Vulnerability analysis of shield tunnels under surcharge loading

Abstract

Accidental surcharge, a type of uncertain manmade hazard, poses a huge threat to the safe operation of shield tunnels. In this regard, a vulnerability assessment framework is proposed in this paper to evaluate the damage state of a shield tunnel subjected to sudden extreme surcharges, accounting for the effect of soil uncertainty and tunnel burial depths. A two-dimensional numerical model of the shield tunnel in soft soil under surcharge loading is established and verified by the field monitoring data. Then, joint opening and horizontal convergence of the shield tunnel are chosen as damage indices, and the corresponding fragility curves and vulnerability curves are established based on the Monte Carlo calculation. The influences of surcharge loading and buried depths on the vulnerability are discussed. Finally, the proposed vulnerability assessment framework is applied in a real case in Shanghai to make a quick judgement on the dangerous sections of shield tunnels. The research results show that the vulnerability of shield tunnels increases with the surcharge loading. Deep shield tunnels have higher initial vulnerability but are not sensitive to surcharge loading. The study sheds light on the robust design, post-hazard decision-making, and rapid risk identification for shield tunnels subjected to surcharge loads.

Graphical abstract

Keywords

shield tunnel / vulnerability assessment / accidental surcharge / uncertainty / failure probability

1 Introduction

With the exponential growth of urban populations, major cities such as Beijing and Shanghai are compelled to address the escalating pressure on urban traffic. One effective solution to this challenge involves the construction of subway systems, designed to streamline citizen’s travel. However, the development of underground spaces necessitates various engineering and construction activities in close proximity to existing subway tunnels, introducing the risk of surcharge loading. Such loading has the potential to induce horizontal convergence, structural deformations, leakage, and dislocations among tunnel rings, leading to critical structural defects [1–6].

An illustrative case is a metro shield tunnel in Shanghai, China, which experienced unexpected extreme surcharge, resulting in a maximum horizontal convergence of approximately 22 cm, constituting 3.55% of the tunnel diameter. Additionally, the radial joint openings reached 20 mm, surpassing the limits prescribed for joint openings in steel bolts yielding criteria [7]. The Fuzhou Metro Line 5 in China also encountered substantial settlement, with a maximum value of 50 mm at Ring 200, accompanied by dislocations at ring joints and 22 leakage points due to an abrupt surcharge [8]. In light of these challenges, ensuring the safety and serviceability of tunnels necessitates the establishment of a robust method for evaluating the structural vulnerability of shield tunnels.

Vulnerability analysis, an approach integrating influencing factors and their uncertainties, emerges as a pivotal methodology to assess structural damage, identify weak points, and evaluate structural safety under diverse uncertainty conditions [9–12]. In recent years, the application of vulnerability analysis theory has gained traction in tunnel engineering [13–19], with research efforts directed toward analyzing tunnel vulnerability under various hazards. Noteworthy examples include vulnerability assessments for tunnels exposed to debris flow, blast loading, and seismic events [17,20–23].

Despite these advancements, existing studies predominantly focus on the failure probability of tunnels under seismic hazards, leaving a significant research gap in evaluating tunnel vulnerability induced by extreme surcharge loading, particularly considering uncertainties related to soil parameters and tunnel burial depths. Moreover, the reliance on traditional single damage indicators in vulnerability assessments may yield inaccurate evaluations, highlighting the necessity to adopt multiple structural damage indicators for a comprehensive evaluation of shield tunnel vulnerability.

To bridge these research gaps, this paper conducts a vulnerability analysis of shield tunnel structures subjected to surcharge loading, taking into account uncertainties related to soil parameters and tunnel burial depths. Initially, a robust framework for vulnerability assessment of shield tunnel under extreme surcharge is proposed, encompassing the definition of vulnerability curves, the selection of damage indices, and the classification of damage levels. Subsequently, a reliable numerical model is developed, and extensive numerical simulations are conducted by incorporating uncertain soil parameters. Based on this, the vulnerability curves for shield tunnels are constructed, and the influence of tunnel depth on the analysis outcomes is discussed. The proposed methodology for vulnerability analysis is then applied to a real case within Shanghai Metro Line 2, facilitating the swift identification of high-risk tunnel sections and the proposition of targeted deformation control strategies. Finally, the paper concludes by presenting key findings, along with the limitations of this research.

2 Probabilistic vulnerability evaluation framework

2.1 Definition of fragility curves and vulnerability curves

The fragility curve generally represents the relationship between the probability of a structure exceeding a certain damage state and the disaster intensity level, and the vulnerability curve represents the relationship between the expectation of structural damage level and the disaster intensity level [24–26]. The fragility curve and vulnerability curve are shown schematically in Fig.1.

In this paper, the fragility curve describes the probability of the shield tunnel structure exceeding a certain level of damage under a certain surcharge loading, which can be calculated by Eq. (1).

