Interface failure of segmental tunnel lining strengthened with steel plates based on fracture mechanics

Abstract

Segmental tunnel lining strengthened with steel plates is widely used worldwide to provide a permanent strengthening method. Most existing studies assume an ideal steel-concrete interface, ignoring discontinuous deformation characteristics, making it difficult to accurately analyze the strengthened structure’s failure mechanism. In this study, interfacial fracture mechanics of composite material was applied to the segmental tunnel lining strengthened with steel plates, and a numerical three-dimensional solid nonlinear model of the lining structure was established, combining the extended finite element method with a cohesive-zone model to account for the discontinuous deformation characteristics of the interface. The results accurately describe the crack propagation process, and are verified by full-scale testing. Next, dynamic simulations based on the calibrated model were conducted to analyze the sliding failure and cracking of the steel-concrete interface. Lastly, detailed location of the interface bonding failure are further verified by model test. The results show that, the cracking failure and bond failure of the interface are the decisive factors determining the instability and failure of the strengthened structure. The proposed numerical analysis is a major step forward in revealing the interface failure mechanism of strengthened composite material structures.

Graphical abstract

Keywords

segmental tunnel lining / steel plate strengthening / connecting interface / cohesive-zone model / extended finite element method

1 Introduction

Shield tunnels, which have become the primary structural system in urban rail transit projects, are mainly assembled with circular lining segments and bolts to form a multi-hinge discontinuous circular arch structure, which heavily relies on lateral arch thrust [1]. For weak, complex, and changeable strata, the lateral pressure coefficient, as an essential characterization parameter, directly determines the stability of the segmental lining and has a significant impact on the load distribution and long-term mechanical response of the segmental lining structure. When the tunnel is located in complex strata, changes in the environment lead to cracking and damage to the segmental lining structure [2], as well as staggered deformation [3] of the segmental lining structure. In severe cases, the segmental lining structure will be in a stress state that involves nonlinear behavior and large displacements [4], threatening the safe operation of the shielded tunnel [5,6]. Therefore, it is necessary to incorporate effective reinforcing measures for deteriorated shield tunnels.

There are many ways to strengthen the segmental lining structure and improve its bearing capacity and durability. The steel plate strengthening method can improve the overall stiffness of the segmental lining structure [7,8], and it is widely used in practical engineering, as shown in Fig.1. Zhai et al. [9] performed a series of physical model tests under 1-g and plane strain conditions, and compared the strengthened and unstrengthened segmental linings to study the effect of steel plates on over-deformed segmental lining. The method for strengthening segmental lining by assembling a steel plates-concrete composite structure was proposed by Liu et al. [10], who carried out a full-scale test of the entire ring under surcharge to explore the mechanical performance of the strengthened structure. Liu et al. [11] conducted a 1:5 scale test model to study the bearing capacity, load-displacement response, and failure mode of the bonded steel-strengthened structure under different deformations. They analyzed the bonded steel plates strengthening effect and reasonable strengthening time. Compared with expensive experimental tests, numerical simulations enable an easy adjustment of model parameters to carry out systematic analyses, which to a certain extent can substitute for multiple large-scale tests. Zhang et al. [12] designed a general framework for a robust retrofit of segmental tunnel linings using bonded steel plates, and proposed a two-dimensional finite element model capable of simulating their deformation. Zhai et al. [13] formulated a numerical model of the annular joints of segmental tunnel linings strengthened with steel plates, which revealed the mechanism responsible for improving the shear performance of the joints. Liu et al. [14] proposed a three-dimensional finite element model considering rubber water-stop pads, hand holes, connecting bolts, epoxy resin bonding and steel plates. The strengthened structure’s strength, deformation characteristics, and failure modes were examined. Zhao et al. [15] prepared a simplified model based on fiber-beam elements to simulate segmental lining strengthened with steel plates. The model was highly versatile and allowed further analysis of the influence of the steel plate thickness and strengthening timing on the strengthening effectiveness.

