Dynamic Response of Adjacent Tunnels to Deep Foundation Pit Excavation: A Numerical and Monitoring Data-Driven Case Study

Abstract

Urban deep excavations conducted near operational tunnels necessitate stringent deformation control. This study investigates the Baiyun Station excavation by employing a three-dimensional finite-element model based on the Hardening Soil Small-strain (HSS) constitutive law, calibrated using Phase I field monitoring data on wall deflection, ground settlement, and tunnel displacement. Material parameters for the HSS model derived from the prior Phase I numerical simulation were held fixed and used to simulate the Phase II excavation, with peak errors of less than 5.8% for wall deflection and less than 2.9% for ground settlement. The model was subsequently applied to evaluate the impacts of Phase II excavation. The key contribution of this study is a monitoring-driven HSS modeling framework that integrates staged excavation simulation with field-based calibration, enabling quantitative assessment of tunnel responses—including settlement troughs, bow-shaped wall deflection patterns, and the distance-decay characteristics of lining displacement—to support structural safety evaluations and protective design measures. The results demonstrate that the predicted deformations and lining stresses in adjacent power and metro tunnels remain within permissible limits, offering practical guidance for excavation control in densely populated urban areas.

1. Introduction

In the rapid expansion of urban underground space, deep foundation-pit construction underpins critical infrastructure—subways, deep drainage systems and other subterranean developments—yet it still faces unresolved technical challenges. Excavation disturbs the soil’s in situ stress equilibrium, causing deformation, pore-water pressure changes and surface settlement. Consequently, numerical simulations have become the principal tool for probing support-structure performance, stratum behavior and impacts on adjacent tunnels.

Early applications of the finite-element method in the 1970s identified two key phenomena: a pronounced “soil-arching effect” around deep pits [1], and markedly improved base stability when stiff strata underlay the pit or when diaphragm walls penetrated deeply into softer layers [2]. Subsequent three-dimensional elastoplastic models clarified that diaphragm-wall construction sequence significantly alters pre-excavation shaft geometry, ground displacement and lateral wall stresses [3]. More recent laboratory and field studies have refined these findings. Centrifuge tests combined with onsite monitoring show that in soft clay, increasing the gap between nested pits reduces inner-wall displacement, while widening the inner pit increases it—and that outer-pit deformation governs overall settlement [4]. Two-dimensional axisymmetric models validated by centrifuge experiments have quantified how shaft diameter, soil friction, cohesion and support stiffness control lateral earth pressures [5]. Three-dimensional simulations under various surcharge conditions have further elucidated soil-wall-load interactions [6], and case studies at major stations demonstrate that staged excavation and targeted reinforcement can curb tunnel heave, convergence and lining-stress fluctuations [7]. Field-validated simulations in diverse soil types confirm optimal support strategies under real-world constraints. In gravelly sand, pile-anchor systems merge prestress diffusion with passive soil-nail action to limit lateral-wall movement [8]. Conversely, excessive strut spacing has been shown to drive pile displacements beyond allowable limits, prompting tighter strut layouts and symmetric excavation schemes [9]. Optimized sequences—excavating peripheral pits before central ones—have proven effective in reducing lateral deformation and settlement rebound [10], and integrating real-time monitoring into numerical models further enhances deformation control and structural safety [11].

Robust predictive tools are required for complex stress redistribution when pits are near tunnels. Simplified analytical models that combine elastic-multilayer theory with beam-on-foundation concepts enhance deformation forecasts by incorporating stratification and shear effects [12]. Further, treating soil movement as an external load on an elastic-foundation beam streamlines Mindlin’s integrals and aligns with field data—underscoring the critical role of excavation sequencing [13]. Case studies show that excavating zones closest to tunnels first and installing diaphragm walls several times deeper than the excavation limits both tunnel movement and crack propagation, with hypoplastic constitutive models accurately capturing small-strain responses [14]. In fault-zone and layered deposits, elastic modulus proves the most sensitive parameter, enabling linear empirical formulas to predict peak tunnel deformation [15]. Fieldwork in Wuhan confirms that properly designed support systems maintain tunnel deformations within safety thresholds and primarily influence bending moments rather than axial or shear forces [16]. Optimization based on the minimum-potential-energy principle—such as reducing support-pile length asymmetrically by 2 m—has demonstrated both deformation reduction and cost savings in wide-pit projects [17]. Numerical–experimental studies further refine these insights. Finite-element limit analysis (FELA) of a pit adjacent to a double-shield tunnel quantifies how construction sequence, tunnel-pit geometry, and key design parameters govern global stability, yielding safety-factor equations and a risk-classification framework for adjacent zones [18]. FLAC3D simulations and field monitoring of simultaneous Beijing Metro pit and channel excavations reveal that channel work disrupts soil structure, increases pile displacement, and should precede pit excavation with at least twice-depth spacing to minimize interaction [19]. Midas GTS NX simulations and scaled model tests comparing pile-anchor with double-row-pile support demonstrate that double-row piles better control vertical settlement, whereas pile-anchor systems induce larger horizontal shifts—guiding support-system selection for metro pits [20]. Orthogonal-test Midas GTS NX studies confirm that rectangular pits oriented with long sides perpendicular to tunnels, shallower depths, and increased clearances consistently minimize tunnel deformation [21].