(1)P\left[\mathrm{D}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\ge D_i\right]=1-\sum _{J=1}^{i-1}\frac{N\left(D_J\right)}{n},

where P[\mathrm{D}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\ge D_i] represents the probability of the segmental lining structure exceeding damage level D_i; n is the total number of samples considering the uncertainty of soil parameters for each surcharge loading level, taken as 1000; N (D_J) represents the number of samples with a damage level of D_J for the segmental lining structure.

The vulnerability curve determines the expectation of the damage to the shield tunnel structure under a certain surcharge loading, which can be obtained from the fragility curve as follows:

(2){\mu }_{\mathrm{D}}=\sum P_i{D}_i,

where {\mu }_{\mathrm{D}}, i (the value of i in this article is 0–4), D_i, P_i represent the expectation of damage level, serial number of damage level, structural damage level and the probability that the damage level exceeds D_i, respectively.

2.2 Damage index

The identification of the damage index is an essential part of the process of establishing fragility and vulnerability curves. The damage index is mainly determined according to the research object and the type of disaster [27,28]. Six damage states are defined based on the degree of loss in a debris flow vulnerability assessment [29]. Horizontal ground strain is used to assess building damage in subsidence regions [30]. The ratio of the developing moment to the moment resistance of the critical cross-section [31,32], the ratio of induced stresses in the shotcrete liner to its strength capacity [13,14,33,34], and the ratio of the lateral displacement (between the top and bottom of the tunnel) and height of the tunnel [35] are selected as damage indices for different types of tunnel structures under seismic action.

For shield tunnel under surcharge loading, a lot of mechanical responses of segmental lining structure can be considered as potential damage indices, such as the stress of concrete segments, the stress of bolts, the width and length of cracks on the segments, the area and velocity of leakage water, the horizontal convergence of the lining rings, the opening of joints, etc. Unfortunately, the stress of concrete segments and bolts is difficult to be monitored on-site, especially for the concrete segments in contact with the soil. While, compared to the deformation simulation, the efficiency and accuracy for direct cracking and leakage prediction are lower. Due to the convenience and accuracy of measuring the horizontal convergence and joint opening of the segmental lining structure, it has been widely used in the safety monitoring of the shield tunnel in the operating stage [36]. In addition, horizontal convergence and joint opening are easily obtained by numerical simulation and verified by field monitoring data. Therefore, in this paper, the horizontal convergence and the joint opening are adopted as the damage indices for shield tunnel under surcharge loading.

2.2.1 Horizontal convergence

Horizontal convergence of the lining structure as a performance indicator can be easily obtained through on-site monitoring and needs to be controlled within safe limits. Different countries and regions have their specific criteria for the convergence of shield tunnel lining. The British Tunnel Standard [37] adopts 2% as the upper limit of normalization convergence \mathrm{\Delta }D/D (where D and ∆D denote the outer diameter and horizontal convergence of a shield tunnel, respectively). The Chinese code [38] for rail transit system sets the \mathrm{\Delta }D/D as the serviceability criteria at 3‰–4‰, while the Shanghai local regulation [39] increases this criteria of \mathrm{\Delta }D/D to 5‰ [36]. Generally, metro operating companies take 5‰, 10‰, and 15‰ of the diameter of shield tunnel lining rings as horizontal convergence indicators. When the construction of shield tunnel is completed, the acceptable horizontal convergence is less than 5‰ of the diameter of shield tunnel lining and 10‰ of its diameter is often considered as a warning value for horizontal convergence. When the horizontal convergence of shield tunnel lining reaches 15‰ of its diameter, emergency measures must be taken. Huang et al. [7] conducted a statistical analysis of horizontal convergence for Shanghai Metro line 2. It was shown that the probability of horizontal convergence exceeding 8‰ of diameter (4.96 mm), exceeding 10‰ of diameter (6.2 mm), exceeding 12‰ of diameter (7.44 mm) and exceeding 14‰ of diameter (8.68 mm) is 0.1, 0.01, 0.001, and 0.0001, respectively. In this paper, 8‰, 10‰, 12‰, and 14‰ of the lining structure diameter are selected as the cut-off points for the damage state in terms of the horizontal convergence, as shown in Tab.1.

2.2.2 Joint opening

Joint is the weak and critical link in the lining structure, and its opening is an important index for the vulnerability and waterproof ability of shield tunnels. Excessive joint opening may decrease the pressure on the elastic gasket of the joints and reduce the water pressure resistance, resulting in tunnel leakage. In addition, the increase of joint opening is usually accompanied by local yielding or even failure of the joint concrete, as well as yielding of joint bolts. Therefore, the opening of joint can well reflect the stress status of the entire lining structure.

