Bending failure performance of a shield tunnel segment based on full-scale test and numerical analysis

Abstract

This study focuses on the bending failure performance of a shield tunnel segment. A full-scale test was conducted to investigate deformation and failure characteristics. During the loading, the bending failure process can be divided into four stages: the elastic stage, working stage with cracks, failure stage, and ultimate stage. The characteristic loads between contiguous stages are the cracking, failure, and ultimate loads. A numerical model corresponding to the test was established using the elastoplastic damage constitutive model of concrete. After a comparative analysis of the simulation and test results, parametric studies were performed to discuss the influence of the reinforcement ratio and proportion of tensile longitudinal reinforcement on the bearing capacity. The results indicated that the change in the reinforcement ratio and the proportion of tensile longitudinal reinforcement had little effect on the cracking load but significantly influenced the failure and ultimate loads of the segment. It is suggested that in the reinforcement design of the subway segment, the reinforcement ratio and the proportion of tensile longitudinal reinforcement can be chosen in the range of 0.7%–1.2% and 49%–55%, respectively, allowing the segment to effectively use the reinforcement and exert the design strength, thereby improving the bearing capacity of the segment.

Graphical abstract

Keywords

shield tunnel / bearing capacity / failure mechanism / segment reinforcement

1 Introduction

With the rapid development of urban rail transit, shield technology has become the preferred choice for urban subway construction owing to its safety, rapid construction, and minimal disturbance to the surrounding environment [1–4]. The lining of a shield tunnel is a prefabricated assembly structure composed of several arc segments [5–7]. As the main structural framework of a tunnel, the bearing performance of a segment is critical to the project quality. At the same time, the cracking and damage of the segment during tunnel use can seriously affect the safety of subway operations [8–11]. Therefore, it is necessary to study the bending failure performance of these segments.

Many experimental studies have been conducted to clarify the bearing capacity and failure mechanism of the segments. Blom [12] studied the damage characteristics and crack development forms of prototype segments under different external pressures; however, the bearing capacity of the segment was not evaluated. To accurately evaluate the factors affecting the bearing capacity, Luttikholt [13] conducted many bending failure tests on prototype segments in a soft soil tunnel. By analyzing the deformation laws and failure mechanism of the segments, it was found that the reinforcement ratio and strength of concrete in the tensile zone significantly affected the bearing capacity of the lining. Fang et al. [14] studied the mechanical properties of segment linings under different water and earth pressures by conducting model tests and summarized the changes in the internal forces of the lining. To clarify the failure mechanism of the segment lining, Wang et al. [15] used model tests to analyze the overall instability process of the segment lining structure under extreme load conditions and divided the failure process of the segment into three stages: initial elastic stage, local damage stage, and overall instability failure stage. It was proposed that the critical instability state of the shield tunnel lining manifested as segment regional fragmentation and loss of bearing capacity.

With the increasing applications of shield tunnels in complex environmental conditions, stricter requirements have been proposed for the bearing capacity of segments [16–20]. Zhang et al. [21] investigated the applications of composite segments in shield tunnels. The composite segments were made of steel plates attached to the concrete filling by shear connectors and significantly improved the bearing capacity of the lining. Owing to the difficulty in making composite segments, Zhang and Koizumi [22] applied the latest technology for strengthening circular segment tunnel linings with steel-concrete composite (SCC) materials and proved that the strengthening effect of SCC on segments was considerable. In addition to strengthening the segment with steel plates, some scholars have proposed methods to strengthen the concrete in the segments. Ding et al. [23] analyzed the effects of composite steel fibers and reinforcement on the mechanical behavior of tunnel segments. The results indicated that adding steel fibers could improve the ultimate bearing capacity of the lining and significantly reduce the strain in the longitudinal reinforcement. Liu et al. [24] conducted experimental and theoretical research on reinforced concrete segments with steel and synthetic fibers. The analysis showed that adding synthetic and steel fibers could significantly improve the yield moment, ultimate moment, and 0.2 mm crack width moment. However, most previous studies have focused on reinforcing segments and improving the mechanical properties of concrete. There are few related studies on improving the bearing capacity by optimizing the reinforcement design of segments. Therefore, it is necessary to study the influence of the segment reinforcement on the bearing capacity.

This study combined a full-scale test and series of numerical simulations to investigate the bending failure performance of a segment in subway shield tunnels. By analyzing the bending capacity of the segments under different working conditions, an appropriate selection range of the reinforcement ratio and proportion of tensile longitudinal reinforcement was proposed, providing a theoretical basis for optimizing the reinforcement design.

