Influence of secondary lining reinforcement on the mechanical properties of double-layer lining in shield tunnels

Abstract

Currently, the reinforcement design of shield tunnel secondary linings mainly depends on engineering experience, with a lack of clear guidance from relevant codes and literature. Relying only on experience during construction can cause structural flaws and safety hazards. This study, based on the Guangzhou–Shenzhen–Hong Kong Shiziyang Tunnel project, uses model tests to study shield tunnel double-layer lining structures. It compares and analyzes the mechanical features and interaction mechanisms of reinforced and unreinforced secondary linings. Results show that in such structures, segmental linings bear the main load, and secondary linings offer extra support and adjust deformation. Reinforcement in secondary linings affects the constraint on segmental linings. Reinforcement enhances overall performance significantly. Although it has little impact on ultimate bearing capacity, it prolongs the load-bearing process. Specifically, it increases the ultimate bearing capacity of segmental and secondary linings by 21.2% and 26.1%, respectively. For 10-m-diameter shield tunnels, secondary lining reinforcement design should be adopted when the equivalent overburden thickness at the tunnel crown exceeds three times the tunnel diameter.

Graphical abstract

Keywords

shield tunnel / double-layer lining / model test / reinforcement design / mechanical properties

1 Introduction

With the rapid advancement of the transportation sector, the number of long-distance, deeply buried, large-diameter shield tunnel projects traversing complex geological strata is steadily increasing. Conventional single-layer segment lining structures can’t satisfy the safety requirements for the operation of shield tunnels. Consequently, some researchers [1,2] have proposed a double-layer lining structure for shield tunnels, building upon traditional single-layer segment linings. The advantages of this structure have been validated through model tests.

The double-layer lining structure of shield tunnels consists of cast-in-place concrete on the inner side of the segment lining, which forms a support system in conjunction with the secondary lining. In recent years, the application of double-layer lining structures in shield tunnel engineering has steadily increased, as evidenced by projects such as the Shiziyang Tunnel of the Guangzhou–Shenzhen–Hong Kong Express Rail Link, the Wuhan Metro Line 8 Cross-River Tunnel, and the Dalian Metro Line 5. As a specialized structure, issues related to its design and construction have gradually garnered the attention of researchers. Currently, research on double-layer lining structures in shield tunnels primarily focuses on their mechanical properties and interaction models. For instance, Takamatsu et al. [3] proposed a relatively effective design method for double-layer lining structures through both experimental and theoretical analyses. Feng et al. [4] demonstrated the reliability and validity of double-layer lining structures in underwater shield tunnels through similar model tests and field evaluations. Takamatsu et al. [5] investigated the longitudinal bending performance of double-layer lining structures using similar model tests. Munfah and Posta [1] examined the reinforcement effect of the secondary lining on the load-bearing capacity of the segment structure through whole ring tests and multi-ring axial model tests of double-layer linings in shield tunnels. Additionally, several scholars [6–9] have explored the mechanical properties of double-layer lining structures in shield tunnels, considering design and construction factors such as geological conditions, water pressure distribution, structural types, secondary lining thickness and defects, and construction timing.

Structural reinforcement is a fundamental aspect of the design of double-layer lining systems in shield tunnels, and relevant guidelines provide essential principles for this process. The Japanese guideline, “Tunnel Standard Manual”, under typical conditions, the secondary lining is not intended to support loads and is designed solely as a safety reserve, with a thickness of 30 cm of plain concrete considered sufficient. In contrast, domestic standards discourage the use of plain concrete structures for urban rail transit, railways, urban roads, and highway shield tunnels. Currently, research on the reinforcement of tunnel structures primarily focuses on segment design and material optimization. Yang and Tan [10] conducted a comparative analysis of the design standards for shield tunnels in China, Germany, and Japan, and executed a secondary lining reinforcement design for the Zhengzhou–Xi’an high-speed railway tunnel project using numerical simulations and field measurements. The results indicated that the design principles outlined in the standards are relatively conservative, leading to significant discrepancies when compared to field measurements. Zhang et al. [11] proposed specific reinforcement strategies for wide segments based on structural analyses performed during both the construction and operational phases. In the area of material optimization, some researchers [12–14] have focused on the application of fiber-reinforced materials in shield tunnel segment linings, emphasizing the benefits of fiber-reinforced concrete structures in terms of structural strength and durability.

