Deformation control criterion of shield tunnel under lateral relaxation of soft soil

Abstract

Metro shield tunnels under the lateral relaxation of soil (LRS) are susceptible to significant lateral deformations, which jeopardizes the structural safety and waterproofing. However, deformation control standards for such situations have not been clearly defined. Therefore, based on a specific case, a model test is conducted to realize the LRS of a shield tunnel in a sandy stratum to reveal its effect on segment liners. Subsequently, a deformation control criterion is established. The LRS is simulated by linearly reducing the loads applied to the lateral sides of the segment structure. During lateral unloading, the lateral earth pressure coefficient on the segment decreases almost exponentially, and the structural deformation is characterized by horizontal expansion at the arch haunches and vertical shrinkage at the arch vault and arch bottom. Based on the mechanical pattern of the segment structure and the acoustic emission, the deformation response of a segment can be classified into three stages: elastic and quasi-elastic, damage, and rapid deformation development. For a shield tunnel with a diameter of approximately 6 m and under the lateral relaxation of sandy soil, when the ellipticity of the segment is less than 2.71%, reinforcement measures are not required. However, the segment deformation must be controlled when the ellipticity is 2.71% to 3.12%; in this regard, an ellipticity of 3% can be used as a benchmark in similar engineering projects.

Graphical abstract

Keywords

shield tunnel / lateral relaxation of soil / deformation control criterion of segment structure / model test

1 Introduction

A reduction in the lateral loads of shield tunnels can result in a significant lateral deformation on the segment structure [1]. This may trigger several tunnel problems, such as cracks in the segment and water leakage in the tunnel. In severe cases, structural damage may occur, which jeopardizes the efficiency and safety of train operations [2–4].

Factors contributing to the reduction in lateral loads on tunnels are complex; however, two primary aspects are typically involved: a) the excavation and unloading of lateral soils around the shield tunnel, and b) the stress relaxation of strata due to the time factor. However, some undesirable factors, such as groundwater flow, landslides, and karst cavities, can amplify the reduction in lateral loads.

In terms of a), the foundation excavations and constructions of other underground structures near an existing tunnel can disturb the soil around the tunnel, thus releasing the lateral soil stress and causing lateral deformation on the shield tunnel lining structure [2,5,6]. Researchers have extensively investigated the unloading of lateral soil caused by the construction of surrounding subsurface structures and their effect on shield tunnels. Results of numerical simulations show that when the horizontal distance between a tunnel and the foundation was small, the maximum horizontal displacement of the shield tunnel exhibited a linear positive relationship with that of the enclosure structure, whereas the effect of soil unloading on the existing tunnel was insignificant when the deep excavation was more than 10 m away from the tunnel axis center [7,8]. The mechanical responses of a metro tunnel under the effect of a significant excavation nearby were investigated via finite element method, and the performances of several protective measures, such as the longitudinally divided excavation, were assessed [3]. Additionally, a structural model based on the Timoshenko beam [9] was employed to investigate the longitudinal mechanical behavior of shield tunnels when excavation projects were performed nearby, where the shearing dislocation between segment rings was prioritized. Furthermore, closed-form solutions for the mechanical response of shield tunnels under arbitrary loads were derived using the Vlasov foundation model [10]. The deformation characteristics in the tunnel cross-section and the dislocation at the segment joints during a single lateral excavation were observed via a discontinuous three-dimensional model of the shield tunnel [11].

Meanwhile, in terms of b), the generation of lateral relaxation of soil (LRS) may decrease the lateral loads in shield tunnels, even in the absence of soil unloading disturbance. To the best of our knowledge, a metro tunnel in China was monitored for lateral outward deformations on both sides, even in the absence of lateral excavation. Because the operation time of the tunnel was short and the tunnel did not indicate structural aging, the relaxation effect of soil at both sides was regarded as the cause of lateral outward deformation. The LRS is primarily characterized by the lateral stress relaxation of soil (the internal stress of soil decreases over time under stable deformation [12]). Previously, stress relaxation tests were performed at different strain levels using soft soil samples from different areas [13,14]. Meanwhile, triaxial compression tests under low and high confining stresses on dense Virginia Beach sand specimens were performed [15], which revealed the effect of deviatoric stress on the stress relaxation of soil. Based on the test results, modified constitutive models were proposed to reflect the aging characteristics of soft soil, such as stress relaxation [16,17]. Wang and Xia [18] and Xu et al. [19] investigated the stress relaxation response of soil particles and the corresponding development of the microstructure using discrete element method. In addition, the characteristics and patterns of the lateral stress relaxation of soil were analyzed using an odometer [20,21]. Notably, these studies focused on the soil, i.e., where the relaxation effect occurred. Studies that investigate the effect of the LRS on shield tunnel structures, particularly those based on model tests, are few. More importantly, unlike lateral excavation, the LRS is uncontrollable, and its relaxation range as well as lateral load reduction values are unpredictable. Whether shield tunnel deformation control criteria or displacement prediction methods [11,22] are available for the LRS is yet to be determined. Therefore, a control criterion for segment deformation with regard to the lateral soil relaxation of shield tunnels must be established. This criterion will not only allow changes in the tunnel status to be evaluated directly under the condition of no lateral excavation, but also allows measures to be implemented to ensure tunnel stability at a reasonable instant. The contributions of this study are as follows.

