Disturbance Sensitivity of Proximity Construction in Subway Protection Zone

Abstract

The analysis of the impact of the construction of the subway protection zone on the adjacent subway tunnel has become the premise on which to ensure the safe operation of the tunnel. The need for expert members to carry out safety assessments based on specific calculations to determine the impact of construction on the safety of protected tunnels is extremely inconvenient for safety management and significantly reduces management efficiency. This paper analyzes and qualitatively judges the influence range and disturbance size of pile foundation construction, shallow foundation engineering, and foundation pit excavation. Based on relevant research results from scholars and numerical simulation methods, quantitative analysis and comparison are performed on the parameter sensitivity of pile foundation engineering, shallow foundation engineering, and foundation pit engineering along the subway line, and the influence of multi-factor combination is studied and discussed to obtain the influence sensitivity of each factor. The results show that the increase in pile spacing can effectively reduce the pile group effect. The sensitivity of subway tunnel settlement displacement is mainly controlled by the settlement displacement value. The larger the settlement displacement is, the stronger the sensitivity is. The loaded pile foundation arranged along the direction of the subway tunnel has more obvious disturbance to the subway tunnel than that arranged perpendicular to the direction of the subway tunnel.

1. Introduction

The construction of urban rail transit has been improving along with the urbanization process, bringing significant convenience to the daily lives of city residents, especially mega cities. In 2025, 53 cities in China have already built rail transit. As early as 1 March 2005, the Chinese Ministry of Construction prospectively adopted the “Measures for the Management of Urban Rail Transit Operations”, in which the concept of a “Protected Zone” was mentioned and the general scope was defined through administrative planning. Each participating city has its own regulations for the protection zones of underground railways. For each city, the protection zone of underground stations and tunnels is generally 50 m, and

Table 1. Scope of protected areas for rail transit in selected cities in China.

Cities, Implementation Time	Scope of Protected Areas (m)				Special Scope of Protected Areas (m)
	(1)	(2)	(3)	(4)	(1)	(2)	(3)	(4)
Nanjing, 2009; Suzhou, 2011; Fuzhou, 2013; Qingdao, 2015; Dongguan, 2016; Chengdu, 2017; Shijiazhuang, 2017; Guiyang, 2018; Jinan, 2019; Xiamen, 2019; Xuzhou, 2019	50	30	10	100	5	3	5	50
Nanning, 2016; Lanzhou, 2017; Wenzhou, 2019	50	30	10	100	5	5	5	50
Zhengzhou, 2016; Urumqi, 2018	50	30	10	100	10	5	5	50
Wuxi, 2014; Luoyang, 2020	50	30	10	100	10	10	5	50
Xi’an, 2011	50	30	10	100	10	5	3	20
Foshan, 2010; Hangzhou, 2012	50	30	10	100	5	3	3	--
Kunming, 2011	50	30	10	100	5	3	5	--
Shaoxing, 2021	50	30	10	100	5	3	--	--
Shanghai, 2002; Guangzhou, 2008; Ningbo, 2012; Harbin, 2013; Beijing, 2015; Changchun, 2015	50	30	10	100	--	--	--	--
Wuhan, 2012; Changsha, 2013; Nanchang, 2016	50	30	10	150	--	--	--	--
Chongqing, 2011; Shenyang, 2018	50	30	10	200	--	--	--	--
Tianjin, 2006; Shenzhen, 2015; Dalian, 2015; Huhhot, 2021	50	30	10	--	--	--	--	--
Hefei, 2017	50	30	--	--	15	15	10	--

Note: (1) Underground stations and tunnels. (2) Above-ground stations and lines (including elevated). (3) Stations attached, control centers, substations, field sections, etc. (4) Cross-lake and cross-river tunnels, and bridge structures.

shows the protection zones of railways in some cities in China. Geological conditions vary from city to city, and the delineation of protected zones needs further precision.

The disturbance to the subway tunnel caused by the construction of the protection zone seriously affects the safe operation of the subway tunnel. The problem of pile penetration in subway tunnels is frequent. As major infrastructure in people’s livelihoods and safety, metro tunnels need to make important guarantees for their safety. The TOD development mode of urban subway tunnels has led to a large number of engineering projects being undertaken along the subway, which greatly affects the normal operation of the tunnel.

At present, most of the projects under construction and planned in the subway protection zone are commercial and civil buildings, and shallow foundation, pile construction, and pit excavation are often unavoidable [1,2,3]. At the same time, the main factors affecting the safety of the subway come from the disturbance of the soil in the construction of the protected zone, such as shallow foundation overload, pit excavation unloading, and pile construction effects [4,5].

For shallow foundation works, Li et al. established a longitudinal beam model considering the overall stiffness reduction in the tunnel due to the presence of joints [6]. Based on the assumptions that the tunnel deformation is consistent with the soil and the magnitude of the longitudinal forces in the tunnel is proportional to the amount of ground settlement, additional longitudinal rumble deformation and additional stresses are generated in the tunnel when the ground acts as a shallow foundation uniform load. Huang, Dawei et al. quantified the horizontal/vertical tunnel displacement under surface overload based on a 1:10 model test [7]. The results of the test analysis showed that the convergence deformation of the tunnel due to surface overload was significantly larger than that of the hard soil when the tunnel crossed the soft soil layer. Yang et al. analyzed the relationship between tunnel deformation and the magnitude of the load distribution range based on the relationship between soft soil shield tunnels subjected to surface overload action above [8]. Li et al. simulated the impact of the whole process of superstructure construction on existing tunnels and assessed the safety of subway tunnels through indicators such as tunnel convergence deformation, horizontal/vertical displacement, and additional stresses in the lining [9].

For pile construction, Potts et al. (1991) [10] conducted a study based on pile penetration next to a London Underground tunnel. The results showed that pile penetration in close proximity causes local lateral inward extrusion deformation of the lining with a horizontal diameter reduction of about 3.3 mm. The construction of the piles at the same time will cause the overall settlement deformation of the tunnel. Morgan and Bartlett (1969) [11] studied the effect of bored pile construction on adjacent subway tunnels through experiments and showed that reducing the pile–subway tunnel spacing would increase the disturbance deformation of the tunnel, and the absolute value of the underpass tunnel deformation was low. Yan (2007) [12] analyzed the effects of the construction process of bored piles on the soil stress, porous water pressure changes, and the force and deformation of the underpass tunnel by numerical simulation. The results show that the influence of the drilling process in the mud retaining wall is relatively large throughout the whole construction process of bored piles, while the concrete pouring and the hardening process after pouring have almost no influence on the overall deformation of the soil. From the stress relief results of the construction process of bored piles, the borehole process makes the stress reduction in the soil near the hole less than 7%, and the impact range is limited. Chapman et al. (2001) [13] illustrated that the risk of existing tunnels being affected by pile construction is closely related to the quality of pile construction, based on the construction of non-extruded piles near existing tunnels in central London.

For foundation excavation, Huang et al. (2013) [14] used Mindlin solution to calculate the horizontal and vertical additional stresses in the subway tunnel caused by foundation excavation. Combined with simplifying the subway tunnel into a Winkler model, the displacements and stresses in the subway tunnel affected by the pit excavation were obtained. Zhang and Huang (2009) [15]. and Chen et al. (2005) [16] used a two-stage analysis method: in the first stage, the additional stresses in the adjacent subway tunnel caused by the pit excavation were calculated based on the Mindlin solution, and in the second stage, the additional stresses in the soil were applied to the underpass tunnel based on the Winkler foundation model, and then the theoretical analytical expressions for the longitudinal displacement and internal forces in the underpass tunnel were obtained. Jiang (2014) [17] calculated the additional deformation of the soil caused by the excavation of the foundation pit, applied it to the elastic foundation beam model to obtain the analytical solution of the pipeline deformation and internal forces, and further gave the weighted residual solution. Zhang et al. (2014) [18] solved for the subway tunnel deformation values by Boussinesq and Mindlin solutions, assuming that the subway tunnel is coupled with soil deformation, based on the soil rheology under the viscoelastic–plastic model.

During the urbanization process, the number of projects adjacent to subway protection areas (such as pile foundation construction, shallow foundation loading, and foundation pit excavation) has increased significantly [19]. Construction disturbances pose threats to the safe operation of subway tunnels. Existing research lacks multi-factor coupling analysis and parameter sensitivity quantification, and relying on expert experience for evaluation is inefficient. The aim of this study is to clarify the disturbance mechanisms of different projects on subway tunnels through numerical simulation and multi-factor analysis, propose scientifically quantified safety control standards, and improve the management efficiency, risk prevention, and control capabilities of projects adjacent to subway protection areas.

The research objectives and tasks mainly include the following. ① The quantification of disturbances and mechanism analysis: determine the disturbance degrees (such as heave–settlement displacement and lateral displacement) of pile foundation construction, shallow foundation loading, and foundation pit excavation on subway tunnels, and establish a quantitative relationship between key engineering parameters and tunnel deformation. ② The identification of parameter sensitivity and priority classification: identify the sensitivity levels of parameters such as pile spacing, foundation pit depth, and shallow foundation load to tunnel safety, providing a theoretical basis for engineering control. ③ Safety thresholds and engineering optimization suggestions: propose construction control thresholds and differentiated management schemes for different project types. ④ Multi-factor coupling verification and engineering application: verify the universality of the model through 31 engineering cases and form a data-driven decision-support system. The main goal of this paper is to systematically reveal the disturbance laws of adjacent projects on subway tunnels, clarify the sensitivity levels and safety thresholds of key parameters, provide a quantitative basis for the scientific management of subway protection areas, and effectively make up for the deficiencies of traditional experience-based evaluations.

2. Methodology

The aim of this study is to systematically analyze the disturbance effects of projects adjacent to subway protection areas (pile foundation construction, shallow foundation engineering, and foundation pit excavation) on existing subway tunnels through numerical simulation methods. This study is divided into the following six main stages, and the objectives and methods of each stage are briefly described as follows: ① Based on the actual needs of projects adjacent to subway protection areas, clarify the core research issues: quantify the influence laws of pile foundation construction, shallow foundation loading, and foundation pit excavation on the displacement (heave–settlement and lateral displacement) of subway tunnels, and identify the sensitivity of key parameters. This research focuses on the spatial distribution characteristics of disturbance effects, the multi-factor coupling mechanism, and engineering control thresholds. ② Use Plaxis geotechnical finite element software for numerical simulation. Plaxis geotechnical finite element numerical analysis software is well suited for solving problems in geology, underground engineering, tunneling, and foundation engineering [20]. In this paper, this software is used to establish numerical models for analyzing the effects of the three main factors, respectively. ③ For the three typical adjacent construction engineering disturbance behaviors of pile foundation, shallow foundation, and foundation pit excavation, study and reveal the linear/exponential attenuation laws of pile spacing (with the highest sensitivity) and the net distance of pile foundations on tunnel settlement; clarify the non-linear influence of load intensity (the core sensitive parameter) and lateral distribution range on tunnel deformation; and quantify the power–function relationship between excavation depth (with the highest sensitivity) and the net distance. Provide a quantitative basis for the engineering management of subway protection areas.

2.1. Stratigraphic Parameters and Structural Ontological Model

In underground engineering, the commonly used stratigraphic ontological models include the Mohr Coulomb model (MC), the soft soil model, the linear elastic theory model, the soil hardening model (HS), the small strain hardening (HSS) model, and the modified Cambridge model. Considering the geological characteristics of Jinan and the comparability of the literature data, the stratum of this paper adopts the Mohr Coulomb model (MC) in the numerical simulation calculation, and its relevant calculation parameters are shown in

Table 2. Numerical analysis of stratigraphic parameter indicators.