Although interactions between deep excavations and adjacent tunnels have been widely studied, important gaps remain in monitoring-calibrated, sequence-explicit validation and in disentangling the roles of proximity/depth alignment versus lining stiffness. Many prior studies do not calibrate the full construction sequence against field data using transparent, comparable fit criteria, making it difficult to quantify how tunnel offset and depth alignment compare with lining stiffness in governing deformation and stress.

To address these gaps, this study develops a monitoring-calibrated PLAXIS 3D–HSS framework that simulates staged excavation, support installation, and groundwater control, and then evaluates the evolving responses of the diaphragm wall, surrounding ground, and two adjacent tunnels with contrasting lining properties and offsets. Calibrated with Phase I monitoring, the framework captures the characteristic wall deflection shape, the funnel-like settlement pattern with base heave, and the decay of tunnel displacement with distance from the pit. The analyses indicate that proximity and depth alignment dominate the tunnel response, while lining stiffness acts as a moderating factor; shear stress tends to migrate toward the stiffer lining as soil support reduces. The calibrated model is subsequently used to forecast the next construction phase, offering a practical basis for sequencing, protection strategies, and risk control in dense urban settings.

2. Project Overview

2.1. Basement Excavation

Baiyun Station, situated between Guangzhou Railway Station (to the south) and Xiamao Station (to the north) on the Fangcun–Baiyun Airport intercity railway line, functions as the third key node within the Guangdong–Hong Kong–Macao Greater Bay Area. Designed as a major interchange hub, the station will integrate Metro Line 12 (under construction), the planned Line 24, Foshan Line 6, a reserved future metro line, and the existing Line 8. The station’s main structure, which spans 751.161 m in length, along with its auxiliary facilities, is part of the Tangxi transportation hub reservation. The primary concrete framework has already been completed.

To the north, a double-track stabling yard turnout extends over a length of 489.389 m, from YDK 76 + 730.030 to YDK 77 + 219.419. The cut-and-cover construction is divided into two phases: Phase 1 is currently under excavation, while Phase II, measuring 85.53 m in length (YDK 76 + 854.000 to YDK 76 + 939.534), is separated by a central wall. Trench widths range from 26.5 to 27.2 m, expanding to 34.7 m at the terminal sections. The final ground elevation is approximately 8.836 m, resulting in excavation depths of about 34.3 m, reaching up to 38.4 m in the widened areas. The construction utilizes a sequential cut-and-cover approach, supported by diaphragm walls and internal bracing systems.

The station site lies immediately west of the Tangxi hub and east of Shilu Road, adjacent to Tangxi Interchange and Shilaxi Road. Minimum horizontal clearances to nearby infrastructure are 57 m to the Line 8 tunnel, 44 m to the power tunnel, and 127 m to the Beijing–Guangzhou Railway embankment. No major pipelines traverse the surrounding area.

2.2. Geological Conditions

Constrained by traffic diversion and temporary land-use conditions, the open-cut connecting section is planned and implemented in two phases, as illustrated in Figure 1a. Phase I corresponds to the excavation stage, while Phase II, which is separated from Phase I by a central partition wall, has not yet begun. Phase II has a total length of 85.53 m. The standard widths of the structure are 27.2 m, 26.5 m, and 26.8 m, with the widened end bay measuring 34.7 m. The finished ground elevation after grading is approximately +8.836 m. In the standard section, the base slab is 1.4 m thick and is constructed over a 0.2 m thick blinding layer. The excavation depth reaches approximately 34.3 m in the standard section and 38.4 m in the widened end section. The construction follows the bottom-up cut-and-cover method, with a diaphragm wall combined with internal struts serving as the primary retaining system.

During Phase I excavation, real-time monitoring was conducted using a variety of instruments, including 40 wall displacement monitoring points (ZQT) installed at 1 m intervals, 40 ground settlement monitoring points (DBC) at 2 m intervals, and 28 tunnel displacement measurement sections spaced every 10 m, as illustrated in Figure 1a.

According to the geotechnical investigation, the site is located on the alluvial plain of the Pearl River Delta at the regional scale, and exhibits typical floodplain and alluvial plain landforms at the local level. Figure 1b shows the excavation layout along with the stratigraphic column. Within the depth range affected by the excavation, the subsurface sequence, from top to bottom, consists of the following layers: artificial fill (0–6 m), silty clay (6–16 m), silty medium to coarse sand (16–23 m), medium to coarse sand (23–34 m), gravelly sand (34–49 m), angular gravel (49–60 m), silty clay (60–70 m), plastic silty clay (70–78 m), plastic clay (78–92 m), hard-plastic to hard residual soil derived from carbonaceous limestone (92–108 m), completely weathered carbonaceous limestone (108–118 m), and highly weathered carbonaceous limestone (118–142 m).

