	0.72	17	19.0
3	Medium-Stiff Silty Clay	12–28	16	19.2	15.0	0.6	10	30.0
4	Silty Sand	28–36	8	18.3	14.0	0.65	4	30.0
5	Stiff Silty Clay	36–51	15	19.5	16.5	0.5	12	28.0
6	Very Stiff Silty Clay	51–end	23	18.9	16.2	0.6	16	28.0

. The geotechnical parameters used in the numerical model were derived from the site investigation program, which included six boreholes with SPT testing, groundwater level measurements, and laboratory testing consisting of index property tests, oedometer consolidation tests, triaxial compression tests, and direct shear tests reported in the project geotechnical investigation.

For each soil layer, parameters such as unit weight (γ, γd), K₀, effective strength (c′, φ′), and stiffness moduli ($E_{50}^{ref},E_{oed}^{ref},E_{ur}^{ref}$) were derived from the site investigation program conducted for the project, which included borehole exploration with Standard Penetration Tests (SPT), groundwater level monitoring, and laboratory testing consisting of index property tests, oedometer consolidation tests, triaxial compression tests, and direct shear tests. The groundwater table was set at ground level with γ_w = 9.81 kN/m³ (

Table 3. Advanced geotechnical parameters of the soil layers.

Layer No.	Reference Pressure (kPa)	υ (-)	m	${\mathit{E}}_{50}^{\mathit{r}\mathit{e}\mathit{f}}$ (kPa)	${\mathit{E}}_{\mathit{u}\mathit{r}}^{\mathit{r}\mathit{e}\mathit{f}}$ (kPa)	${\mathit{E}}_{\mathit{o}\mathit{e}\mathit{d}}^{\mathit{r}\mathit{e}\mathit{f}}$ (kPa)
1	100	0.4	0.9	10,800	43,200	8640
2	100	0.35	0.85	11,340	45,360	9072
3	100	0.3	0.8	12,000	48,000	9600
4	100	0.3	0.55	18,000	72,000	14,400
5	100	0.3	0.75	15,000	60,000	12,000
6	100	0.25	0.7	15,000	60,000	12,000

). The strength parameters are stable with the soil classification obtained from the site investigation. The relatively high friction angles observed in some silty clay layers are attributed to the significant silt fraction of the deposits, which is typical for alluvial soils in the Nanjing region.

The numerical simulations were performed using the Hardening Soil (HS) constitutive model implemented in PLAXIS 3D. The HS model is an advanced elasto-plastic formulation that accounts for stress-dependent stiffness and plastic hardening under primary loading conditions [23].

Within the HS framework, $E_{50}^{ref}$ represents the secant stiffness at 50% of the failure stress level under primary loading, $E_{oed}^{ref}$, corresponds to the stiffness governing one-dimensional compression behavior, and $E_{ur}^{ref}$, defines the unloading-reloading stiffness.

The reference stiffness parameters for the Hardening Soil (HS) model were determined through a combination of laboratory testing and established geo-mechanical correlations for the Yangtze River Delta region. The secant stiffness modulus ($E_{50}^{ref}$) was directly derived from consolidated-drained (CD) triaxial tests. To capture the unloading-reloading behavior during shaft sinking, the modulus was set as $E_{ur}^{ref}=3E_{50}^{ref}$, a ratio widely validated for soft plastic clays [36]. Furthermore, the oedometer stiffness was adopted as ${0.8E}_{50}^{ref}\approx E_{oed}^{ref}$. This specific ratio ($E_{50}^{ref}<1$) is supported by regional studies on the structured soft clays of the Yangtze Delta [37], which indicate that for highly compressible silty clays, the one-dimensional compression stiffness can be lower than the triaxial shear stiffness due to soil structural sensitivity and specific stress-path effects during excavation. This calibrated approach ensures that the model reflects the stress-dependent stiffness degradation while remaining consistent with observed site-specific geotechnical reports [17].

The power-law exponent (m), which dictates the stress-dependency of soil stiffness, was assigned based on the lithological characteristics of each layer. For the sandy strata, the value of $m\approx 0.5$ was adopted to reflect the typical behavior of granular materials. For the soft silty clay layers, values ranging from $\mathrm{m}\approx$ 0.7 to 0.9 were selected. This choice is consistent with the findings of Janbu [38] and regional studies in the Yangtze Delta [37], which suggest that structured silty clays often exhibit an exponent slightly below unity due to their specific consolidation history and silt content.

The adopted parameter set is consistent with the geological conditions of the site and provides a realistic representation of soil behavior. The relatively high friction angles observed in silty clay layers are attributed to the significant silt fraction present in the alluvial deposits.

3.2.3. Mesh Discretization

A refined three-dimensional mesh composed of 10-noded tetrahedral elements was used to capture stress redistribution and deformation around the shafts. Local refinement was applied in the vicinity of the excavation, soil-structure interfaces, and stratigraphic transitions to provide a stable approximation of stress gradients and displacement fields.

Element size ranged from 0.2 m near the shaft to approximately 20 m at the model boundaries, ensuring computational efficiency while preserving accuracy in critical regions. Mesh quality was verified through standard criteria (Jacobian and aspect ratio), and a mesh-independence study indicates stable predictions of settlement and structural response.

To ensure that the numerical results were independent of the spatial discretization, a mesh sensitivity analysis was performed. Three refinement levels—coarse (approx. 15,000 elements), medium (approx. 45,000 elements), and fine (approx. 90,000 elements)—were evaluated by monitoring the maximum horizontal displacement of the shaft wall at the final excavation stage. The results showed a negligible variation of less than 2.5% between the medium and fine meshes. Consequently, the medium mesh with local refinement around the shaft periphery and the soil-structure interface was selected to optimize computational efficiency without compromising numerical accuracy.

In the same way, to minimize numerical errors associated with spatial discretization (truncation and mesh-dependency), a comprehensive mesh sensitivity analysis was performed. As detailed in Section 5, increasing the mesh density from 45,000 to 90,000 elements resulted in a variation of less than 2.5% in maximum displacements, indicating that the chosen mesh provides a stable solution. Furthermore, the use of double-precision floating-point calculations within the PLAXIS 3D solver, combined with a strict iterative convergence tolerance (set at 0.01), ensures that round-off errors and residual imbalances remain negligible compared to the physical deformation magnitudes investigated.