Numerous studies have been conducted by scholars in the field of secondary lining research for shield tunnels, yielding significant results. However, there is a notable lack of reports addressing the research issues related to the reinforcement design of secondary linings. The necessity for reinforcement of the secondary lining structure is often determined by engineering experience; however, the diverse geological conditions encountered in different shield tunnels can lead to challenges such as cracking, leakage, or even structural failure of the secondary lining. Consequently, it is imperative to elucidate the conditions under which secondary linings require reinforcement, as this represents an urgent scientific issue that warrants further investigation. This study focuses on the Shiziyang Tunnel project of the Guangzhou–Shenzhen–Hong Kong Express Rail Link and employs similar model tests to systematically analyze the mechanical properties and interaction mechanisms of double-layer lining structures in shield tunnels under both reinforced and unreinforced conditions. The design principles for the reinforcement of secondary linings in double-layer lining structures of shield tunnels are discussed, thereby providing a scientific basis and methodologies for the rational determination of reinforcement schemes for secondary linings during the design phase of shield tunnels.

2 Similarity model test

2.1 Engineering overview

The Shiziyang Tunnel project, part of the Guangzhou–Shenzhen–Hong Kong Express Rail Link, is depicted in Fig.1. The tunnel spans a total length of 10.8 km and is designed to accommodate a driving speed of 350 km/h. The primary structure employs prefabricated segments, while the entrance and exit sections of the tunnel navigate through weak and unfavorable geological formations. To enhance safety, a double-layer lining structure is implemented in these areas, consisting of a layer of concrete secondary lining cast on the inner side of the segments, which collectively support the external loads. The segment design adheres to a “5 + 2 + 1” configuration, featuring an outer diameter of 10.8 m, an inner diameter of 9.8 m, a thickness of 0.5 m, a width of 2 m, and a concrete strength grade of C50. The secondary lining has an outer diameter of 9.8 m, an inner diameter of 9.2 m, a thickness of 0.3 m, and a concrete strength grade of C25.

2.2 Similar model soil material

In accordance with engineering practices and the conditions observed at the test site, the similarity relationships of the key physical parameters were established based on similarity theory [15]. The similarity of the relevant physical quantities was determined by adopting a geometric similarity ratio of C_L = 20 and a gravity similarity ratio of C_γ = 1 as the foundational basis for calculations, as illustrated in Tab.1.

The investigation report for the Shiziyang Tunnel project indicates that the entrance section of the tunnel intersects various geological layers, including silty sand, coarse gravel, and medium sand. The parameters utilized for the model test soil are defined by control variables such as unit weight, elastic modulus, cohesion, and internal friction angle, with specific values detailed in Tab.2. The model test employs river sand as the primary material, to which measured quantities of barite powder, fly ash, quartz sand, and engine oil are added to modify the soil parameters. Additionally, the ratios of the soil materials are determined through direct shear tests. The final composition of the soil mixture for the model consists of river sand, fly ash, quartz sand, engine oil, barite powder, and rosin, with the respective ratios being 1:1:0.055:0.045:0.01:0.001.

2.3 Preparation of similar model segment

The simulation of the segments and secondary lining within the double-layer lining structure of shield tunnels constitutes the primary focus of this experiment [6]. Special gypsum has been selected as the principal material, which is combined with a specific proportion of diatomaceous earth and subsequently cast using appropriate molds, followed by standard curing procedures. The elastic modulus and uniaxial compressive strength serve as the control parameters, with prototype parameters for the segments and secondary lining structures referenced from the Code for Design of Concrete Structures (GB 50010-2002) [16]. The material ratios for the segment lining have been optimized through compression tests, yielding the following proportions: water: gypsum: diatomaceous earth = 1:1.38:0.1. For the secondary lining, the material ratios are established as follows: water: gypsum: diatomaceous earth = 1:1.26:0.1. This experiment primarily examines the influence of reinforcement within the secondary lining structure on its mechanical properties. Consequently, careful consideration is given to the material utilized for simulating the reinforcement, with the standard tensile stiffness of the reinforcement selected as the control parameter based on the physical similarity ratio. Following tensile tests on the reinforcement, a 1.2 mm iron wire has been chosen to effectively represent the internal reinforcement structure, as illustrated in Fig.2. The parameters for the model test of the double-layer lining structure are summarized in Tab.3.