3. Prior HSS Parameterization and Phase I Calibration for Phase II Prediction

Since Phase II excavation has not yet commenced and early monitoring data is unavailable, this study utilizes the adjacent Phase I project for model calibration. The displacement of the diaphragm wall, surface settlement measured near the tunnels, and displacement of the two adjacent tunnels during Phase I excavation are used as reference data. This approach ensures that the numerical simulation of Phase II excavation closely aligns with real-world conditions and enhances construction safety. A numerical model of the Phase II pit was developed, and soil and support parameters were calibrated by matching simulated outputs to field measurements. This calibration improved the model’s accuracy and reliability, enabling it to predict the behavior of Phase II excavation. To assess the potential impact of Phase II excavation on adjacent tunnels, trends in diaphragm wall deflection with depth, surface settlement with distance, and tunnel displacement with distance were compared across both phases. This comparison facilitates early risk identification and ensures that Phase II will produce comparable impacts on adjacent tunnels.

Material parameters for the HSS model in the prior numerical simulation during Phase I were obtained from multiple sources to enhance realism and ensure reproducibility. First, site-specific geotechnical investigation data—including index tests, oedometer compression tests, and consolidated triaxial tests—were used to establish baseline ranges for effective strength parameters $\phi$ and c, compressibility modulus $E_{oed}^{ref}$, and stress dependence exponent $m$. Second, the small-strain shear stiffness $G_0^{ref}$ was calculated from shear wave velocity Vs using the relation $G_0=\rho V^2$, and the resulting values were verified against local experience. Third, published correlations and recommended values from Reference [22] were employed to fill data gaps and define prior estimates for parameters with limited test coverage, such as $E_{50}^{ref}$, $E_{ur}^{ref}$, ${\gamma }_{0.7}$.

Figure 2 presents real-time measurements together with the priori numerical model, enabling a comparison between field data and numerical predictions for wall deflection, ground settlement, and tunnel displacements around the excavation.

In Figure 2a, the simulated diaphragm-wall deflections align closely with the field measurements: the peak deflection of −63.45 mm occurs at a depth of 17 m in the field data, while the model predicts a slightly deeper peak of −67.14 mm at 20 m, resulting in a 5.8% discrepancy. Below 20 m, the displacement decreases sharply and exhibits fluctuations, likely due to variations in soil stratigraphy and the influence of internal struts.

Figure 2b illustrates that both measured and simulated settlements form a trough-shaped profile extending approximately one excavation depth from the pit. The peak settlement values are 61.81 mm (measured) and 63.62 mm (simulated), occurring at approximately 0.5 H and 15 m, respectively.

Figure 2c,d indicate that tunnel displacements are greatest near the excavation and diminish with increasing distance. The power tunnel exhibits a maximum displacement of 5.12 mm, which is higher than the 2.24 mm observed for Metro Line 8, reflecting both its closer proximity to the excavation and its relatively lower lining stiffness.

To quantify the consistency between the computed displacement $u_c\left(z\right)$ and the measured displacement $u_m\left(z\right)$, we first align the computed curve to the set of measured depths ${\left\{z_i\right\}}_{i=1}^n$ without modifying any measured data. Specifically, for each measured depth $z_i$, we obtain the interpolated computed displacement by linear interpolation between the two neighboring computed depths $z_j$ and $z_{j+1}$:

$${\hat{u}}_c\left(z_i\right)=u_c\left(z_j\right)+{\displaystyle \frac{z_i-z_j}{z_{j+1}-z_j}}[u_c\left(z_{j+1}\right)-u_c\left(z_j\right)], z_j\le z_i\le z_{j+1}$$

(1)

This yields paired observations ${(u}_m\left(z_i\right),{\hat{u}}_c\left(z_i\right))$ for $i=1,\dots ,n$.

We then evaluate pointwise agreement using two complementary metrics: the error-based coefficient of determination and the root-mean-square error (RMSE), together with the mean absolute error (MAE):

$$R_{err}^2=1-{\displaystyle \frac{{{\sum }_{i=1}^n\left(u_m\left(z_i\right)-{\hat{u}}_c\left(z_i\right)\right)}^2}{{{\sum }_{i=1}^n\left(u_m\left(z_i\right)-{\hat{u}}_m\right)}^2}}$$

(2)

$${\overline{u}}_m={\displaystyle \frac{1}{n}}\sum _{i=1}^n{u}_m\left(z_i\right)$$

(3)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(u_m\left(z_i\right)-{\hat{u}}_c\left(z_i\right)\right)}^2}$$

(4)

$$MAE={\displaystyle \frac{1}{n}}\sum _{j=1}^n\left|u_m\left(z_i\right)-{\hat{u}}_c\left(z_i\right)\right|$$

(5)

Here, $R_{err}^2$ measures the fraction of the variance in the measurements explained by the model on a point-by-point basis, while RMSE and MAE summarize the typical magnitude of pointwise errors, with RMSE placing greater emphasis on larger deviations.

The second metric assesses trend agreement between the curves while explicitly allowing an overall offset and scale difference. We fit an ordinary least-squares (OLS) relation between the measured displacements and the interpolated computed displacements,

$$u_m\left(z_i\right)=a+b{\hat{u}}_c\left(z_i\right)+{\epsilon }_i$$

(6)

and evaluate the associated coefficient of determination,

$$R_{reg}^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\epsilon }_i^2}{{\sum }_{i=1}^n{\left(u_m\left(z_i\right)-{\hat{u}}_m\right)}^2}}={\rho }^2{(u}_m\left(z_i\right), {\hat{u}}_c\left(z_i\right))$$

(7)

Thus, $R_{reg}^2$ equals the square of the Pearson correlation, emphasizing shape/trend consistency without over-penalizing a uniform bias or an amplitude mismatch.