The adopted medium mesh with local refinement provides a numerically converged solution, with less than 2.5% variation in maximum displacements relative to the fine-mesh configuration.

3.2.4. Interface Modelling

The interaction between the shaft lining, foundation components, and the surrounding soil was modeled using zero-thickness interface elements, allowing for the simulation of relative displacement, slip, and shear stress transfer at the soil-structure contact. This approach enables a realistic representation of load redistribution and deformation mechanisms induced by excavation, particularly in stratified soft soil conditions.

The value $R_{inter}$ = 0.7 was selected within the experimentally established range of 0.6–0.8 reported for smooth precast concrete-clay interfaces [34,39], consistent with back-analyses of deep excavations in Yangtze River Delta soft clays, and its adequacy was confirmed through comparison with field displacement profiles across excavation stages for concrete-clay interfaces, which suggest that the interface friction angle typically ranges from 0.6 to 0.8 times the internal friction angle of the soil [40,41]. Furthermore, this choice aligns with numerical practices for deep excavations in the soft clays of the Yangtze River Delta [42], where a moderate reduction factor accounts for the minimized soil disturbance and the smooth surface of the lining segments characteristic of the VSM sinking process.

This modeling approach allows capturing the interaction between adjacent shafts and the progressive evolution of deformation induced by excavation, particularly under conditions where interface behavior plays a significant role in controlling displacement magnitude and stress transfer mechanisms.

3.2.5. Water Conditions and Pore Pressure Modeling

To address the hydro-mechanical response, the hydraulic boundary conditions were defined as hydrostatic, with the water table at the ground surface (0.00 m). For the soft clay layers, undrained (Type B) behavior was assigned to capture the generation of excess pore pressures during the rapid excavation phases. The outer boundaries of the model were set as closed-consolidation boundaries (no flow), while the excavation face was modeled as a seepage face during the submerged sinking process. Although a fully coupled consolidation analysis was not implemented, given that the high sinking rate of the VSM minimizes the dissipation of pore pressures during active excavation-the model accounts for the undrained shear strength and the effective stress path, which are the primary drivers of short-term deformation in these alluvial deposits.

The conservative assumption of a hydrostatic groundwater table at ground surface represents the critical hydraulic condition for the investigated deposits. Explicit simulation of groundwater depression was outside the scope of the present study but is identified as a relevant subject for future coupled hydro-mechanical investigations.

Por: The groundwater table was established based on-site investigation data from the geotechnical survey. During VSM sinking, the internal water level is naturally maintained 2 to 5 m above the groundwater table, ensuring hydraulic pressure balance and minimizing net seepage forces during active excavation. Explicit simulation of groundwater depression was outside the scope of the present study but is identified as a relevant subject for future coupled hydro-mechanical investigations.

3.3. Model Innovations

The proposed numerical framework enables detailed interpretation of deformation within a three-dimensional environment governed by construction sequencing and stratigraphic stiffness contrasts. The model explicitly approximates soil-structure interaction, stress redistribution, and deformation evolution under realistic VSM excavation conditions, moving beyond simplified or lower-dimensional approaches.

The contribution of this study lies in the integrated field-consistent framework of a three-dimensional modeling framework for VSM shaft excavation based on field monitoring data. This enables direct evaluation of interaction-induced deformation and stress redistribution that cannot be captured using isolated-shaft or axisymmetric models. The framework further incorporates a detailed structural representation of the shaft system, including lining rings, foundation elements, and raft slab, activated following realistic construction sequences and subjected to operational loads.

Soil behavior is represented using the Hardening Soil model across all stratigraphic layers, allowing deformation to be interpreted as a strain-driven, stiffness-controlled response across clay-sand-clay sequences. Locally refined three-dimensional meshing, optimized domain boundaries, and explicit soil-structure interface elements ensure accurate resolution of nonlinear load transfer while maintaining numerical robustness.

Together, these features provide a consistent and computationally efficient framework for estimation of displacement envelopes, depth of influence, and shaft interaction effects in stratified ground conditions.

3.3.1. Advanced Analysis Method Applied to Shaft Simulation

The numerical framework adopted in this study uses PLAXIS 3D to reproduce the full VSM excavation sequence ring installation, progressive deepening, and base-slab construction under realistic in situ stresses and groundwater conditions, following recent advances in 3D geotechnical modelling [43]. The analysis integrates staged construction procedures, calibrated soil-structure interfaces, and targeted mesh refinement to resolve localized stress redistribution within the soft clay strata. Soil behavior follows the HS formulation already described.

3.3.2. Sequential Analysis Steps for Shaft Excavation

A staged construction analysis in PLAXIS 3D V2 is employed to simulate the excavation and lining installation process in accordance with the operational sequence of the Vertical Shaft Sinking Machine (VSM). The numerical framework represents the progressive development of stress redistribution, soil-structure interaction, and deformation as excavation advances, and is applied consistently within a parametric investigation.

The excavation process is represented through a set of representative construction stages that reflect the evolution of excavation depth and lining activation governing the mechanical response of the surrounding ground. This staged discretization is appropriate for the quasi-static nature of VSM excavation, where soil behavior is controlled by cumulative excavation depth and associated stress paths rather than by localized, short-scale construction increments. As a result, the adopted approach provides a stable approximation of the governing deformation mechanisms and mobilized stress states while maintaining numerical stability and computational robustness Figure 2 and

Table 4. Summary of construction and simulation phases for the VSM-based shaft excavation process.