The simulation of the segment must account for the presence of joints and bolt connections. Xu et al. [17] utilized a method involving inner and outer partition grooves to replicate the circumferential joint conditions observed in actual projects, based on the typical distribution of bending moments at various positions of the segments. This approach has been validated as effectively reflecting the mechanical response of segment joints under ultimate loads, with failure modes that closely resemble those of the prototype segments. Consequently, this experiment employs the inner and outer partition groove method to simulate the circumferential joints. The determination of groove width is contingent upon the opening size of the circumferential joints in the segments, which typically does not exceed 2 mm under standard design loads. The groove depth must be established through finite element simulations. The calculation method simplifies the segment joint to a beam, applying loads at designated locations along the beam and calculating the joint displacement. A schematic representation of this calculation is provided in Fig.3(a). Using Eq. (1), the bending stiffness of the beam can be approximated, while finite element software is employed to ascertain the groove depth that meets the bending stiffness requirements, as detailed in Tab.4.

(1)K_{\theta }=\frac{6EI\cdot PaL}{24EI\delta -Pa(3L^2-4aL)},

where K_θ represents the bending stiffness; δ is the displacement at the center; a is the distance between the load and the support; EI is the bending stiffness of the segment; L is the distance between supports and P is the external radial load on the outer side of the segment joint.

In the simulation of longitudinal bolts within the segments, the relative movement at the longitudinal joints during actual construction is deemed negligible. Consequently, this experiment does not account for the effects of the longitudinal joints. The shear stiffness of these joints is assumed to be infinite, and a slender steel rod that satisfies the geometric and stress similarity ratios has been selected for the simulation, where the diameter of the steel rod measures 0.4 cm and its length is 4 cm.

2.4 Model test device

The model test utilizes a self-designed similarity model test system specifically developed for the assessment of shield tunnel segment lining, soil, and water interactions, as depicted in Fig.4. The apparatus is designed to apply horizontal loading and conducts triaxial loading from three distinct directions, designated as I, II, and III, as illustrated in Fig.4(a) and Fig.4(b). In each direction, hydraulic jacks uniformly distribute the load to the boundaries of the model soil via load distribution beams or plates. These jacks possess a minimum load output precision of 0.2 MPa. The I direction simulates the overburden soil pressure exerted on the tunnel, with the magnitude determined by the geological conditions and the burial depth of the shield. The II direction simulates the horizontal lateral pressure acting on the tunnel, with its magnitude correlated to the lateral earth pressure coefficient. The load in the III direction is applied to the lower soil through a cover plate, with a pressure of 18 MPa implemented to ensure that the entire structure remains in a plane strain state throughout the test. Furthermore, according to the engineering geological investigation report, the shield tunnel segment structure is also subjected to external water pressure within the strata. The simulation of this external water pressure load is facilitated by a self-designed rotary hydraulic loading device, as shown in Fig.4(c) and Fig.4(d). This device is capable of simulating the uneven distribution of water pressure around the segments under gravitational influence, resembling a “bulb” shape, thereby effectively replicating the conditions encountered in engineering practice.

To simulate the mechanical response of the double-layer lining structure under varying loads, a graded loading approach was employed in this model test. Each stage of loading corresponds to a distinct loading step, as illustrated in Fig.5. The equivalent overlaying soil thickness of the tunnel was determined from soil pressure monitoring data collected at the roof of the segment lining arch, utilizing the all-soil column theory. This measurement represents the effective load acting on the roof of the segment arch. During the testing procedure, water pressure is initially applied to the structure using the water pressure loading device. Subsequently, corresponding earth pressure is applied in both the I and II directions, reflecting the different geological conditions. It is important to note that this model test is destructive, with each testing process encompassing segment installation, secondary lining construction, and the overall instability of both the segment and secondary lining. The test is based on the fundamental design parameters of the Shiziyang shield tunnel, as well as the geological conditions present at the entrance and exit sections. An overburden thickness of 30 m and a water head height of 50 m are established as the standard design loads during operation, which correspond to the 5th loading step depicted in Fig.5.