Based on the present dataset—where the computed values are first linearly interpolated to the measured depths—the results are as follows. For pointwise agreement, the error-based metrics yield $R_{err}^2=0.677$, $RMSE=10.53 \mathrm{m}\mathrm{m}$, and $MAE=8.62 \mathrm{m}\mathrm{m}$. Normalizing RMSE by the measured peak displacement magnitude $\left|u_{max}\right|\approx 16.7\%$ gives ${NRMSE}_{peak}\approx 10.53/63\approx 16.7\%$, which indicates a good level of fit: the overall pattern is captured with some amplitude bias. For trend agreement, the regression-based metric gives $R_{reg}^2=0.851$ with regression coefficients $a=-0.868$ and $b=0.807$, indicating that applying 0.81 scale adjustment together with −0.9 m offset brings the computed trend into close agreement with the measurements.

In summary, $R_{err}^2$ together with RMSE/MAE quantifies pointwise error, whereas $R_{reg}^2$ captures overall trend consistency. Reporting both perspectives—optionally supplemented by RMSE over key depth intervals and residual-versus-depth plots—provides a comprehensive and interpretable assessment of how well the model reproduces the measured displacement profile.

4. Numerical Modeling

4.1. Model Dimensions and Boundary Conditions

The second-phase excavation of the foundation pit was simulated using PLAXIS 3D (version 2024.1.0.1060) finite element software. The excavation geometry—85.53 m in length, 27.28 m in standard width, and 34.70 m at the widened sections, with a maximum depth of 38 m—was used to define the model dimensions.

According to Saint-Venant’s principle, which indicates that the influence zone extends approximately 2.5 times the excavation depth, the overall model size was set to 300 m (X) × 250 m (Y) × 142 m (Z) to minimize boundary effects. As shown in Figure 3, the soil domain was discretized using a swept mesh with “fine” element density to ensure computational accuracy, resulting in a total of 17,584 elements and 70,377 nodes. All four vertical boundaries were constrained against normal displacement, while the base of the model was fixed in all directions to simulate realistic boundary conditions. The relative position between the excavation pit and the tunnels is presented in Figure 4.

4.2. Constitutive Models

Given the site’s stratigraphy and the soil’s nonlinear elastic behavior, this study employs the Hardening Soil model with small-strain stiffness (HSS) to simulate ground and retaining system deformations. The HSS model builds upon the conventional HS formulation by incorporating a small-strain stiffness law that relates shear stiffness to shear strain, as well as stress-dependent moduli for primary loading, oedometric compression, and unloading–reloading cycles, thereby enhancing its applicability in deep excavation analyses. Compared to the 11 parameters required in the HS model, the HSS model introduces two additional small-strain parameters: the reference shear strain ${\gamma }_{0.7}$ and the reference small-strain shear modulus $G_0^{ref}$.

For this project, parameters are identified from laboratory testing and monitoring-based back-analysis. The model captures the evolution of shear and compressional moduli and more accurately reflects small-strain stiffness changes, resulting in predictions consistent with field observations in soft soils and at excavation depths exceeding 15 m. Accordingly, we recommend using the HSS model under these conditions to obtain reliable deformation forecasts and support safe construction planning.

The Hardening Soil Small (HSS) model is a constitutive model developed based on the Hardening Soil (HS) model, with the additional consideration of soil behavior under small strain conditions. While the HS model effectively captures shear and compressive hardening characteristics of soil, it does not account for soil behavior under small strain. Benz [23] proposed the HSS model by integrating the modified Hardin-Drnevich shear modulus relationship with the HS model, incorporating the effects of soil strain history and the multi-axial expansion of the yield surface. As a result, the HSS model retains all the features of the HS model while also capturing the nonlinear variation in soil stiffness under small strain conditions. Except for the parameters that describe small strain behavior, all other parameters are identical to those used in the HS model. In the HSS model, two key small-strain characteristic parameters—the initial shear modulus ($G_0$) and the threshold shear strain (${\gamma }_{0.7}$)—must be determined.

The shear yield function of the HSS model in principal stress space is expressed as follows:

$$f_{12}^s={\displaystyle \frac{2q_a}{E_{50}}}{\displaystyle \frac{({\sigma }_1-{\sigma }_2)}{q_a-({\sigma }_1-{\sigma }_2)}}-{\displaystyle \frac{2\left({\sigma }_1-{\sigma }_2\right)}{E_{ur}}}-{\gamma }^p$$

(8)

$$f_{13}^s={\displaystyle \frac{2q_a}{E_{50}}}{\displaystyle \frac{({\sigma }_1-{\sigma }_3)}{q_a-({\sigma }_1-{\sigma }_3)}}-{\displaystyle \frac{2\left({\sigma }_1-{\sigma }_3\right)}{E_{ur}}}-{\gamma }^p$$

(9)

In this context, ${\gamma }^p$ denotes the accumulated plastic shear strain, $q_a$ is the soil’s asymptotic strength, $E_{50}$ represents the secant stiffness corresponding to 50% of that strength, and $E_{ur}$ is the unloading–reloading secant modulus.