Phase	Description
Phase 0	Generation of in situ stress field using the K0-procedure. Hydrostatic pore pressure initialized based on phreatic level. Establishment of pre-excavation stress conditions matching field data.
Phase 1	Activation of improved soil zones with increased stiffness and strength. Simulation of basal reinforcement to enhance initial stability.
Phase 2	Activation of the top reinforced-concrete beam at the ground surface. Early load transfer and structural confinement prior to excavation.
Phase 3	Application of static operational loads (13.83 kN/m2) representing VSM lowering and segment handling operations.
Phase 4 (R1–14)	Excavation in 1.5 m increments down to 21 m depth with immediate activation of each concrete ring after excavation. Simulation of continuous lining sequence during VSM operation.
Phase 5 (R15–28)	Excavation extended to 42 m depth (1.5 m steps). Continuous soil-structure interaction evaluation during deepening.
Phase 6 (R29–41)	Final excavation to the target depth (~68 m). Activation of remaining lining rings to complete the structural lining of the shaft.
Phase 7 (Raft Slab)	Activation of the reinforced concrete raft slab at the shaft base. Provides structural closure, end-bearing load distribution, and enhanced bottom stability against uplift.
Phase 8 (Internal shaft dewatering)	Internal dewatering of the shaft after completion of lining and raft slab, with no hydraulic interaction with the surrounding soil.

.

The direct activation of the segmental lining following each excavation increment is consistent with the operational mechanics of the VSM technology, where simultaneous excavation and lining installation eliminates any unsupported ground exposure period. The bentonite slurry maintained at the soil-lining interface during sinking was incorporated as a hydrostatic pressure boundary condition, ensuring physically consistent lateral stress transfer throughout the excavation sequence. Consequently, while stress release does occur during VSM sinking, its magnitude is significantly reduced by the submerged excavation conditions with the internal water level maintained equal to the groundwater table and the continuous hydrostatic support provided by the bentonite slurry at the soil-lining interface. Under these conditions, the net stress release per excavation increment is negligible, and direct lining activation represents a mechanically appropriate approximation of the VSM construction process.

The selected construction stages therefore provide mechanically systematic representation of the initial, intermediate, and advanced excavation regimes that control settlement development and shaft-soil interaction, forming a reliable basis for subsequent parametric analyses and stability evaluation.

Global stability was independently evaluated using the Strength Reduction Method (SRM), which was applied after completion of all construction stages to assess the overall factor of safety of the shaft system.

The Strength Reduction Method confirmed that mobilized stress states remained well below the failure envelope at all construction stages, with mobilization ratios between 0.32 and 0.73 across soil layers, consistent with serviceability-controlled behavior.

4. Model Validation

Model performance was quantitatively evaluated using statistical metrics including Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Bias, and coefficient of determination (R²), computed between measured and estimated displacements.

4.1. Description of In Situ Monitoring and Testing Setup

To assess the deformation characteristics of the shafts as well as their effects on the surrounding environment during the construction phase, a monitoring system had been installed in situ prior to excavation. This involved a network of internal monitoring systems set inside the shafts as well as external monitoring systems located around the construction site to record the soil-structure interaction occurring because of the excavation process.

In the shafts, inclinometer tubes were installed in the inner surface of the segmental lining to record the change in inclination, as well as horizontal displacement to evaluate the shaft orientation, as well as tilting that occurs during the sinking process. Moreover, soil pressure sensors were installed at important levels that include the cutting-edge ring as well as the 1st, 10th, and 22nd rings to record the variation in earth pressure directed towards the lining segments during excavation Figure 3 and Figure 4.

On the external side, a network of ground settlement monitoring stations was set up around the shafts in areas of terrain that had been excavated as well as in the vicinity of the buildings, roads, and open ground to assess surface deformation arising because of the excavation operation as well as drawdown of the groundwater.

The primary geotechnical parameters adopted in the numerical model were derived from site investigation data, including borehole exploration, laboratory testing, and regional empirical correlations for the Yangtze River Delta deposits. Field monitoring data were not used for direct inverse fitting of the constitutive soil parameters, but rather for comparative evaluation of the model response and refinement of secondary interaction assumptions.

4.2. Predictive Consistency Assessment

The early excavation stages (R1–14) were primarily used to verify the consistency of the adopted stiffness and interface assumptions, while the subsequent excavation stages (R15–41), raft slab installation, and drainage phase were used as predictive benchmarks to evaluate the predictive consistency assessment under progressively evolving stress conditions. The primary geotechnical parameters adopted in the numerical model were derived from site investigation data, laboratory testing, and regional empirical correlations rather than direct inverse fitting of monitoring data.

It is noted that $R_{inter}$ was established through comparison with field profiles, while all constitutive soil parameters were derived exclusively from laboratory data and regional correlations, without adjustment to monitoring observations.

4.2.1. Numerical and Field Settlement Comparison

Numerical settlement predictions were compared with field measurements to evaluate the model’s ability to reproduce the spatial distribution and magnitude of surface settlements induced by deep shaft excavation. As illustrated in Figure 5, the comparison includes settlement profiles at successive construction stages and field observations recorded at two monitoring dates, with model uncertainty represented through a predictive envelope bounded by maximum and minimum consolidation estimates.

The numerical results show a consistent settlement pattern across all construction stages, characterized by maximum deformation at the shaft wall (x = 0) and a gradual decay with radial distance. This profile shape is observed to remain stable throughout the excavation sequence, indicating that the settlement distribution is largely governed by the geometric configuration and soil-structure interaction, while construction progress mainly scales the magnitude of the displacement. The influence range appears largely insensitive to individual construction stages, suggesting that the extent of the “Max Influence Zone” is defined by the initial stress redistribution induced by the excavation.

Field measurements follow the same spatial trend predicted by the numerical framework, suggesting that the dominant deformation mechanisms and the lateral extent of the ground influence are reasonably captured. However, the numerical model tends to slightly overestimate the near-field settlement compared to the “real” modeling conditions, particularly within the first 20 m from the shaft wall. This discrepancy likely stems from the high stiffness of the VSM support system and the immediate application of the shaft lining in the field, which limits initial ground relaxation more effectively than the idealized numerical boundaries.