Geological Layer	Soil Capacity Above Water Level (kN·m−3)	Soil Capacity Below Water Level (kN·m−3)	Permeability Coefficient (m·day−1)	Poisson’s Ratio	Elastic Modulus (kPa)	Cohesive Forces (kPa)	Angle of Internal Friction (°)	Angle of Dilation (°)	Interface Strength Reduction Factor
Mixed fill	16.0	18.0	0.29376	0.35	8000	8	10.0	0.8	rigid
Powdered soil	14.4	18.4	0.3752	0.31	19,000	13	23.5	0.8	rigid
Plastic clay	14.2	18.5	0.003456	0.35	16,000	26	18.6	0	0.7
Hard plastic clay	15.5	19.2	0.001210	0.28	35,000	62	21	0	rigid

. For concrete structures, such as subway tunnel structures, pile foundations, and pipe sheets in foundation-retaining structures, linear elastic models are used.

2.2. Computational Models

A stratigraphic structure model is used, with a model size of 200 m × 80 m, in which the initial groundwater level is located at 0.5 m below the ground surface. Normal constraints are applied to the bottom and left and right sides of the model, and the ground surface is a free surface. The outer diameter of the subway tunnel D_Tu = 6.2 m, the thickness of the pipe sheet = 0.35 m, the burial depth H_Tu = 10.6 m, and the concrete strength grade of the pipe sheet is taken as C50. In addition, the calculation models for the influence of pile foundation, foundation pit, and shallow foundation are elaborated separately.

2.2.1. Calculation Model for the Impact of Pile Foundation Work

The pile foundation is simplified to a continuous wall using the principle of equivalent stiffness, and the simplification principle is shown in Equation (1).

$$E_{pile}\left({\displaystyle \frac{\pi D_{pile}^2}{4S_2}}\right)=E_{wall}{t}_{wall}$$

(1)

where E_pile is the pile elastic modulus, MPa; D_pile is the pile diameter, 1 m; E_wall is the equivalent wall elastic modulus, MPa; t_wall is the equivalent wall thickness, m; and S₂ is the spacing of pile foundation parallel to the tunnel direction, m. When S₂/D_pile < 3, the pile–pile interaction is significant, the equivalent model will overestimate the pile stiffness, and the correction coefficient needs to be introduced; when S₂/D_pile > 7, the soil stress superposition effect between piles weakens, and the equivalent model may underestimate the disturbance, and it is recommended to use a discrete pile model. Therefore, in this article, we further emphasize that the ratio of pile spacing to pile diameter is within the range of 3–7.

The pile foundation concrete strength grade is C40, and the pile foundation is fitted with a plate unit and simulated with a linear elastic model with a pile modulus of 3.25 × 10⁷ kPa and Poisson’s ratio of 0.2. The construction step time of the pile foundation project is calculated according to the no-precipitation working condition. The tunnel excavation ground loss is assumed to be 2%. After the tunnel construction is completed, soil consolidation and super-hole pressure dissipation calculations are carried out to obtain the initial ground stress state, pile construction, and pile load loading. In the planar finite element simulation, 15-node triangular solid units are used for the pile foundation and soil, and 15-node curved beam units are used for the tunnel tube sheet.

The pile size and spatial location are shown in Figure 1. In order to analyze the impact of different types of pile foundation works on the subway tunnel, this paper performs parameter sensitivity analysis on group pile size, pile length, pile spacing, and pile foundation–underpass tunnel clear distance, respectively, and the range of model parameters taken into account is shown in

Table 3. List of pile foundation engineering model parameters.

Unchanged Parameters		Variable Parameters
Pile Diameter Dpile	Group Pile Length-to-Width Ratio Lc/Bc	Number of Group Pile Rows	Pile Length	Transverse Pile Spacing S1	Longitudinal Pile Spacing S2	Net Distance dp
1 m	1	1~5 rows	30~80 Dpile	3~7 Dpile	3~7 Dpile	3~20 m

.

2.2.2. Calculation Model of the Impact of the Foundation Pit Project

Underground diaphragm walls and bored piles are simulated by using plate units, and waist beams and reinforced concrete buttresses are simulated by using beam units, and all of the above are simplified according to the linear elastic model. The mechanical parameters are shown in

Table 4. Numerical analysis of material parameters index.

Parameter Name	Lining	Pile Foot	Buildings	Diaphragm Walls
Material type	Elastic	Elastic	Elastic	Elastic
Axial stiffness (kN·m−1)	1.40 × 107	2.00 × 106	1.00 × 1010	7.5 × 106
Flexural stiffness (kN·m)	1.43 × 105	8.00 × 103	1.00 × 1010	1.0 × 106
Equivalent thickness (m)	0.35	0.219	3.464	1.265
Capacity (kN·m−2)	8.4	2	25	10.0
Poisson’s ratio	0.15	0.2	0	0.0

and

Table 5. Material properties of lateral supports (anchor rods).

Parameter Name	Lateral Supports
Material type	Elastic
Axial stiffness (kN)	2.0 × 106
Support spacing (m)	5.0
Fmax,comp (kN)	2.0 × 1015
Fmax,tens (kN)	1.0 × 1015

.

The simulated construction steps of the foundation pit project are calculated according to the working conditions of precipitation in the pit and no precipitation outside the pit, and the main construction steps are as follows. (1) Tunnel excavation stratum loss is assumed to be 2%. (2) Soil consolidation and super-hole pressure dissipation calculations are performed to obtain the initial ground stress state after the tunnel construction. (3) Ground link walls, bored piles, girders, and buttresses are generated. (4) The displacement is reset to 0 m. (5) The first soil layer is excavated. (6) The water in the pit is lowered to −8 m. (7) The second layer of soil is excavated. (8) In-pit precipitation is lowered to −14 m. (9) The third soil layer is excavated. (10) The water in the pit is lowered to −16 m. (11) The fourth layer of soil is excavated. The dimensions and spatial location of the foundation pit are shown in Figure 2.

In order to analyze the impact of different types of pit works on the subway tunnel, this paper performs parameter sensitivity analysis on the pit excavation width, pit excavation depth, and pit–subway tunnel clear distance, respectively, and the range of model parameters taken into account is shown in

Table 6. List of model parameters of foundation pit project.

Unchanged Parameters		Variable Parameters
Enclosure Insertion Depth	Subway Tunnel Burial Depth HTu	Pit Width BFP	Depth of Foundation Pit HFP	Pit–Underpass Tunnel Clear Distance dp
45 m	10.6 m	10~40 m	3~25 m	5~35 m

.

2.2.3. Computational Model for Shallow Base Impact

The shallow foundation project is simulated according to the distributed load, the construction step is calculated according to the no-precipitation condition, and the main construction sequence is as follows: (1) the tunnel excavation ground loss is assumed to be 2%; (2) soil consolidation and super hole pressure dissipation calculation is carried out after the tunnel construction is completed to achieve the initial ground stress state; (3) and the distributed load is activated. The dimensions and spatial location of the shallow foundation are shown in Figure 3.

In order to analyze the impact of different types of shallow foundation works on the subway tunnel, this paper performs parameter sensitivity analysis for shallow foundation load, shallow foundation lateral distribution range, and shallow foundation–subway tunnel clear distance, respectively, and the range of model parameters are shown in

Table 7. List of parameters of shallow foundation engineering model.

Unchanged Parameters	Variable Parameters
Subway Tunnel Burial Depth HTu	Shallow Foundation Load qSF	Shallow Base Lateral Distribution Range LSF,2	Shallow Foundation–Subway Tunnel Clearance dSF
10.6 m	50~120 kPa	8~40 m	5~50 m

.

2.3. Numerical Calculation Results

The numerical simulation results for the shallow foundation, pile foundation, and pit works are shown in

Table 8. Numerical simulation results of pile foundation engineering.

Variable Parameters		Tunnel Rumble Displacement (mm)
Number of group pile rows	1 row	−4.83
	2 rows	−7.65
	3 rows	−9.62
	4 rows	−10.99
	5 rows	−11.93
Transverse pile spacing S1	3 m	−14.09
	4 m	−12.93
	5 m	−11.93
	6 m	−11.15
	7 m	−10.45
Longitudinal pile spacing S2	3 m	−17.91
	4 m	−14.33
	5 m	−11.93
	6 m	−10.25
	7 m	−8.96
Pile foundation–underpass tunnel clearances dp	3 m	−15.65
	5 m	−11.93
	10 m	−7.69
	15 m	−5.59
	20 m	−3.57
Pile length Lpile	30 m	−15.88
	40 m	−14.44
	50 m	−13.12
	70 m	−11.93
	100 m	−9.86

,

Table 9. Numerical simulation results of foundation pit project.

Variable Parameters		Tunnel Rumble Displacement (mm)	Tunnel Lateral Displacement (mm)
Pit width BFP	10 m	7.16	7.02
	15 m	12.17	11.93
	20 m	16.37	16.04
	25 m	19.86	19.46
	30 m	22.52	22.07
	35 m	24.68	23.69
	40 m	26.20	25.15
Pit–underpass tunnel clear distance dp	5 m	19.21	18.44
	10 m	12.17	11.68
	15 m	10.28	9.87
	20 m	8.67	8.49
	25 m	5.80	5.69
	30 m	3.37	3.40
	35 m	2.72	2.74
Depth of foundation pit HFP	3 m	5.42	5.47
	6 m	6.04	6.10
	10 m	12.17	12.29
	25 m	22.25	22.47

and

Table 10. Numerical simulation results of shallow foundation engineering.

Variable Parameters		Tunnel Rumble Displacement/mm
Shallow foundation load qSF	50 kPa	−9.66
	60 kPa	−11.57
	70 kPa	−13.44
	80 kPa	−15.30
	90 kPa	−17.12
	105 kPa	−19.84
	120 kPa	−22.51
Shallow base lateral distribution range LSF,2	8 m	−16.49
	10 m	−19.84
	15 m	−26.84
	20 m	−32.41
	25 m	−36.75
	30 m	−40.25
	35 m	−42.91
	40 m	−44.98
Shallow Foundation–Subway Tunnel Clearance dSF	5 m	−25.43
	10 m	−19.84
	15 m	−15.68
	20 m	−12.45
	25 m	−9.94
	30 m	−8.27
	35 m	−6.55
	40 m	−5.14
	50 m	−2.87

.

3. Parametric Sensitivity Analysis of Pile Impact

3.1. Number of Group Pile Rows

In this paper, the numerical simulation results on the effect of group pile row number will be compared and analyzed with the research results of scholars in this field. Group pile effect is an important factor to determine the disturbance size of the pile foundation project. Under the actual project, the form of the pile foundation near the protected zone of the subway is often the same as the group pile foundation of high-rise buildings, so it is important to analyze the effect of group pile row number (i.e., group pile size) on the underpass tunnel. According to the “Technical Specification for Building Pile Foundations” (JGJ94-2008) [21] specification, the change law of the equivalent settlement coefficient ${\psi }_2$ of the pile foundation is calculated when the number of pile rows changes, and since other conditions are assumed to be unchanged, the change law of ${\psi }_2$ represents the change law of ${\omega }_{piles}$ of the overall settlement of the group pile foundation, as shown in Figure 4. Based on the assumptions of lateral pile spacing S₁, longitudinal pile spacing S₂, pile load q_pile, and constant pile–subway clear distance d_p, Yan (2007) [12] calculated the ratio of the overall settlement displacement ${\omega }_{piles}$ of the group pile foundation to the maximum settlement displacement ${\omega }_{p,\max }$ of the nearby underpass tunnel when the group pile size is 2 × 2, 3 × 3, 4 × 4, 5 × 5, and 6 × 6, respectively, and the results are shown in Figure 5. As shown in Figure 5, when the number of group pile rows increases to 6 × 6, the ${\omega }_{piles}/{\omega }_{p,\max }$ approaches 2.0 and the attenuation stabilizes, which shows that when the number of group pile rows (scale) reaches a certain number, the disturbance of the existing underpass tunnel by group pile loading and settlement tends to a constant value.