The HSS model further accounts for the stress-dependency of both $E_{50}$ and $E_{ur}$, expressed as follows:

$$E_{50}=E_{50}^{ref}{\left({\displaystyle \frac{c^{\prime }{\cot \phi }^{\prime }-{\sigma }_3}{c^{\prime }{\cot \phi }^{\prime }-{\sigma }^{ref}}}\right)}^m$$

(10)

$$E_{ur}=E_{ur}^{ref}{\left({\displaystyle \frac{c^{\prime }{\cot \phi }^{\prime }-{\sigma }_3}{c^{\prime }{\cot \phi }^{\prime }-{\sigma }^{ref}}}\right)}^m$$

(11)

In this context, $E_{50}^{ref}$ and $E_{ur}^{ref}$ are the secant stiffness and unloading–reloading secant modulus evaluated at the reference stress ${\sigma }^{ref}$, and $m$ denotes the stress exponent.

The initial shear modulus under small strain conditions is derived from the reference initial shear modulus and is determined using a power-law relationship that allows for the calculation of the initial shear modulus at any given stress state:

$$G_0=G_0^{ref}{({\displaystyle \frac{c cos\phi + {\sigma }_3 sin\phi }{c\cos \phi + p^{ref} sin\phi }})}^m$$

(12)

In this context, $G_0^{ref}$ represents the initial shear modulus corresponding to the reference pressure $p^{ref}$.

The threshold shear strain 0.7$\gamma$ is the shear strain at which the shear modulus G decreases to 0.7 times the initial shear modulus 0.7$G_0$ as the strain increases. Benz also suggests that the threshold shear strain 0.7$\gamma$ should be taken as (1~2) × 10⁻⁴.

The small-strain stiffness is a well-known parameter in the dynamic analysis of soil, but it is rarely considered in the static analysis of soil. In soil dynamic problems, the most commonly used small-strain model is the Hardin-Drnevich [24] relationship.

$${\displaystyle \frac{G}{G_0}}={\displaystyle \frac{1}{1+\left|\frac{\gamma }{{\gamma }_r}\right|}}$$

(13)

In this context, ${\gamma }_r={\tau }_{max}/G_0$ (${\tau }_{max}$ is the maximum shear stress at failure).

When applying Formula (13), the error is relatively large. Hardin-Drnevich also suggested replacing ${\gamma }_{\mathrm{r}}$ with ${\gamma }_{0.7}$, that is, the shear strain corresponding to the initial shear modulus reduced to 70% of its initial value. Formula (13) can then be rewritten as Formula (10).

$${\displaystyle \frac{G}{G_0}}={\displaystyle \frac{1}{1+a\left|\frac{\gamma }{{\gamma }_{0.7}}\right|}} (a=3/7)$$

(14)

Research shows that the stiffness of soil will decrease when it undergoes very small strains. However, if the loading direction changes, the stiffness will recover to its initial value. If the load continues to be applied to the soil, the stiffness will decrease again. In Equation (14), the strain history of the soil and multi-axial expansion should be considered. Benz [23] proposed this relationship and derived the corresponding shear strain ${\gamma }_{\mathrm{h}ist}$, which can be expressed as Equation (15).

$${\gamma }_{hist}=\sqrt{3}{\displaystyle \frac{‖H\Delta e‖}{‖\Delta e‖}}$$

(15)

In this context, $\Delta e$ represents the actual strain increment, and $H$ is a symmetric tensor representing the material’s deviatoric strain history.

The shear modulus of the soil decreases with the increase in strain. In the HSS model, the shear modulus corresponding to the unloading and reloading modulus is taken as the lower limit value for the reduction in the initial shear modulus. At this time, the corresponding shear strain is ${\gamma }_c$. From Equation (14), the expression for the shear strain value at this time can be obtained, which is Equation (16).

$${\gamma }_c={\displaystyle \frac{7}{3}}({\displaystyle \frac{G_0}{G_{ur}}}-1){\gamma }_{0.7}$$

(16)

$$G_{ur}={\displaystyle \frac{E_{ur}}{2(1+V_{ur})}}$$

(17)

The HSS model can reflect the characteristics of soil under small strain mainly because its stiffness matrix determines whether the soil is under small strain based on different strain amplitudes, and calculates the stiffness under different stress states according to the corresponding power index relationship, and solves the deformation of each point inside the soil. The HSS model is an isotropic hardening elastoplastic model, and its isotropic elastic stiffness tensor can be expressed by Equation (18).

$$D_{ijkl}={\displaystyle \frac{2G}{1-2V_{ur}}}(\left(1-2V_{ur}\right){\delta }_{ik}{\delta }_{jl}+{\upsilon }_{ur}{\delta }_{ik}{\delta }_{jl})$$

(18)

$$G=\left\{\begin{array}{c}{G}_0({\displaystyle \frac{{\gamma }_0}{{\gamma }_0+a{\gamma }_{hist}}})\\ {\displaystyle \frac{E_{ur}}{2(1+V_{ur})}}\end{array}\right.$$

(19)

It can also be seen from Equation (16) that when the shear strain is less than ${\gamma }_c$, the small strain of the soil can be considered; when the shear strain is greater than ${\gamma }_c$, it is the deformation characteristics of the soil under large strain.