This paper classifies the degree of joints opening based on their mechanical and waterproof characteristics. A lot of researches focused on the relationship between shield tunnel segment joint opening and mechanical characteristics of joints. When the opening of the joint is less than 2 mm, the concrete and bolts at the joint are in the elastic stage and the bending stiffness of the joint is large, which means the joint is in a safe state [40]. However, when the joint opening exceeds 2 mm, the joint concrete gradually enters the local yield state, and then the bolts begin to yield, resulting in a rapid decrease in joint stiffness. Even worse, when the opening of the joint reaches 8 mm, the concrete at the joint has been partially damaged and the bolts yield, which means the joint is in a dangerous state.

Waterproof performance is also an important indicator for evaluating the service performance of joints [41–43]. Zhao et al. [44] conducted a series of joint waterproofing tests, revealing the relationship among joint opening, surface pressure of elastic sealing pads, and leakage water pressure. Zhou et al. [45] developed a numerical modeling of joint leakage, revealing the variation pattern of joint opening with joint waterproofing capacity. Besides, after shield tunnel is assembled into a ring, the opening of the circumferential joint is allowed to reach 6 mm without water leakage under the external loads [46]. Therefore, based on the above analysis, the joint opening of 2, 4, 6, and 8 mm are selected as the classification indicators of damage states, as shown in Tab.2.

2.3 Procedure for vulnerability assessment of tunnels under surcharge loading

As shown in the flowchart in Fig.2, the vulnerability assessment of shield tunnel structure under surcharge loading considering the influence of uncertain soil parameters and tunnel burial depth can be divided into five steps.

1) Initially, a two-dimensional finite element model of a shield tunnel under surcharge loading was established through ABAQUS platform. The reliability of the numerical model was then verified by comparing the numerical results with the measured data.

2) To account for the uncertainties in the key physical and mechanical parameters of soils, the elastic modulus, friction angle and cohesive force were assigned to 1000 random discrete samples, respectively, using the Monte Carlo method based on their probability density functions.

3) Before developing the fragility curves and vulnerability curves, horizontal convergence and joint opening were adopted as damage indices, as their on-site monitoring data are convenient to be obtained and their corresponding numerical simulation results are accurate and reliable. each surcharge loading intensity.

4) Fragility curves and vulnerability curves of shield tunnel under different burial depths are developed based on a large number of numerical calculations and Monte Carlo calculations. Furthermore, Logistic functions and hyperbolic tangent functions were used to fit the obtained fragility curves and vulnerability curves.

5) The impact of tunnel burial depths on the fragility and vulnerability of shield tunnel under surcharge loading is discussed by comparing the failure probability and vulnerability index at different burial depths. Besides, Rapid identification of high-risk tunnel sections is performed by calculating the vulnerability index for different lining rings.

3 Numerical modeling

3.1 Shield lining structure

The shield tunnels in Shanghai are mostly excavated by the earth pressures balance shield machine. The burial depth (i.e., distance from the ground surface to the tunnel crown) of these shield tunnels is not fixed and has a direct impact on their vulnerability. In this paper, three different burial depths, i.e., 8, 16, and 30 m representing the typical burial depths of shallow, moderately deep and deep tunnel in Shanghai, respectively, are selected to conduct numerical simulations. Meanwhile, a typical circular-shaped cross section of segmental lining is selected, as shown in Fig.3. The outer diameter D of the lining ring is 6.2 m and its wall thickness t is 0.35 m. An integrated ring consists of six concrete segments, denoted as one key segment (F), two adjacent segments (L1 and L2), two standard segments (B1 and B2), and one bottom segment (B). There are six joints in each lining ring, and the positions of the joints directly affect their stress condition and vulnerability. Extensive studies have revealed that under combined soil and water pressure, the longitudinal joints at tunnel crown are predominantly subjected to positive bending moment, while the longitudinal joints at tunnel springline are generally subjected to negative bending moment and the joints at tunnel invert suffer less bending moment [5,47,48]. These distinct mechanical responses highlight the necessity to distinguish the joints in different positions when conducting vulnerability analysis. The joints between the key segment (F) and the adjacent segments (L1 and L2) are named as Joint 1, i.e., the joints at tunnel crown. The joints between the adjacent segments (L1 and L2) and the standard segments (B1 and B2) are named as Joint 2, i.e., joints at tunnel springline. The joints between the standard segments (B1 and B2) and the bottom segment (B) are named as Joint 3, i.e., joints at tunnel invert.

3.2 Geological profile

For simplicity, the geological profile is simplified to a layered formation with three layers, as shown in Fig.4. These three layers of soils, namely soil type L1, L2, and L3, are typical soft clay in Shanghai, with depth of 11 m, 18 m and 19 m, respectively. The soil parameters of the simplified layered formation including void ratio (e), saturated soil unit weight (γ), friction angle (φ), cohesive force (c), and elastic modulus (E_{\mathrm{C}}) are illustrated in Tab.3.