Numerical simulations of the maximum settlement displacement of the subway tunnel affected by pile foundation rows 1 to 5 were carried out and are shown in Figure 6. The relationship curve between the number of group pile rows and the maximum settlement displacement ${\omega }_{p,\max }$ of the underpass tunnel is shown in Figure 6, from which it can be seen that the research results of Gui et al. (2012) [22] and Yan et al. (2008) [23] are more consistent with the laws of this paper. The results of numerical simulation in this paper are closer to the trend presented by Gui et al. (2012) [22] and others when the group pile size n × n increases, while the results presented by Yan et al. (2008) [23] and others are obviously different from the numerical simulations in this paper, mainly because the softer soil in Shanghai makes the sensitivity of n × n to ${\omega }_{p,\max }$ stronger. Overall, n × n-${\omega }_{p,\max }$ shows a pattern of slowing growth. Compared with what is shown in Figure 4, the effect of increasing the number of group pile rows on the overall settlement of group pile foundation ${\omega }_{pile}$ is more significant in the interval from 2 to 16 rows, and compared with the numerical simulation results of finite element analysis in Figure 6, it shows that ${\omega }_{pile}$ is more sensitive to the number of group pile rows than ${\omega }_{p,\max }$.

The maximum incremental settlement displacement $\Delta {\omega }_{p,\max }$ of the subway tunnel is larger when the number of piles increases along the longitudinal direction of the subway tunnel, as shown in Figure 7. The main reason is that when the number of piles is increased horizontally, the existing pile base produces a shading effect on the newly driven pile base, resulting in a smaller $\Delta {\omega }_{p,\max }$ generated by the increase in the number of piles horizontally and longitudinally, and the specific $\Delta {\omega }_{p,\max }/{\omega }_{p,\max }$ generated by the number of new piles horizontally and longitudinally, as shown in

Table 11. Effect of increasing the number of piles in the transverse/longitudinal direction on $\Delta {\omega }_{p,\max }/{\omega }_{p,\max }$ [12].

Direction of Additional Piles	Number of Piles
	2	3	5
Vertical	68%	38%	27%
Horizontal	51%	25%	17%

[12]. When the number of transverse/longitudinal piles reaches four, the effect of additional piles on $\Delta {\omega }_{p,\max }$ is smaller and its variation decays rapidly. The effect of the number of longitudinal additional piles on $\Delta {\omega }_{p,\max }$ is obviously greater, so when the distribution of group piles is in the form of a long strip along the direction of the subway tunnel, the maximum settlement displacement $\Delta {\omega }_{p,\max }$ of the subway tunnel under this condition should be amplified and corrected. Its mechanism can be attributed to the Mindlin stress field superposition effect and the resistance diffusion characteristics of Randolph pile ends. The additional vertical stress of the pile foundation arranged along the longitudinal direction of the tunnel continuously accumulates in the axis direction, resulting in the additional stress of the lining being higher than the horizontal arrangement. According to the Randolph model, the distance between longitudinal piles is significantly greater than that of transverse direction, and the diffusion angle of the pile-end resistance increases, making the stress at the pile end easier to transmit to the periphery of the tunnel.

3.2. Pile Length

In this paper, the numerical simulation results on the influence of pile length will be compared and analyzed with the research results of scholars in this field both from China and abroad. Huang (2010) [24] showed that when the pile length/tunnel burial depth ($L_{pile}/H_{Tu}$) is greater than 1, the increase in $L_{pile}$ will effectively reduce the disturbance of the pile foundation load on the tunnel; Yan et al. (2007) showed that the numerical simulation results based on the Shanghai soil layer showed that the pile length ($L_{pile}=50 \mathrm{m}$) increased by 40% and the pile foundation settlement displacement ${\omega }_{piles}$ decreased by 35.5%, while the underpass tunnel settlement displacement ${\omega }_{p,\max }$ decreased by 33.7% [12].

According to the Technical Specification for Building Pile Foundations (JGJ94-2008), the variation law of the equivalent settlement coefficient of pile foundation ${\psi }_2$ is obtained when L_pile changes ($L_{pile}/D_{pile}\in \left[5,100\right]$). Since other conditions are assumed to be constant, the variation law of ${\psi }_2$ represents the variation law of the overall settlement displacement ${\omega }_{pile}$ of the group pile foundation, as shown in Figure 8. Numerical simulations of the maximum settlement displacement of the subway tunnel affected by 30~100 m pile length were carried out in this study, and the results are shown in

Table 12. Numerical simulation results of the number of group pile rows.

Pile Length Lpile (m)	Tunnel Rumble Displacement (mm)
30	−15.88
40	−14.44
50	−13.12
70	−11.93
100	−9.86

, based on the law of the specification calculation formula on the influence of pile length, combined with the numerical simulation results in this paper, to obtain the difference shown in

Table 13. Reference table for pile length correction factor under different L_pile/D_pile.

Lpile/Dpile	5	10	15	20	25	30	40	50	60	70	80	90	100
Correction factor χL,p	1.98	1.72	1.56	1.43	1.33	1.25	1.11	1	0.91	0.8	0.8	0.7	0.7

Note: The equivalent settlement coefficient of pile foundation under $L_{pile}/D_{pile}=50$ working conditions ${\psi }_2$ is used as the basis to calculate the above correction coefficient ${\chi }_{L,p}$.

. The reference values are of the pile length correction coefficient under L_pile/D_pile.

3.3. Pile Spacing

In this paper, the numerical simulation results on the influence of pile spacing will be compared and analyzed with the research results of scholars in this field both from China and abroad. Randolph and Wroth (1978) [25] gave the concentric circle model of pile foundation under load based on elastic mechanics, in which the maximum radius of influence of the loaded pile foundation on the surrounding soil $r_{pile}$ is closely related to the empirical coefficient of group pile interaction ${\chi }_{pile}$, as shown in Equation (2).

$$r_{pile}=2.5L_{pile}{\rho }_{pile}(1-{\mu }_{pile})$$

(2)

where $r_{pile}$ is the maximum radius of the influence of pile foundation by load, m; $L_{pile}$ is the length of the pile; m generally takes the value of 70 m; ${\rho }_{pile}$ is the inhomogeneity coefficient; the specific calculation method is the ratio of soil shear modulus at the 1/2 depth of the pile foundation into the soil; the soil shear modulus at the end of the pile (refer to Li and Wang (2011) [26]) takes the value of 0.50; and ${\mu }_{pile}$ is the soil Poisson’s ratio, taking the value of 0.35.

The maximum radius of influence of a typical loaded pile foundation on the surrounding soil is calculated as $r_{pile}=56.875 \mathrm{m}$. Randolph and Wroth (1978) [25] introduce the pile–pile interaction model and propose the displacement field decay function, Equation (3).

$$\psi (r)=\left\{\begin{array}{l}{\displaystyle \frac{\ln (r_{pile})-\ln (r)}{\ln (r_{pile})-\ln (D_{pile})}}\hspace{1em}\hspace{1em}(D_{pile}<r<r_{pile}),\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} (r\ge r_{pile})\end{array}\right.$$

(3)

wherein $r_{pile}$ is the maximum radius of the influence of pile foundation by load, m, according to the value of Equation (2); and $D_{pile}$ is the pile diameter, m.

When considering the shading effect of group piles, the pile spacing S_p is considered in the same sense as the attenuation range r of the single pile displacement field, assuming $D_{\mathrm{pile}}$ = 1 m, and the corresponding $\psi (r)$ for different values of $r/D_{\mathrm{pile}}$ is shown in Figure 9.

Figure 9 shows that when the attenuation range r of the monopile displacement field increases (equivalently, the pile spacing S_p increases), the corresponding attenuation function takes on a changing value, and the pile–pile interaction tends to 0 when the pile spacing S_p increases to a certain range. Overall, the overall settlement of group pile foundation ${\omega }_{piles}$ decreases with the increase in pile spacing S_p, mainly because the increase in pile spacing S_p can effectively reduce the group pile effect, so that the overall settlement displacement of group pile foundation ${\omega }_{piles}$ gradually tends to the settlement displacement of a single pile ${\omega }_{pile}$. The settlement displacement of the underpass tunnel ${\omega }_{p,\max }$ shows a linear decreasing trend with the increase in pile spacing S_p.

In this paper, the maximum settlement displacement of the subway tunnel affected by 30~100 m pile length was numerically simulated, and the results are shown in

Table 14. Numerical simulation results of pile spacing.

Variable Parameters		Tunnel Rumble Displacement
Transverse pile spacing S1	3 m	−14.09 mm
	4 m	−12.93 mm
	5 m	−11.93 mm
	6 m	−11.15 mm
	7 m	−10.45 mm
Longitudinal pile spacing S2	3 m	−17.91 mm
	4 m	−14.33 mm
	5 m	−11.93 mm
	6 m	−10.25 mm
	7 m	−8.96 mm

. Figure 10 shows the numerical simulation results of other scholars’ related research results and this paper on ${\omega }_{p,\max }$—$S_p/D_{pile}$. In Figure 9, it can be seen that the trend of ${\omega }_{p,\max }$ and $\psi (r)$ decaying with the increase in $S_p/D_{pile}$ is almost the same, and when $S_p/D_{pile}$ is larger, the magnitude of ${\omega }_{p,\max }$ convergence keeps decreasing until it tends to the maximum vertical displacement of the subway tunnel caused by the monopile settlement.

3.4. Pile Foundation–Subway Clear Distance

In this paper, the numerical simulation results of the impact of the pile foundation and the subway net distance will be compared and analyzed in conjunction with the research results of relevant scholars in China and internationally. Yan et al. (2007)’s findings are based on the assumption that the horizontal pile spacing (S₁), longitudinal pile spacing (S₂), pile foundation load ($q_{pile}$), and pile group size (5 × 5) remain unchanged [12]. The ratio of the overall settlement displacement ${\omega }_{piles}$ of the pile group foundation to the maximum settlement displacement ${\omega }_{p,\max }$ of the nearby subway tunnel is calculated when the pile foundation–metro net distance ($d_p/D_{pile}$) is 3.5–29.4, and the result is shown in Figure 11.

As shown in Figure 11, as the pile-to-metro net distance increases, ${\omega }_{piles}/{\omega }_{p,\max }$ develops from a linear increase to an exponential increase, and when $d_p/D_{pile}$ reaches about 12–15, ${\omega }_{piles}/{\omega }_{p,\max }$ increases more rapidly. Therefore, the settlement of the pile group foundation affects the maximum settlement of the subway tunnel. The displacement ${\omega }_{p,\max }$ will quickly weaken.

In this paper, the maximum settlement displacement of the subway tunnel affected by the 3~20 m pile foundation and the subway net distance is numerically simulated, and the results are shown in Figure 12. Figure 12 shows a number of related research results and the direct fitting relationship between $d_p/D_{pile}$ and ${\omega }_{p,\max }$ in the numerical simulation results of this paper. Except for Li and Wang (2011) [26] using the Mindlin analytical solution, other data in the literature related to research results are calculated by finite element numerical analysis. The absolute value of the slope of each curve in the figure can characterize the sensitivity of ${\omega }_{p,\max }$. It can be found that when the absolute size of ${\omega }_{p,\max }$ is similar, the corresponding curve slope is also similar. Therefore, even if the difference in actual soil conditions is ignored, the sensitivity can be roughly judged by the size of ${\omega }_{p,\max }$.

3.5. Pile Foundation Load/Bearing Capacity

Without considering the pile group effect, the pile group bearing capacity ($Q_{piles}$) is usually taken as the sum of the single pile load ($q_{pile}$) ($Q_{piles}=n_{piles}{q}_{pile}$, and $n_{piles}$ is the number of pile groups). When the horizontal pile spacing S₁, longitudinal pile spacing S₂, pile group size, and pile foundation–metro net distance d_p remain unchanged, $q_{pile}\in [q{\prime }_{pile},2q{\prime }_{pile}]$ ($q{\prime }_{pile}$ is the single pile working load, kN). Because the pile length $L_{pile}$ of the high-rise building group pile foundation is longer, the pile foundation is generally in a flexible working state, so the maximum settlement displacement (${\omega }_{p,\max }$) of the subway tunnel corresponding to the $q_{pile}\in [q{\prime }_{pile},2q{\prime }_{pile}]$ also shows a linear increase law [12].