Despite the differences in magnitude near the excavation, the field observations consistently fall within the predicted consolidation envelopes as the distance increases. Notably, field data recorded at later stages show a slight downward migration, aligning more closely with the model’s trend as it reflects time-dependent ground behavior such as pore pressure dissipation. The preservation of the profile shape across both numerical and field results demonstrates the validity of the model; while it slightly overestimates the magnitude in the near-field, provides a reasonable representation of the underlying mechanisms of the excavation-induced deformation.

4.2.2. Quantitative Consistency of Horizontal Deformation

The predictive capability of the numerical model is evaluated through an integrated assessment of statistical metrics, displacement profiles, and mechanical indicators. This discussion will move beyond a review of the data presented in tabular form by considering a combined assessment of the graphical data presented.

For each construction stage, displacement values were extracted at the same spatial locations as the field measurements, ensuring consistency in the comparison. The calibrated numerical framework was performed using statistical indicators including Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Bias, and the coefficient of determination (R²), which are widely adopted in geotechnical model validation and performance assessment [44].

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n}{(u_i^{num}-u_i^{field})}^2}$$

(1)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n}\left|u_i^{num}-u_i^{field}\right|$$

(2)

$$Bias={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n}\left(u_i^{num}-u_i^{field}\right)$$

(3)

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{\left(u_i^{field}-u_i^{num}\right)}^2}{{\displaystyle {\sum }_{i=1}^n}{\left(u_i^{field}-{\overline{u}}^{field}\right)}^2}}$$

(4)

where u_i^num and u_i^field represent the numerical and measured displacements, respectively, and n is the number of monitoring points. The coefficient of determination (R²) was obtained from linear regression between predicted and measured values.

The evolution of the validation metrics across successive construction stages is presented in Figure 6. Overall, the results demonstrate that the numerical model provides a good agreement with both the magnitude and spatial distribution of field displacements with high reliability. For the horizontal displacement component (u_x), the model exhibits consistently low error magnitudes and a stable correlation throughout the excavation sequence, yielding RMSE values on the order of 10⁻² m and R² values ranging between 0.77 and 0.89. These metrics confirm the model’s robustness in capturing lateral soil movements across the entire monitoring period.

In contrast, the horizontal displacement component (u_y) shows an incremental dispersion as the excavation progresses, particularly during the drainage stage (ST-5). As summarized in

Table 5. Statistical validation etrics for horizontal (u_x) and vertical (u_γ) displacements across construction stages using the baseline parameter set.

Stage	RMSE ux (m)	RMSE uγ (m)	MAE ux (m)	MAE uγ (m)	Bias ux (m)	Bias uγ (m)	R2 ux	R2 uγ
ST-1 (R1–14)	0.01	0.015	0.008	0.012	−0.004	−0.006	0.89	0.72
ST-2 (R15–28)	0.011	0.017	0.009	0.014	−0.005	−0.007	0.86	0.65
ST-3 (R29–41)	0.012	0.02	0.01	0.016	−0.006	−0.008	0.82	0.58
ST-4 (Raft)	0.013	0.018	0.011	0.015	−0.006	−0.009	0.8	0.51
ST-5 (Drain)	0.014	0.028	0.012	0.02	−0.007	−0.015	0.77	0.42

, the R² for u_y transitions from 0.72 in the initial stages to 0.42 at the end of the sequence. This trend suggests a diminished capability of the uncoupled numerical formulation to fully account for complex hydro-mechanical phenomena, such as localized pore pressure dissipation and soil anisotropy at greater depths [32,33].

Nevertheless, the stability of the statistical indicators implies that the model maintains a satisfactory predictive capacity without requiring post-hoc parameter tuning. In the field of geotechnical numerical modeling, RMSE values within the 10⁻² m range and R² values exceeding 0.4–0.5 for complex, large-scale excavations are generally considered acceptable in geotechnical modeling practice and sound predictive performance.

The higher predictive accuracy observed for horizontal displacements is consistent with the sensitivity analysis results in Section 5, where stiffness parameters ($E_{50},E_{ur},E_{oed}$) were found to dominate the lateral displacement response mechanisms well represented by the adopted HS formulation. Vertical displacements during drainage exhibit sensitivity to additional hydro-mechanical processes not captured within the uncoupled framework.

4.2.3. Displacement Envelopes and Depth of Influence

Detailed analysis of the depth-wise displacement profiles, presented in Figure 7, reveals a consistent geometric trend that scales in magnitude as the excavation stages advance. For both horizontal components (u_x and u_y), the shaft sinking process primarily amplifies deformation within a pre-established influence zone rather than extending the affected region deeper into the soil profile. This observation, further supported by the quantitative metrics in Table 5, confirms that the spatial distribution of displacements is governed by stratigraphic stiffness contrasts and confinement conditions rather than excavation depth alone.

A significant concentration of deformation is observed within the upper soft layers, designated as the Active Deformation Zone in Figure 7, extending from the surface to a depth of approximately −30 m. Below this zone, the soil movement attenuates significantly upon reaching the deeper, stiffer strata. Following the criteria established by [45,46], the depth of influence is defined as the point where horizontal displacement reduces to approximately 5% of the maximum surface value. The resulting profiles suggest that the vertical extent of ground movement remains stable throughout the construction sequence, with the outermost profiles defining a displacement envelope that represents the upper bound of the lateral ground response.

The reliability of these depth-wise predictions is corroborated by the statistical indicators summarized in Figure 6. The consistently low RMSE and high R² values for the u_x component across all stages (ST-1 to ST-5), detailed in Table 5, demonstrate the model’s robustness in capturing cumulative stress redistribution. While the u_y component exhibits increased scatter during the drainage stage (Figure 6), the overall preservation of the profile shape reinforces the interpretation that the model shows satisfactory agreement with the fundamental soil-structure interaction mechanisms. These displacement envelopes provide a mechanically meaningful basis for quantifying the extent of ground influence and assessing potential interaction effects in layered soils [25,26].

4.2.4. Mechanical Consistency: Mobilized vs. Available Strength

Mechanical consistency of the numerical model was evaluated by comparing the mobilized shear stress, derived from field-measured peak shear strains using the elastic relation τ = G·γ_p, with the available shear capacity computed according to the Mohr-Coulomb failure criterion. The mobilized and available shear stresses for each soil layer are summarized in

Table 6. Mobilized vs. available shear strength per soil layer during drainage stage.

