Introduction

In recent years, with the rapid development of urban and underground engineering, a large number of projects such as subway tunnels, foundation pit support, and pile foundation construction have emerged1,2,3. Issues related to operation and maintenance caused by the construction of foundation pits adjacent to subway tunnels have become increasingly common4,5,6. Underground expressways constructed by open-cut methods above subway tunnels induce excessive displacement and deformation in the segment structures of underlying existing tunnels due to excavation unloading, narrow-width and long-span open-cut foundation pits adjacent to operational tunnels will likely increase in future urban developments.

In engineering, tunnels are commonly located on side of foundation pits. Liu et al.7 investigated the deformation patterns of subway tunnels positioned laterally adjacent to foundation pit and above the pit bottom through 42 case studies. The results indicated that horizontal displacements of the tunnels experience horizontal displacement toward the pit, while vertical displacements manifest as settlement during foundation pit excavation. Liu et al.8 analyzed 33 case studies and revealed that when subway tunnels are positioned laterally adjacent to a foundation pit and beneath the foundation pit bottom, the tunnels undergo horizontal displacements toward the excavation zone, while vertical displacements resulting in uplift during partial unloading. Another common situation involves tunnels positioned beneath foundation pits9. When a tunnel is located directly below the centerline of the pit and equidistant from the retaining walls, the soil pressure induced by partial unloading remains symmetrically distributed, resulting in negligible lateral movement of the tunnel. However, if the tunnel is asymmetrically positioned below the pit, an eccentric unloading condition arises. It generates a horizontal additional force on the tunnel, ultimately leading to displacement toward the retaining wall on the eccentric unloading side.

Partial excavation and unloading inevitably triggers soil disturbance and alters stress states, thereby imposing structural impacts on adjacent subway tunnel segments that manifest as deformation. Typically, foundation pit excavation reduces the confining pressure surrounding nearby tunnel. Furthermore, the internal forces and deformation patterns of tunnel segments exhibit intricate variations depending on their relative spatial configurations to the excavation zone10,11. The foundation pit excavation exerts distinct disturbance ranges on surrounding ground and tunnels, termed the excavation influence zone. Many scholars have studied the impacts and extent of disturbance caused by pit excavation. Ng et al.12 investigated the relationship between tunnel deformation and tunnel-pit relative spacing through centrifuge tests, the results same as conclusions drawn from analysis by Shi et al.13, which indicated that heave deformation occurred in tunnels within the excavation coverage area. Fan et al.14 analyzed the effects of excavation depth and tunnel positions on the displacement distribution of underlying tunnels during partial unloading. Based on tunnel displacement contours and three-tier displacement control thresholds (20 mm, 10 mm, and 5 mm), the sub-pit zone was classified into: primary influence zone (I), secondary influence zone (II), general influence zone (III) and slight influence zone (IV). Meng et al.15 identified three distinct sub-pit regions affected by excavation: loosened zone, stress-transfer zone, and stable zone. When excavation of the foundation pit partly removes the soil within the soil-arch effect zone, the pit excavation unloading is partial unloading for tunnel. Tunnel under partial unloading will present a new state of both internal force and deformation. Excavation-induced unloading can induce significant deformation in subjacent tunnel, with tunnel heave causing track irregularity that compromises train operational speed. Underground expressways are typically retrofitted from existing roads, predominantly built using open-cut methods due to shallow burial depths and other constraints. This approach differs significantly from the partial unloading effects induced by large-scale foundation pit excavation on tunnels5,16. For narrow-width open-cut foundation pits, the partial unloading on tunnel segment structures is partially constrained. Consequently, significant stress heterogeneity develops in the surrounding soil after partial unloading, leading to novel mechanical response patterns in the segment structures. Additionally, most subway tunnels are constructed via shield tunneling, where prefabricated segments are assembled using bolted connections. Prefabricated segment structure reduces the stiffness of the tunnel and compromises structural safety17,18.