(1) Based on an actual case, we perform a model test to simulate the relaxation effect of soft soil on both sides of a shield tunnel and analyze its effect on the segment deformation pattern.

(2) The deformation control criterion under the lateral relaxation of soft soil established in this study can be applied to similar engineering projects.

2 Mechanism of shield tunnel response to LRS

The LRS is primarily related to the rheological properties of soil. As time progresses, the weakening and relaxation of the structural strength between particles within the soil reduce the internal stresses [23]. This deteriorates the stability of the interaction between the underground structure and soil, which adversely affects the forces on the structure. Typically, this phenomenon occurs on tunnels built on soft ground [24]. Herein, the effect of the lateral relaxation of soft soils on shield tunnels is elaborated from a macro view (Fig.1) based on actual projects such as subway tunnels in Shanghai [25].

Ideally, long-term stable strata with favorable geological conditions would be present above and below the subway shield tunnel structure, whereas the lateral stratum would be susceptible to stress relaxation. After the tunnel is completed, the LRS is initially insignificant; therefore, the segment structure is less affected, and the stability between the segment structure and stratum is maintained. However, as operation progresses, the LRS develops further. From a load-structure system perspective, the lateral stratum resistance as a passive load decreases, resulting in insufficient lateral restraints on the segment structure. In addition, the lateral earth pressure may decline during the LRS, resulting in a further reduction in lateral loads on the segment. In such cases, the segment lining is susceptible to large lateral deformations, which implies that the arch haunches will expand horizontally outward and the shield tunnel will gradually transform into an oval-shaped structure from a circular structure.

In addition to the strata properties, factors such as insufficient backfill grouting and train vibration may induce or aggravate the LRS of subway shield tunnels [26,27]. An uncompacted grouting circle provides relaxation and deformation space for soft soils around the tunnel, and the frequency vibration of the train may result in internal changes to the lateral stratum (which exhibits rheological properties). In general, although the micro-development laws and principles pertaining to the LRS of shield tunnels are still being investigated, the macroscopic effect of the LRS on shield tunnels should not be disregarded. Therefore, this study primarily focuses on simulating the LRS of shield tunnels based on a similar model test. Subsequently, the deformation and force changes of the structure to prevent or address relevant tunnel defects are analyzed.

3 Test scheme

3.1 Background of project

The model test is based on the Wuhan Metro Line 2 Project, which is a typical subway shield tunnel constructed in soft soil in China. A portion of this metro line is located on the banks of the Yangtze River, which is classified under the Grade I terrace region of the Yangtze River. The upper and lower layers of the river are primarily composed of soft clay and sandy soils, respectively. The relaxation effect of soft ground is more significant and sensitive than that of rocky strata.

Fig.2 shows the segment design of the shield tunnel. The external and internal diameters of the tunnel were 6.2 and 5.5 m, respectively. The segment width and thickness were 1.5 m and 35 cm, respectively. The segment lining ring was composed of three standard blocks (B1, B2, and B3), two adjacent blocks (L1 and L2) and one capping block F. The central angles of block F, L, and B were 21.5°, 68°, and 67.5°, respectively. In addition, the segment was prefabricated with concrete of C50 strength grade and reinforced steel rebar.