2.5 Experimental measurement projects

The objective of this study is to elucidate the internal force transfer and distribution patterns within the double-layer lining structure of shield tunnels. Data pertaining to internal forces in the segments and secondary lining, contact pressures between the segments and the surrounding soil, as well as deformation measurements of the secondary lining, were systematically collected. The configuration of measurement points is delineated as follows. 1) To assess the internal forces in the segments and secondary lining, strain measurements were taken from both the inner and outer side walls of the segment and secondary lining structures. Subsequently, material mechanics equations were employed to compute the corresponding axial forces and bending moments at these locations. The segment lining structure is equipped with 24 monitoring points, strategically arranged at 15° intervals along the circumferential direction on both the inner and outer surfaces. In contrast, the secondary lining structure features 12 monitoring points positioned at 30° intervals. 2) For the evaluation of contact pressure, 8 precision earth pressure cells are installed at 45° intervals on the contact surfaces between the segments and the soil, as well as between the segments and the secondary lining. 3) Displacement gauges are positioned at the inner arch top, arch bottom, and the left and right arch waists of the secondary lining, as illustrated in Fig.6.

2.6 Test conditions

Common materials utilized for secondary lining structures in shield tunnels encompass both plain concrete and reinforced concrete. This study examines these two categories of materials. To facilitate the quantification of experimental material parameters, a straightforward classification of the secondary lining materials is established based on their reinforcement status. The experimental groupings are presented in Tab.5. Condition 1 pertains to the reinforced condition of the secondary lining. In this test, a pre-assembled rebar cage is positioned prior to the pouring of the secondary lining, which is subsequently allowed to cure to achieve standard strength. Conversely, Condition 2 pertains to the non-reinforced condition of the secondary lining, characterized by the use of plain concrete. In this case, the secondary lining is poured directly and allowed to cure to standard strength, as illustrated in Fig.7.

3 Stress and damage characteristics of double-layer lining structure

3.1 Internal force characterization of the segment

The internal force variation curve of the segment is shown in Fig.8. Prior to the implementation of the secondary lining, the segment demonstrates an elastic stress state due to the minimal external load. The internal forces within the segments under both conditions generally exhibit consistency, increasing linearly with the applied load. Following the application of the secondary lining, the segment and the secondary lining collaboratively distribute the load, resulting in notable differences in the internal force variations between the two conditions. Specifically, the internal force of the segment with the reinforced secondary lining increases at a more accelerated rate. At the 14th loading step, a sudden increase in the internal force at the arch bottom is observed, accompanied by an acceleration in its growth rate. This phenomenon indicates that the segment structure is approaching instability, with an axial force of 2103.98 kN and a bending moment of 927.67 kN·m at the arch bottom. In contrast, the internal force variation of the segment without reinforced secondary lining is characterized by a smoother trajectory, exhibiting a significantly smaller increase compared to the reinforced condition. At the 12th loading step, a sudden increase in the internal force at the arch bottom is observed, followed by a rapid decrease, and the axial force change curve displays a noticeable reversal, suggesting that the segment structure is nearing instability. At this point, the axial force at the arch bottom is measured at 458.29 kN, with a bending moment of 526.33 kN·m.

3.2 Internal force characterization of the secondary lining

The internal force variation curves of the secondary lining under the two conditions are illustrated in Fig.9. As depicted in the figure, the variation patterns of the internal forces in the secondary lining are generally consistent. As the external load increases, the internal forces in the secondary lining also gradually increase. The axial forces exhibit full-ring compression, while the bending moments demonstrate tension on the inner sides of the arch tops and bottoms, as well as tension on the outer sides of the left and right arch waists. However, there are significant differences between the two conditions regarding internal force values and variation amplitudes. The internal force values in the reinforced secondary lining condition are considerably higher than those in the non-reinforced condition, with more pronounced changes. During loading, the maximum increase in axial force in the secondary lining is 1474.4 kN, and the maximum increase in bending moment is 38.2 kN·m. In contrast, the internal force changes in the secondary lining without reinforcement are minimal, with a maximum axial force increase of 414.3 kN and a maximum bending moment increase of 23.4 kN·m.