Composite shield tunnels strengthened with steel plates suffer a complex deterioration process under soil resistance and load. The reduction of load-carrying capacity starts with the microscopic damage to the material. The gradual accumulation of this damage on the material microstructure and the resulting deformation eventually translate into damage at the macroscopic level. To systematically study the mechanical properties and crack propagation mechanism of the composite shield tunnel, a numerical model that considers the discontinuous deformation characteristics at the interface is necessary. Based on an experimental characterization of the discontinuous deformation at the interface, Yang and Thouless [16] used the Cohesive Zone Model (CZM) to formulate a constitutive relationship for the discontinuous deformation cohesive zone. A CZM was also used by Belytschko et al. [17] to analyze crack propagation in concrete members. Liu et al. [18,19] proposed a topology optimization method considering the mechanical properties of multi-material and structural interfaces, incorporating fixed-shape components based on a CZM and the extended finite element method (XFEM) under a fixed grid. This method considered the possible opening and failure of the interface and designs the structure with stiffness as the goal. Lin [20] conducted an in-depth study on the numerical simulation of steel-concrete composite bridge girders focusing on the critical issue of discontinuous deformation at the steel-concrete interface. A multi-polyline CZM simulated the interface’s slippage, opening, and detachment. A three-dimensional numerical analysis model was defined for the push-out test of the composite structure, and the shear slip between the interfaces and the distribution of the opening displacement along the average direction of the interface were analyzed. Finally, Yan et al. [21] evaluated the characteristics of the dynamic response of segmental linings, including crack distribution, width, and propagation, and maximum principal stress of the loop interposer bolt, when subjected to the action of an impact load caused by the derailment of a high-speed railway train.

Most of the previous researches focus on the effectiveness of steel plate strengthening and the stress response of the strengthened segmental tunnel lining, primarily based on an ideal interface assumption, ignoring discontinuous deformation at the steel-concrete interface. The influence of cracking and failure of the connection on the bearing capacity of the structure still needs further study. In this study, interfacial fracture mechanics of composite material was applied to the segmental tunnel lining strengthened with steel plates. A three-dimensional solid nonlinear numerical model of the segmental tunnel linings strengthened with steel plates was formulated, combing a CZM with the XFEM to account for the discontinuous deformation at the strengthened structure’s steel-concrete interface. Lastly, the interface failure mode and position of the strengthened structure were further verified by model test.

2 Numerical analysis

2.1 Prototype structure

The test model represents the segmental lining of an urban subway tunnel. The lining segments are made of grade C55 concrete and have an outer diameter of 6.2 m, a width of 1.2 m, and a thickness of 0.35 m, as shown in Fig.2. A full segmental tunnel lining ring is divided into one key block (F), two adjacent blocks (L1 and L2), two standard blocks (B1 and B2), and one bottom block (D), which span central angles of 16°, 65°, 65°, and 84°, respectively. Each pair of segmental lining blocks is connected by two M30 grade 5.8 circumferentially curved bolts.

2.2 Analysis model of the strengthened structure

Based on the finite element analysis software ABAQUS, considering the steel−concrete segment interface’s discontinuous deformation characteristics, this section describes a three-dimensional finite element model of the strengthened structure. In addition, the correctness of the numerical model is verified by experiments.

The CZM is widely used in fracture mechanics to describe the interface of composite structures. The mechanical properties of the interface material are described by the cohesive force-opening displacement relationship, which can accurately simulate the slippage, failure, and the debonding of the interface, making the simulation of the separation process of the interface material concise. This paper uses the bilinear CZM to characterize the relationship between the bonding force and the opening displacement between the steel plate and the segmental lining concrete interface, as shown in Fig.3.

The maximum nominal stress criterion was applied to define the damage initiation criterion of interface failure. Under this criterion, damage initiates when the maximum nominal stress ratio reaches a value of 1. The criterion is expressed with the following equation [22]:

(1)\mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{⟨t_{\mathrm{n}}⟩}{t_{\mathrm{n}}^0}\frac{t_{\mathrm{s}}}{t_{\mathrm{s}}^0}\frac{t_{\mathrm{t}}}{t_{\mathrm{t}}^0}\right\}=1,

where t_{\mathrm{n}}^0, t_{\mathrm{s}}^0, and t_{\mathrm{t}}^0 are the tensile stress and the shear strength of each of the two tangential directions, respectively, and ⟨⟩ is the Macaulay bracket.