The typical strata distributions along this metro line are shown in Fig.3, which were derived from the geological survey report of the Wuhan Metro Line 2. The shield tunnel featured a buried depth of 20–25 m and primarily passed through a silty fine sand layer and a medium-coarse gravel and sand layer. The composition from the surface downward is as follows: (2-1) silty fine sand, (2-2) medium sand, (2-1a) silty clay, (2-1b) silty soil, (2-3) silty fine sand, (2-4) medium-coarse sand with gravel, (2-5) silty fine sand, (2-6) medium-coarse sand with gravel, and (20a-2) medium-weathered mudstone.

3.2 LRS simulation and test apparatus

The relaxation effect of the soil is long-term. The LRS is difficult to reproduce artificially via laboratory tests owing to the test duration and laboratory conditions. However, the end state of the LRS and its effect on the shield tunnel are prioritized of this study. The reduction in the lateral earth pressure and lateral stratum resistance, as a result of the LRS, can be directly reflected by the lateral loading value. Thus, in this study, the LRS was simulated by decreasing the lateral loads on the tunnel structure step by step, which allows the test time to be reduced. However, this does not significantly affect the accuracy.

To analyze the effect of lateral sandy soil relaxation on the segment lining, a model similar to the shield tunnel segment lining–stratum system (Fig.4) was applied in the test, and it measured 2 m × 1.5 m × 1 m (horizontal × vertical × longitudinal). The model was independently developed by the Southwest Jiaotong University and primarily comprised two sections: a test body and a loading/unloading device.

(1) Test body: The segment lining structure was the object of this test. The model segment and its surrounding soil were placed in a large model box. Specifically, the segmental lining structure traversed the interior of the test box, and the remaining space was filled with the model soil.

The sides and bottom of the test box were constructed using steel plates to form the constrained boundaries. Two holes were created on the left and right sides to allow the loading/unloading device to enter the test box. A steel plate was fixed onto the surface of the model soil to allow the uniform application of vertical loads. Two unfixed L-shaped steel plates were installed on the left and right sides to allow the uniform application of lateral loads.

(2) Loading/unloading device: Hydraulic jacks (with a minimum accuracy of 0.2 MPa) were used to apply and remove the load during the test. Three sets of hydraulic jacks were installed on the three reaction frames. The jacks on the top were used to apply vertical loads, whereas those on the left and right sides were used to apply horizontal loads.

The load of the hydraulic jack can be applied to model the soil evenly using the three additional steel plates mentioned above to satisfy the demand of soil pressure in the test. The lateral load was adjusted by controlling the pump pressure of the horizontal jacks. Specifically, when the lateral loads on the model soil decreased with the horizontal pump pressure, the lateral earth pressure transmitted to the lining structure decreased, thus achieving the simulation of the LRS.

3.3 Design of similarity ratio

Based on a previous method used in similar model tests [28,29], the geometry similarity ratio and bulk density similarity ratio are regarded as basic similar ratios, and the similarity ratios of other physical and mechanical parameters between the prototype and model tunnels can be obtained based on the similarity principle.

A geometry similarity ratio C_L = 20 and bulk density similarity ratio C_γ = 1 were used in this model test. Therefore, the Poisson’s ratio, strain, and friction angle similarity ratio were C_μ = C_ε = C_φ = 1, and the strength, stress, cohesion, and elastic modulus similarity ratio were C_R = C_σ = C_c = C_E = 20.

The geometric sizes of the segmental lining structure adopted in the model test are listed in Tab.1.

3.4 Materials and preparation

3.4.1 Materials of model soil

The design and selection of stratum materials for similar model tests are particularly important. The model soil was constructed based on the geological information of the selected section, as shown in Fig.2. To establish the model soil, the weight, cohesion, internal friction angle, and elastic modulus were considered as the control indexes. Meanwhile, parameters corresponding to the prototype soil were calculated based on the similarity principle, geological report, and related specifications (Tab.2).

River sand was used as the main material of the model soil, whereas fly ash, quartz sand, motor oil, barite powder, and rosin were used as auxiliary materials for adjusting the elastic modulus, cohesion, and internal friction angle of the model soil. All these materials were mixed homogeneously to form the model soil. The mass ratios of the main and auxiliary materials in the model soil were determined via geotechnical tests, as shown in Tab.3, and Tab.4 lists the corresponding mechanical parameters of the model soil.

3.4.2 Similar material of segmental lining structure

The physical and mechanical parameters of the prototype segment were obtained based on the Code for Design of Concrete Structures in China [30], whereas the corresponding parameters of the model segment were calculated (Tab.5).