Excavation unloading in the foundation pit can cause severe deformation of the underlying tunnel, leading to tunnel uplift and irregularities in the track. Structural damage to tunnel segments may occur, ultimately affecting subway operational safety19. When the displacement of the tunnel exceeds the specification requirements, appropriate reinforcement measures should be used to control the tunnel deformation according to the actual conditions. Many reinforcement methods have been proposed to control deformation in tunnel due to pit excavation. Sun et al.20 achieved control of tunnel deformation due to pit excavation by changing the Young’s modulus of the soil at the bottom of the pit; Liu et al.21 adopted micro-disturbance grouting to correct the deformed tunnel; Wei et al.22 reinforced shield tunnel using channel steel to reduce the deformation effects on the tunnels caused by deep excavation. If foundation pit excavation is partial unloading, the difficulty in controlling tunnel deformation will be unpredictable.

Investigation is required to evaluate the impact of partial unloading on the prefabricated segment structure. Based on the underground expressway reconstruction project over Zhengzhou subway, this study proposes criteria for partial unloading assessment, and establishes a numerical analysis model for prefabricated segment structures. The optimal reinforcement method is found through comparative analysis of multiple methods. This paper addresses the mechanical response mechanisms of prefabricated segments subjected to partial unloading caused by strip foundation pit excavation above the subway tunnel, and the optimal reinforcement is proposed to minimize the effect of partial unloading.

Engineering background

The underground expressway was reconstructed with fully supported foundation pits. These pits positioned above the shield subway tunnel of Zhengzhou metro. The pit support system primarily adopted a pile-bracing structure, composed of retaining piles and multiple layers of internal struts. The retaining piles used bored cast-in-place piles, while the struts consist of concrete supports and steel supports. The overlapping section between the expressway and the underlying shield subway tunnel spans approximately 593 m. Initially, the underground expressway alignment intersects the existing tunnel at an oblique angle before transitioning to a parallel configuration, as shown in Fig. 1A. The K2+ 180 section represents the most critical scenario in the project. This section not only exhibits the maximum excavation width and deepest excavation depth, but also experiences a significantly higher unloading ratio in the surrounding soil mass of the tunnel. Therefore, the K2+ 180 section is selected as the study case, with its geometric configuration and parameters detailed in Fig. 1B. The subway tunnel has an outer diameter of 6.2 m, inner diameter of 5.5 m, segment thickness of 0.35 m, and ring width of 1.5 m, as shown in Fig. 1C.

Fig. 1
Fig. 1
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Investigation site [(A) Alignment of the subway; (B) Vertical profile of the investigation tunnel; (C) Layout of lining ring.]

Within the exploration depth of 70.0 m at the site, aside from shallow fill soil, the predominant soil layers consist of holocene quaternary silt, fine sand layers, and upper pleistocene quaternary silty clay and silt layers with field drilling, in-situ testing, and laboratory geotechnical tests. These soil layers within the exploration depth were classified into eight engineering geological units based on their different origins, ages, and physic-mechanical properties. The standard values of shear strength parameters for each soil layer were determined using direct shear tests, as shown in Table 1.

Table 1 Statistical summary of shear strength parameters for soil layers.
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Influence mechanism of partial unloading on tunnels

Based on the calculation method for boundaries of the strong and weak influence zones proposed by He23, a comparison with the excavation width reveals that the foundation pits in the studied cross-sections are categorized as small-scale pits. Due to the corner effect at the pit bottom and the restraint effect of retaining piles, the soil unloading beneath these pits is incomplete. When analyzing the impact of excavation on underlying shield tunnel segments, the influence range of small-scale pit excavation on the segments is limited, and its dominant mechanism is governed by partial unloading effects. The differential impacts of excavation on tunnel at varying positions are illustrated in Fig. 2.

Fig. 2
Fig. 2
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Effect of partial unloading on tunnel. [I: complete unloading region; II: partial unloading region.]

In Fig. 2, the tunnel radius is R. Considering that the soil at the pit bottom experiences active unloading and the corner effect at the pit’s foot, two lines are drawn at \((45^\circ + \varphi /2)\) angles to the pit bottom on both sides (where \(\varphi\) represents the internal friction angle of the soil). These lines divide the soil into two regions: namely, Region I and Region II. In Region I, the soil primarily undergoes free unloading and rebound deformation. In Region II, soil deformation is restricted due to the displacement of the supporting piles and partial unloading of the overlying soil. This results in limited unloading and smaller deformation of the soil. However, the rate of displacement change near the lines remains relatively large. When tunnel is located within Region I, its deformation patterns typically transition from a “horizontal ovalization” to a “vertical ovalization”, exhibiting similar characteristics to full-scale foundation pit excavation. However, when the tunnel segment structure intersects the line or lies within Region II, significant differences in the soil stress state and displacement field around the tunnel segment occur. This leads to substantial variations in the internal forces and deformation responses of different tunnel segments, with slight discontinuities often occurring at the segment joints. The deformation of the tunnel segment structure during excavation transitions from a “horizontal ovalization” to an “oblique ovalization,” representing the partial unloading effects on the tunnel segment structure due to pit excavation. For the tunnel in Fig. 2, the excavation width of the pit is small, and the pit is located above and to the left of the tunnel segment structure. The unloading degree of the surrounding soil varies, resulting in differences in the response patterns at different positions of the tunnel.