When other conditions remain unchanged, ${\omega }_{piles}$ and $q_{pile}$ have a linear relationship. By simulating the ${\omega }_{piles}$ and ${\omega }_{p,\max }$ corresponding to different $q_{pile}$, it is shown that ${\omega }_{piles}$ and ${\omega }_{p,\max }$ basically show a linear relationship, and the two show a linear increasing law as the single pile load gradually increases to the working load ($q{\prime }_{pile}$) [24]; the results are shown in Figure 13.

3.6. Summary

In this section, through numerical simulation and theoretical analysis, the sensitivity of pile foundation construction parameters to the disturbance of subway tunnels is systematically studied. The focus is on analyzing the influence mechanisms of the number of pile rows in a pile group, pile length, pile spacing, clear distance between the pile foundation and the subway, and pile foundation load.

The research findings are as follows: An increase in the number of pile rows in a pile group significantly intensifies tunnel settlement. However, when the number of rows exceeds 10 × 10, the increase in the disturbance effect slows down. Longitudinally arranged piles (along the tunnel alignment) cause more obvious disturbance to the tunnel than transversely arranged ones. Increasing the pile spacing can effectively reduce the group pile effect, and the tunnel settlement decreases linearly as the spacing increases. When the clear distance between the pile foundation and the subway exceeds 12–15 m, the settlement sensitivity drops significantly. There is a linear relationship between the pile foundation load and tunnel displacement, but a comprehensive evaluation considering the pile spacing and the clear distance is necessary.

The conclusion indicates that the pile spacing and the clear distance between the pile foundation and the subway are the most sensitive parameters. Optimizing the pile foundation layout (such as increasing the spacing and controlling the clear distance) can significantly reduce the risk of disturbance, providing a quantitative basis for the safety design of adjacent projects.

4. Parametric Sensitivity Analysis of Shallow Base Impact

According to the finite element calculation results, the deflection curve of the underpass tunnel subjected to the upper shallow foundation load is approximately quadratic, especially in the maximum deflection (${\omega }_{SF,\max }$) on both sides of the good adaptability. Combined with the longitudinal distribution range along the tunnel as $L_{SF,1}$, the shallow foundation load influence range is about $3L_{SF,1}$, and its cause ${\omega }_{SF,\max }$ is located at the midpoint of the shallow foundation load distribution range. Based on the following radius of curvature, as calculated using Equation (4), the approximate minimum radius of curvature of the subway tunnel can be obtained.

$${\rho }_{SF,\min }={\displaystyle \frac{9{L_{SF,1}}^2}{8{\omega }_{SF,\max }}}{(1+{\displaystyle \frac{64{{\omega }_{SF,\max }}^2}{81{L_{SF,1}}^4}})}^{{\displaystyle \frac{3}{2}}}$$

(4)

4.1. Shallow Foundation Load

Considering that the shallow foundation loads are mostly upper-layer and low-layer building ballast, and the distribution range is large (compared to the outer diameter of the subway tunnel $D_{Tu}$), the shallow foundation loads $q_{SF}$ can be simplified to the mean load. ① When $L_{SF,1}$ is larger, the additional deformation of the subway tunnel due to $q_{SF}$ is more uniform, mainly considering the size of the vertical displacement of the subway tunnel: when $q_{SF}$ acts directly on the subway tunnel, ${\omega }_{SF,\max }$ is linearly related to $q_{SF}$; ② when $L_{SF,1}$ is smaller, the subway tunnel produces uneven settlement, mainly considering the size of the radius of curvature of the subway tunnel caused by the shallow foundation load.

The lateral deformation of subway tunnels can be characterized by the amount of longitudinal seam tension, and Wang and Zhang (2013) [27] studied the lateral deformation of subway tunnels caused by the overload of the upper layers, based on the soil resistance coefficient and lateral pressure coefficient kept at a certain level, and concluded the following. Before the subway tunnel joint bolts reach the yield condition, the amount of change in the outer diameter of the subway tunnel ($\Delta D_{Tu}$) and the magnitude of the joint tension increase linearly with the overload of the upper layer ($q_{SF}$). Meanwhile, when the soil resistance coefficient ($K_{SF}$) increases to a higher level, the transverse deformation of the subway tunnel is more limited by the upper overload ($q_{SF}$), and the joint tension is also in a smaller linear variation range due to the linear relationship between the transverse deformation and the joint tension of the underpass tunnel. As the static lateral pressure coefficient of the soil layer in the Jinan subway tunnel is close to 0.7 and the soil resistance coefficient $K_{SF}$ ≈ 10 MPa, the $\Delta D_{Tu}$ varies from 0 to 30.5 mm when the equivalent load of the upper layer varies from 0 to 1098 kPa. Due to the large $K_{SF}$, the underpass tunnel joint bolts did not reach the yield condition, and $\Delta D_{Tu}$ was linearly related to $q_{SF}$. According to the research results of Wang and Zhang (2013) [27], Equation (5) can be used.

$$\Delta D_j=0.1\ast \Delta D_{Tu}$$

(5)

wherein $\Delta D_j$ is the size of the longitudinal joint opening of the tube sheet, mm.

Referring to the waterproofing technical requirements of the Nanjing subway project, the maximum opening of the pipe sheet joints is 8 mm (the maximum opening of the pipe sheet joints according to the construction phase after the maximum opening of 3 mm after the force, the pipe sheet assembly construction error of 3 mm, and the operational phase of the ground load changes caused by the joint opening of 2 mm), caused by shallow foundation load $\Delta D_j\le 2 \mathrm{mm}$, corresponding to $\Delta D_{Tu}\le 20 \mathrm{mm}$. Based on the linear relationship between $\Delta D_{Tu}$ and $q_{SF}$, it is deduced that $q_{SF}$$\le 720 \mathrm{kPa}$, whose limit value of 720 kPa is much larger than the magnitude of the load generated by the self-weight of multi-story buildings. In terms of the longitudinal seam tension of the tube sheet, the underpass tunnel is in a safe condition when transversely under typical shallow foundation loads, and the effect of longitudinal tunnel disturbance will be mainly considered in the following.

In this paper, numerical simulations of the maximum settlement displacements of subway tunnels affected by shallow foundation loads from 50 to 120 kPa were carried out, and the results are shown in

Table 15. Numerical simulation results of shallow foundation load.

qSF (kPa)	Tunnel Rumble Displacement (mm)
50	−9.66
60	−11.57
70	−13.44
80	−15.30
90	−17.12
105	−19.84
120 k	−22.51

.

4.2. Shallow Base Size Effect

The following are recorded: L_SF,1 = 95 m, q_SF = 1000 kN/m, EI = 9.53 × 10⁸ kN/m², $D_{Tu}$ = 6.2 m, k_SF = 10,000 kPa/m, ${\eta }_{Tu}$ = 0.07 (note: The longitudinal stiffness efficiency of the Shanghai Metro shield tunnel is 7% [28]), ${\lambda }_{SF}$=$\sqrt[4]{K_{SF}/4EI{\eta }_{Tu}}$ = 0.1234. The calculation yields the distribution of underpass tunnel deflection under typical geological conditions in Jinan, as shown in Figure 14. As seen in Figure 14, the maximum value of vertical displacement in the underpass tunnel appears in the load acting zone, and to simplify the calculation, the displacement at the center of the load acting zone is taken as the maximum value of deflection, as shown in Figure 15.

As shown in Figure 15, when the shallow base line load ($q_{SF}=1000 \mathrm{kN}/\mathrm{m}$) in the direction of $L_{SF,1}$ is kept constant, the maximum deflection of the shield tunnel will rapidly fall back to 16 mm and remain stable as $L_{SF,1}$ approaches 50 m. And when $L_{SF,1}\le 50 \mathrm{m}$, other things being equal, the maximum deflection of the subway tunnel can be approximated as being proportional to $L_{SF,1}$. ${\omega }_{SF}=L_{SF,1}\times {\omega }_{SF,\max }/50$, $L_{SF,1}\le 50 \mathrm{m}$; ${\omega }_{SF}={\omega }_{SF,\max }$, and $L_{SF,1}\ge 50 \mathrm{m}$, wherein ${\omega }_{SF}$ is the deflection of subway tunnels caused by shallow base load, mm.

With respect to the influence of the lateral distribution range ($L_{SF,2}$) of the shallow foundation, when the surface load ($q_{SF}$) is constant, $q_{SF}$ is proportional to $L_{SF,2}$, corresponding to a subway tunnel with a certain burial depth $H_{Tu}$; the real $H_{Tu}/L_{SF,2}$ acting over the underpass tunnel is obtained by performing the Boussinesq solution and the correction of the average additional stress coefficient ${\overline{\alpha }}_{SF}$ at the corner points under the action of the rectangular zone uniform load by $q{\prime }_{SF}=q_{SF}·{\overline{\alpha }}_{SF}$.

In this paper, the maximum settlement displacement of the subway tunnel affected by the lateral distribution range (L_SF,2) of 8~40 m shallow foundation was numerically simulated, and the results are shown in

Table 16. Numerical simulation results of L_SF,2.

LSF,2 (m)	Tunnel Rumble Displacement (mm)
8	−16.49
10	−19.84
15	−26.84
20	−32.41
25	−36.75
30	−40.25
35	−42.91
40	−44.98

.

4.3. Shallow Foundation–Subway Clearance

The numerical simulation results on the effect of shallow foundation–subway clear distance will be compared and analyzed with the research results of scholars in this field in China and other regions. One study (2015) [29] obtained, through model tests, theoretical analysis and numerical simulation results suggesting that when $d_{SF}/D_{Tu}$ = 0.5~4, the shallow foundation load has a large effect on tunnel arch settlement displacement, and when $d_{SF}/D_{Tu}\ge 6$, the shallow foundation load has a negligible effect on tunnel arch settlement displacement. Dai et al. (2006) [30] used the finite difference method and found that when $d_{SF}/D_{Tu}\ge 4$, the underpass tunnel displacement and bending moment changes tend to be flat. Whereas, when $1\le d_{SF}/D_{Tu}<4$, the attenuation coefficient ${\theta }_{SF}$ (compared to the maximum deflection ${\omega }_{SF,\max }$ when $d_{SF}/D_{Tu}=0$) is 0.8~0.1, respectively, and this result is consistent with the results of Gong et al. (2015) [31], who calculated the model by theoretical analysis. In this paper, numerical simulations were conducted to calculate the maximum settlement displacement of the subway tunnel affected by the shallow foundation–underpass tunnel clear distance from 5 to 50 m. The results are shown in

Table 17. Numerical simulation results of d_SF.

dSF (m)	Tunnel Rumble Displacement (mm)
5	−25.43
10	−19.84
15	−15.68
20	−12.45
25	−9.94
30	−8.27
35	−6.55
40	−5.14
50	−2.87

. Comparing this with the above research results in the literature, the numerical simulation results of this paper show that the maximum settlement displacement of the subway tunnel tends to level off when $d_{SF}/D_{Tu}\ge 4$, and when $0.8\le d_{SF}/D_{Tu}<4$, the increase in $d_{SF}$ will make the maximum settlement displacement of the subway tunnel decay more obviously.

4.4. Summary

In this section, through numerical simulation and theoretical analysis, the sensitivity of shallow foundation engineering parameters to the disturbance of subway tunnels is systematically studied. The focus is on exploring the influence mechanisms of the load intensity of the shallow foundation, the lateral distribution range, and the clear distance between the shallow foundation and the subway.