For the model test, the segmental lining was fabricated using gypsum as the basic material, and a certain proportion of diatomite was added to achieve a more accurate simulation of the concrete material [31]. Specimens with various material ratios were trial produced and their corresponding uniaxial compressive strength tests were performed. Fig.5 and Fig.6 show the changes in the elastic modulus and uniaxial compressive strength of the specimens with respect to the gypsum–water ratio. Therefore, the material ratio of the segment model was set as water: gypsum: diatomite = 1:1.38:0.1 to satisfy the requirements for the parameters of the model segment.

In addition, a thin wire was used as the main reinforcement for the segment lining structure. Wire meshes with a diameter of 1 mm were symmetrically arranged on the inside and outside of each model segment to simulate the circumferential main reinforcement in the segment lining.

3.4.3 Segment joint processing

Longitudinal and circumferential joints are present in the shield tunnel lining structure. These joints can reduce the stiffness of the segment structure, which adversely affects the mechanical characteristics and failure modes of the segment [32]. Hence, a reasonable method should be adopted to simulate a joint in a similar model test. As an important component in the joint configuration, the bolt connects the segments and affects the mechanical properties of the joints. Therefore, in this test, the simulation of longitudinal and circumferential bolts during segment joint processing was prioritized.

(1) Circumferential joints

Currently, three measures are typically used to simulate circumferential joints in the segment lining structure: external cutting grooves, internal and external cutting grooves, and internal and external zoned cutting grooves [33]. Considering the stress state of the segment lining in the actual load bearing, the internal and external zoned cutting grooves were selected to simulate the circumferential joint of the segment lining in the model test.

The width of the cutting groove was designed as 4 mm owing to the limitations imposed by the size of the model segment. Meanwhile, the groove depth was determined via iterative calculations using the FEM based on the equivalent principle of flexural rigidity [34]. Specifically, the segment circumferential joint was simplified to a straight beam (Fig.7 [35]), the displacement at the cut groove was calculated under a certain load, and the flexural rigidity of the beam was obtained using the formula shown in the figure. The groove depth was adjusted repeatedly such that the calculated flexural rigidity of the beam structure reflected the actual value. Finally, the groove depth in the model segment was obtained based on the similarity ratio. Tab.6 lists the applied internal and external cutting groove depths, which satisfy the requirements for the positive and negative bending rigidity of the segment lining.

(2) Longitudinal joint

The behavior of longitudinal bolts during longitudinal-segment joint processing must be simulated based on the basic elastic similarity ratio. In this study, an I-shaped polyethylene patch was selected [36]. The relevant parameters are presented in Tab.7. The I-shaped patch represents a longitudinal bolt, and the angle spacing between two I-shaped patches is consistent with that in the actual project. The I-shaped patches and assembly model of the segmental lining structure are shown in Fig.8.

Fig.9 shows the circumferential cutting grooves and longitudinal patches of the model segment.

3.5 Monitoring content and arrangement of measuring cells

The test box measured 1 m long in the longitudinal direction, which corresponds to the length of 13 model segment lining rings. The seventh ring was regarded as the measurement object, and the data categories are shown in Tab.8.

The vertical and horizontal convergence values at the center of the lining structure were measured using displacement gauges with an accuracy of 0.001 mm. A pair of symmetric resistance strain gauges was placed inside and outside the model segment. Beginning from the arch vault, a pair of strain gauges was installed clockwise at 45° intervals, resulting in a total of eight pairs. After obtaining the strain values under load changes, the circumferential bending moments and axial forces of the segment structure can be calculated using Eqs. (1) and (2), respectively.

(1)N=\frac{1}{2} E({\epsilon }_{\text{i}} +{\epsilon }_{\text{o}})bh ,

(2)M=\frac{1}{12}E ({\epsilon }_{\text{i}}-{\epsilon }_{\text{o}} )b{h}^2,

where b is the unit length, set as 1 m; h is the segment lining thickness, set as 0.35 m; E is the elastic modulus of the segment lining, set as 34.5 GPa; ε_i is the strain measured by the strain gauges on the inner surface of the segment; ε_o is the strain measured by the strain gauges on the outer surface of the segment.

The earth pressure reflected the actual load applied to the segment. A vibrating-wire pressure cell with a minimum accuracy of 0.0001 MPa and a measuring range of 0–0.3 MPa was used in the test. Beginning from the arch vault, one earth pressure cell was set clockwise at 45° intervals outside the segment lining, resulting in a total of eight cells. A full-digital acoustic emission monitoring instrument (Physical Acoustics Corporation, USA) was used in this test to detect the internal damage to the lining structure. The acoustic emission threshold was 45 dB. Acoustic emission probes were installed at the arch vault, arch bottom, and arch haunches of the lining structure, and Vaseline was applied as a coupling agent between the probe and structural surface.