To better describe the damage evolution of interface materials, a mixed-mode effective displacement is introduced [22]:

(2){\delta }_{\mathrm{m}}=\sqrt{{⟨{\delta }_{\mathrm{n}}⟩}^2+{\delta }_{\mathrm{s}}^2+{\delta }_{\mathrm{t}}^2},

where {\delta }_{\mathrm{m}}^f, {\delta }_{\mathrm{m}}^0, {\delta }_{\mathrm{m}}^{\text{max}} are the parameters of effective displacement at complete failure, at damage initiation, and at the maximum measured value during the loading history, respectively. The degree of damage of the interface is described by damage variable D:

(3)D=\frac{{\delta }_{\mathrm{m}}^{\mathrm{f}}({\delta }_{\mathrm{m}}^{\text{max}}-{\delta }_{\mathrm{m}}^0)}{{\delta }_{\mathrm{m}}^{\text{max}}({\delta }_{\mathrm{m}}^{\mathrm{f}}-{\delta }_{\mathrm{m}}^0)}.

With a bilinear CZM, the mechanical behavior of the interface between the steel plate and the concrete can be divided into three stages: 1) the linear elastic stage, during which the cohesive stress is positively correlated with the relative displacement until the initial condition of damage is reached; 2) the softening stage, after the cohesive force reaches the maximum value and then decreases to zero with the increase of the relative displacement; 3) the debonding failure stage, during which the relative displacement increases and the cohesive stress is always zero.

The damage evolution is described by the mixed-mode B-K fracture criterion, when the critical fracture energy during deformation is the same along the first and second shear directions; the B-K fracture has a good effect when the critical fracture energies during the deformation process are the same along the first and the second shear direction, and the functional expression of the fracture energy G^C [23]:

(4)G^C=G_{\mathrm{n}}^C+(G_{\mathrm{s}}^C-G_{\mathrm{n}}^C){\left\{\frac{G_{\mathrm{s}}+G_{\mathrm{t}}}{G_{\mathrm{n}}+G_{\mathrm{s}}+G_{\mathrm{t}}}\right\}}^{\eta },

where G_n, G_s, and G_t are the work done by the stress in each direction, respectively, G_{\mathrm{n}}^{\mathrm{c}} and G_{\mathrm{s}}^{\mathrm{c}} are the critical fracture energies of the type I and type II fracture modes, and \eta are the interface material parameters.

The CZM is used to describe the bonding failure of the interface, and an analysis model of the strengthened structure considering the discontinuous deformation of the interface under the combined action of tension and shear is established, as shown in Fig.4.

2.3 Crack analysis theory based on extended finite element method

Since the CZM is used in this paper to describe the mechanical properties of the interface, when calculating the structural response, an accurate description of the opening displacement of the interface (that is, the strong discontinuity in the displacement field at the interface) is required. The interface traverses the structured finite element mesh, and additional degrees of freedom are introduced at the nodes of the cut element to interpolate the opening displacement of the interface, as shown in Fig.5.

Combining the CZM with the XFEM, not only can the discontinuous deformation characteristics such as crack propagation and slip failure of the interface of the strengthened structure be considered, but also the bearing performance of the strengthened structure in the entire loading range can be analyzed entirely. The specific calculation process is shown in Fig.6.

2.4 Numerical modeling

2.4.1 Establishment of model

The width of the steel plate is 850 mm, and the thickness is 20 mm. Referring to the prototype segmental lining structure in Chapter 2, a numerical model of the strengthened structure of the steel plate is established, as shown in Fig.6. The joints are connected by circumferential bolts. An eight-node linear brick reduces integration with an hourglass control (C3D8R) elements were used to simulate the segmental lining, and the S4R shell element is used to simulate the steel plate.

The interaction between the segmental lining and the contact surface of the segment includes the normal action and the tangential action. The shear resistance of the tunnel joint is provided by the interaction of the friction force and the bolts. Therefore, defining the contact between segments as hard contact can reasonably simulate the deformation of segment joints. That is, when the contact pressure between the contact surfaces is zero or negative, the contact surfaces are separated, and the contact constraints on the corresponding nodes are released at the same time; the tangential direction contact is the Coulomb friction based on the penalty function method, and the friction coefficient is 0.4.