Criteria for determining partial unloading in tunnels

Taking the left-line tunnel as an example, a single ring of the segmental lining structure is analyzed. When the thickness of the soil cover above the shield tunnel exceeds the outer diameter of the tunnel, the soil-arching effect becomes significant in the upper soil layer. In such cases, the vertical soil pressure on the tunnel should be analyzed using the Tasaki method for loosening earth pressure24. Tunnel stress analysis diagram was shown in Fig. 3. The Terzaghi loosening earth pressure can be expressed as Eqs. (1) and (2).

$$\sigma_{v} = \frac{{B_{1} (\gamma - \frac{c}{{B_{1} }})}}{{K_{0} \tan \varphi }}(1 - e^{{ - K_{0} H\tan \varphi /B_{1} }} ) + p_{0} e^{{ - K_{0} H\tan \varphi /B_{1} }}$$
(1)
$$B_{1} = R_{0} \cot \left( {\frac{\pi }{8} + \frac{\varphi }{4}} \right)$$
(2)

where \(R_{0}\) is the tunnel radius, \(B_{1}\) is half of the excavation influence width, \(H\) is the thickness of the overlying soil, \(\gamma\) is the soil unit weight, \(\varphi\) is the internal friction angle of the soil, \(P_{0}\) is the surface surcharge and \(\sigma_{v}\) is the Tasaki relaxed soil pressure.

Fig. 3
Fig. 3
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Calculation model of loosening earth pressure.

When excavation of the foundation pit completely removes the soil within the soil-arch effect zone, the pit excavation unloading is complete unloading for tunnel. However, if only part of the soil in the soil-arch effect zone is excavated, the pit excavation unloading is partial unloading for tunnel. As shown in Fig. 2, the tunnel has a depth of 19.22 m and a diameter of 6.2 m, with the left-line tunnel positioned 8.89 m from the left edge of the excavation pit. It means the width of the underground expressway foundation pit excavation influence is 8.89 m. The internal friction angle of the soil is 22.8°. Substituting these values into the Eq. (2) yields half of the excavation influence width \(B_{1}\) as 11.56 m, which is greater than 8.89 m. Therefore, for the tunnel beneath the pit, the excavation unloading is classified as partial unloading.

Numerical simulation

This section describes the detailed segmental lining structure modeling process through ABAQUS. The accuracy and reasonableness of the numerical model constructed in this article is verified by comparing and analyzing the results with the full-scale test.

Model settings

Model dimensions

The segmental lining structure of the subway tunnel has an outer diameter of 6.2 m, an inner diameter of 5.5 m, and a ring width of 1.5 m. It includes a crown segment (K) with a central angle of 21.5°, adjacent segments (B1, B2) with a central angle of 68°, and standard segments (A1, A2, A3) with a central angle of 67.5°. Within the segmental lining structure, only the reinforcement bar subjected to tension and compression are modeled. Each segment joint is connected by two M30 direct bolts. The bolts are considered as a single entity, with their diameter and length specified according to actual values. The segmental lining structure is illustrated in Fig. 4.

Fig. 4
Fig. 4
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Structural model of prefabricated segment of shield tunnel [(A) A single ring of tunnel segmental structure; (B) Internal structure of tunnel segmental structure; (C) Bolt].

The foundation pit engineering for the underground expressway employs an internal support system with bored piles. The piles have a diameter of 0.85 m and a length of 21 m, with a spacing of 1.2 m. These piles are equivalently converted to a diaphragm wall based on stiffness equivalence principles, resulting in a wall thickness of 0.43 m. The model measures 160 m (L) × 50 m (W) × 1.5 m (H)., as illustrated in Fig. 5.

Fig. 5
Fig. 5
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Schematic diagram of finite element model.