3.6 Test procedure and loading control

The aim of this test was to simulate the LRS by reducing the lateral loads and to monitor the deformation and forces of the shield tunnel. The main procedures of the test were as follows.

(1) Production of the model segment structure

The model segment ring was fabricated as described in Subsection 3.4. Subsequently, a groove was created on it. The grooved segment lining ring was longitudinally assembled using I-shaped connecting patches. Subsequently, the strain gauges were placed at certain locations in advance, and the earth pressure cells were installed at the corresponding measuring points.

(2) Soil filling and model segment placement in test box

To reduce the deviation caused by soil compaction, layered filling was adopted. The model soil was compacted when the filling reached the bottom of the model segment lining. Subsequently, the model lining structure was fixed, and the remaining space of the box was filled with the model soil. The soil surface was shielded using a steel plate. Finally, displacement gauges and acoustic emission probes were installed.

(3) The loading preparation process

All instruments were connected and debugged to ensure their functionality. Next, loads were applied to the model soil and model segment using hydraulic jacks. The purpose of loading was to place the tunnel structure under normal design loads to ensure the accuracy of the subsequent lateral unloading test. The normal design load was the external load imposed on the tunnel under normal conditions and was derived using the burial depth and strata parameters, as follows.

Normal vertical loads applied on the prototype tunnel: p_{\text{V}}=\gamma h=20\times 25 =500\text{kPa}, where γ is the bulk density of the strata, which was set as 20 kN/m³ based on Tab.2; and h is the burial depth of the tunnel, which was set as 25 m. Subsequently, the normal design loads on the model tunnel were obtained based on the similarity ratio.

Considering the coordination of two-direction loading and the limitations of the equipment used, loading was performed step by step, first vertically and then horizontally. Notably, the ratio of the lateral loads to the vertical loads must remain unchanged during the loading preparation stage. Based on the geotechnical test for the actual project, the coefficient of the lateral earth pressure was 0.4–0.6; therefore, the theoretical coefficient was maintained at 0.587 by adjusting the oil pump pressure. The loading data are listed (Tab.9). Load application was halted in Step 10, where the normal design loads were attained.

(4) Lateral unloading test

After loading to the normal design loads, a lateral unloading test was conducted to simulate the LRS of the shield tunnel. Horizontal hydraulic jacks were used to reduce the horizontal uniform load gradually until the horizontal oil pump pressure decreased to zero. A particular concern was that the load decrease remained constant. Tab.10 lists the unloading schemes.

4 Test results and analysis

4.1 Lateral earth pressure of segment under lateral unloading

Fig.14 shows the variation in the lateral earth pressure during unloading. When the lateral loads applied to the boundary of the model soil decreased, the earth pressures at the arch as well as their reduction rates decreased. In Step 10, the earth pressures on the left and right arch haunches were 11.99 and 10.75 kPa, respectively, which then decreased to 3.35 and 2.99 kPa, respectively, at the last loading step. However, lateral unloading barely affected the earth pressures at the segment arch vault and bottom, i.e., they remained at approximately 19.5 and 25.5 kPa, respectively.

The lateral earth pressure coefficient at the arch haunch was calculated, as shown in Fig.15. The lateral earth pressure coefficient decreased exponentially with the lateral load, and its corresponding declining rate decreased gradually. The fitting curve is shown in Fig.15, and the fitting equation was y=20.145{\text{e}}^{-x/2.578}+0.134. Based on Tab.8, after the horizontal oil pump pressures decreased below 1.4 MPa, both the lateral earth pressure and the corresponding coefficient change only slightly. When the horizontal oil pump pressure decreased to 0, the lateral earth pressures remained on the arch haunches of the lining structure, and the coefficient of the lateral earth pressure coefficient was 0.17. This is attributable to the presence of the model soil in the test box.

4.2 Variation in acoustic emission (AE) characteristics

Fig.16 provides information regarding the changes in the cumulative number of AE events and the rates of the AE events during the model test. Owing to the contact state between the probe and segment, the sensitivity of each probe to elastic waves was different, which may result in differences in the data scale of the AE detected by each probe. However, all probes were able to detect abrupt changes in the AE characteristics during the entire loading procedure, and structural damages can be inferred based on these abrupt changes.