The boundary conditions of the numerical model are constraints in z and y directions on both sides of the haunch (the location of tunnel 90°, clockwise from the tunnel crown) and constraints in z and x directions on both sides of the crown (the location of tunnel 0°) and bottom (the location of tunnel 180°, clockwise from the tunnel crown).

2.4.2 Simulation of the steel-concrete interface

The processing method of the interface between layers directly determines the interaction force characteristics of the strengthened structure. Zhao et al. [15] pointed out that when the shear stress of the interlayer interface is between zero and the peak stress, the strength is provided by epoxy resin combined with bolts; when the shear stress is greater than the yield stress, the strength is provided only by the bolts. By setting the cohesive behavior between the segmental lining and the steel plate, the cohesive behavior is applied to all elements between layers to replace the cohesive performance of epoxy resin and the shear performance of bolts. This enables simultaneous transmission of tensile (compressive) and shear forces between layers, forming a synergistic strengthening structural support system. The steel-concrete interface parameters are: t_{\mathrm{n}}^0 = 2.4 MPa, {\delta }_{\mathrm{n}}^{\mathrm{f}} = 0.8 mm, G_{\mathrm{n}}^{\mathrm{C}} = 960 N/m, t_{\mathrm{s}}^0 = 2.5 MPa, {\delta }_{\mathrm{s}}^{\mathrm{f}} = 4.5 mm, G_{\mathrm{s}}^{\mathrm{C}} = 5625 N/m. k_nn = 144 MPa/mm, k_ss = k_tt = 40 MPa/mm [13], where k_nn, k_ss, and k_tt are normal and shear direction interface stiffness.

2.4.3 Material properties of the model

The cracking behavior of the segmental lining is highly nonlinear, and the concrete adopts an elastic-plastic constitutive model. The constitutive relation of concrete is described as follows [24]:

(5)

where σ₀ is the compressive strength of concrete, ε_c0 is the strain corresponding to the peak stress, and ε_cu is the ultimate strain.

The failure criterion based on the evolution of damage mechanics is adopted, and the damage evolution law based on energy and linear softening is selected. The fracture energies G_If, G_IIf, and G_IIIf of segmental lining cracking are all 80 N·m [14]. The stress-based (maximum principal stress) damage initiation criteria were used for the emergence and propagation of cracks:

(6)f=\left\{\frac{⟨{\sigma }_{max}⟩}{{\sigma }_{max}^0}\right\},

where {\sigma }_{\text{max}}^0 is the maximum tensile stress of concrete, ⟨⟩ is Macaulay bracket, when {\sigma }_{\text{max}}\ge 0, ⟨{\sigma }_{\text{max}}⟩={\sigma }_{\text{max}}^0; {\sigma }_{\text{max}}\text{lt 0}, ⟨{\sigma }_{\text{max}}⟩=0. The crack appears when f>1.

With high strength and good plasticity, HRB335 is used as built-in strengthening for the segmental lining, and the thickness of the protective layer is 50 mm. T3D2 truss elements are used to simulate rebar, and the interaction between bolts and rebar and concrete is established using ‘embedded’ constraints. Longitudinal bolts are M30 high-strength bolts of grade 5.8, which are hardened steel, adopt bilinear constitutively, and have yield strength of 400 MPa. The design parameters of the main components of the steel plate strengthened structure are shown in Tab.1.

2.4.4 Surcharge simulation scheme and test groups

The loading method of numerical calculation is the same as that of the full-scale test [7]. The load is applied by evenly setting 24 points along the segmental lining ring, which are divided into P1, P2, and P3 groups, and the loading protocol is divided into three stages: 1) P1 gradually increases, P2 = λ × P1, where λ is the lateral earth pressure coefficient, P3 = 0.5 × (P1 + P2), load to the strengthening point and strengthening steel plate; 2) P1, P2 continues to increase, P3 = 0.5 × (P1 + P2), until P2 equals 275 kN (passive earth pressure); 3) P2 remains constant at 275 kN, P3 = 0.5 × (P1 + P2), and P1 continues to increase until the ultimate bearing capacity is reached. The strengthening condition is first increased to the strengthening point according to the load step, keeping the load unchanged, and considering the secondary stress mode to strengthen the steel plate. The experiment loading protocol is shown in Fig.8.