3.3 Development of numerical model

A plane-strain finite element model is established using the commercial finite element software ABAQUS as shown in Fig.5. In this model, the width and the depth of the simplified ground model are set to be 55.8 and 47 m, respectively. The shield tunnel is modeled as an assembled segment ring. The model has been simplified to improve the efficiency of the simulation, which is validated in session 3.4. These simplifications include modeling the connecting bolts at the joints as springs that can only be subjected to tension, without considering the details of hand holes, bolt holes, elastic gasket, etc. The stiffness of springs is adopted as 100 kN/mm. A linear elastic perfectly-plastic model with a Mohr-Coulomb failure criterion is adopted to simulate the soil [49–51]. CPE4R (linear plane strain element, reduced integral) is used as the mesh type for the concrete segments and the soil. The surface-to-surface contact is assigned to the interface between different segments and between the tunnel lining structure and the surrounding soil. The tangential behavior of these contact face was defined based on the penalty friction formulation. The coefficient of friction ratio is taken as 0.3 and the normal behavior is a hard contact [52,53].

3.4 Validation of numerical model

The proposed numerical model with the simplifications is validated before it is used to establish fragility curves and vulnerability curves. To verify the accuracy and reliability of the model, the simulated horizontal convergence of tunnel lining under three different surcharge loadings is compared to the in-situ measurements, as shown in Fig.6. When the surcharge loading is 0 kPa, the simulated horizontal convergence of the shield tunnel lining structure is 2.4 cm, which is relatively close to the average horizontal convergence of the Shanghai subway tunnel during normal operation stage (i.e., 1.95 cm) [54]. When the thickness of the soil is 6 m, the horizontal convergence of lining structure obtained from in situ monitoring and numerical simulation is 17.0 and 17.2 cm, respectively. When the thickness of deposited soil reaches the maximum value of 7.5 m in an accident, horizontal convergence obtained by finite element simulation is 24.2 cm. This value is slightly larger than the in situ measured horizontal convergence of 19.8 cm. By comparing the above three sets of data, it is shown that the simulation results are relatively accurate and this numerical model can be used to simulate the shield tunnel lining structure under surcharge loading.

4 Development of fragility curves

On the basis of the damage states defined in Section 2 and numerical model established in Section 3, uncertain soil parameters are introduced to the numerical model and Monte Carlo calculation is performed under different surcharge loadings. Fragility curves for shallow, moderately deep and deep tunnels are then established to describe the probability of tunnel lining structure reaching a certain damage state under a certain surcharge loading and understand the impact of tunnel burial depth on fragility.

4.1 Fragility curves for shallow tunnel

4.1.1 Probability fragility analysis for shallow tunnel

Fig.7 shows the fragility curves for a shallow tunnel under accidental surcharge. As shown in Fig.7(a), when the surcharge loading is less than 50 kPa, the probability of Joint 1 reaching the minor damage state (D₁) is almost zero. However, when the surcharge loading is increased to 100 kPa, the probability of Joint 1 reaching minor damage state is 100%. Collapse state (D₄) is the most dangerous state in this paper. The probability of reaching it needs to be closely watched. When the surcharge is 175 kPa, the probability that the opening of Joint 1 reaches collapse state is 30%.

As shown in Fig.7(b), the probability of the Joint 2 opening exceeding 2 mm (i.e., reaching minor damage state (D₁)) is 100% when the surcharge loading is 60 kPa. Even worse, when the surcharge increases to 175 kPa, the probability of the Joint 2 reaching extensive damage state (D₃) and collapse damage state (D₄) is 100% and 90%, respectively. Under such a large surcharge loading, some measures need to be implemented to repair the waterproofing capacity of the shield tunnel and polyurethane and epoxy resin grouting are widely used in waterproofing treatment of shield tunnels [55].

Joint 3 suffers less deformation damage than Join1 and Joint 2. Only when the surcharge loading exceeds 100 kPa, Joint 3 may reach the minor damage state (D₁) and the probability of joint 3 reaching extensive damage state (D₃) and collapse state (D₄) is almost zero.

Fig.7(d) illustrates the failure probability of lining structure based on the horizontal convergence. When the surcharge loading is less than 70 kPa, the probability of the lining structure reaching minor damage state (D₁) is zero. Even if the surcharge loading is 90 kPa, the lining structure is still in the minor damage state (D₁). It is worth noting that as the surcharge loading increases to 175 kPa, the probability of structural damage reaching extensive damage state (D₃) and collapse state (D₄) are 90% and 80%, respectively.

4.1.2 Fitting functions for fragility curves of the shallow tunnel

To develop the fragility curves, it is necessary to find a suitable function to fit them. The Logistic function (Eq. (3)) has been widely adopted to model fragility curves in the vulnerability assessment of infrastructure under extreme events [56,57].