As illustrated, at the loading stage (loading steps 1–10), the AE events of each probe increased gradually, and mutations in the AE rate coincided with those of the AE events. At the fourth loading step, the number of AE events detected by the four probes changed abruptly, indicating that the stress state of the lining structure may change significantly and that microscopic damage may occur in the structure.

In the lateral unloading stage of Step 14, significant changes in the AE characteristics were detected by the probes placed at the left and right arch haunches (Probes 2 and 4, respectively). Additionally, some macrocracks were observed by the naked eye, indicating that the structural damage was becoming more severe. At this time, the number of AE events at Probe 2 increased to 12000, which was 10 times higher than before, and the AE rate reached a maximum value of 2772 time/s. Similarly, the cumulative AE events and AE rate at Probe 4 increased to 6300 and 584 time/s, respectively. However, the corresponding AE characteristics for Probes 1 and 3 did not vary considerably during this loading step. The number of AE events at Probe 1 was only approximately 250, and the AE rate of Probe 3 was less than 5 time/s. This indicates that the arch vault and arch bottom may be safer and less likely to break during the lateral unloading stage.

As the lateral loads decrease further, the number of AE events at all probes increased, accompanied by the development of cracks. The number of AE events at Probe 1 increased in Step 16, as verified via macrocrack observation at the segment arch vault. However, for the segment arch bottom, its corresponding AE characteristics were more stable than those of the other probes, and no significant cracks were observed even at the final loading step.

4.3 Internal force analysis of segment structure

The strain values of the segment lining structure were measured using the resistance strain gauges adhered onto the structure, and the corresponding internal forces were calculated. Fig.17 shows the internal forces exerting on each section of the prototype tunnel in the lateral unloading stage.

At the end of the loading stage (Step 10), the axial forces of the segment structure were positive, and the entire ring was compressed. The values of the bending moment at the arch vault and bottom were positive, whereas those at the left and right arch haunches were negative.

Throughout the lateral unloading stage, the structural axial forces increased gradually, and the axial forces at the four positions ranked as follows: arch vault > arch bottom > left arch haunch > right arch haunch. At the final loading step, the axial force at the arch vault increased to 2692.73 kN, whereas that at the right arch haunch remained less than 1300 kN.

Meanwhile, the bending moments of the lining structure increased with the unloading of lateral soil. Furthermore, the growth rate of the bending moment at each position in the structure was different. The bending moments at the right arch haunch and arch bottom increased slightly, i.e., by only 71.85 and 40.50 kN·m, respectively. However, the bending moments at the left arch haunch and arch vault changed significantly, and their increments were approximately three to four times those at other positions. At the end of this lateral unloading test, the bending moment at the left arch haunch was the maximum (−451.91 kN·m), whereas that at the arch bottom was the minimum (164.19 kN·m).

In addition, the capping block of the segment lining ring was located at the right arch haunch (Fig.9); therefore, compared with the left arch haunch, the right arch haunch contained more cutting grooves, exhibited greater stiffness reduction, and indicated a smaller bending moment value. In the model test, the bending moment at each section conformed to the stiffness distribution law.

The presence of conspicuous cracks can facilitate the identification of unfavorable areas in the segment. First, cracks occurred at the arch haunches of the segment structure at Step 14, which might be caused by the compressive shearing damage of the segment lining structure under unloading. At this time, the corresponding axial forces of the left and right arch haunches were 1292.03 and 646.01 kN, respectively, and the bending moments were −347.68 and −175.11 kN·m, respectively. Subsequently, the bending moment at every position increased further, although the value at the right arch haunch remained almost unchanged after the 15th loading step. This indicates that plasticity hinges might be present at the right arch haunch crack, i.e., load bearing and deformation were possible therein even though the resistance bending moment no longer increased.

Fig.18 shows the changes in eccentricity (e = bending moment/axial force) at the four positions of the segment lining during unloading.

During lateral unloading, the eccentricity of the segment lining at the two arch haunches exceeded that at the arch vault and bottom, and the eccentricity at the section of the arch vault exceeded that at the arch bottom. This is consistent with the occurrence sequence and location of the macroscopic cracks observed. As the lateral loads decreased in a stepwise manner, the eccentricities at the right arch haunch and arch bottom decreased, whereas a different pattern was indicated at the left arch haunch and arch vault. In general, the eccentricity of the structure did not change significantly, except at the right arch haunch, which decreased considerably after Step 15.