4.3. Layer Parameters

The parameters of the HSS model include the effective cohesion (c), the effective internal friction angle ($\phi$), the reference secant stiffness from triaxial drained tests ($E_{50}^{ref}$), the reference tangent stiffness from oedometer tests ($E_{oed}^{ref}$), the stress-dependent exponent ($m$), the reference unloading/reloading stiffness ($E_{ur}^{ref}$), Poisson’s ratio ($\mu$), and the reference initial shear modulus ($G_0^{ref}$), which corresponds to the shear strain (${\gamma }_{0.7}$) at which the shear modulus degrades to 0.7$G_0$.

In this numerical model, soil layers were defined based on borehole data. The HSS parameters were derived by back-analysis against the prior numerical model presented in Section 3. Accordingly, the parameters adopted in this study follow the calibration results of that prior model. The constitutive parameters for each soil layer are summarized in

Table 1. Geotechnical parameters adopted for the HSS constitutive model by soil layer.

Soil Layer	γ /kN·m−3	${\mathit{E}}_{50}^{\mathit{r}\mathit{e}\mathit{f}}$ /MPa	${\mathit{E}}_{\mathit{o}\mathit{e}\mathit{d}}^{\mathit{r}\mathit{e}\mathit{f}}$ /MPa	${\mathit{E}}_{\mathit{u}\mathit{r}}^{\mathit{r}\mathit{e}\mathit{f}}$ /MPa	$\mathit{\mu }$	${\mathit{G}}_0^{\mathit{r}\mathit{e}\mathit{f}}$ /MPa	γ0.7	c/kPa	φ/°
Fill	19.5	3	3	15	0.30	30	10−4	10	7
Silty soil	16.5	2.5	2.5	12.5	0.30	25	10−4	10.7	9.1
Silty medium-coarse sand	18.7	5	5	25	0.30	40	10−4	1	22
Medium-coarse sand	19.0	13	13	39	0.30	65	10−4	1	32
Gravel-sand	20.0	20	20	60	0.24	100	10−4	1	34
Angular gravel	20.5	28	28	84	0.24	140	10−4	1	35
Silty soil	17.4	3.5	3.5	17.5	0.30	35	10−4	8	10
Plastic silty clay	19.2	5.4	5.4	27	0.30	54	10−4	15.4	13.6
Plastic cohesive soil	19.6	4	4	20	0.30	40	10−4	20	16
Hard-plastic to stiff residual Carbonaceous limestone soil	19.5	7.6	7.6	35	0.28	70	10−4	22	18
Completely weathered Carbonaceous limestone	20.5	5.9	5.9	29	0.27	58	10−4	33	24
Highly weathered Carbonaceous limestone	21	10	10	30	0.23	50	10−4	40	28

.

The linings of the power tunnel and Metro Line 8 tunnel are constructed using C35 and C50 concrete, respectively. A linear elastic constitutive model is adopted in the simulation, and the corresponding material parameters are listed in

Table 2. Material and geometric properties of tunnel linings.

Structure	Materials	γ $/\mathbf{k}\mathbf{N}·{\mathbf{m}}^{-3}$	E /MPa	µ
Power tunnel	C35	24.0	32,500	0.20
Metro Line 8 tunnel	C50	25.0	35,000	0.20

.

Based on on-site engineering records, the excavation support system comprises diaphragm walls, ring beams, steel struts, and internal columns. In PLAXIS 3D, diaphragm walls and secant-pile walls are typically modeled using plate elements, whereas ring beams, struts, and columns are simulated using beam elements. The material properties for these support components are summarized in

Table 3. Constitutive parameters of diaphragm-wall ring beams, steel struts, and columns in the numerical model.

Structure	Material Elements	γ /kN·m−3	Cross-Sectional Dimensions/m	Properties	E /MPa	µ	G /MPa
Diaphragm Wall	Plate	25	-	Isotropic	36,000	0.2	15,000
Ring beam	Beam	25	0.5 × 0.5	Isotropic	25,000	0.2	-
Steel strut	Beam	78.5	0.6 × 0.6	Isotropic	30,000	0.2	-
Column	Beam	78.5	0.6 × 0.6	Isotropic	30,000	0.2	-

.

4.4. Numerical Modeling Procedure

In the field, the diaphragm wall was constructed to a depth of 44 m and was modeled in PLAXIS 3D using plate elements. Interface elements were applied along its perimeter to simulate the contact and frictional behavior, thereby enabling a more realistic representation of wall-soil interaction and stress evolution. The excavation was carried out using the open-cut and top-down method. To ensure an accurate simulation of the construction process, the numerical model incorporated the complete sequence of excavation, support installation, and dewatering, all of which were consistent with the actual site schedule and technical specifications. At each stage, the model assessed the stress distribution, deformation characteristics, and stability of the support system. The staged construction feature in PLAXIS 3D was utilized to define each construction scenario, as summarized in

Table 4. Numerical modeling scenarios for the excavation.