Material parameters of the model

The Modified Cambridge Model was adopted to represent the soil behavior. Guo25 has adopted a single soil constitutive model for numerical simulations of silt strata in Zhengzhou. Based on the field investigation report, both the excavation area of the foundation pit and the overlying soil of the tunnel consist of silt. Therefore, the soil is simplified to a single silt type by applying a thickness-weighted average. Referring to the research results of Li26, the relevant physical and mechanical parameters of the soil are presented in Table 2.

Table 2 Soil parameters.
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The segmental lining structure is modeled using the concrete plastic damage model. In practical engineering, the spacing between reinforced concrete supports and steel supports in sections with excavation are 9.0 m and 4.5 m respectively while the longitudinal length of the model is 1.5 m. To ensure that the effect of the foundation pit retaining structure in resisting earth pressure matches reality, while keeping the support cross-section unchanged, the elastic modulus of the two types of supports were reduced to one-sixth and one-third of their original values respectively. The constitutive models and mechanical parameters selected for each component are detailed in Table 3.

Table 3 Tunnel lining and relevant parameters.
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Interaction and boundary settings

The interaction relationships between the various elements of the segmental lining structure in the tunnel are detailed in Table 4 and Fig. 6. The surface-to-surface contact was implemented by setting the normal direction of the contact interface to hard contact and the tangential direction to Coulomb friction contact.

Table 4 Interaction settings of different part.
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Fig. 6
Fig. 6
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Interaction relationship of segment structure [(A) Segment-Segment; (B) Reinforcement bar-Segment; (C) Bolt-Segment; (D) Bolt-Bolt hole].

The soil boundary conditions are shown in Fig. 7A. Normal constraints are applied to the sides of the overall soil model, with X = 0. The bottom surface is fully fixed, while the upper surface is free. The tunnel and support boundary conditions are shown in Fig. 7B, C. Longitudinal normal constraints are applied to the segmental lining structure and the support structure, with Y = 0.

Fig.7
Fig.7
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Boundary conditions [(A) Soil; (B) Tunnel; (C) Support].

Mesh generation and properties

Three-dimensional solid elements C3D8R are used to model the soil. Three-dimensional solid elements C3D8 are employed for the segmental lining structure, support piles, bolts, concrete supports, and steel supports. Three-dimensional truss elements T3D2 are used to model the reinforcement bar. The mesh generation of the model are shown in Fig. 8. To achieve more accurate simulation results and ensure continuity in mesh size between different components, the mesh of the soil in contact with the segmental lining structure and support piles is refined.

Fig. 8
Fig. 8
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Model mesh [(A) Support model; (B) Tunnel model; (C) Whole model].

Excavation step settings

A surface surcharge of 20 kPa is applied on both sides of the foundation pit. The excavation process of the foundation pit is simulated according to the actual construction steps, as outlined in Table 5.

Table 5 Simulation process of pit excavation.
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Validation of the tunnel numerical model

This study validates the accuracy of the numerical model by comparing it with the results of full-scale tests conducted by Liu et al.28.

A set of loads is first applied around the segmental lining structure to simulate soil pressure. By varying the magnitude of these loads and adjusting their positions, an overloaded state is simulated, as illustrated in the figure. The loads applied to the segmental structure are divided into 24 concentrated forces, with the angular spacing between adjacent load centers set at 15°. This includes 6 loads (P1), 10 loads (P2), and 8 loads (P3), all directed towards the center of the segmental ring, as shown in Fig. 9.

Fig. 9
Fig. 9
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Model loading scheme [(A) Loading position; (B) Loading process].

The convergence deformation of the tunnel segment structure is shown in Fig. 10A, where the simulation results closely align with the results from full-scale testing. A comparison of the bolt strain in the model and in the full-scale test is presented in Fig. 10B. Due to the absence of bolt prestress in the simulation, there are some differences between them at the initial stage. However, the bolt strain is basically consistent with the full-scale test results under loading. The deformation patterns of both the segments and bolts matched the full-scale test results, confirming the accuracy of the model.

Fig. 10
Fig. 10
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Analysis diagram of deformation of tunnel segment structure [(A) Compression tunnel diameter numerical simulation and full-scale test; (B) Comparison of bolt strain numerical simulation and full-scale test].