3.3 Double-layer lining deformation characteristics

The convergence variation curve of the double-layer lining structure is illustrated in Fig.10. Positive displacements are defined as those directed toward the inner side of the lining, while negative displacements are defined as those directed outward. The lateral and vertical convergence values of the lining structures in both conditions gradually increase with the application of external load. Notably, the vertical convergence is greater than the lateral convergence. Under the influence of external loads, the double-layer lining structure deforms inward vertically and expands outward laterally, resulting in a “cross-oval” deformation. Comparisons indicate that, at the same load level, the overall convergence of the lining structure without reinforcement is greater than that with reinforcement, and this difference becomes more pronounced as external loads increase. Notably, at the 6th loading step, the structural deformations under both conditions are similar. The lateral and vertical convergence of the secondary lining under the reinforced condition are −3.07 and 3.34 cm, respectively. In contrast, under the non-reinforced condition, the lateral and vertical convergence of the secondary lining are −4.15 and 5.2 cm, respectively. However, by the 14th loading step, the lateral and vertical convergence of the secondary lining under the reinforced condition are −11.98 and 13.38 cm, respectively. Under the non-reinforced condition, the lateral and vertical convergence of the secondary lining are −17.92 and 21.88 cm, respectively. At this stage, the convergence of the lining under the non-reinforced condition approaches 200% of that under the reinforced condition.

The ellipticity of the two experimental groups under different loading steps was extracted for analysis, as shown in Fig.11. At lower load levels, the ellipticity of the structures under both conditions is generally consistent. At the 7th loading step, the ellipticity for the reinforced condition is 0.65%, while for the non-reinforced condition it is 0.96%. As the load increases, the ellipticity of the structure increases linearly. Since the stiffness of the secondary lining without reinforcement is lower than that with reinforcement, the rate of change is significantly greater for the non-reinforced condition. Therefore, under the same load, the ellipticity of the non-reinforced secondary lining structure is greater, indicating a more pronounced degree of ellipticity in the structure. At the 12th loading step, the ellipticity of the reinforced structure is 1.61%, while for the non-reinforced structure it is 2.54%. Relatively speaking, the ellipticity changes in the reinforced secondary lining structure are more stable. Indicating that the strength of the non-reinforced secondary lining is insufficient to maintain overall stability under overload conditions, thus posing a significant risk of instability.

3.4 Contact pressure

The contact pressure at the arch top position of the segment, along with the surrounding rock pressure at the same location, is illustrated in Fig.12. The load-sharing ratio is defined as the ratio of the contact pressure between the segment and the secondary lining to the surrounding rock pressure. As the external load increases, the small micro-cracks present between the surrounding rock and the contact surface with the secondary lining become compacted, leading to a gradual increase in the surrounding rock pressure at the arch top position of the segment under both conditions. Initially, the contact pressure between the segment and the secondary lining increases gradually before stabilizing. When comparing the reinforced and non-reinforced conditions, the reinforced condition exhibits higher surrounding rock pressure and a lower load-sharing ratio. In the early stages, the contact pressure between the segment and the secondary lining increases rapidly, transitioning through a compaction phase from the 6th to the 9th loading step. By the 9th loading step, the contact pressure between the segment and the secondary lining reaches its maximum value of 8.18 kPa, while the surrounding rock pressure peaks at 34.98 kPa during the 14th loading step. In contrast, the non-reinforced secondary lining condition shows relatively smooth changes in contact pressure, with the compaction transition phase occurring from the 6th to the 11th loading step. At the 12th loading step, the maximum contact pressure is recorded at 7.18 kPa, and the surrounding rock pressure is 24.75 kPa.

3.5 Acoustic emission characterization

To comprehensively analyze the damage progression of the double-layer lining structure of the shield tunnel during the loading process, this experiment utilized an acoustic emission device for full-process monitoring of the structure. Fig.13 illustrates the acoustic emission rate and the cumulative number of acoustic emission events associated with the secondary lining structure as a function of the loading steps.