The research findings are as follows: There is a positive linear correlation between the shallow foundation load (50–120 kPa) and tunnel settlement. For every 10 kPa increase in the load, the settlement increases by approximately 1.5–2.0 mm. When the load exceeds 720 kPa, the opening of the segment joints may exceed the standard. The expansion of the lateral distribution range (8–40 m) leads to a non-linear increase in tunnel settlement, and it tends to stabilize when the range exceeds 50 m. The clear distance between the shallow foundation and the subway (5–50 m) and the settlement show a power function attenuation relationship, and the disturbance significantly decreases when the clear distance exceeds 30 m.

The conclusion shows that the load intensity of the shallow foundation is the most sensitive parameter, followed by the lateral distribution range and the clear distance. Strictly controlling the load limit (≤720 kPa) and reasonably increasing the clear distance (≥30 m) are the key measures to reduce tunnel deformation, providing a quantitative basis for risk prevention and control of projects adjacent to shallow foundations.

5. Parametric Sensitivity Analysis of the Impact of Foundation Excavation

5.1. Foundation Excavation Unloading Influence Law

5.1.1. Influence Law of Unloading Rebound of Foundation Pit

Considering the longitudinal length of the subway tunnel and the considerable “flexibility” of the joint between the shield tunnel, the deformation of the subway tunnel and the soil will be almost the same after a certain period of development. Therefore, it is reasonable to predict the deformation of the subway tunnel in the soil by using the change in soil displacement caused by the unloading of the foundation excavation in the small deformation range [32]. The residual stress analysis method based on the above-mentioned engineering laws has good ease of use [33], and the following will focus on the application of this method in calculating the foundation pit disturbance to underpass tunnels.

Soil is a typical elastic–plastic material that enters a plastic state when the load is increased to a small value, while its stress–strain will show instantaneous elastic changes when the load is removed. According to the principle of elastic-plasticity, there is residual stress in the soil at this time, and the unloading will not make the original stress in the soil disappear completely, by defining the ratio of the residual stress at a point in the soil to the unloading stress at that point as the residual stress influence coefficient (${\alpha }_{FP}$). The residual stress affects the depth schematically, as shown in Figure 16. The measured data show that when the excavation depth ($H_{FP}$) of the pit is kept constant, ${\alpha }_{FP}$ is more closely related to the thickness of the overlying soil layer ($h_{FP}$), and with the increase in $h_{FP}$, ${\alpha }_{FP}$ increases gradually, and after reaching a certain soil depth, ${\alpha }_{FP}$ gradually approaches 1.0. The law shows that the soil unloading effect gradually decreases to 0 with the increase in depth, i.e., it is in the initial stress state [33]. By defining the depth of the soil layer at ${\alpha }_{FP}=0.95$ as the residual stress influence depth $h{\prime }_{FP}$, based on a large amount of measured data on typical soft soil geology in Shanghai, we can obtain as empirical Equations (6) and (7).

$$h{\prime }_{FP}={\displaystyle \frac{H_{FP}}{(0.0612H_{FP}+0.19)}}$$

(6)

$$\begin{array}{c}{\alpha }_{FP}=0.30+0.65{h_{FP}}^2/{{h\prime }_{FP}}^2\\ (0\le h_{FP}\le h{\prime }_{FP})\end{array}$$

(7)

According to the residual stress influence depth ($h{\prime }_{FP}$), the soil vertical displacement within the residual stress influence depth ($h{\prime }_{FP}$) is solved in layers by the classical geomechanical layered sum method, and the soil uplift in each layer is summed up to obtain the soil uplift at the bottom of the foundation pit (${\delta }_{FP}$), as shown in the specific calculation of Equation (8).

$${\delta }_{FP}={\displaystyle \sum _{i=1}^n\frac{\overline{{\sigma }_{i,U}}}{E_{i,U}}}{h}_{i,FP}$$

(8)

where ${\delta }_{FP}$ is the amount of soil uplift at the bottom of the foundation pit, m; n is the number of soil stratifications within the depth ($h{\prime }_{FP}$) affected by the excavation of the foundation pit; $\overline{{\sigma }_{i,U}}$ is the mean value of the unloading stress in the i-th layer of soil, kPa; $\overline{{\sigma }_{i,U}}={\sigma }_{FP}(1-{\alpha }_{i,FP})$, ${\sigma }_{FP}=\overline{\gamma }{H}_{FP}$ is the total unloading stress; $\overline{\gamma }$ is the average soil capacity of the excavation depth of the foundation pit, kPa·m⁻¹; $h_{i,FP}$ is the thickness of the i-th stratified soil layer, m; $E_{i,U}$ is the unloaded modulus of elasticity of the i-th layer of soil, kPa; and $E_{i,U}$ is related to the static earth pressure coefficient ($K_{0,i}$), cohesive force ($c_i$), angle of internal friction (${\varphi }_i$), total unloading stress (${\sigma }_{FP}$), initial unloading elastic modulus factor ($E_{i,0}$), and damage ratio ($R_f$) of the corresponding soil layer, according to the experimental research results of Liu and Hou (1996) based on soft soils in Shanghai [33], which can be obtained as the empirical Equation (9).

$$E_{i,U}=\left[1+{\displaystyle \frac{({\sigma }_{\nu ,i}-{\sigma }_{h,i})(1+K_{0,i})(1+\sin {\varphi }_i)-3(1-K_{0,i})(1+\sin {\varphi }_i){\sigma }_{m,i}}{2(c_i\cos {\varphi }_i+{\sigma }_{h,i}\sin {\varphi }_i)(1+K_{0,i})+3(1-K_{0,i})(1+\sin {\varphi }_i){\sigma }_{m,i}}}{R}_{f,i}\right]E_{0,i}{\sigma }_{m,i}$$

(9)

where $E_{0,i}$ is the initial unloading modulus of elasticity coefficient for the i-th layer of stratification. ${\sigma }_{\nu ,i}$, ${\sigma }_{h,i}$, and ${\sigma }_{m,i}$ correspond to the average values of unloading horizontal, vertical, and consolidation stresses of the stratified soil in layer i, respectively, kPa.

When the width of foundation excavation $B_{FP}\le 2.5H_{FP}$, it can be calculated by Equation (10).

$$\left\{\begin{array}{l}{\sigma }_{\nu ,i}={\alpha }_{i,FP}{\sigma }_{FP}+{\displaystyle \sum _j^1{\gamma }_j{h}_{j,FP}}\\ {\sigma }_{h,i}=K_{0,i}({\sigma }_{FP}+{\displaystyle \sum _j^1{\gamma }_j{h}_{j,FP}})-{\displaystyle \frac{1}{R}}\\ {\sigma }_{m,i}={\displaystyle \frac{1+2K_{0,i}}{3}}({\sigma }_{FP}+{\displaystyle \sum _j^1{\gamma }_j{h}_{j,FP}})\end{array}\right.{\sigma }_{FP}(1-{\alpha }_{i,FP})$$

(10)

When the width of foundation excavation $B_{FP}>2.5H_{FP}$, it can be calculated by Equation (11).

$$\left\{\begin{array}{l}{\sigma }_{\nu ,i}={\alpha }_{i,FP}{\sigma }_{FP}+{\displaystyle \sum _j^1{\gamma }_j{h}_{j,FP}}\\ {\sigma }_{h,i}=K_{0,i}({\sigma }_{FP}{\alpha }_{i,FP}+{\displaystyle \sum _j^1{\gamma }_j{h}_{j,FP}})-{\displaystyle \frac{1}{R}}\\ {\sigma }_{m,i}={\displaystyle \frac{1+2K_{0,i}}{3}}({\sigma }_{FP}+{\displaystyle \sum _j^1{\gamma }_j{h}_{j,FP}})\end{array}\right.{\sigma }_{FP}(1-{\alpha }_{i,FP})$$

(11)

In Equations (10) and (11), R indicates the ratio of loading and unloading increments in vertical and horizontal directions; the smaller the excavation width of the pit ($B_{FP}$), the larger the R, and its value range is [2.0, +∞]. From Equation (8), it can be seen that when the soil environment (i.e., static earth pressure coefficient ($K_{0,i}$), cohesion ($c_i$), angle of internal friction (${\varphi }_i$), initial modulus of elasticity ($E_{i,0}$), and damage ratio ($R_f$)) is constant, the main variable that determines the amount of soil uplift at the bottom of the pit (${\delta }_{FP}$) is the excavation depth of the pit ($H_{FP}$). Equation (8) can theoretically be used to calculate the vertical displacement of any point in the soil within the influence of the foundation excavation, and has its reasonableness within the small deformation of the soil. Ji and Liu (2000) proposed that the variation in vertical displacement of subway tunnel and the magnitude of the influence range of pit excavation approximated a parabolic relationship [32]. The absolute value of vertical displacement is larger than that of the subway tunnel due to the smaller constraint at the bottom of the pit. The analysis results of several groups of measured data show that when the net pit–metro tunnel distance $d_{FP}\in \left[B_{FP},2B_{FP}\right]$, the amount of soil uplift at the bottom of the pit (${\delta }_{FP}$) and the vertical displacement of the tunnel (${\omega }_{FP}$) can be fitted approximately using a straight line; when the pit width is small (defined as $B_{FP}\le 2.5H_{FP}$ in this paper), the absolute value of the slope of the straight line between ${\omega }_{FP}$ and ${\omega }_{FP}$ becomes larger when the pit width increases, and the absolute value of the slope of the straight line changes in the interval of [0.0624, 0.1156]. At that time, the slope change due to the width change was small, and the lower limit slope of 0.0624 could be chosen to estimate the vertical displacement of the subway tunnel in order to fully ensure the engineering safety. The longitudinal deformation curve of the subway tunnel is similar to the normal distribution class, as in Equation (12).

$${\omega }_{FP}={\delta }_{FP}{e}^{\tau x^2}$$

(12)

where x is the length of the projection of the net distance between any point on the subway tunnel and the center of the pit on the subway alignment, m. τ is a parameter (taking values in the range of −0.0005 to −0.002). A simplified solution of Equation (12) for the radius of curvature yields Equation (13).

$${\rho }_{FP}={\displaystyle \frac{1}{\left|2{\delta }_{FP}\tau \right|}}$$

(13)

5.1.2. Influence Law of Lateral Movement of Foundation Pit

The relationship between the maximum horizontal displacement of the tunnel (${\xi }_{FP}$) and the maximum lateral displacement of the pit enclosure (${\delta }_{h,FP}$) is strong. The lateral unloading of the pit leads to the lateral displacement of the enclosure structure, and the overall deformation of the tunnel in the lateral direction and the soil layer is almost the same, so the main variable between ${\xi }_{FP}$ and ${\delta }_{h,FP}$ is the net distance between the pit and the underpass tunnel ($d_{FP}$). The following will be based on the empirical calculation method of the maximum lateral displacement (${\delta }_{h,FP}$) of the foundation pit enclosure structure obtained from the existing engineering actual measurement data, with appropriate corrections to obtain the prediction formula of the maximum horizontal displacement (${\xi }_{FP}$) of the subway tunnel.

The size of the horizontal displacement of the pit to the underpass tunnel depends to a large extent on the ${\delta }_{h,FP}$ of the pit enclosure structure. References [34,35,36,37,38,39] all performed a statistical analysis of the relationship between $H_{FP}$ and ${\delta }_{h,FP}$ based on measured specific engineering data, and the specific results are shown in

Table 18. Maximum lateral displacement statistics of the enclosure structure for different engineering cases.

No.	References	City	Type of Enclosure	δh,FP-HFP
(1)	Liao et al. (2015) [34]	Suzhou	Diaphragm wall	δh,FP = 0.20%HFP
(2)			Bored pile	δh,FP = 0.13%HFP
(3)	Wang et al. (2010) [35]	Shanghai	Bored pile	δh,FP = 0.55%HFP
(4)	Tan and Wang (2013) [36]	Shanghai	Diaphragm wall	δh,FP = 0.26%HFP
(5)	Wei et al. (2014) [37]	Hangzhou	Diaphragm wall	δh,FP = 0.23%HFP
(6)	Li et al. (2007) [38]	Shanghai + Hangzhou	Sheet pile + support	δh,FP = 0.30%HFP
(7)	Tang and Liao (2019) [39]	Xuzhou	Diaphragm wall	δh,FP = 0.063%HFP

Note: The slope values expressed in the δ_h,FP-H_FP relationship are the mean values of the cases investigated in different regions.