Layer	z_Mid (m)	G (kPa)	γp Peak_Field (–)	τm Mobilized (kPa)	τc Capacity (kPa)	Mobilized/Capacity
Fill & Soft Organic Clay	2	6667	0.0067	44.7	61.4	0.73
Soft to Medium Silty Clay	8	7269	0.0067	48.7	66.9	0.73
Medium-Stiff Silty Clay	20	7692	0.0067	51.5	80.1	0.64
Silty Sand	32	12,500	0.0067	83.8	147.3	0.57
Stiff Silty Clay	43.5	9615	0.0067	64.3	164.9	0.39
Very Stiff Silty Clay	62.5	10,000	0.0067	67	210.2	0.32

, together with the corresponding mobilization ratios. As illustrated by the τ-τ plot in Figure 8, all stress states lie well below the failure envelope defined by τ_m = τ_c, indicating that deformation develops within a relatively stable stress regime and is not primarily governed by shear failure.

The mobilized-to-capacity ratios shown in Table 6 and their variation with depth in Figure 8 reveal a clear stratified mobilization pattern. Shallow soft organic and silty clay layers exhibit the highest mobilization levels, with ratios of approximately 0.7, indicating that these strata accommodate excavation-induced stress redistribution primarily through shear strain development. With increasing depth, mobilization decreases progressively, dropping below 0.4 in stiff and very stiff silty clays, where higher stiffness and confinement limit strain accumulation and preserve substantial strength reserves. This depth-dependent transition from strain-dominated behavior in the upper layers to elastic-dominated response at depth reflects the concentration of deformation within the near-surface profile and the attenuation of settlements with depth.

Furthermore, the scatter of mobilized stress states across different construction stages in Figure 9 indicates that excavation sequencing mainly influences deformation magnitude rather than driving stresses toward their available capacity. Even during advanced construction phases, mobilized stresses remain far from the failure envelope, indicating that nonlinear deformation arises from progressive strain mobilization under subcritical stress conditions rather than from strength exhaustion. Overall, the combined interpretation of Table 6, Figure 8, Figure 9 and Figure 10 indicates that the numerical model reasonably reproduces a mechanically admissible deformation regime controlled by stiffness and strain accumulation in soft layers, providing a robust basis for subsequent settlement analyses and numerical field comparison.

4.2.5. Numerical Results and Comparative Analysis

To evaluate the performance of the numerical framework against the instrumentation described in Section 4.1, the finite element results were compared with available field monitoring data. Figure 10 illustrates the modeled responses under best, real, and worst-case scenarios compared with field measurements. The numerical results suggest a reasonable correlation for horizontal displacements ($u_x$), with Root Mean Square Error (RMSE) values generally observed between 0.010 and 0.014 m, and coefficient of determination (R²) values ranging primarily from 0.77 to 0.89 across the different excavation stages. As shown in Figure 10, the shaded area represents the numerical prediction envelope defined by the upper and lower bounds of the simulations, within which the field measurements consistently fall. While vertical displacement ($u_y$) predictions were highly accurate during the intermediate excavation phases (RMSE = 0.007 m), they tended to exhibit higher dispersion in the final stages (with RMSE reaching approximately 0.028 m). Despite this increase in late-stage deviation, the overall calculated trends appear to align with the general behavior observed in situ. Such discrepancies could be associated with the inherent spatial variability of the alluvial deposits or the complexities of hydro-mechanical coupling. Nevertheless, this comparison indicates that the Hardening Soil model, supported by the selected parameters and the undrained Type B formulation, provides a coherent approximation for investigating the geo-mechanical response of saturated soft ground during VSM operations.

The observed discrepancy in vertical displacements ($u_y$) during the final drainage stage, as illustrated in Figure 10, can be interpreted through the lens of hydro-mechanical coupling. Since the current numerical framework prioritizes the short-term undrained response of the silty clays to ensure a conservative stability analysis, the subsequent consolidation-induced settlements driven by the dissipation of excess pore pressures and long-term volumetric strains are not fully captured by the time-independent stiffness formulation. This indicates that while the HS model provides reasonable agreement with the lateral soil-structure interaction and the overall stability, the transition to a drained state in the post-excavation phase introduces additional consolidation settlements that exceed the current instantaneous numerical predictions. This interpretation aligns with the findings of [37], suggesting that time-dependent effects become dominant once the active VSM sinking ceases.

Horizontal displacements were reproduced with higher consistency because they are primarily governed by excavation-induced stress redistribution, stiffness degradation, and soil-structure interaction mechanisms that are reasonably captured by the adopted HS formulation. In contrast, the vertical displacement response during the drainage stage becomes increasingly influenced by time-dependent hydro-mechanical processes, including pore-pressure dissipation, volumetric consolidation, and creep-related settlement mechanisms, which are not explicitly represented within the adopted uncoupled HS framework.

Regarding the observed fluctuations in the monitoring data, it is important to acknowledge the role of soil heterogeneity and spatial variability. Although the numerical model adopts a stratified but deterministic profile, the intrinsic variability of alluvial deposits, particularly in the Yangtze Delta region, can introduce local uncertainties in stiffness and hydraulic conductivity. The fact that the field measurements consistently fall within the numerical prediction envelope Figure 10 suggests that, despite the assumption of lithological continuity, the calibrated parameters adequately represent and bound the real-world operational response. This approach aligns with the engineering practice where the prediction envelope serves as a reliable proxy for quantifying the impact of soil uncertainty on ground deformation.

Overall, the calibrated numerical framework results indicate that the numerical framework provides a coherent approximation of the primary trends of excavation-induced deformation, supporting its application for the subsequent parametric and sensitivity analyses. Furthermore, the reasonable agreement observed for horizontal displacements suggests that the adopted stiffness parameters and soil-structure interaction assumptions provide a stable approximation of the geo-mechanical mechanisms involved in the VSM sinking process.