At the final loading step, eccentricities of the segment were 0.28 m at the left arch haunch, 0.17 m at the right arch haunch, 0.14 m at the arch vault, and 0.06 m at the arch bottom. Based on the AE characteristics and macrocrack observations, the stress states of the structural arch vault and arch bottom were better than those of the arch haunches, and the segment was more likely to be damaged at the arch haunches.

4.4 Analysis of segment structure deformation

During the lateral unloading stage, the convergence of the segment structure was measured using a displacement gauge. The vertical and horizontal convergent deformations and the structure ellipticity as unloading progressed are shown in Fig.19 and Fig.20, respectively, and the ellipticity is calculated using Eq. (3).

(3)\alpha \text{=}\left(D_{max}-D_{min}\right)/D_{\text{o}} \times 100\mathrm{\%} ,

where \alpha is the ellipticity of the segment structure, D_o is the initial outside diameter of the segment structure (m), and D_max/min is the maximum/minimum outside diameter of the segment structure measured at each loading step (m).

During the entire lateral unloading phase, the radial convergent deformation and ellipticity of the segment lining structure increased as the lateral loads decreased, and the structural deformation was indicated by horizontal outward expansion at the arch haunches and vertical inward shrinkage at the arch vault and arch bottom.

The increments in the structural deformation and ellipticity were significant at the beginning of unloading. At the 13th loading step, the vertical and horizontal radial convergence values of the segment were 8.20 and −8.60 cm, respectively, which were 3.9 and 4.8 times higher than those in Step 10. Additionally, the ellipticity in this step was 2.71%. This is because prior to Step 10, the surrounding soil was compacted, and the segment structure was in a stable load-bearing condition. As the lateral loads decreased abruptly in this loading step, the lateral soil restraints on the segment lining weakened, which allowed for greater structural deformations.

After Step 13, the average increase rates in the radial convergence values and ellipticity began to decrease (based on comparing K_S.A and K_S.B in Fig.20). In addition, cracks were clearly observed at the left and right arch haunches of the segment lining in Step 14, followed by cracks at the arch vault in Step 16. At the end of the 16th loading step, the vertical and horizontal radial convergence values of the segment structure were 12.50 and −11.54 cm, respectively, with an ellipticity of 3.88%.

After macrocracks emerged at the structure arch vault, more lateral unloads occurred and the amount of structural deformation increased at a significantly greater growth rate than before. The structural ellipticity reached 4.89% in Step 17, and the vertical and horizontal radial convergence values of the segment structure were 16.07 and −14.27 cm, respectively.

In summary, based on the changes in the lateral earth pressure coefficient during unloading (Fig.15) and the structural AE characteristics (Fig.16), the LRS of the shield tunnel can be classified into three stages: Stage A, elastic and quasi-elastic stage (Steps 10–13); Stage B, damage stage (Steps 14–16); Stage C, rapid deformation development stage (after Step 16).

In Stage A, the lateral earth pressure coefficient decreased significantly with the lateral loads. Although the amount of segment radial deformation increased significantly, the AE characteristics show that no microcracks within the structure was possible at this time. In Stage B, the lateral earth pressure coefficient decayed more gradually, and the radial convergence values and ellipticity of the lining structure increased slightly. However, the number of AE events and the AE rate increased abruptly, which indicates that microdamage occurred within the structure, and some of them evolved into macrocracks. After entering Stage C, the deformation of the segment structure developed rapidly until the lateral unloading test was completed.

5 Deformation control criterion for segment structure under LRS

Among the many structural responses of shield tunnels caused by the LRS, the deformation of the segment lining structure was the most direct, stable, and easy to observe and control in actual projects. Therefore, this paper presents the deformation control criterion for a shield tunnel under the LRS, specifically for tunnels approximately measuring 6 m in diameter. During the daily monitoring of similar projects, a rational assessment of the structural status should be conducted and the optimal timing for deformation control should be determined. When the tunnel deformation satisfies the criterion, then the structure is considered safe and reinforcement maintenance is not necessary, thus reducing operating costs. However, when the tunnel deformation reaches a control threshold, the appropriate measures should be implemented timely to avoid irreversible structural damage as well as increasing difficulties and costs for subsequent maintenance activities.

Fig.21 shows the relationship between the segment radial convergence and structural ellipticity of the segment lining, whereas Fig.22 shows that between the structural ellipcity and lateral earth pressure coefficient under the LRS.