(3)P=\frac{1}{1+{\mathrm{e}}^{(aQ+b)}},

where P represents the probability that the shield tunnel exceeds a certain level of damage, a and b are the two parameters of the Logistic function, and Q is the level of surcharge loading.

The errors sum of squares (SSE) and the coefficient of determination (R^2) are widely utilized metrics for assessing the performance of curve fitting in statistical analysis [58,59]. An SSE value approaching 0 and an R^2 value close to 1 are indicative of a good fit. Due to the limited space, this paper only demonstrates the fitting parameters of fragility curves in terms of the opening of Joint 1 and horizontal convergence for the shallow tunnel, as shown in Tab.4 and Tab.5.

4.2 Fragility curves for moderately deep tunnel

4.2.1 Probability fragility analysis for moderately deep tunnel

Fig.8 illustrates the fragility curves of a moderately deep tunnel under accidental surcharge. When there is no surcharge loading, the probability of Joint 1 exceeding minor damage state is less than 20% (Fig.8(a)). Meanwhile, the probability of Joint 1 reaching the moderate, extensive and collapse damage state is basically 0. This indicates that, in the absence of surcharge loading, the opening of Joint 1 is impossible to exceed 4 mm. With the increase of surcharge loading, the fragility curve of the minor damage state (D₁) rises rapidly. When the surcharge reaches 100 kPa, the probability of Joint 1 reaching minor damage state (D₁) is close to 100%. Starting from a surcharge loading of 50 kPa, the fragility curve in terms of collapse state (D₄), representing the most dangerous sate, gradually rises. When the surcharge loading is 100 kPa, the growth rate of fragility curve in terms of collapse state (D₄) reaches the maximum. The probability of Joint 1 reaching collapse damage state (D₄) is 100% with the surcharge loading equal to 175 kPa, indicating that Joint 1 has been completely destroyed at this time.

As shown in Fig.8(b), when there is no surcharge loading, there is a 30% probability that the opening of Joint 2 will reach minor damage state (D₁), a 10% probability that it will reach moderate damage state (D₂) and a 0% probability that it will reach extensive damage state (D₃) or collapse state (D₄). In other words, in the case of no surcharge, the opening amount of the Joint 2 is basically impossible to exceed 6 mm. However, when the surcharge loading reaches 70 kPa, the probability of the joint 2 reaching minor damage state is close to 100%. It should be noted that starting from a surcharge loading of 20 kPa, the fragility curve representing the most dangerous state (D₄) gradually rises. The probability is 100% for Joint 2 reaching collapse state at surcharge loading of 150 kPa.

The probability of Joint 3 reaching minor damage state, moderate damage state, extensive damage state, and collapse state is basically 0 in the cases without surcharge loading (Fig.8(c)). This indicates the opening of Joint 3 cannot exceed 2 mm in this condition. Even when the surcharge loading is 25 kPa, Joint 3 still does not reach the minor damage state. However, when the surcharge loading reaches 150 kPa, the probability of joint 3 opening exceeding 2 mm is close to 100%. Close monitoring is needed, when collapse state (D₄), the most dangerous state, is reached. When the surcharge loading exceeds 75 kPa, the probability of the Joint 3 reaching collapse state is no longer 0. The probability of collapse state increases to its maximum growth rate when the surcharge loading is 150 kPa. By comparing the fragility curves of Joint 1, Joint 2, and Joint 3, Joint 2 has a higher failure probability than Joint 3 and Joint 1 under the same surcharge lading. This observation is consistent with the result of an experimental investigation conducted by Liu et al. [60] that the longitudinal joints at tunnel springline often collapse earlier with large joint opening.

Fragility curves using horizontal convergence as damage index are shown in Fig.8(d). When there is no surcharge loading, the horizontal convergence of the shield tunnel lining structure is basically impossible to exceed 8‰ of its diameter (i.e., reaching minor damage state), while when the surcharge loading reaches 125 kPa, the probability of horizontal convergence exceeding 8‰ of the horizontal diameter is close to 100%. When the surcharge loading increases to 25 kPa, the probability of horizontal convergence reaching 14‰ of the horizontal diameter (i.e., reaching collapse state) is no longer 0. When surcharge loading reaches 175 kPa, the probability of the horizontal convergence of the lining structure reaching 14‰ of its diameter is 100%, indicating that the shield tunnel has been completely damaged at this time. Urgent actions are needed to reinforce tunnels with large horizontal convergence. Bonding of the Aramid Fiber Reinforced Polymer can effectively limit the horizontal convergence and the reinforcement efficiency basically increases linearly for shield tunnels with the normalization convergence (\mathrm{\Delta }D/D) less than 1.6% [61]. For severely distorted tunnel rings, the bonding of steel plates as a strong secondary lining is an alternative [62,63]. In addition to reinforcing the tunnel structure to limit horizontal deformations, injecting the grouting into the soil on both sides of the tunnel not only controls further deformation but also reduces the existing large horizontal convergence [2].