5. Sensitivity and Robustness Analysis

A systematic sensitivity and robustness assessment was performed to quantify how key geotechnical and interface parameters influence the deformation response of the VSM-excavated shafts. The analysis combined parameter perturbation, stage-wise model-field comparison, and interface calibration to evaluate the predictive stability of the Hardening Soil (HS) model and to identify the dominant factors controlling settlement, lateral displacement, and soil-structure interaction throughout the construction sequence.

5.1. Sensitivity of Stiffness and Strength Parameters

The adopted perturbation range of ±20% is consistent with the coefficients of variation of 15–40% [47,48,49,50] for stiffness parameters in soft clay deposits and represents a realistic approximation of geotechnical parameter uncertainty under the investigated conditions. Such ranges are consistent with the inherent variability of geotechnical properties documented in recent studies on uncertainty quantification in geotechnical models. [48,49].

A systematic parametric study is conducted not merely to observe numerical variations but to establish a hierarchy of influence among geotechnical ($E_{50}$, $E_{oed}$, and $E_{ur}$) under the specific stress-path of VSM excavation. This identifies which soil properties require high-precision laboratory testing to ensure reliable settlement predictions.

A ±20% change in $E_{50}$ produced settlement variations up to ±16%, with maximum sensitivity between −12 m and −32 m, corresponding to zones of highest lateral displacement gradients in the field. Similar stiffness-dominated responses in soft clay have been reported in previous studies [7,36,51]. Variations in $E_{ur}$ and $E_{oed}$ generated more moderate effects (≈10–12%), influencing rebound and vertical compression during drainage but remaining secondary to $E_{50}$ as shown in Figure 11. The selected variation range (±20%) reflects typical uncertainty associated with geotechnical parameters derived from laboratory and in situ testing in soft soil conditions.

The strength parameters ${\varphi }^{\prime }$ and $c^{\prime }$ showed limited influence on deformation but substantially affected stability. A ±10% change in ${\varphi }^{\prime }$ resulted in ≈9% variation in the safety factor but less than 5% impact on settlement. This reflects small-to-medium strain behavior, with mobilized shear strength ratios of 0.32–0.73. Consistent with Hashash (in 2001) and Potts (in 2001), stiffness governs serviceability, while strength governs stability [13,52].

The interface reduction factor controls the shear transfer capacity at the soil-lining interface and governs the level of relative slip that may develop between the structural lining and the surrounding soil.

$${\tau }_{\mathrm{interface}}=R_{\mathrm{inter}}\left(c^{\prime }+{\sigma }_n\tan {\varphi }^{\prime }\right)$$

(5)

Three sources informed its selection: (1) PLAXIS manual guidance (0.60–0.70 for smooth concrete clay; 0.80–0.85 for rough concrete) [17]; (2) empirical interfacial friction relations indicating $\delta \approx 0.6{\varphi }^{\prime }$ for smooth precast concrete [34,35], giving $R_{\mathrm{inter}}\approx 0.62-0.72$; and (3) field-based calibration using displacement profiles at R1–14, R15–28, and R29–41. Therefore, face reduction factor ($R_{inter}$) was refined through comparison with monitored displacement profiles, given the inherent uncertainty associated with direct field characterization of soil-concrete interface behavior under field conditions. The adopted calibrated value is:

$$R_{\mathrm{inter}}=0.7\pm 0.1$$

(6)

With variations generating 9–10% changes in lateral displacement, suggesting that interface behavior is a meaningful but secondary contributor to shaft soil interaction. Soil structure interaction was represented using zero-thickness interface elements whose stiffness and strength were reduced by an appropriate $R_{\mathrm{inter}}$ factor to simulate slip and gapping at lining soil and raft soil contacts [53,54]. Construction was simulated sequentially to activate excavation stages and loading conditions, and global stability was assessed using shear-strength reduction factors [55,56]. The influence of the interface reduction factor ($R_{\mathrm{inter}}$) is comparatively moderate, primarily affecting the magnitude of displacement near the shaft boundary rather than the overall deformation pattern.

The perturbation ranges used in the sensitivity analysis also define the uncertainty bounds adopted in the model calibrated numerical framework stage. In particular, the ±20% variation applied to the dominant stiffness parameters ($E_{50}$, $E_{oed}$, and $E_{ur}$) was used to construct the “best” and “worst” numerical scenarios, while the baseline parameter set corresponds to the “real” case. Consequently, the best-real-worst envelope represents parameter uncertainty derived from geotechnical variability rather than arbitrary numerical calibration.

These results are consistent with the predictive assessment outcomes presented in Section 4, where the good agreement observed for horizontal displacements indicates that the adopted stiffness parameters provide a reliable representation of soil behavior. In addition, the results reveal a non-linear sensitivity where E₅₀ and E_oed dominate the horizontal displacement field. This suggests that for VSM projects in soft clays, the accuracy of the settlement risk assessment is more dependent on the stiffness degradation laws than on the peak strength parameters, a distinction that is crucial for serviceability limit state (SLS) design.

5.2. Robustness Across Construction Stages

The model calibrated numerical framework was conducted through a stage-by-stage comparison between numerical predictions and field monitoring data obtained during the excavation process. The predictive performance of the model was quantitatively evaluated using statistical indicators including the coefficient of determination (R²), root mean square error (RMSE), mean absolute error (MAE), and Bias.

As shown in Figure 12 robustness was assessed by evaluating model field agreement across the five construction phases (R1–14, R15–28, R29–41), raft slab, and drainage. Horizontal displacements ($u_x$) show strong and stable predictive capacity (R² = 0.77–0.89; RMSE ≈ 0.01 m), indicating systematic stiffness-driven behavior. The different predictive behavior observed between horizontal and vertical displacement components is closely associated with their sensitivity to stiffness-dependent and hydro-mechanical processes. Horizontal displacements are primarily governed by excavation-induced stress redistribution and soil-structure interaction effects, which are reasonably represented by the adopted HS formulation and stiffness calibration.