In the process of numerical calculation, the problem of calculation convergence should also be given attention regarding the steel plate strengthened structure under surcharge. Since the strengthening of the steel plate in the second step is conducted under continuous loading, the segmental lining itself becomes severely deformed, and there is a strain difference between the steel plate and the original segmental lining concrete; that is, there is a phenomenon of “strain lag”. Therefore, in the simulation process, the material parameters of the steel plate should at first be set to almost zero. The steel plate will be deformed by the deformation of the segmental lining under the load. However, due to the minimal set of material parameters, the influence of the deformation of the segmental lining is negligible. When the steel plate needs to be strengthened, the original material parameters of the steel plate are activated to complete the numerical calculation.

During excavation of a shielded tunnel, the upper soil pressure is mainly borne by the shield shell, the segmental lining gradually falls out of the shield tail. At that moment, part of the soil pressure is transferred to the segmental lining structure. However, due to the specific nature of the topography, geology, and hydrological conditions where the segmental tunnel lining structure is located, the earth pressure transfer after the tunnel excavation is variable. When the shielded tunnel begins to operate, the surface excavation and filling, the fluctuation of the groundwater level and the external load changes will also lead to significant heterogeneity of the strata characters. Based on the above considerations, this experiment evaluates the response of the steel-plate-strengthened structure under surcharge for lateral pressure coefficients of 0.55, 0.65, and 0.75[1].

2.5 Results analysis

2.5.1 Bearing capacity of the strengthened structures

In the process of full-scale test, the hydraulic loading system is used for test loading. The load value at all points in the group is equal, and the loading is completely synchronized. The radial displacement around the structure is measured by setting a displacement sensor every 15° clockwise along the tunnel circumference from the key block. The curves between the load of the strengthened structure and the changing values in the vertical convergence are shown in Fig.9. The abscissa represents the diameter-change values of the segmental lining structure in the direction from 0° to 180° (convergence deformation). Strengthening the steel plate when the lateral pressure coefficient is 0.65 and the diameter-change of the structure is 120 mm (19.35‰D, D is the outer diameter of the segmental lining). By comparison, the structural deformation calculation result of the numerical simulation is consistent with the load-displacement curve in the full-scale experiment, which verifies the correctness of the model.

Fig.9–Fig.11 indicate that the failure of the strengthened structure is divided into the following seven key steps. A crack initiates at the outer surface of the segmental lining (S1), strengthening point (S2), cracks at the arch crown (bottom) interface (S3), bolt yielding at the joint (S4), bonding failure of interface (S5), the yield of the steel plate and an inner row of rebars (S6) and overall instability failure of strengthened structure (S7). Its destruction chain is as follows. 1) Before the steel plate is used for strengthening, the outer side of the haunch of the segmental lining becomes cracked; the diameter change increases linearly with the load before the crack occurs. 2) After the segmental lining cracks, the rigidity of the strengthened structure is affected, and the slope of the load-displacement curve decreases. When the change values in the vertical convergence of the strengthened structure reach the strengthening point, the steel plate is used to strengthen. 3) After strengthening, the overall stiffness of the structure is significantly improved, and the interface of the crown (bottom) cracks. 4) As the load increases, the cracking degree of the segmental lining gradually increases, the cracks propagate and penetrate, the overall stiffness of the strengthened structure is further reduced, and the degree of debonding at the interface of the arch crown (bottom) also intensifies. The convergence deformation increases but still increases approximately linearly with the load and the bolts near the arch crown joint yield. 5) As the structural cracks propagate and penetrate, and the degree of debonding of the interface intensifies, bond failure occurs between the steel plate and the segmental lining. 6) The strengthened structure carried initially as a whole has failed, and the stress mode of the steel plate and the segmental lining has gradually changed from cooperative stress to independent stress. In these failure stages, bolt yield (S4), interface bonding failure (S5) and steel plate yield (S6) are consistent with those described in the experiments [7]. Stress-displacement curves of structural components in the strengthened structure is shown in Fig.10. It can be seen from Fig.10 that the stress of the bolt and the inner row of rebars increases approximately linearly with the vertical convergence before the steel plate strengthened. After strengthening, the structural stiffness is increased, and the slope of the stress curve of the bolt is reduced. With the intensification of the concrete crack and bonding failure, the overall stiffness of the structure is rapidly reduced, and the bolt stress gradually increases and enters the yield state. After the bonding failure, the stress of the 73° and 287° steel plates increases rapidly, and finally the steel plate bears the stress alone and enters the yield state.