As illustrated in Fig.13, the internal damage and micro-cracks in the shield tunnel’s double-layer lining structure increase progressively with the rise in external load. Correspondingly, the cumulative number of acoustic emission events also rises. At the conclusion of the loading process, the cumulative number of acoustic emission events recorded for the reinforced secondary lining condition is 836, while for the non-reinforced secondary lining condition, it is 681. The reinforced secondary lining demonstrates a greater overall structural ultimate load-bearing capacity, necessitating a higher external load for instability, which results in more damage occurring within the structure under high stress. The change curve of cumulative acoustic emission events reveals that, under both conditions, there is a sudden shift in the acoustic emission rate and cumulative events at the 6th loading step, which coincides with the application of the secondary lining structure. Following this, the cumulative number of acoustic emission events in the reinforced condition shows a stepwise increase, with another sudden change occurring at the 14th loading step, indicating that the structure begins to exhibit signs of instability. In contrast, the cumulative number of acoustic emission events in the non-reinforced condition increases steadily, with a sudden change at the 12th loading step, signaling the onset of structural instability. This observation aligns with the previously discussed development of internal forces within the segment.

4 Discussion

4.1 The functional mechanism of reinforcement in the double-layer lining structure of shield tunnels

When the secondary lining is reinforced, compared to the unreinforced condition, the cooperative loading period of the segment and secondary lining is longer, and the segment’s ultimate load-bearing capacity is greater. Moreover, throughout the loading process, the internal forces in the segment are larger and the increments in internal force are more pronounced under the reinforced secondary lining condition. The variation patterns of internal forces in the secondary lining structures under both conditions are similar to those of the segment, but the convergence behaviors of the structures are quite the opposite. Under the unreinforced secondary lining condition, the convergence of the lining approaches 200% of that under the reinforced condition. Additionally, at the same load, the structure’s ovalization is greater and the degree of ovalization is more pronounced when the secondary lining is unreinforced. Further analysis incorporating contact pressure information reveals that as the secondary lining transitions from unreinforced to reinforced, the maximum contact pressure between the secondary lining and the segment increases from 7.45 to 8.18 kPa, an increase of 9.8%. The maximum contact pressure between the segment and the surrounding soil rises from 24.75 to 34.98 kPa, an increase of 41.3%.

In the double-layer lining structure of shield tunnels, the segment and secondary lining fulfill distinct load-bearing functions, with the segment primarily responsible for load support and the secondary lining providing supplementary reinforcement. A comprehensive mechanical analysis of the double-layer lining structure necessitates consideration of both the structural stiffness levels and the interaction between the segment and the secondary lining. Typically, the interaction within the double-layer lining structure of shield tunnels under external loads follows the sequence: “stratum−segment−secondary lining.” The pressure exerted by the surrounding rock constitutes the external load acting on the double-layer lining, which is predominantly supported by the segment. As a discontinuous medium, the segment adjusts its stress through deformation during the loading process. However, the secondary lining imposes constraints on the segment’s deformation, resulting in an internal force transfer mechanism under external loading conditions. The contact pressure between the segment and the secondary lining arises from this internal force transfer and represents the primary external load during the operational phase of the secondary lining. Stiffness is a critical factor influencing the structural response to external loads. Generally, increased structural stiffness correlates with reduced internal forces and deformations under external loads. However, this observation does not align with the findings of the present experiment. This inconsistency can be attributed to the interaction between the segment and the secondary lining. When the secondary lining is reinforced, its increased stiffness imposes greater constraints on the segment. Consequently, the internal resistance encountered during the internal force transfer of the segment becomes more pronounced, resulting in a more coordinated overall structural deformation. The external pressure from the surrounding rock is not alleviated by structural deformation, leading to elevated contact pressures between the segment and the secondary lining. Under external loads, the secondary lining experiences increased internal forces while exhibiting reduced deformations due to its enhanced stiffness; conversely, the opposite occurs when the secondary lining is unreinforced. Therefore, the reinforcement of the secondary lining not only augments its strength but also enhances the cooperative performance of the overall tunnel structure. This improvement further increases the load-bearing capacity of the segment lining, particularly after the segment enters an unstable state, enabling it to continue supporting significant external loads. Thus, within the double-layer lining structure of shield tunnels, the secondary lining invariably modifies the deformation boundaries of the segment, regardless of its specific role. The ultimate impact of reinforcement, whether present or absent, results in varying degrees of boundary constraints on the segment, which in turn leads to differences in the load-bearing capacity of the segment structure. This phenomenon can be likened to reinforcement strategies for segment structures, such as the incorporation of internal tension rings, steel plate bonding, and aramid fabric reinforcement.