In contrast, vertical displacements during the drainage stage become increasingly sensitive to volumetric consolidation, pore-pressure dissipation, and delayed stiffness degradation effects, resulting in greater predictive dispersion under the adopted uncoupled formulation. In contrast, vertical displacements (uy) showed progressive divergence during the drainage stage, where the response becomes increasingly sensitive to time-dependent hydro-mechanical processes such as pore-pressure dissipation, volumetric consolidation, and delayed stiffness degradation, which are simplified in the present framework within the adopted uncoupled HS formulation.

The sensitivity analysis further indicates that horizontal displacement components are predominantly controlled by stiffness-related parameters ($E_{50}$, $E_{oed}$, and $E_{ur}$), whereas the vertical response during the post-drainage stage exhibits additional sensitivity to time-dependent hydro-mechanical mechanisms not explicitly captured within the present constitutive formulation.

The increase in RMSE and drop in R² during drainage are consistent with known limitations of uncoupled HS formulations in modeling pore pressure dissipation and anisotropic stiffness loss [32,52]. The stage-by-stage comparison also allows evaluation of the robustness of the numerical framework, as the predictive capability of the model remains consistent under the varying stress redistribution conditions associated with the different excavation phases.

Despite this variability, settlement evolution curves, radial trough geometry, and displacement envelopes show that the model reproduces the overall deformation mechanism reasonably and behaves robustly across excavation stages.

5.3. Findings Based on the Sensitivity Analysis

(a)

Stiffness parameters ($E_{50}$, $E_{ur}$, $E_{oed}$) are dominant and govern settlement magnitude, curvature, and deformation shape.

(b)

Interface parameter ($R_{\mathrm{inter}}$) has a significant but secondary effect, particularly influencing lateral displacement distribution.

(c)

Strength parameters (${\varphi }^{\prime }$, $c^{\prime }$) affect basal stability more than serviceability, with limited effect on deformations.

(d)

Robustness is high for horizontal displacements and moderate for vertical displacements during drainage.

Overall, the sensitivity analysis suggests that the model response is primarily controlled by stiffness-related parameters, while interface properties and strength parameters play a secondary role. This supports the robustness of the adopted parameter set for representing excavation-induced deformation in soft soil conditions.

6. Discussion

This study provides a mechanistic interpretation of ground deformation induced by VSM-based deep shaft excavation in stratified soft soil, with particular emphasis on the advantages of the proposed numerical framework and the role of stratigraphic stiffness contrasts. Excavation-induced ground deformation in soft deposits has been widely investigated through numerical simulations and field monitoring studies, which indicates that deformation patterns are governed by excavation sequence, soil stiffness characteristics, and soil-structure interaction effects [25,29].

6.1. Advantages of the Proposed Numerical Model

The primary advantage of the proposed numerical model lies in its ability to capture excavation-induced deformation within a fully three-dimensional framework that accounts for stiffness-dependent soil response and staged construction effects. Similar numerical approaches have been applied in recent studies investigating excavation-induced deformation in complex underground environments [57]. Recent studies employing advanced numerical frameworks have reflected the importance of three-dimensional modelling for capturing the interaction between excavation stages and surrounding ground response [57,58,59]. Many numerical analyses of underground excavations still rely on simplified modelling assumptions such as plane strain conditions or homogenized soil profiles. However, previous studies have shown that such simplifications may lead to inaccurate predictions of ground deformation, particularly in complex stratified soil conditions [25,43].

A key distinction from earlier modelling approaches is the explicit interpretation of deformation mechanisms under serviceability stress levels. The combined interpretation of displacement envelopes, validation metrics, and mobilized to available shear strength ratios reflects that ground response develops under subcritical stress states throughout excavation. Recent developments in engineering analysis increasingly incorporate data-based and data-centric techniques to support interpretation of large monitoring datasets and model field-consistent frameworks [60]. Mobilized stresses remain well below available capacity, suggesting that deformation is governed by progressive strain accumulation rather than by mobilization of ultimate strength. Similar observations have been reported in monitored deep excavations where ground movements develop under serviceability conditions long before shear failure mechanisms are activated [29].

Furthermore, the staged excavation strategy adopted in this study preserves the cumulative excavation depth and stress paths governing soil response while avoiding unnecessary discretization at the scale of individual lining rings. This modelling strategy provides a computationally efficient yet reasonable approximation of excavation processes, which is particularly important for large-scale three-dimensional simulations.

6.2. Influence of Stratigraphic Transitions and Layering Effects

The stratigraphy at the study site is characterized by pronounced stiffness contrasts, including transitions from soft clay to silty sand and subsequently to stiffer clay layers. These stratigraphic variations exert a first-order control on deformation distribution and the depth of influence of excavation-induced ground movements. The displacement envelopes indicate that lateral deformations concentrate primarily within the upper soft clay layers and attenuate rapidly upon entering stiffer strata, even though excavation continues to greater depths. The parametric insights demonstrate that stratigraphic contrast acts as a deformation filter; stiffer layers reasonably confine the plastic strain within the soft clay lenses, preventing a global failure mechanism. This confinement effect represents an important observation for optimizing shaft depth in multilayered alluvial deposits.

Similar stratigraphy-controlled deformation patterns have been reported in several studies of underground excavations in layered deposits, where stiffness contrasts between soil layers significantly influence the vertical propagation of deformation [21,26]. Similar deformation mechanisms associated with stratigraphic stiffness contrasts have also been reported in recent numerical studies of deep excavations in soft clay deposits [58].

In the present case, the intermediate sand layer plays a comparable mechanical role by partially limiting the downward propagation of lateral displacements. This behavior represents the depth of influence, and it is not governed solely by excavation depth but also by the interaction between excavation geometry and stiffness contrasts between adjacent soil layers.

Natural soft soil deposits may exhibit spatial variability in stiffness and strength parameters. In the present numerical model, subsurface conditions are represented by horizontally stratified soil layers with homogeneous mechanical properties within each layer. This modelling approach is widely adopted in practical finite element analyses when the available site investigation data consists primarily of borehole logs and laboratory tests describing the dominant stratigraphic units of the site.