Construction Step	Corresponding Actions	Construction Step	Corresponding Actions
Case 1	Initial stress state balancing	Case 6	Activation of the third internal support; hydraulic conditions inside the pit set to dry
Case 2	Activation of existing pipelines and tunnels, with replacement of tunnel lining materials by C35 and C50 concrete	Case 7	Activation of the fourth internal support; hydraulic conditions inside the pit set to dry
Case 3	Activate the diaphragm walls and columns	Case 8	Activation of the fifth internal support; hydraulic conditions inside the pit set to dry
Case 4	Activate the first internal support; set hydraulic conditions inside the pit to dry	Case 9	Activation of the sixth internal support; hydraulic conditions inside the pit set to dry
Case 5	Activation of the second internal support; hydraulic conditions inside the pit set to dry

.

5. Results and Discussion

5.1. Diaphragm-Wall Deformation and Bending Moments

Figure 5 presents the horizontal displacement contours of the diaphragm wall in both the X and Y directions after excavation. The wall exhibits a relatively uniform inward deformation pattern, with the maximum horizontal displacement observed at the mid-height section. Specifically, the peak displacements along the long and short sides of the excavation are 66.49 mm and 61.36 mm, respectively, accounting for only 0.018% of the excavation depth. These values are considerably below the deformation control limit of 0.2% H specified in the Technical Code for Building Foundation Pit Support (DB11/489-2016) [25] and the Code for Design of Urban Rail Transit Engineering (DB11/995-2013) [26]. The results demonstrate that the diaphragm wall ensures structural safety and maintains stability throughout the excavation process.

Figure 6a presents the deformation profiles of the diaphragm wall adjacent to the tunnel at various excavation stages. Negative values indicate inward displacement toward the excavation pit. As the excavation proceeds, the maximum lateral displacement increases, reflecting a progressive accumulation of deformation. All displacement curves exhibit a “bow-shaped” pattern. The maximum horizontal displacement reaches 66.42 mm, primarily occurring at depths between approximately 10 m and 20 m. In comparison, the displacements at the top and bottom of the wall are relatively smaller, which can be attributed to the lower active earth pressure and the partial restraint provided by the stiff internal support system.

Figure 6b illustrates the bending moment distribution along the diaphragm wall after excavation. The highest bending moments are observed along the long sides of the pit, corresponding to the larger structural span and greater lateral earth pressure. As excavation and support installation progress, peak bending moments develop in the mid-sections of the wall, following a consistent pattern associated with each construction stage. At the base of the wall, bending moments remain relatively high due to the reaction forces exerted by the underlying soil layers. The maximum bending moment increases from 2169 kN·m after the first excavation stage to 2857 kN·m after the second stage, and subsequently decreases to 1917 kN·m upon completion of excavation, as the support system provides increased stiffness to the wall.

5.2. Ground Settlement Patterns

Figure 7a presents the settlement contours of the surrounding soil after excavation. The maximum settlement occurs near the excavation edges, particularly around the diaphragm walls, forming a characteristic funnel-shaped profile with a peak value of approximately 79.14 mm. At the excavation base, soil heave also exhibits a funnel-like shape, reaching a maximum of about 177.69 mm and gradually decreasing with distance from the pit center. These observations indicate that vertical deformation is primarily concentrated at the excavation boundaries and base, respectively, with both settlement and uplift diminishing progressively outward, reflecting the localized influence of excavation on soil movement.

Figure 7b illustrates the ground settlement curves along the tunnel-adjacent side at various excavation stages, based on vertical displacement data collected within 80 m from the pit wall. Upon completion of excavation, the main settlement zone extends outward to a distance approximately equal to the excavation depth. All curves exhibit a trough-shaped profile, with the maximum settlement of approximately 63.62 mm occurring at about 15 m from the pit edge. Settlement magnitudes increase progressively with each excavation stage, developing slowly during the initial phases and intensifying as excavation depth increases. This evolution aligns with previous studies, confirming the typical influence of deep excavation on ground settlement.

5.3. Dynamic Response of Adjacent Tunnels During Excavation

Figure 8 and Figure 9 present the displacement cloud map and overall displacement curves of the existing tunnels after excavation is completed. In Figure 9, the horizontal axis is referenced from the point on the tunnel closest to the excavation site. These figures demonstrate that the foundation pit excavation induces displacement in both the power tunnel and Metro Line 8 tunnel. Moreover, the overall displacement of both tunnels increases progressively with excavation depth.

The maximum tunnel displacement occurs at the central section, decreasing toward both ends. During excavation stages 4 to 6, overall displacements for both the power tunnel and Metro Line 8 remained below 1.5 mm, indicating minimal impact at shallower depths. Displacements increased significantly during stages 7 to 9, peaking at 3.77 mm for the power tunnel and 2.63 mm for Metro Line 8 in stage 9, as deeper soil unloading intensified the effects. Upon completion of the excavation, the power tunnel exhibited greater total displacement than Metro Line 8, primarily due to its lower lining stiffness and closer proximity to the excavation pit.

Overall, although the excavation influences the existing tunnels, the resulting deformation remains within a controllable and acceptable range, ensuring the safety and structural stability of the adjacent tunnel systems.

Figure 10a–f demonstrates that as the excavation depth increases, the maximum shear stress in the Metro Line 8 tunnel lining—nearest to the excavation pit—rises from −237.19 kPa to −548.64 kPa. Similarly, in the power tunnel lining, shear stress increases from −126.57 kPa to −383.94 kPa. These variations indicate progressive soil stress relief and intensified shear deformation near the excavation area.