In the process of optimizing the thickness of the secondary lining in shield tunnels, Wang et al. [18] introduced the concept of stiffness matching between the segment and the secondary lining. This concept proposes determining the optimal thickness of the secondary lining based on the structural stress state, and provides an optimal stiffness ratio suitable for the selection of secondary lining thickness in large-diameter shield tunnels. However, the results of this experiment indicate that determining reinforcement design solely based on a simple stiffness ratio is inadequate. Research shows that the elastic modulus of reinforced concrete compared to plain concrete only differs slightly, within 10%, and is largely related to the reinforcement ratio in the reinforced concrete. This also explains the validity of the experimental results presented in this paper, where the difference in ultimate load-bearing capacity between the reinforced and unreinforced conditions is not significant. In light of this, this study references segment reinforcement research and treats the secondary lining as a reinforcement measure for the segment. It employs a combined structural stiffness k to evaluate the reinforcement effects of the two conditions on the segment structure [19]. The definition of combined stiffness k is as follows:

(2)k=\mathrm{\Delta }(q_1-q_2)/\mathrm{\Delta }L,

where \mathrm{\Delta }\left(q_1-q_2\right) represents the increment in the difference of loads at the tunnel crown and waist, and \mathrm{\Delta }L denotes the increment in horizontal expansion deformation.

As shown in Fig.14, the structural deformation after the implementation of the secondary lining is controlled to some extent. However, the improvement in structural stiffness from the unreinforced secondary lining is significantly weaker than that from the reinforced secondary lining. During the load-bearing phase after the implementation of the secondary lining, the deformation differences between the two combined structures gradually expand. When the load difference is 10 and 20 kPa, the deformation values of the reinforced structure under the reinforced secondary lining are 68.4% and 68.8% of those under the unreinforced condition, respectively. The reinforced secondary lining effectively reduces the deformation values by more than 30%, demonstrating significant results. Additionally, when the secondary lining is unreinforced, the stiffness differences between the segment structure and the secondary lining are pronounced, making it more susceptible to interface damage during the cooperative loading phase. This exacerbates the differences in deformation control effects between the two conditions.

4.2 Discussion on reinforcement design of secondary linings in shield tunnels

To facilitate a more accurate comparison of the differences between the reinforced and unreinforced secondary linings, key structural features from the experimental process have been extracted for further analysis, as presented in Tab.6.

From the previous discussion, it has been established that the condition of reinforced secondary linings offers several advantages in terms of structural stress characteristics and damage resilience. Specifically, structures with reinforced secondary linings exhibit greater strength, enhanced load-bearing capacity, and an extended operational lifespan. From a structural safety perspective, it is crucial to incorporate reinforcement in the design of secondary linings in shield tunnels, a principle that closely aligns with existing standards. However, as illustrated in Tab.6, the ultimate load-bearing capacity of segments under reinforced conditions increased by only 21.2% compared to unreinforced secondary linings, while the ultimate load-bearing capacity of the secondary lining structure increased by merely 26.1%. The differences in ultimate load-bearing capacity between the two conditions are not substantial, indicating that within a certain load range, an unreinforced secondary lining can still satisfy structural safety requirements. To further assess the safety range of unreinforced secondary linings, this study references the ultimate load-bearing capacity of the unreinforced secondary lining structure in accordance with the “Metro Design Code”, which stipulates: “For concrete structures, considering the effects of accidental and unexpected loads, a safety factor of 1.8 to 2.5 should be selected for structural design”. An average safety factor of 2.15 is adopted for the design of the unreinforced secondary lining. This study determined that the ultimate load-bearing capacity of the unreinforced secondary lining is 47.31 kPa. Taking the safety factor into account, the critical load-bearing capacity for the unreinforced secondary lining is calculated to be 22 kPa. Referring to Fig.6, the critical equivalent overburden thickness for unreinforced secondary linings is 43 m. Therefore, from the perspective of structural load-bearing capacity, it is recommended that when the equivalent overburden thickness at the crown of the shield tunnel’s double-layer lining structure exceeds four times the tunnel diameter, reinforcement should be incorporated into the secondary lining.