Simulation results

The relative positioning of the left and right tunnels with respect to the excavation pit was identical, with the same embedded depth of the support piles on both sides and a symmetrical excavation shape. Therefore, the impact pattern of partial unloading from the pit excavation on both tunnel segment structures was similar. To avoid redundant analysis, only the force and deformation characteristics of the left tunnel segment structure were analyzed in the simulation results.

Displacement analysis of the tunnel

The deformation of the left tunnel segment structure at section K2+ 180 before and after pit excavation is illustrated in Fig. 11A, B. Vertical displacement is defined as positive in the upward direction, while horizontal displacement is positive toward the right and negative toward the left.

Fig. 11
Fig. 11
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Tunnel segment structure deformation nephogram (deformation scaling factor: 100) [(A) Deformation contour map of the tunnel segmental structure before foundation pit excavation; (B) Deformation contour map of the tunnel segmental structure after completion of foundation pit excavation (The arrows are the direction of displacement, with red representing large displacements and blue representing small displacements.)].

Before excavation, the tunnel displayed a “horizontal ovalization” shape under the surrounding hydrostatic and soil pressures, causing horizontal expansion and vertical convergence of the segment structure. Notably, there was a significant deformation difference between the tunnel’s crown and invert, with the crown exhibiting greater deformation than the invert, reaching a maximum difference of 3.31 mm. Deformation along the two sides of the tunnel waist was relatively symmetrical.

Following excavation, the left tunnel shifted toward the pit center, with greater deformation on the right side than on the left, reaching a maximum displacement of 13.10 mm at adjacent segment B2. This induced an overall shift toward a “diagonal ovalization” deformation pattern for the tunnel.

Vertical displacement

The vertical displacement changes in the left-line tunnel segment structure are shown in Fig. 12A. Following the construction of the tunnel and support piles, the segment structure displayed vertical deformation characterized by subsidence at the crown, with a maximum settlement of 1.22 mm, and heave at the invert, reaching a maximum uplift of 3.46 mm. Vertical displacements at both haunches were identical. When the excavation depth reached 7.65 m, the tunnel exhibited an overall upward trend, with a maximum uplift of 7.71 mm occurring at a position 176° clockwise from the tunnel crown. The right side showed slightly greater vertical displacement changes compared to the left. As excavation continued to 12.73 m, both uplift magnitude and rate increased further, with the right side’s vertical displacement surpassing the left. The highest vertical displacement, 13.04 mm, occurred at 40° clockwise from the crown. The section of the tunnel closer to the excavation midline experienced more significant vertical displacement due to soil movement, causing a larger uplift on the right side compared to the left.

Fig. 12
Fig. 12
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Displacement of left tunnel [(A) Vertical displacement; (B) Horizontal displacement (①–⑨ refer to steps in Table 5)].

Horizontal displacement

Figure 12B shows the horizontal displacement changes in the left-line tunnel. Upon completion of the tunnel and excavation support pile construction, the largest horizontal displacements occurred at the haunches, while the crown and invert experienced negligible horizontal movement. During excavation unloading, the left-line tunnel shifted toward the excavation midline, with notable differences in horizontal displacement on either side. Near the excavation midline, the right side of the tunnel showed minimal horizontal soil displacement, while the left side, being farther from the midline, experienced substantial horizontal soil displacement. This displacement aligned with the direction of horizontal movement induced by the excavation’s shape-restoring effect on the tunnel segment structure. The combined influence of these two effects led to a more pronounced horizontal displacement on the left side of the tunnel.

Tunnel internal force

The bending moment of the tunnel segment structure was defined as positive when the exterior side was in tension and the interior side in compression, while axial force was defined as positive in compression and negative in tension.

Bending moment

The bending moment distribution of the left-line tunnel segment structure is shown in Fig. 13A. Prior to excavation, the crown and invert of the tunnel experienced tension on the interior side and compression on the exterior, whereas the haunches experienced compression on the interior side and tension on the exterior. When the excavation depth reached 7.65 m, the absolute values of the segment structure’s bending moments decreased compared to pre-excavation levels, with notable differences between the left and right haunches. The left haunch bending moment was negative, while the right haunch exhibited a positive bending moment, reaching a maximum of 47.9 kN m. Upon excavation to the pit bottom, the bending moment distribution of the tunnel segment structure shifted toward the midline of the excavation, consistent with the deformation trend of the tunnel.