4.2.2 Fitting functions for fragility curves of the moderately deep tunnel

To obtain the approximate failure probability at any surcharge loading, fragility curves of the moderately deep shield tunnel are also fitted using a Logistic function. The fitting parameters of the fragility curves for the moderately deep tunnel are demonstrated in Tab.6 and Tab.7.

4.3 Fragility curves for deep tunnel

4.3.1 Probability fragility analysis for deep tunnel

Fig.9 shows the fragility curves of the deep shield tunnel. Fig.9(a)−Fig.9(d) show the development of damage states of shield tunnels with increasing surcharge loading using the opening of Joint 1, Joint 2, Joint 3, and horizontal convergence as damage index, respectively.

In the initial state where surcharge loading is not applied, there is a 50% probability for the opening of Joint 1 reaching minor damage state, while the probability of Joint 2 reaching minor damage state is close to 100%. For the moderate damage state, when the surcharge loading is less than 40 kPa, the probability of the Joint 1 and horizontal convergence reaching this state is almost zero. However, when the surcharge loading is 0 kPa, the probability of Joint 2 reaching the moderate damage state is already 10%. It should be noted that the probability of Joint 2 and horizontal convergence reaching collapse damage state at surcharge loading of 175 kPa is close to 20% and 10%, respectively. Even if the surcharge loading is very large, the probability of Joint 1 in the extensive damage state or the collapse state is small and Joint 3 will not even reach a minor damage state.

The fragility analysis for the deep tunnel also indicates that damage of risk of Joint 2 is higher than that of Joint 1 and Joint 3. In addition, due to the impact of soil and water pressure on deep tunnels, their failure probability is relatively high even when the surcharge loading is small. However, with the increase of surcharge loading, the enhancement of failure probability is not obvious.

4.3.2 Fitting functions for fragility curves of the deep tunnel

Similar to the fragility curves of the shallow tunnel and the moderately deep tunnel, the fragility curves are fitted with Eq. (3), except for the fragility curves of the collapse state (D4). When the surcharge loading is less than 200 kPa, the probability for Joint 1 reaching collapse state (D4) remains 0. Tab.8 and Tab.9 show the fitting parameters of the fragility curves for the deep tunnel.

4.4 Effects of tunnel burial depth on fragility

The burial depth of urban subways is not fixed and the fragility of shield tunnels under surcharge loading is closely related to their burial depth. To take different burial depths of shield tunnels into account, the following compares the probability of reaching moderate damage state for tunnels with different burial depths (i.e., shallow tunnel, moderately deep tunnel and deep tunnel). The cases with surcharge loading of 50, 100, and 150 kPa are discussed. The horizontal convergence, the opening of Joint 1, Joint 2, and Joint 3 are used here as damage indices, as shown in Fig.10.

When the horizontal convergence is adopted as the damage index, the probability for shallow tunnel reaching moderate damage state is almost 0% under a surcharge loading of 50 kPa. However, when the surcharge loading is increased to 150 kPa, the probability of moderate damage state for shallow and moderately deep tunnel is close to 100%, which is 40% higher than that of deep tunnel. When the opening of Joint 1 is used as the damage index, the probability of reaching moderate damage state is almost the same as the probability when the horizontal convergence is used as the damage index. It is worth noting that when the surcharge loading is 150 kPa, the probability of achieving moderate damage state for shallow, moderately deep and deep tunnel using Joint 2 opening as moderate damage state is close to 100%. When the opening of Joint 3 is used as the damage index, the probability of reaching moderate damage state for moderately deep tunnel under a surcharge loading of 150 kPa is 0.88, which is approximately 2.5 times of that for shallow tunnel. While, the probability for deep tunnel to reach moderate damage state is 0. Fig.10(a) to Fig.10(d) shows that moderately deep tunnels have a higher probability of reaching moderate damage state than shallow and deep tunnel under the same surcharge loading.

5 Vulnerability curves

After an accidental disaster, there is an urgent need to determine the damage states of the shield tunnel in order to take the appropriate emergency measures. On the basis of fragility curves, the vulnerability index {\mu }_{\mathrm{D}}, the expectation of damage state under different surcharge loadings, can be calculated by Eq. (2). To study the impact of surcharge loading and tunnel burial depth on vulnerability, vulnerability curves for shallow, moderately deep and deep shield tunnels are developed here.