2.5.2 Analysis of cracking characteristics of strengthened structures

Diagrams of the cracking propagation process for the structure strengthened by the steel plate, obtained through numerical simulation, are as shown in Fig.11. PHILSM is the specified displacement function used to describe the crack surface. The XFEM mainly uses the value of PHILSM to locate the path of a crack in an element. The crack tip can only be on the boundary, approximating the crack to an approximately straight line within a cell. With the Python scripting program, a secondary development is performed in ABAQUS to extract the PHILSM zero position on each crack edge obtained by extended finite element calculation. The polygon formed by all the zero positions is the crack surface, and the integration method is used to calculate the PHILSM value. Then, the area of a single rupture unit is obtained, and after accumulation, it is the area of the entire crack, and the total area of the crack opening at the crack part of the strengthened structure is obtained (For example, F represents the total area of cracks in the key block), as shown in Fig.12.

Fig.12 indicates that during the failure process, the total crack opening area shows a noticeable step-by-step growth feature. This is because concrete is a brittle material, and when the elastic strain energy stored in the segmental lining is greater than the surface energy required for cracking to form a new surface, cracks begin to propagate. The driving force of crack propagation is the elastic strain energy released by the concrete, and the crack propagation is the cyclic process of energy accumulation to release, which is stepped.

Fig.13 shows the maximum principal stress of the strengthened structure under surcharge. According to the crack propagation nephogram (Fig.11), when cracks occur at the interface (S3), as shown in Fig.13(a), the inner side of the segment arch crown and arch bottom and the outer side of the waist bear greater tensile stress, and the maximum principal stress is located at the interface of the arch bottom, with a wide distribution area. When the bonding failure occurs at the interface, as shown in Fig.13(b), the maximum principal stress area at the arch bottom is further increased. The green area at the outer edge of the haunch and the inner edge of the adjacent block of the crown are gradually expanded and continuously distributed, and the cracks at the arch bottom are propagated and penetrative, resulting in a large area of network cracks.

When the lateral pressure coefficient is 0.65, longitudinal cracks (B2, B1) first appear in the segmental lining near the outer surfaces of the right haunch at 90° and the left haunch at 270°. The final crack opening area is 0.099 and 0.306 m², respectively, and the numbers of cracks are 2 and 1, respectively. After strengthening (S2), longitudinal cracks (D) appeared near the 180° interface of the arch bottom. When the changing value in the vertical convergence of the segmental lining is 21.01‰D, the overall deformation of the strengthened structure is temporarily stagnant. Then the opening area and the number of cracks increase sharply. A wide range of cracks are formed at the interface of the segment arch bottom and penetrate the hand hole, and finally extended to the concave-convex falcon and form an overall network of cracks (Fig.14), with a crack opening area of 1.341 m². In the crack propagation process at the bottom of the arch, the general trend is a long stability. This is because, with the increase of the lateral pressure coefficient, the strengthened structure’s overall stiffness and bearing capacity increase, and more strain energy needs to be accumulated for structural cracking failure. In addition, at the stage of structural instability failure, the cracks at the interface significantly propagate rapidly, indicating that the propagation and penetration of the cracks here are the decisive factors leading to the instability failure of the structure, and the local cracks in the remaining positions have relatively little influence on the overall stability of the structure.

3 Similarity model tests

3.1 Similarity relation

The model size is determined according to the geometric similarity ratio. After comprehensively considering reliability, economy, and flexibility, the model test was designed based on a 1:12 geometric similarity ratio and a 1:1 bulk density similarity ratio. According to the three similarity theorems, the experimental similarity constants are obtained as follows:

(7)C_v=C_{ϵ}=C_{\gamma }=1,C_E=C_{\sigma }=C_{\delta }=12,

where C_v, C_{\gamma }, C_{ϵ}, C_E, C_{\sigma }, and C_{\delta } are the similarity ratios of Poisson ratio, unit weight, strain, Young’s modulus, stress and displacement, respectively.