Fig. 13
Fig. 13
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Internal force of the left tunnel [(A) Bending moment; (B) Axial force. (①–⑨ refer to steps in Table 5)].

Axial force

The axial force distribution of the left-line tunnel is shown in Fig. 13B. Before excavation unloading of the pit, the tunnel segment structure exhibited the highest axial forces, with the greatest values occurring at the haunches and relatively lower forces at the crown and invert. As excavation progressed to a depth of 7.65 m, the axial force within the tunnel segments substantially decreased, particularly on both sides where the reduction rate was faster. The axial force distribution pattern remained with higher values at the haunches and lower values at the crown and invert. With the foundation pit was excavated to the bottom, the axial force of the segments continues to decrease, and the position of the maximum axial force shifting to the right arch shoulder and the left arch foot. This pattern and variation in axial force distribution were due to the lateral expansion, which increased the soil resistance on both sides of the tunnel, thereby elevating the axial forces at the haunches. However, partial unloading due to pit excavation reduced the surrounding soil pressure on the tunnel, resulting in a gradual decrease in axial forces within the segment structure as excavation advanced.

Bolt axial force

In practical engineering, pretension is typically applied to bolts to strengthen the connection between adjacent tunnel segments. In this numerical simulation, the bolts were preloaded with a force of 30 kN, while the cushioning layer of the segment structure was omitted. The variation in bolt axial force during partial unloading due to excavation is shown in Fig. 14, where the axial force at the center cross-section of a selected bolt at the segment joint was extracted. The initial three steps had minimal impact on bolt axial force and were therefore disregarded.

Fig.14
Fig.14
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Bolt axial force in left tunnel.

For the bolt at the 10.75° position, a substantial decrease in axial force was observed during the excavation depth range of 0.8 m to 7.65 m. This reduction was attributed to the segment structure returning to a more circular shape, leading to compression on the inner side of the segment joint and a rapid decrease in joint bending moment. By the end of excavation, the bolt axial force at this location had decreased to zero.

The bolts located at 78.75° and 281.25° exhibited a similar pattern of axial force change. With the initial pretension, the inner sides of these segment joints were already under considerable compression, ensuring a tight fit. Excavation caused the tunnel to shift from an “elliptical” to a more circular shape, which reduced the pressure on the inner side of these joints while bending moments at these locations led to tensile stress on the inner segment structure. The combined effect resulted in an increase in bolt axial force at these positions.

However, the bolts at 146.25° and 213.75° showed distinctly different patterns of axial force variation due to the differential deformation of the tunnel toward the excavation center. This differential behavior also explains why the decrease in axial force at the 349.25° bolt was slower than at the 10.75° bolt. Partial unloading from excavation had a significant effect on bolt axial force, which varied from the initial 30 kN to a maximum of 103.6 kN and a minimum of 0.

Reinforcement simulation

The partial unloading from excavation of the foundation pit caused significant uplift deformation in the underlying prefabricated tunnel segment structure, with an increasing trend in internal forces at certain segments. The uplift of the tunnel segment structure disrupts track smoothness, affecting train speeds, and in severe cases, may result in structural damage to the segments, posing a safety risk to subway operations. Consequently, when excavation-induced unloading results in excessive deformation of the lower tunnel’s segmental lining structure, appropriate reinforcement measures must be implemented based on the specific circumstances.

Reinforcement methods

In actual conditions, at section K2+ 180, the segmental lining structure of the tunnel exhibited the maximum vertical displacement after the completion of excavation. Reinforcement measures should be implemented based on the observed displacement to mitigate its impact on tunnel deformation. The reinforcement methods include: ① no reinforcement; ② full under-pit reinforcement, ③ portal frame reinforcement, ④ portal anti-uplift reinforcement, ⑤ grouting reinforcement, and ⑥ portal frame reinforcement combined with counter-pressure29. In ④, the uplift piles were constructed as bored cast-in-place piles with a diameter of 0.85 m, a spacing of 1.5 m, and a length of 30 m. The tunnel reinforcement methods are presented in Table 6.

Table 6 Reinforcement methods and range.
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The soil, support, and tunnel segment structure parameters are consistent with those described earlier. The finite element modeling of each reinforcement scheme is shown in Fig. 15. In the anti-uplift frame reinforcement scheme, the anti-uplift piles are spaced 1.5 m apart, with the model’s longitudinal dimension set to 1.5 m, so the anti-uplift piles are positioned at the center along the model’s length. In the frame reinforcement with counter-pressure scheme, the counter-pressure load, based on the method in Yu29, is applied at the bottom of the tunnel segment structure with a load value of 60 kPa.