5.1 Vulnerability curves for shallow tunnel

Fig.11 shows the vulnerability curves of a shallow shield tunnel, using the opening of Joint 1, Joint 2, Joint 3, and the horizontal convergence of the lining structure as damage indices. When the surcharge loading is less than 20 kPa, the damage indices based on Joint 1 opening, Joint 2 opening, Joint 3 opening and horizontal convergence are all zero. This indicates the shield tunnel is in a safe operating condition. It should be noted that the vulnerability index of the shield tunnel almost reaches 4, when the surcharge loading reaches 175 kPa and the opening of the Joint 2 is used as the damage index. If the horizontal convergence is used as the damage index, the vulnerability index of the shield tunnel falls to 3.6. However, the vulnerability index of the shield tunnel using Joint 3 as the damage index is less than 1, which is much smaller than that of the shield tunnel with the opening of Joint 2 or Joint 1 as the damage index. In other words, Joint 3 is less vulnerable than Joint 1 and Joint 2 under the same load.

A hyperbolic tangent function (Eq. (4)) is adopted as the fitting function for the obtained vulnerability curves [24].

(4){\mu }_{\mathrm{D}}=a[b+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(cQ+d)],

where {\mu }_{\mathrm{D}} represents the expectation of damage level. a, b, c, and d are the four parameters in the fitting function, and Q is the surcharge loading level. Tab.10 shows the fitting results and evaluates the fitting performance. It shows that the hyperbolic tangent function can well characterizes the vulnerability curves with the SSE close to 0 and R^2 close to 1.

5.2 Vulnerability curves for moderately deep tunnel

The vulnerability curves of a moderately deep tunnel are shown in Fig.12. When the surcharge loading exceeds 100 kPa, the vulnerability index using the opening of Joint 2 as damage index first reaches 4. This indicates that the shield tunnel lining structure is highly vulnerable and in a very dangerous state. Besides, the vulnerability index based on the opening of Joint 2 is significantly greater than that based on the opening of Joint 1 and Joint 3 under the same surcharge loading. Therefore, the monitoring of Joint 2 has the priority over the monitoring of other joints and appropriate protections should be performed for Joint 2. The vulnerability index of the horizontal convergence evolves faster than those of other damage indices. When the surcharge loading reaches 75 kPa, the vulnerability index of the horizontal convergence increases rapidly. If the surcharge loading increases to 150 kPa, the horizontal convergence reaches the most dangerous damage state (i.e., the collapse state).

To establish a functional relationship between vulnerability index and surcharge loading, vulnerability curves of the moderately deep shield tunnel are also fitted using hyperbolic tangent function. The fitting results and its performance are presented in Tab.11.

5.3 Vulnerability curves for deep tunnel

As shown in Fig.13, when the surcharge loading is 0 kPa, the vulnerability of the deep shield tunnel using the opening of Joint 1, Joint 2, Joint 3, and horizontal convergence as damage index are equal to 0.5, 1, 0.1, and 0, respectively. It is shown that the vulnerability indices based on the opening of Joint 1, Joint 2, and horizontal convergence rise slowly with the increasing surcharge loading. They finally reach approximately 1.8, 2.8, and 2.2, respectively, when the surcharge loading is increased to 180 kPa. Nevertheless, the vulnerability index based on the opening of Joint 3 remains zero, even if the surcharge loading approaches to 180 kPa. When the surcharge loading is small, the vulnerability index based on the horizontal convergence is smaller than the vulnerability index based on the opening of Joint 1. while the vulnerability index of the horizontal convergence becomes larger than that of the opening of Joint 1 with the increase of surcharge loading. This also can be observed in the vulnerability curves of shallow and moderately deep tunnels.

Similarly, the vulnerability curves are fitted with Eq. (4). Tab.12 shows the fitting parameters and the fitting performance.

5.4 Effects of tunnel burial depth on vulnerability analysis

This section presents a comparison of vulnerability indices to reveal the impact of tunnel burial depth on the vulnerability of shield tunnels. The vulnerability indices for shallow, moderately deep and deep shield tunnel under the surcharge loading of 0, 50, 100, and 150 kPa are compared in Fig.14. Horizontal convergence, the opening of Joint 1, Joint 2, and Joint 3 are adopted as vulnerability indices in Fig.14(a)–Fig.14(d), respectively. The initial vulnerability index (i.e., the vulnerability index when the surcharge loading is 0 kPa) of deep tunnel is larger than that of shallow and moderately deep tunnel, due to the larger earth pressure and water pressure. When the surcharge loading is increased from 0 to 150 kPa, the vulnerability index of the shallow and moderately deep tunnel based on the horizontal convergence rises to 3.18 and 3.94, respectively. However, the vulnerability index for the deep tunnel under the same conditions only increases from 0.09 to 1.73. When using the joint opening as the damage index, the increase in vulnerability index of the deep tunnel remains minimal with surcharge loading increasing from 0 to 150 kPa. It can be concluded that the vulnerability of deep tunnels is not highly sensitive to the increase of surcharge loading.