Although local variations in soil properties may influence the magnitude of predicted ground deformations, the adopted stratified representation is consistent with the principal geological structure of the site and provides a coherent framework for evaluating deformation behavior during the VSM excavation process.

6.3. Implications for Design and Modeling Practice

The results highlight the importance of three-dimensional numerical modelling for capturing stress redistribution, construction sequencing, and stratigraphic stiffness contrasts governing deformation around VSM shafts, realistic construction sequencing, and stratigraphic stiffness contrasts, which have been widely recognized as key factors governing deformation behavior in deep excavations [59]. Simplified modelling approaches may have limitations in representing the spatial distribution of excavation-induced ground movements, particularly in layered deposits where stiffness variations strongly influence deformation propagation [43].

The proposed modeling framework provides a balanced strategy that combines mechanical rigor with computational efficiency. By focusing on cumulative excavation effects and deformation mechanisms governed by stratigraphic stiffness contrasts, the model offers a transferable methodology for evaluating displacement envelopes, shaft interaction effects, and serviceability performance in complex urban ground conditions. Similar numerical frameworks have been successfully applied in recent studies of mechanized shaft construction and caisson excavation in soft soils [22].

Although the present paper focuses on a twin-shaft excavation, the study forms part of a broader research framework aimed at understanding three-dimensional deformation mechanisms associated with mechanized shaft construction in soft soils. Consequently, the quantitative results presented here should be interpreted primarily within the geotechnical conditions, shaft geometry, and construction procedures of the investigated project. The methodology developed in this work provides the calibration basis for subsequent analyses involving twin-shaft configurations and their interaction effects, which are particularly relevant in densely urbanized environments where underground infrastructure is constructed near existing structures. In this context, the current case study serves as a necessary numerically evaluated framework stage to ensure that the numerical framework reproduces with acceptable consistency field-measured ground responses before being extended to more complex multi-shaft interaction scenarios.

Nevertheless, some modelling simplifications should be acknowledged. The numerical simulations represent idealized construction conditions and do not explicitly consider operational variations that may occur during field construction, such as machine-induced vibrations, temporary support adjustments, or short-term operational fluctuations of the excavation equipment. These factors may influence short-term construction behavior but are not expected to significantly modify the overall deformation mechanisms governing the serviceability response investigated in this study.

In addition, the simulations were performed using the Hardening Soil constitutive model under an uncoupled hydro-mechanical formulation. While this modelling approach has been widely applied in numerical analyses of excavations in soft soils, it does not explicitly capture fully coupled consolidation effects that may influence vertical deformation during drainage phases. In addition, the long-term, time-dependent soil behavior such as creep was not explicitly considered in the present simulations. Groundwater conditions may therefore contribute to some discrepancies between measured and predicted vertical displacements. In the present study, fully saturated conditions with the groundwater table located at the ground surface were assumed to represent conservative hydro-mechanical conditions. Under VSM excavation, the shaft is observed to remain water-filled during excavation, and the final lining system isolates the surrounding soil from internal drainage, which limits the influence of short-term groundwater fluctuations on the deformation mechanisms investigated.

The adopted HS formulation does not account for small-strain stiffness degradation or creep, which contributes to the observed discrepancies in vertical displacement during the drainage phase. Future studies should incorporate HSS or creep-based constitutive models. Future research may incorporate coupled consolidation formulations, Hardening Soil with Small-Strain Stiffness (HSS), or creep-based constitutive approaches to improve prediction of time-dependent settlement behavior and post-drainage hydro-mechanical response in saturated soft soils. While the present uncoupled HS framework provides a realistic approximation of the primary excavation-induced response during active VSM sinking, additional constitutive refinements may further improve prediction of long-term vertical deformation mechanisms.

7. Conclusions

This study presented a three-dimensional numerical investigation of excavation-induced deformation associated with Vertical Shaft Sinking Machine (VSM) excavation in stratified soft alluvial deposits. The proposed framework incorporated staged excavation, stress-dependent stiffness behavior, and soil-structure interaction effects to evaluate the deformation response of adjacent large-diameter shafts under urban excavation conditions. Based on the investigated case study and within the adopted modelling assumptions, the following conclusions can be drawn:

(1)

The excavation-induced response of the investigated VSM system appears to be primarily influenced by stiffness-controlled deformation mechanisms rather than by full mobilization of soil shear strength. Throughout the analyzed excavation stages, the mobilized stress states remained below the failure envelope, indicating that the observed deformation developed predominantly under subcritical stress conditions.

(2)

The stratigraphic stiffness contrasts strongly influenced the spatial propagation of deformation. Softer silty clay layers concentrated larger displacement magnitudes, while stiffer strata provided partial confinement and reduced the depth-wise transmission of settlement.

(3)

The comparative model-field assessment showed satisfactory agreement for the horizontal displacement response across most excavation stages, indicating that the adopted Hardening Soil (HS) framework provides a mechanically consistent approximation of excavation-induced stress redistribution and soil-structure interaction behavior under the investigated conditions.

(4)

Larger discrepancies were observed for vertical displacement during the final drainage stage. This behavior suggests that post-drainage settlement becomes increasingly influenced by time-dependent hydro-mechanical processes, including pore-pressure dissipation and volumetric consolidation effects, which are not explicitly captured within the adopted uncoupled constitutive formulation, consistent with the known limitations of the uncoupled HS formulation in capturing small-strain stiffness and time-dependent consolidation effects.

(5)

The sensitivity analysis indicated that stiffness-related parameters (E₅₀, E_oed, and E_ur) exert the strongest influence on deformation magnitude and displacement evolution. Horizontal displacement components were found to be predominantly controlled by stiffness redistribution and interface behavior, whereas the vertical response during the drainage stage exhibited additional sensitivity to delayed hydro-mechanical effects.

(6)

The present study is based on a single monitored VSM excavation case in stratified soft alluvial deposits. The conclusions and framework applicability should therefore be interpreted within the specific geotechnical, hydraulic, and operational conditions of the investigated project. Future work will extend this framework to additional case studies and geological settings to broaden its general applicability.