Initially, shear stress is concentrated on the power tunnel lining. However, as excavation proceeds deeper, the high-stress zones gradually transfer into and expand within the Metro Line 8 tunnel lining. This transition occurs because the excavation process reduces the original soil support, and the relatively stiffer Metro Line 8 lining attracts a greater share of the redistributed load, thereby intensifying the shear stress around that tunnel.

Simulation results further show that peak shear stresses are localized at the mid-height sections of both tunnel sidewalls. Due to its smaller diameter and more flexible structural system, the power tunnel experiences lower lateral shear stress and a narrower stress distribution compared to Metro Line 8.

6. Conclusions

This paper investigates the super-deep excavation project at Baiyun Station within the Guangdong–Hong Kong–Macau Greater Bay Area. Using PLAXIS 3D in conjunction with the Hardening Soil Small (HSS) constitutive model, the study simulates the entire construction sequence, including staged excavation, support installation, and groundwater control, while assessing the potential impacts on adjacent tunnels. The model’s reliability is verified through comparison with field monitoring data. The key findings of the research are summarized as follows:

• The PLAXIS 3D model with the HSS constitutive law, calibrated against Phase I monitoring, reproduces both the magnitude and pattern of excavation-induced deformations. The peak diaphragm-wall deflection is captured within 5.8% (measured −63.45 mm at −17 m vs. simulated −67.14 mm at −20 m), and the settlement trough within 2.9% (measured 61.81 mm vs. simulated 63.62 mm). Over the full depth/offset ranges, goodness-of-fit metrics indicate $R_{err}^2=0.677$, $RMSE=10.53 \mathrm{m}\mathrm{m}$, $MAE=8.62 \mathrm{m}\mathrm{m}$, and strong trend agreement $R_{reg}^2=0.851$ with ${NRMSE}_{peak}\approx 16.7\%$. The model also captures the proximity-dominated tunnel response (power tunnel > Line 8). Balancing accuracy with computational efficiency, we retain the calibrated HSS parameters unchanged for Phase II prediction, and consider the model adequate for forecasting excavation-induced deformations.

2

The diaphragm wall exhibited a bow-shaped horizontal deformation pattern, with maximum displacements of 66.49 mm on the long side and 61.36 mm on the short side, accounting for only 0.018% of the excavation depth. The peak bending moments reached 1917 kN·m, which were significantly below the safety thresholds. The soil settlement around the excavation pit displayed a funnel-shaped distribution, with maximum settlement of 79.14 mm near the wall and 177.69 mm of heave observed at the base. The ground settlement profiles were trough-shaped, with the maximum settlement of 63.62 mm occurring within one excavation depth.

3

The maximum tunnel displacements were observed closest to the excavation site and decreased with increasing distance. The power tunnel experienced a peak displacement of 3.84 mm, which exceeded that of Metro Line 8 (2.63 mm), primarily due to its closer proximity to the excavation and lower lining stiffness. All measured displacements remained within acceptable limits. Excavation activities led to an increase in shear stresses, with maximum values reaching −548.64 kPa in the Metro Line 8 tunnel and –383.94 kPa in the power tunnel. As the soil support weakened, the high-stress zones gradually shifted toward the stiffer metro tunnel lining.

Overall, this study demonstrates that a rigorously calibrated numerical model can accurately simulate the complex soil–structure interactions occurring during ultra-deep excavation. The findings provide a solid foundation for optimizing support systems and mitigating potential risks in similar urban foundation pit projects.

7. Limitations

This study utilizes a monitoring-calibrated HSS (HS-Small) framework to predict the response during Phase II excavation; however, several underlying assumptions and simplifications limit the generalizability of the findings.

Groundwater representation. A constant groundwater table, consistent with the monitored period, is assumed, and internal pit conditions during staged excavation are modeled as dry, in accordance with the construction plan. Seasonal or long-term variations in piezometric levels, as well as transient seepage, are not considered. As a result, the potential effects on basal heave stability, effective stress distribution, and ground settlements under prolonged water level changes are not evaluated in this analysis.

Parameter uncertainty and spatial variability. The initial parameter values are derived from site investigations and the published literature, and subsequently calibrated using data from Phase I monitoring. While the model captures key responses within acceptable engineering accuracy, formal quantification of uncertainty (e.g., posterior distributions) is not conducted. Soil layer properties are assumed to be piecewise homogeneous; small-scale heterogeneity, anisotropy, and random spatial variability are not explicitly accounted for in the model.

Boundary conditions and discretization. Far-field boundaries are positioned at distances designed to minimize boundary effects, and a mesh convergence check was performed. Nevertheless, residual sensitivity to domain size, boundary type, and local mesh refinement cannot be entirely ruled out.

Implications for practice and future work. Within these limitations, the calibrated model offers decision-ready predictions for Phase II under the as-designed construction sequence. Future research will aim to expand the framework by (i) incorporating transient groundwater analyses and seasonal fluctuation scenarios, (ii) integrating uncertainty quantification and spatial variability (e.g., random-field HSS parameters), and (iii) assessing transferability through multi-site datasets and targeted parametric studies linked to code-level triggering thresholds.