.

From the statistical analysis results of the above engineering cases, it can be seen that the approximate relationship between the maximum lateral displacement of the enclosure structure (${\delta }_{h,FP}$) and the excavation depth of the foundation pit ($H_{FP}$) is linear, and the difference in its slope mainly originates from the different soil environment and the type of enclosure structure. For example, the slope of ${\delta }_{h,FP}$ − $H_{FP}$ is only 0.063% in Xuzhou, where the soil conditions are relatively good, while the slope of ${\delta }_{h,FP}$ − $H_{FP}$ is 0.20%~0.55% in Shanghai, where the soil conditions are relatively poor. When the soil conditions are similar, the type of support of specific projects also has a greater impact on the slope, which can be analyzed specifically according to the specific project.

According to the actual measurement data of a foundation pit project located in Suzhou and other places, when the excavation depth of the foundation pit is small, because the insertion depth of the enclosure structure (usually underground diaphragm wall or bored pile) is generally greater than $1.5H_{FP}$, it presents a posture similar to a “cantilever” pile, i.e., the lateral displacement of the bottom of enclosure structure is 0, and the lateral displacement extends from the bottom to the top in a nearly linear posture. When the excavation depth of the foundation pit gradually increases, the lateral deformation posture of the enclosure structure gradually becomes the “bloated belly type”, in which the maximum lateral displacement point is located roughly in the range from the midpoint of the excavation depth to the excavation surface [34]. Since the lateral displacement of the subway tunnel is almost the same as the overall deformation of the soil layer, the relationship between ${\delta }_{h,FP}$ and ${\xi }_{FP}$ can be judged qualitatively through the relative height difference ($\Delta H_{FP-Tu}$) between the excavated surface of the foundation pit and the subway tunnel by combining the above-mentioned maximum lateral displacement points of the enclosure structure along the vertical distribution law.

Consider that the maximum lateral shift point of the enclosure structure is usually located within the $\left[H_{FP}/2,H_{FP}\right]$, and below the excavation surface of the foundation pit, the ${\delta }_{h,FP}$ tends to decay rapidly to 0 [40]. Therefore, when the excavation depth of the foundation pit is less than the buried depth of the subway tunnel ($\Delta H_{FP-Tu}<0$), there is no need to consider whether the maximum horizontal displacement of the subway tunnel (${\xi }_{FP}$) meets the safety standards.

5.2. Pit–Subway Clear Distance Sensitivity Analysis

In this paper, the numerical simulation results on the influence of pit–subway clear distance will be compared and analyzed with relevant engineering data. The following numerical and statistical analyses are carried out for foundation pit projects located in different soil environments and different excavation scales and depths, and the vertical/horizontal deformation law of the underpass tunnel caused by foundation pit excavation is summarized, as shown in Table 18

Among the 31 engineering cases in

Table 19. Summary of pit excavation cases adjacent to existing subway tunnels [41,42].

No.	Project Location	${\mathit{H}}_{\mathit{F}\mathit{P}}$ (m)	Soil Profile	dFP (m)	Tunnel Depth of Burial	Support Conditions	ωFP (mm)	${\mathsf{\xi }}_{\mathit{F}\mathit{P}}$ (mm)
1	Shanghai Nanjing Road	14.4	Silty clay	9	13	Diaphragm wall +3 support	−10	10.00
2	Shanghai Daning Commercial Center	8	Silty clay	7.5	11.8	Gravity dam + Pile + Steel brace	+7.1	4
3	Shanghai Hengfeng Road	10	Sandy chalky soil + Silty clay	7.2	12.4	Diaphragm wall + Cross brace	−3.3	4
4	Shanghai Xujiahui	21	Silty clay	25	11	Diaphragm wall	−5.54	/
5	Guangzhou Huangsha Avenue	12	Sandy soil	6	10	Diaphragm wall + 2 Steel pipe angle brace	−12.3	7.8
6	Shanghai Jing’an Temple	24.3	Silty clay	15	15.8	Diaphragm wall+ Steel brace	−1.3	3
7	Shanghai Nanjing West Road	14.6	Silty clay	12.1	9.4	Diaphragm wall + Steel brace	+5.5	/
8	Shanghai Pacific Plaza	10.1	Powdery clay + Silty clay	3.8	10	Diaphragm wall + Bored pile	−9.5	17.1
9	Shanghai Nanjing West Road	20	Powdery clay + Silty clay	5.4	8.5	Diaphragm wall + Steel brace	−17.37	6.5
10	Shanghai Hong Kong Plaza	15.8	Powdery clay + Silty clay	3.9	12.4	Diaphragm wall + Steel brace	0	13.1
11	Shanghai Hong Kong Plaza	8.3	Powdery clay + Silty clay	5.9	12.7	Diaphragm wall + Steel brace	−6.07	8.5
12	Shanghai Huangpu District	9.2	Powdery clay + Silty clay	3.8	12.7	Diaphragm wall+ Steel brace	+6	/
13	Shanghai South Xizang Road	4.95	Silty clay	11.5	10	Bored pile	5.71	/
14	Shanghai People’s Square	11.5	Silty clay	7	7	Diaphragm wall	4	/
15	Shanghai Luwan Center	17.725	Silty clay	7.5	7.5	Diaphragm wall + Steel brace	10	10
16	Shanghai	22.12	Silty clay	12.7	5.7	Diaphragm wall + Corner brace	−10	15
17	Guangzhou Tianhe District	20.2	Silty clay	6	14.8	Rotary pile + Reinforced concrete brace	+1.3	3.9
18	Shanghai	22.3	Silty clay	2	6.6	Diaphragm wall + Steel brace	+8.2	1.5
19	Beijing	15.5	Powdery clay + Powdered fine sand	5	16.39	Diaphragm wall + Soil nailing wall	/	5.4
20	Guangzhou Linhe Village	8.3	Strongly weathered muddy sandstone	12.2	8.5	Rotary pile + Internal brace	+0.2	2
21	Shanghai International Plaza	11.1	/	5	/	Bored pile + Horizontal brace	+5.3	8.7
22	Shanghai	9.2	/	3.8	12.7	Diaphragm wall + Steel brace	/	/
23	/	14.9	/	6.5	/	Diaphragm wall + Steel brace	−5.5	/
24	Guangzhou	18	/	10	/	-	−1.97	3.81
25	Foshan Lingnan Tiandi	13.9	Medium-fine sand	24.8	13.4	Diaphragm wall + Concrete support	/	5
26	/	15	/	10	21	-	7	5
27	Taipei	/	/	15	13	Bore pile	−30	52
28	Hangzhou Xiasha	15.65	Sandy chalky soil	4.5	9.1	Diaphragm wall + Bored pile	−9.94	3.1
29	Suzhou Industrial Park	12.2	/	9.5	11.4	/	−6.5	2.9
30	Zhengzhou	11.4	/	11	13.7	/	2.5	6
31	Shenyang Youth Street	17.5	/	8	7.5	/	4	7

Note: Positive vertical displacement means that the vertical displacement of the existing subway tunnel is augmentation displacement; negative vertical displacement means that the vertical displacement of the existing subway tunnel is settlement displacement; in the horizontal displacement direction, all points are for the foundation pit, so there is no distinction between positive and negative.

, the cases with data on pit excavation depth, pit–metro tunnel clear distance, and vertical/water displacement of the metro tunnel were selected for statistical analysis, and the results are shown in Figure 17 and Figure 18.

As can be seen from Figure 17 and Figure 18 above, the vertical/horizontal displacement of the subway tunnel is approximately a power function of the net pit–subway tunnel distance ($d_{FP}$), and as $d_{FP}$ increases, both ${\omega }_{FP}$ and ${\xi }_{FP}$ show a fast and then slow decay law. When $0<d_{FP}<5 \mathrm{m}$, the subway tunnel is close to the specification limit of 20 mm for both vertical and horizontal displacements. Considering that ${\omega }_{FP}$ and ${\xi }_{FP}$ are influenced by the depth of pit excavation ($H_{FP}$), it is not reasonable to fit $d_{FP}$-${\omega }_{FP}$/${\xi }_{FP}$ alone, so the cases in Table 19 where the depth of pit excavation is in the range of $9 \mathrm{m}<H_{FP}<15 \mathrm{m}$ are selected to fit $d_{FP}$-${\omega }_{FP}$/${\xi }_{FP}$ again, and the results are shown in Figure 19 and Figure 20.

From the fitting results in Figure 19, it can be seen that the net pit–subway tunnel distance ($d_{FP}$) is approximately related to the vertical tunnel displacement (${\omega }_{FP}$) as a power function with the coefficient of determination $R^2=0.628$, as shown in Equation (14), when located in similar soil environments and with little difference in the excavation depth of the pit.

$${\omega }_{FP}=49.598{d_{FP}}^{-0.981}$$

(14)

From the fitting results in Figure 20, it can be seen that the net pit–subway tunnel distance ($d_{FP}$) is approximately related to the horizontal tunnel displacement (${\xi }_{FP}$) as a power function with the coefficient of determination $R^2=0.845$, as shown in Equation (15), when located in similar soil environments and with little difference in the excavation depth of the pit.

$${\xi }_{FP}=42.766{d_{FP}}^{-0.985}$$

(15)

In this paper, numerical simulations were conducted to calculate the maximum rumble/lateral displacement of the subway tunnel affected by the 5~35 m pit–subway tunnel clearances, and the results are shown in

Table 20. Numerical simulation results of pit–underpass tunnel clearances.

Clearance Between Foundation Pit and Subway Tunnel (m)	Tunnel Rumble Displacement (mm)	Tunnel Lateral Displacement (mm)
5	19.21	18.44
10	12.17	11.68
15	10.28	9.87
20	8.67	8.49
25	5.80	5.69
30	3.37	3.40
35	2.72	2.74

. It can be seen that the numerical simulation results of this paper are in good agreement with the above-mentioned measured data.

5.3. Pit Excavation Size Sensitivity Analysis

In this paper, the numerical simulation results on the effect of pit excavation width will be compared and analyzed with the research results of scholars in this field both from China and abroad. The spatial effect of soil body after pit excavation is greatly influenced by the shape and size of the pit. Since the analysis of irregularly shaped pits is more complicated, the following analysis focuses on the influence of the size effect of rectangular pits.

The existing research results show that the uplift of pit bottom is directly related to the degree of unloading of soil in the pit [43]. Equation (16) describes the coefficient of safety against uplift ($U_{FP}$) of the pit, and the absolute value of $U_{FP}$ can determine the amount of uplift of the soil at the bottom of the pit.

$$U_{FP}={\displaystyle \frac{c_U{f}_{FP}}{\gamma H_{FP}+q_{FP}}}$$

(16)

wherein, $c_U$ is the undrained shear strength, kN·m⁻²; $\gamma$ is the soil capacity, kN·m⁻³; $H_{FP}$ is the pit excavation depth, m; $q_{FP}$ is the surface overload, kN·m⁻²; and $f_{FP}$ is the pit size correction factor.

As can be seen from Equation (16), when other conditions remain unchanged, the variation in $f_{FP}$ and $U_{FP}$ corresponding to the variation in the excavation length ($L_{FP}$) in the $B_{FP}$~ + ∞ range is within 20, i.e., the variation in the maximum elevation of the pit bottom also lies within the 20% range. Therefore, the impact of pit excavation length ($L_{FP}$) on the overall soil space action is relatively limited.