Fig. 15
Fig. 15
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Reinforcement 3D model. [① No reinforcement; ② Full under-pit reinforcement; ③ Portal frame reinforcement; ④ Portal anti-uplift reinforcement; ⑤ Grouting reinforcement; ⑥ Portal frame reinforcement combined with counter-pressure].

Reinforcement simulation results

Without reinforcement measures, the maximum vertical displacement of the tunnel segment structure caused by partial unloading from the excavation reached 13.04 mm, exceeding the 10 mm vertical displacement warning threshold outlined in the technical code for protection structures of unban rail transit30. Meanwhile, the maximum horizontal displacement of the tunnel segment structure was 2.45 mm, significantly below the threshold for horizontal displacement specified in the guidelines. Therefore, controlling vertical displacement was prioritized as the primary criterion for evaluating the effectiveness of the reinforcement schemes.

Vertical displacement

Displacement data was taken from the central location of the outer surface of the left tunnel segment structure, and the vertical displacement of the tunnel segment structure after excavation was plotted for each reinforcement scheme, as shown in Fig. 16A. With the use of full base reinforcement, frame reinforcement, anti-uplift frame reinforcement, grouting reinforcement, and frame reinforcement with counter-pressure, the tunnel segment structure’s vertical displacement decreased by 1.23 mm, 4.37 mm, 4.68 mm, 3.84 mm, and 6.08 mm, respectively, compared to the unreinforced case. The respective reduction percentages were 9.4%, 33.5%, 34.9%, 29.4%, and 46.7%. Among these, the frame reinforcement with counter-pressure scheme demonstrated the best control over vertical displacement, allowing the tunnel segment structure to maintain a nearly circular shape.

Fig. 16
Fig. 16
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Comparison of displacement of the left tunnel with different reinforcement methods [(A) Vertical displacement; (B) Horizontal displacement. (①–⑥ refer to Table 6.)].

Horizontal displacement

After excavation, the horizontal displacement of the tunnel segment structure under different reinforcement methods is shown in Fig. 16B, with positive values indicating movement to the right and negative values indicating movement to the left. Following excavation, the tunnel segment structure displayed a tendency to shift to the right due to lateral compression from surrounding soil pressing towards the excavation area. Compared to the non-reinforced case, all reinforcement measures were effective in controlling the tunnel segment structure’s horizontal deformation. Among them, the grouting reinforcement method provided the best control over horizontal displacement, followed closely by the portal reinforcement with counter-pressure.

In summary, using the portal reinforcement with counter-pressure scheme significantly reduced both vertical and horizontal displacements and maintained minimal differential displacement across the tunnel segment structure, enhancing overall stability. Therefore, for this scenario, the portal reinforcement with counter-pressure is the optimal reinforcement measure.

Conclusions

In this study, a refined numerical analysis model for prefabricated segment tunnel structures was developed. The mechanical response of subway tunnel to partial unloading induced by the excavation of foundation pit was investigated. A comparative analysis of multiple reinforcement methods leading to the identification of an optimal reinforcement strategy. The following conclusions were drawn based on this investigation:

  1. 1.

    The underlying tunnel demonstrates global uplift under partial unloading effects, with the maximum vertical displacement occurring at the crown. Notably, the vertical displacement on the side adjacent to the pit centerline is significantly larger, and the peak displacement is predominantly localized at segment joints of the lining structure.

  2. 2.

    During the excavation and unloading process, the bolted connections at precast segmental lining joints introduce discontinuities in bending moments and axial forces due to the mechanical behavior of the bolts, which primarily resist tensile forces. As partial unloading progresses in the upper foundation pit, the bending moments and axial forces acting on the tunnel segments exhibit a decreasing trend.

  3. 3.

    The structural deformation of the tunnel is affected by a variety of factors, and the control of the structural deformation of the tunnel should start from the improvement of the mechanical properties of the soil around the tunnel as well as the reduction of the impact of the proximity construction on the tunnel. For this project, the portal reinforcement with counter-pressure is the optimal reinforcement to control the displacements generated by the partial unloading.