In the model test, the lateral earth pressure coefficient at the segments was 0.53 under normal design loads, whereas that at the final step of the lateral unloading stage decreased to 0.17. The lateral earth pressure coefficient decayed the fastest in Stage A. Subsequently, its declining rate decreased in Stage B. Although the radial convergence values and ellipticity of the segment lining increased linearly, successive macroscopic cracks were observed by the naked eye at the arch haunches and arch vault, in addition to numerous released elastic waves.

The sensitivity of the radial convergence value and ellipticity of the segment structure to changes in the lateral pressure coefficient increased substantially in Stage C. As the lateral pressure coefficient decreased, the two parameters above increased significantly, indicating a state of nonconvergence.

Hence, for in-service shield tunnels similar to that on which this study is based, i.e., shield tunnels with a diameter of approximately 6 m in soft sandy strata, we propose a deformation control criterion under the LRS.

When the ellipticity of the segment lining was less than 2.71%, the structure was in the elastic and quasi-elastic states and thus did not require strengthening. However, the further development of the LRS can increase the structural ellipticity, thus causing the structure to damage gradually as well as some of the internal microcracks to expand and evolve into macrocracks. The emergence of macrocracks can change the original structural system of the segment lining and adversely affect the structural forces.

From a safety perspective, segment deformation control should be performed before the occurrence of large macrocracks. Based on the test results, when the ellipticity of the structure is 2.71%–3.12%, the deformation monitoring of the shield tunnel should be strengthened and the appropriate measures must be implemented timely to prevent the further development of segment deformation. Otherwise, more significant damage will occur, e.g., cracks at the arch vault due to tensile failure, thus rendering the maintenance of the shield tunnel more difficult and costly. Conspicuous cracks may indicate a more prominent weakening effect on the load-bearing capacity of the structure.

Considering the effects of some uncontrollable factors on the test results and practical reference value, the structural ellipticity of shield tunnels under the lateral soil relaxation effect of soft soil should be maintained within 3%, which is approximately the deformation limit for some transverse sections of shield tunnels in soft soils. In terms of Shanghai Rail Transit shield tunnels for instance, structural strengthening and reinforcement are urgently required when the radial deformation reaches 2% tunnel external diameter (approximately 120 mm) [37,38].

6 Conclusions

(1) The LRS of a shield tunnel in soft ground can be simulated via model tests by adding a load followed by unloading. During lateral unloading, the lateral restraints on the segment lining structure attenuated, whereas the lateral earth pressure coefficient decayed in an approximately exponential function.

(2) Based on an analysis of the test results, for a shield tunnel measuring approximately 6 m in diameter in soft sandy soil, the corresponding LRS can be classified into three stages.

1) Elastic and quasi-elastic stages: The declining rate of the lateral earth pressure coefficient was the highest, and the internal forces and deformation of the segment structure increased linearly and significantly as loading progresses. The structural ellipticity in this stage was less than 2.71%; additionally, a few internal microdamages and no macrosurface damage were observed.

2) Damage stage: The decay of the lateral earth pressure coefficient was slower than that in the previous phase. The internal forces and deformation of the segment structure increased further, although their growth rates were lower. In addition, the AE characteristics became more prominent, with observable macrocracks appearing first at the segment arch haunches.

3) Rapid deformation development stage: As the lateral loads decreased, the lateral earth pressure coefficient decreased marginally, but the structural ellipticity and radial convergence values increased rapidly. Further development of the LRS might result in structural failure.

(3) The development of the LRS of shield tunnels in soft soils would render the segment structure susceptible to significant transverse elliptical deformations. When the structure ellipticity was less than 2.71%, the segment lining was stable in the elastic and quasi-elastic stages. Meanwhile, when the ellipticity is 2.71%–3.12%, the appropriate measures should be implemented to effectively control the deformation of the shield tunnel. Moreover, to avoid test deviation and fully consider the applicability of ellipticity as an engineering reference, an ellipticity of 3% was specified as the benchmark value for controlling the deformation of a 6 m-diameter shield tunnel under the abovementioned scenario.

The origins of transverse deformation in metro shield tunnels are complex, and theoretical transverse deformation control criteria for shield tunnels in operation are yet to be developed systematically. The LRS contributes to the lateral elliptical deformation of the segment structure. Therefore, the findings reported herein may benefit studies pertaining to the transverse deformation control of segments.

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