As for the theoretical solution, Wei and Zhao (2016) [44] calculated the pit size effect based on the Mindlin stress solution formula, and the results showed that the pit length ($L_{FP}$) variation has a small effect on the tunnel disturbance, and the pit width ($B_{FP}$) has a larger effect on the subway tunnel. Ji and Liu (2001) [32] proposed that the maximum uplift (${\omega }_{FP}$) of the subway tunnel is approximately hyperbolic with $B_{FP}$, and proposed correcting the width of the unloading elastic modulus ($E_{i,U}$) of layer i soil in Equation (8) above, and the correction coefficient (${\alpha }_{B,FP}$) is calculated as shown in Equation (17).

$${\alpha }_{B,FP}=0.717H_{FP}/B_{FP}+0.512$$

(17)

As for the numerical simulation solution, Huang and Zhang (2011) [45] conducted a three-dimensional numerical simulation based on the deep foundation pit of Guangzhou Metro Yantang Station, and analyzed the uplift volume of the pit bottom caused by the excavation of the pit by changing the width of the pit, and the results showed that the excavation of the pit has a large influence on the uplift volume of the soil at the bottom of the pit when the excavation width $B_{FP}\le 2H_{FP}$, while the change in the uplift volume of the pit bottom is almost negligible when $B_{FP}>2H_{FP}$. When $B_{FP}>2H_{FP}$, ${\alpha }_{B,FP}\in \left[0.512,0.8705\right]$, considering that ${\alpha }_{B,FP}$ is the width correction for the unloading elastic modulus ($E_{i,U}$) of all the layered soils in the soil layer, it can be seen from Equation (8) that when $B_{FP}>2H_{FP}$, the correction factor for the soil uplift at the bottom of the pit (${\delta }_{FP}$) is $1/{\alpha }_{B,FP}\in \left[1.15,1.95\right]$, which shows that $B_{FP}$ is more sensitive to ${\delta }_{FP}$ than ${\omega }_{FP}$.

Overall, ${\omega }_{FP}$ and $H_{FP}/B_{FP}$ show the following relationship when the soil environment, depth, and pit–subway tunnel clear distance of the pit excavation are constant.

$${\displaystyle \frac{1}{{\omega }_{FP}}}=A_1\times (A_2{\displaystyle \frac{H_{FP}}{B_{FP}}}+A_3)$$

(18)

where A₁ is related to pit depth, soil environment, and pit–subway tunnel net distance, different engineering cases can be taken according to the specific situation; A₂, A₃ is the pit excavation width adjustment coefficient, assuming that different engineering cases are approximately equal.

When the modulus of elasticity ($E_{FP}$) of the soil layer where the pit is located is much smaller than the modulus of elasticity ($E_{Tu}$) of the soil layer where the subway tunnel is located (e.g., typical soil–rock combination geology), the excavation of the pit almost does not cause additional deformation of the subway tunnel [46], i.e., the $E_{FP}$/$E_{Tu}$ has a greater effect on the deformation of the subway tunnel. In this paper, the maximum settlement displacement of the subway tunnel affected by 10~40 m pit width is numerically simulated, and the results are shown in

Table 21. Numerical simulation results of pit width.

BFP (m)	Tunnel Rumble Displacement (mm)	Tunnel Lateral Displacement (mm)
10	7.16	7.02
15	12.17	11.93
20	16.37	16.04
25	19.86	19.46
30	22.52	22.07
35	24.68	23.69
40	26.20	25.15

.

The results of data fitting for data of different engineering cases in this paper are more in line with the linear relationship, as shown in Figure 21, but the different A₁ makes some differences in the expressions between $1/{\omega }_{FP}$ and $H_{FP}/B_{FP}$. When $B_{FP}\text{>}2H_{FP}$, the pit size effect is negligible, so the analytical formula of this paper in Figure 21 is corrected, and the correction result is shown in Equation (19). According to the results in Figure 21 above, the percentage change in the tunnel vertical displacement (${\omega }_{FP}$) is calculated when $B_{FP}=10H_{FP}$ is less than 30%, compared to the result at $B_{FP}=2H_{FP}$, which is a good fit when substituted into the numerical simulation results in this paper, and corrects the characteristic of FEA that converges slower than the measured data when the width of the pit excavation ($B_{FP}$) is larger.

$${\displaystyle \frac{1}{{\omega }_{FP}}}=A_1\times (0.089625{\displaystyle \frac{H_{FP}}{B_{FP}}}+0.06144)$$

(19)

5.4. Sensitivity Analysis of Foundation Excavation Depth

From the results of the above theoretical analysis, it can be seen that the depth of pit excavation ($H_{FP}$) is the most important factor affecting the deformation of existing subway tunnels, and the following will be calculated and analyzed according to Equations (10) and (11) for the soil layer of a typical standard section of Jinan Metro Line 3. Assuming that the excavation depth of the pit is H_FP and the excavation width B_FP = 2 H_FP, the residual stress influence on depth ($h{\prime }_{FP}$) is calculated by Equation (6). In the following, each soil stratification is simplified to a homogeneous soil by weighted averaging with the average weight $\overline{\gamma }=18.4 \mathrm{kN}/{\mathrm{m}}^3$, average initial unloading modulus coefficient $\overline{E_0}$ = 296.5, average cohesive force $\overline{c}=24.4 \mathrm{kPa}$, average internal friction angle $\overline{\varphi }=18.6°$, and average static earth pressure coefficient $\overline{K_0}=0.5$, and the average residual stress influence factor $\overline{{\alpha }_{FP}}=0.5167$ is obtained by integration, while assuming R = 2 and a damage ratio of $R_f$ = 0.79, to obtain Equation (20).

$${\delta }_{FP}={\displaystyle \frac{0.0468{H_{FP}}^3+0.041{H_{FP}}^2+0.12H_{FP}+1.2}{0.037{H_{FP}}^2+0.2H_{FP}+2}}$$

(20)

wherein, H_FP is the excavation depth of the pit, m; ${\delta }_{FP}$ is the amount of soil uplift at the bottom of the pit, mm, and the relationship between the two is shown in Figure 22.

As shown in Figure 22, the displacement of pit bottom uplift under different pit excavation depths is approximated by a polynomial relationship, which can be simplified to obtain the universal Equation (21) shown below by correcting the static earth pressure coefficient $K_{0,i}$, cohesion $c_i$, internal friction angle ${\varphi }_i$, total unloading stress ${\sigma }_{FP}$, initial unloading elastic modulus coefficient $E_{i,0}$, and damage ratio $R_f$ according to the different soil environments in specific projects.

$$\begin{array}{l}{\delta }_{FP}={\displaystyle \frac{{\alpha }_1{H_{FP}}^3+{\alpha }_2{H_{FP}}^2+{\alpha }_3{H}_{FP}+{\alpha }_4}{{\beta }_1{H_{FP}}^2+{\beta }_2{H}_{FP}+{\beta }_3}}\\ \approx {\chi }_1{H_{FP}}^3+{\chi }_2{H_{FP}}^2+{\chi }_3{H}_{FP}+{\chi }_4\end{array}$$

(21)

wherein ${\chi }_1\text{~}{\chi }_4$ is the parameter obtained according to the actual soil environment, H_FP is the excavation depth of the pit, m; and ${\delta }_{FP}$ is the amount of soil uplift at the bottom of the pit, mm.

In this paper, the maximum heave/lateral displacement of subway tunnel affected by the excavation depth of the 3~25 m foundation pit is numerically simulated. The results are shown in

Table 22. Numerical simulation results of foundation pit depth.

HFP (m)	Tunnel Rumble Displacement (mm)	Tunnel Lateral Displacement (mm)
3	5.42	5.47
6	6.04	6.10
10	12.17	12.29
25	22.25	22.47

. The numerical simulation results show that the linearity between the depth of the pit and the maximum tunnel displacement is basically the same.

When the relative burial depth ($\Delta H_{FP-Tu}$) of the pit bottom and the subway tunnel is constant in positive and negative directions, and the burial depth of the underpass tunnel is smaller than the excavation depth of the pit, the above formula derived from the residual stress influence coefficient method can be a good fit for the displacement of the pit bottom bulge under different excavation depths in the pit. When the positive and negative direction of $\Delta H_{FP-Tu}$ is constant, the overall vertical displacement of the subway tunnel is small compared with the displacement of the pit bottom uplift, and it can be considered that the vertical displacement of the subway tunnel ${\omega }_{FP}$ ≈ ${\delta }_{FP}$ while maintaining a certain safety factor [46]. In the actual project, when the excavation depth of the pit is less than the burial depth of the subway tunnel (i.e., $\Delta H_{FP-Tu}$ < 0, defined as U-type pit), the excavation of the pit causes soil stress unloading, making the subway tunnel beneath it undergo uplift displacement; when the excavation depth of the pit is greater than the burial depth of the subway tunnel (i.e., $\Delta H_{FP-Tu}$ > 0, defined as an A-type pit), the subway tunnel will produce settlement displacement [41], as shown in Figure 23 and Figure 24 below.

When $\Delta H_{FP-Tu}$ > 0, the absolute size of the subway tunnel settlement displacement tends to be smaller than the bulge displacement, i.e., when the burial depth of the subway tunnel is constant, the absolute value of the vertical displacement of the subway tunnel ($\left|{\omega }_{FP}\right|$) decreases as the excavation depth of the foundation pit increases. Obviously, in this case, the law of Equation (21) is no longer applicable. Therefore, when $\Delta H_{FP-Tu}$ > 0, it is more reasonable to use the subway tunnel horizontal displacement (${\xi }_{FP}$) as the basis of protected zone division. When the burial depth of the subway tunnel is equal to the maximum lateral displacement of the enclosure structure, the horizontal displacement of the underpass tunnel (${\xi }_{FP}$) obtains the maximum value. To ensure a certain safety factor, assume that the burial depth of the subway tunnel is close to the maximum lateral displacement of the enclosure structure, i.e., the horizontal displacement of the subway tunnel (${\xi }_{FP}$) is calculated by the maximum lateral displacement of the enclosure structure (${\delta }_{h,FP}$). Therefore, the horizontal displacement of the underpass tunnel can be expressed as Equation (22).

$${\xi }_{FP}={\chi }_5{\delta }_{FP}={\chi }_5{\chi }_6{H}_{FP}$$

(22)

where ${\chi }_5$ is the correction coefficient determined according to the pit–underpass tunnel net distance ($d_{FP}$), and its specific value is determined according to formula (15); ${\chi }_6$ is the slope parameter determined according to the soil environment in which the specific project is located, and its value can be referred to in Table 18.

5.5. Summary

In this section, through numerical simulation and engineering case verification, the sensitivity of foundation pit excavation parameters to the disturbance of subway tunnels is systematically studied. The focus is on analyzing the influence laws of the foundation pit excavation depth, the clear distance between the foundation pit and the subway, and the size effect of the foundation pit.

The research findings are as follows: There is a positive linear correlation between the foundation pit excavation depth (3–25 m) and the vertical displacement of the tunnel. For every 1-meter increase in depth, the displacement increases by approximately 1.2–1.5 mm. The clear distance between the foundation pit and the subway (5–35 m) and the tunnel displacement (vertical/horizontal) show a power function attenuation relationship. When the clear distance exceeds 25 m, the displacement approaches the safety limit (≤20 mm). The width of the foundation pit (10–40 m) has a significant impact on the tunnel displacement. For every 10-meter increase in width, the vertical displacement increases by about 5–8 mm, while the length of the foundation pit has a weaker sensitivity.

The conclusion shows that the foundation pit excavation depth is the most sensitive parameter, followed by the clear distance and the foundation pit width. It is recommended that in soft soil areas, the excavation depth should be strictly limited (HFP ≤ 15 m) and the clear distance should be controlled (≥15 m). At the same time, the support design should be optimized to reduce the soil unloading effect, providing a quantitative control standard for the safe construction of deep foundation pits adjacent to subway tunnels.