3.2 Segmental lining model and similar materials

The reference concrete grade and elastic modulus are C55 and 35.5 GPa, respectively. From appropriate calculations, a material with an elastic modulus close to 2.96 GPa is found to be needed for the model. According to Refs. [25,26] and direct research, MC nylon was finally selected as the material of the segmental lining model. This material has an elastic modulus of 2.7 GPa, which is similar to the elastic modulus of the concrete material and close to the target value, and offers stable physical and chemical properties that can prevent the performance of the segmental lining model from varying during the test simply due to environmental factors. In the model test, incisions with depths of 5.49, 4.27, 5.52 cm were cut at each joint position along the lining at a ring 0°−180° clockwise to simulate the longitudinal joint.

According to Ref. [9], the interface failure position of the strengthened structure, which is the critical index in this test, mainly depends on the size and deformation of the specimen. Studies have shown that MC nylon material can accurately reflect the deformation law of segmental lining [26]. The model test focuses on the interface failure position determined by the deformation degree of the strengthened structure, and ignores the influence of material strength. So the interface failure position of the model test has a certain reference value and significance.

3.3 Experimental cases

The model test workflow, which is shown in Fig.15, comprises the following steps. 1) First, manufacture of the test loading device, and use of the test loading method referred to the Subsubsection 2.4.4. 2) According to the flexural stiffness of the segmental lining and the internal force at the joint, the outside of the model segmental lining is slotted, and then subjected to loading. 3) When the vertical diameter of the segmental lining model changes by 15‰D, the aluminum plates are installed and fixed to the segmental lining model with screws. 4) the epoxy resin is evenly poured into the interface between the segment and the aluminum plate and allowed to solidify for 24 h. At this point, the strengthening process is completed, and the subsequent loading of the model test is conducted.

When the lateral pressure coefficient is 0.65, the bonding failure process of the interface of the steel plate strengthened structure is as shown in Fig.16. It can be seen from Fig.16 (CSDMG is the damage variable value of interface failure), when the load P1 reaches its ultimate value, the bond failure begins to appear near 40° of the interface of the arch crown, and then the damage range gradually expands. Finally, failure is mainly located in two areas: 0° to 50° of the interface of the arch crown and 205° to 235° of the arch bottom (clockwise from the tunnel crown). Moreover, it is consistent with the model test (Fig.16(b)) and the full-scale test (Fig.16(c)) phenomenon.

4 Conclusions

In this study, interfacial fracture mechanics of composite material was applied to the segmental tunnel lining strengthened with steel plates and was used to analyze the interface failure. The detailed positions of the interface bonding failure of the strengthened structure are verified by model test. The following conclusions can be drawn.

1) The proposed three-dimensional solid nonlinear model of the strengthened shield tunnel, based on the CZM and XFEM, and considering the discontinuous deformation at the steel−concrete interface, can accurately describe the crack propagation process. The model was in good agreement with the results from the physical tests performed, making it appropriate not only to dynamically simulate deterioration due to sliding and material cracking at the steel−concrete interface, but also to fully analyze the bearing capacity of the strengthened structure across the entire loading range.

2) The failure mechanism of the strengthened structure under surcharge can be divided into four stages: cracking initiates at the segmental lining, cracking at the interface after strengthening, yielding of the main internal components, and bond failure at the interface.

3) Concerning the failure modes of the strengthened structure, the first failure occurs at the left and right haunches, mainly caused by longitudinal cracks. The decisive factor leading to structural instability is the bond failure and crack penetration at the interface between the steel plate and concrete. With the increase in lateral pressure, the cracks in the strengthened structure tend to gradually multiply, exhibiting a more extensive and uniform distribution along the circumferential direction. The crack widths increase and the bearing capacity of the cracked structure improves within a specific load range.

4) Combined with the results of model test and numerical calculation of steel plate strengthened structure, it is determined that the bonding failure is mainly located in two areas: 0° to 50° and 205° to 235° from the arch crown of the interface(clockwise from the tunnel crown).

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