6 Application of the proposed vulnerability assessment method

The shield tunnel of Metro line 2 in Shanghai suffered an extreme surcharge induced by the dumped soils (Fig.15). Significant horizontal convergence was observed in the surcharged area with a length of 400 m and a width of 120 m. As shown in Fig.16, the horizontal convergence of most of the lining rings in the surcharged area exceeds 10 cm. Among them, the horizontal convergence of ring Nos. 370 to 410 exceeds 15 cm, and the horizontal convergence of ring No. 580 even reaches 20 cm.

To quickly determine which tunnel sections are more vulnerable in the surcharge area, a vulnerability assessment for the shield tunnel under surcharge loading is conducted. Fig.17 presents the procedure of the vulnerability assessment.

1) The vulnerability index for each segmental lining ring is calculated based on the fragility curves and vulnerability curves obtained from extensive numerical simulations and Monte Carlo calculations.

2) According to the vulnerability index, the vulnerability levels of the segmental lining rings are classified into low vulnerability (i.e., vulnerability index is 0 to1), moderate vulnerability (i.e., vulnerability index is 1 to 2), high vulnerability (i.e., vulnerability index is 2 to 3) and extreme vulnerability (i.e., vulnerability index is 3 to 4). The dangerous shield tunnel sections are then determined based on the vulnerability level and the on-site measured data (i.e., horizontal convergence).

3) The specific measures are suggested for shield tunnel sections with different vulnerability levels.

The vulnerability index of each lining ring is calculated using the vulnerability assessment procedure (Fig.17). It is shown in Fig.16 that from ring Nos.350 to 390 and from ring Nos. 550 to 590 are the two most dangerous areas of the shield tunnel, which is consistent with the on-site monitoring data. Among them, the vulnerability indices of ring Nos. 565 to 585 and ring Nos. 335 to 380 are close to 4 and 3.5, respectively. More attention should be given to these two areas during subway operation, and effective reinforcement measures should be taken when it is necessary.

Different measures were taken for lining rings according to their vulnerability levels and horizontal convergence. Fiber Reinforce Plastic was used in those lining rings with a convergence-to-diameter ratio (\mathrm{\Delta }D/D) larger than 1.6% and steel plate was attached to those lining rings with a convergence-to-diameter ratio (\mathrm{\Delta }D/D) larger than 2.42%. Soil grouting on both sides of lining rings with a convergence-to-diameter ratio (\mathrm{\Delta }D/D) ranging from 1.62% to 2.75% was conducted to decrease the convergence after three years [3]. Fortunately, the convergence was dramatically reduced after multiple repair measures were implemented, as shown in Fig.16.

7 Conclusions

This study aimed to quantify the vulnerability of a shield tunnel under surcharge loading, incorporating considerations of soil uncertainty and tunnel burial depths. A finite element (FE) model for a shield tunnel subjected to surcharge loading was established and validated against on-site measured data. Fragility curves and vulnerability curves were derived through Monte Carlo simulation utilizing the developed FE model. The vulnerability assessment employed joint opening and horizontal convergence as damage indicators to analyze the local and global damage states of the shield tunnel. Subsequently, a rapid vulnerability assessment for a real case in Shanghai identified tunnel sections with elevated vulnerability. The key conclusions drawn from this research are as follows.

1) Joint 2 (at the tunnel springline) exhibited a higher failure probability under the same surcharge loading compared to Joint 3 (at the tunnel invert) and Joint 1 (at the tunnel crown). Consequently, Joint 2 is deemed more vulnerable, necessitating the implementation of effective monitoring and protection measures, such as epoxy resin/polyurethane grouting and spalling remediation.

2) When the surcharge loading exceeds 50 kPa, the vulnerability index of shield tunnels with moderate burial depth is higher than that of shallow and deep tunnels, and the probability of moderately deep tunnels reaching a moderate damage state is also greater under identical surcharge loading conditions

3) For small surcharge loadings, the vulnerability index based on horizontal convergence is smaller than the vulnerability index based on the opening of Joint 1. However, as surcharge loading increases, the vulnerability index of horizontal convergence surpasses that of Joint 1.

4) Deep tunnels exhibit higher initial vulnerability than shallow tunnels due to increased pressure from soil and water. However, the vulnerability of deep tunnels demonstrates limited sensitivity to surcharge loading escalation.

It is crucial to acknowledge the simplifications incorporated into the numerical modeling approach in this work, such as the representation of joints between segments using springs. Despite the consistency of numerical simulation results with on-site monitoring data, these simplifications possess inherent limitations, notably in failing to capture the shear characteristics of joints. In future research, a more refined model could be developed, incorporating detailed configurations such as connecting bolts, bolt holes and elastic gasket, to accurately capture the deformation characteristics of the lining ring under surcharge loading. Additionally, the spatial variability of soil parameters could be further considered using random field theory to reflect the uncertainties in geological environment.

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