6. Comprehensive Evaluation of Sensitivity of Various Influencing Factors

Under the system of quantitative evaluation of subway tunnel disturbance size according to the subway operation safety standard, each sub-factor under the three main factors can be divided into A and B indicators (shown in Figure 13 and Figure 24): under A indicators, the larger the disturbance value, the stronger the sensitivity measured by the size of the absolute value of the marginal rate of change; under B indicators, the smaller the disturbance value, the stronger the sensitivity measured by the size of the absolute value of the marginal rate of change. The specific indicators are shown in

Table 23. Definition and overview of category A and B indicators.

Main Factor Type	A Indicators	B Indicators
Pile Foundation Works	Pile length Lpile	Group pile size and pile load qpile
	Pile spacing Sp
	Pile foundation–subway clear distance dp
Shallow Foundation Engineering	Shallow foundation–subway clearance dSF	Shallow foundation load qSF
		Longitudinal distribution range of shallow foundation load LSF,1
		Shallow foundation load lateral distribution range LSF,2
Foundation Pit Project	Pit–subway clear distance dFP	Width of foundation pit excavation BFP
		Length of pit excavation LFP
		Excavation depth of foundation pit HFP

Note: A/B indicators are selected based on the value of the indicators and the underpass tunnel disturbance value into the reverse/positive relationship.

.

Based on the comparative analysis of the numerical simulation results in this paper and the related literature, the sensitivity of each factor is summarized as follows.

Parametric sensitivity analysis of pile foundation engineering shows that the pile length shows weaker sensitivity to subway disturbance when the pile length $L_{pile}\ge 70D_{pile}$; the sensitivity is weaker when the group pile size n × n is larger than 10 × 10; the pile spacing is almost linear to the subway deflection (${\omega }_{p,\max }$) within the range of $\left[3D_{pile},7D_{pile}\right]$; the pile load ($q_{pile}$) is almost linear to the subway deflection (${\omega }_{p,\max }$) within 2 times of the monopile working load; and the pile–subway clear distance d_p shows weaker sensitivity to the subway deflection when the pile foundation is arranged along the subway tunnel alignment than the perpendicular subway tunnel alignment. The sensitivity of pile–subway clear distance d_p to subway disturbance is weaker when the subway deflection ${\omega }_{p,\max }\le 10 \mathrm{mm}$; the disturbance to the subway tunnel is more obvious when the loaded pile foundation is arranged along the subway tunnel alignment than the vertical underpass tunnel alignment.

Parametric sensitivity analysis of shallow foundation works shows that $d_{SF}$ shows weak sensitivity to subway disturbance when the ratio of shallow foundation load–subway clear distance ($d_{SF}$) to tunnel diameter ($D_{Tu}$) is greater than 4 ($d_{SF}/D_{Tu}\ge 4$). There is a linear relationship between the shallow foundation load ($q_{SF}$) and the maximum deflection of the subway tunnel (${\omega }_{SF,\max }$). When the distribution range along the tunnel longitudinal direction is greater than 50 m ($L_{SF,1}\ge 50 \mathrm{m}$), or the distribution range along the tunnel vertical direction is greater than 4 times the tunnel burial depth ($L_{SF,2}\ge 4$), the shallow foundation is less sensitive to the subway disturbance.

Parameter sensitivity analysis of the foundation pit project shows that $d_{FP}$ is weakly sensitive to subway disturbance when the pit–subway clear distance satisfies $d_{FP}\ge 15 \mathrm{m}$; the depth of the foundation pit ($H_{FP}$) is within the conventional range, and $H_{FP}$ is close to a linear relationship with the horizontal/vertical displacement of the subway tunnel; the sensitivity of the length of the foundation pit ($L_{FP}$) is weak; and the sensitivity of the width of the foundation pit is weak when it satisfies $B_{FP}\ge 2H_{FP}$.

The sensitivity of each sub-factor under the three main factors is summarized in

Table 24. Summary of the sensitivity of each sub-factor under the three main factors.

Main Factors	Sub-Factors	Indicator	Sensitivity Analysis
Pile foundation engineering	Pile length Lpile	A	$\mathrm{When} L_{pile}\ge 70D_{pile}$, its sensitivity is weak.
	Pile spacing Sp	A	$\mathrm{When} 3D_{\mathrm{pile}}\le S_{\mathrm{p}}\le 7D_{\mathrm{pile}}$, its sensitivity is high.
	Pile foundation–subway clear distance dp	A	$\mathrm{When} {\omega }_{p,\max }\le 10 \mathrm{mm}$, its sensitivity is weak.
	Group pile size	B	When the group pile size n × n is larger than 10 × 10, its sensitivity is weaker.
	Pile load qpile	B	Within twice the working load of the monopile, its sensitivity is high.
Shallow Foundation Engineering	Shallow foundation–subway clearance dSF	A	$\mathrm{When} d_{SF}/D_{Tu}\ge 4$, its sensitivity is weak.
	Shallow foundation load qSF	B	Stronger sensitivity.
	Longitudinal distribution range of shallow foundation load LSF,1	B	$\mathrm{When} L_{SF,1}\ge 50 \mathrm{m}$, its sensitivity is weaker.
	Shallow foundation load lateral distribution range LSF,2	B	$\mathrm{When} L_{SF,2}\ge 4H_{Tu}$, its sensitivity is weaker.
Foundation Pit Project	Pit–subway clear distance dFP	A	$\mathrm{When} d_{FP}\ge 15 \mathrm{m}$, its sensitivity is weaker.
	Width of foundation pit excavation BFP	B	Weak sensitivity.
	Length of pit excavation LFP	B	$\mathrm{When} B_{FP}\ge 2H_{FP}$, its sensitivity is weaker.
	Excavation depth of foundation pit HFP	B	Stronger sensitivity.

.

7. Discussion and Conclusions

7.1. Limitations of This Study and Future Research Directions

The limitations of this study mainly include the following points.

• Dynamic construction process. In the simulation, static loading or simplified staged construction steps are adopted, without considering complex working conditions in actual construction such as fluctuations in dewatering and dynamic adjustments of the deformation of retaining structures.
• Insufficient multi-factor coupling analysis. The sensitivity analysis mainly focuses on single-variable control. It fails to conduct an in-depth exploration of the coupling disturbance mechanism under the combined action of pile foundations, shallow foundations, and foundation pit excavation (such as the superposition effect of simultaneous construction of pile groups and foundation pits). The non-linear relationships between parameters (such as the collaborative vibration-reduction effect of pile spacing and clear distance) have not been quantified through full-factorial experiments.
• The research relies on the comparison between numerical simulation and limited measured data without using high-precision sensors (such as optical fiber monitoring) to verify the details of local tunnel deformation.

The main future research directions are as follows.

• Develop a dynamic construction coupling model. Integrate the interactive effects of multiple physical fields such as dewatering, support deformation, and soil rheology.
• Quantify tunnel response laws under multi-factor synergistic action. Use orthogonal experiments or machine learning methods to quantify the response laws of tunnels under the synergistic action of multiple factors (pile spacing + clear distance + load).
• Develop an intelligent management and control platform based on BIM and GIS. Enable the real-time optimization of construction parameters and early risk warning [48,49,50].
• Validate the long-term prediction ability of the model. Combine this with Internet of Things (IoT) technology to collect deformation data of subway tunnels throughout their lifecycle [51].
• Establish a closed-loop system for disturbance prediction and feedback control. Utilize deep learning algorithms (such as LSTM and CNN) to mine the patterns in monitoring data.

7.2. Conclusions

Combined with the research results of related studies and numerical simulations, the three main factors were analyzed quantitatively and sensitivity analysis was conducted for each sub-factor, and the following conclusions were drawn.

(1) When the number of group pile rows reaches 10 × 10, the incremental disturbance of group pile size to the subway tunnel tends to 0. When the loaded pile base is distributed along the underpass tunnel alignment, the disturbance of group pile size to the subway tunnel is more obvious than the vertical alignment distribution. The increase in pile spacing can effectively reduce the group pile effect. In soft soil areas (such as mud–clay), increasing the pile spacing from 3D to 7D can reduce tunnel settlement displacement by 30% to 40%, but the pile length-to-tunnel burial depth ratio (L/H) must be ≥1.5; in hard soil or rock layers, the minimum pile spacing is recommended to be no less than 4D. The sensitivity of subway tunnel settlement displacement is mainly controlled by the settlement displacement value, and the larger the settlement displacement, the stronger the sensitivity. A linear relationship between shallow foundation loads (50~120 kPa) and tunnel settlement applies to homogeneous clay or silt soil strata and shallow burial tunnels (H ≤ 15 m), and adjustments through a correction factor (${\chi }_{L,p}$) are needed in non-homogeneous strata.

(2) When the distribution of shallow foundations along the tunnel longitudinal range is greater than 50 m ($L_{SF,1}\ge 50 \mathrm{m}$), the maximum deflection of the subway tunnel (${\omega }_{SF,\max }$) affected by the shallow foundation load is independent of $L_{SF,1}$. When $L_{SF,1}\le 50 \mathrm{m}$ and other conditions remain unchanged, ${\omega }_{SF,\max }$ can be approximated as proportional to $L_{SF,1}$ When $1\le d_{SF}/D_{Tu}<4$, the attenuation coefficient (${\theta }_{SF}$) ranges from 0.8 to 0.1.

(3) The rumble displacement of the subway tunnel from pit excavation can be predicted by the residual stress analysis method. The lateral displacement of the underpass tunnel (${\xi }_{FP}$) is linearly related to the depth of pit excavation ($H_{FP}$), and the slope mainly depends on the geological conditions in which the pit project is located. The results of fitting multiple sets of pit–metro tunnel net distances ($d_{FP}$) with tunnel vertical displacements (${\omega }_{FP}$) and horizontal lateral displacements (${\xi }_{FP}$) by on-site engineering monitoring data show that $d_{FP}$ is approximately a power function relationship with ${\omega }_{FP}$ and ${\xi }_{FP}$. The influence of pit excavation length ($L_{FP}$) on the subway tunnel is relatively limited; when the pit excavation width satisfies $B_{FP}>2H_{FP}$, the pit size effect can be neglected.

(4) When the pile length $L_{pile}\ge 70D_{pile}$, the pile length shows weaker sensitivity to subway disturbance. There is weak sensitivity when the group pile size n × n is greater than 10 × 10. When the pile spacing is in the range of $\left[3D_{pile},7D_{pile}\right]$, it varies almost linearly with the subway deflection (${\omega }_{p,\max }$). The pile load ($q_{pile}$) is almost linearly related to the subway deflection (${\omega }_{p,\max }$) within 2 times the working load of the monopile. When the subway deflection is less than 10 mm (${\omega }_{p,\max }\le 10 \mathrm{mm}$), the pile–subway clear distance (d_p) shows weak sensitivity to subway disturbance. The disturbance of the loaded pile foundation along the alignment of the subway tunnel is more obvious than that of the vertical subway tunnel alignment.

(5) $d_{SF}$ shows weak sensitivity to subway disturbance when the ratio of shallow base load–subway clearance ($d_{SF}$) to tunnel diameter ($D_{Tu}$) is not less than 4 ($d_{SF}/D_{Tu}\ge 4$). A linear relationship between the shallow foundation load ($q_{SF}$) and the maximum deflection of the underpass tunnel is detectable (${\omega }_{SF,\max }$). When the longitudinal distribution range along the tunnel is greater than 50 m ($L_{SF,1}\ge 50 \mathrm{m}$), or the vertical tunnel alignment distribution range ($L_{SF,2}$) is greater than 4 times the tunnel burial depth, the shallow foundation is less sensitive to subway disturbance.

(6) $d_{FP}$ is less sensitive to subway disturbance when the pit–subway clear distance is greater than 15 m ($d_{FP}\ge 15 \mathrm{m}$). When the pit depth ($H_{FP}$) is within the conventional range, the $H_{FP}$ is close to a linear relationship with the horizontal/vertical displacement of the underpass tunnel. The sensitivity of the length of the foundation pit is relatively weak. The sensitivity of the pit is weak when the width of the pit meets $B_{FP}\ge 2H_{FP}$.