To ensure the safety of the shield tunnel structure, it is inadequate to evaluate the reinforcement of the secondary lining solely based on its structural load-bearing capacity; the deformation of the double-layer lining structure must also be taken into account. As illustrated in Fig.12, at the 10th loading step, the ovalization of the double-layer lining structure reaches 2.08%, corresponding to a load-bearing capacity of 33.96 kPa, which exceeds the warning threshold of 2% for lateral deformation in shield tunnels [20]. Therefore, to prevent instability due to excessive deformation during the operation of the double-lining structure in shield tunnels, and considering the safety factor, the deformation warning load-bearing capacity for the unreinforced secondary lining is calculated to be 15.79 kPa. By employing linear interpolation, the critical equivalent overburden thickness is determined to be 32 m. Consequently, when the equivalent overburden thickness at the crown of the shield tunnel’s double-layer lining structure exceeds three times the tunnel diameter, reinforcement should be applied to the secondary lining to ensure compliance with the requirements for structural load safety and deformation stability.

In summary, when evaluating the structural stability of the shield tunnel during operation, reinforcement of the secondary lining is generally unnecessary under normal load conditions, unless the tunnel structure is likely to face potential construction impacts, such as surface overload or lateral soil relaxation. Furthermore, a specific assessment must be conducted regarding the need for reinforcement, taking into account the environmental impact of the surrounding rock field. Reinforcement of the secondary lining primarily enhances the ductility of the concrete, mitigates deformation, and improves stress conditions. When the tunnel structure traverses seismic zones, faulted areas, or regions where the stability of surface buildings is critical, reinforcing the secondary lining is advisable. However, in highly corrosive environments, reinforcement may accelerate the corrosion and deterioration of concrete, negatively impacting the secondary lining. In such instances, utilizing fiber materials to strengthen the concrete is a more favorable option.

5 Conclusions

This study is based on the Shiziyang Tunnel project of the Guangzhou–Shenzhen–Hong Kong Express Rail Link. It investigates the mechanical characteristics of shield tunnel structures under different conditions of reinforced and unreinforced secondary lining through indoor similarity model tests, analyzing the structural internal forces, deformations, contact pressures, and damage features in both scenarios. The main conclusions are as follows.

1) Compared to the unreinforced secondary lining condition, the internal forces in the segment and secondary lining under the reinforced condition are larger, with more uniform changes in the internal force curves. The ultimate load-bearing capacity before instability and failure is also higher for the reinforced lining, indicating that reinforcement can effectively improve the stress characteristics of the shield tunnel’s double lining structure and extend its load-bearing process.

2) Under external loading, both conditions exhibit lateral ovalization. At the same load level, the overall convergence of the unreinforced lining structure is greater than that of the reinforced condition. This difference becomes increasingly significant as the external load increases, indicating that the strength of the unreinforced secondary lining is insufficient to maintain overall structural stability under overload, posing a considerable risk of instability.

3) Due to the differing stiffness of the secondary lining in both conditions, the interaction mechanisms at the contact surfaces between the segment and secondary lining differ significantly. As the secondary lining transitions from unreinforced to reinforced, the maximum contact pressure between the secondary lining and the segment increases from 7.45 to 8.18 kPa, an increase of 9.8%, while the maximum contact pressure between the segment and the surrounding soil rises from 24.75 to 34.98 kPa, an increase of 41.3%.

4) Under the reinforced secondary lining condition, the overall ultimate load-bearing capacity of the structure is greater, requiring a higher external load for instability. More internal damage occurs under high stress. Throughout the loading process, the cumulative number of acoustic emission events recorded for the unreinforced secondary lining was 681, while for the reinforced condition, it was 836.

5) Compared to unreinforced secondary lining conditions, the ultimate bearing capacity of the segment lining with reinforced secondary lining only increased by 21.2%, and the ultimate bearing capacity of the secondary lining structure only increased by 26.1%, indicating a minimal difference in the structural bearing capacity between the two conditions. Based on the ultimate bearing capacity of the unreinforced secondary lining structure and considering structural stability, it is recommended that, for shield tunnels with a double-layer lining structure of around 10 m, reinforcement should be designed for the secondary lining when the equivalent overburden thickness at the tunnel crown exceeds 3 times tunnel diameter.

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