2 Full-scale test

2.1 Test overview

A shield segment of the city subway was selected for testing. The outer diameter of the shield tunnel was 6200 mm, the inner diameter was 5500 mm, and the width was 1200 mm. Each segment ring consisted of one capping block (segment F) with a central angle of 20° , two adjacent blocks (segments L1 and L2) with a central angle of 68.75° , and three standard blocks (segments B1, B2, and B3) with a central angle of 67.5° . Segment B3 was selected as the study object for a full-scale test of the shield segment, and a three-dimensional schematic diagram is shown in Fig.1.

The method of graded loading, with a load of 65 kN for each grade, was adopted for the test. When the concrete on the outer arc surface was crushed or the concrete near the outer arc surface showed a transverse crack, the segment reached the ultimate failure state, and the test was terminated immediately.

As shown in Fig.2, the test loading system included a reaction frame, a hydraulic jack, a load-distributing beam, two load cushions, and two horizontally movable supports. The vertical load was provided by the hydraulic jack, acted on the load-distributing beam, and was then applied to the segment through the load cushions. The setting of the load cushions can prevent the local crushing of concrete owing to the small contact area between the load distribution beam and the segment, making the vertical load finally applicable to the segment in the form of symmetric surface loads.

The test measured concrete strain, crack width, and deflection at the midpoint of the segment between load cushions. The layout of the measurement points is illustrated in Fig.3. The instrument parameters used in the measurements are listed in Tab.1.

2.2 Analysis of test results

2.2.1 Segment deflection

The load−deflection variation relationships at the midpoint of the segment between load cushions are shown in Fig.4. According to the deflection variation laws and the failure characteristics of the test segment, the bending failure process can be divided into four stages.

Stage I (elastic stage): At this stage, the change in the deflection was not apparent, and there were no obvious cracks on the surface of the segment. Therefore, this stage was defined as the elastic stage.

Stage II (working stage with cracks): When the load reached 317 kN, the first crack appeared on the inner arc surface of the segment, and the bending test entered the stage of working with cracks. The initial load at this stage was defined as the cracking load (expressed as F_c). The section neutral axis moved upward, and the segment underwent plastic deformation with increasing load. Specifically, the number of cracks in the tensile zone increased continuously and cracks developed along the width of the segment, and the segment deflection increased linearly.

Stage III (failure stage): When the load reached 962 kN, all through-cracks, distributed symmetrically on both sides of the segment, were formed along the width of the inner arc surface as shown in Fig.5(a). The maximum width of the crack reached 0.2 mm, which is the allowable crack width under the normal service limit state; therefore, the segment was judged to have reached the normal service limit state. Subsequently, the bending test of the segment entered the failure stage, and the initial load of this stage was defined as the failure load (expressed as F_f). At this stage, the stress on the tensile reinforcement increased significantly and the crack width of the concrete expanded rapidly, resulting in considerable plastic deformation of the segment and a rapid increase in the deflection.

Stage IV (ultimate stage): When the load reached 1095 kN, the maximum width of the crack on the inner arc surface reached 2 mm (i.e., the allowable width of the crack in the bearing capacity limit state), as shown in Fig.5(b). Subsequently, the bending test entered the ultimate stage, and the initial load entering this stage was defined as the ultimate load (expressed as F_u). As the load increased, the segment deflection increased sharply, and cracks rapidly developed toward the outer arc surface. When the cracks developed in the sealing ring near the outer arc surface, a transverse crack occurred, indicating that the part above the sealing ring of the outer arc surface was about to be pulled and peeled off, and the segment could no longer bear the load. The final failure state of the segment is shown in Fig.5(c).

The segment exhibited prominent failure characteristics at each stage. For convenience of subsequent research, the initial load in each stage is collectively referred to as the characteristic load. The load and segment failure characteristics of each stage are listed in Tab.2.

2.2.2 Concrete strain on segment surface

(1) Concrete strain in outer arc surface

The outer arc surface monitoring points W04, W05, and W06 were selected and the strain variation curve is shown in Fig.6(a). The three positions are located on the same longitudinal section of the segment, and the loads acting on each position are the same; thus, the three strain variation curves in the figure overlap significantly. The strain curves at the three locations can be divided into four stages, the same as the four stages mentioned in Subsubsection 2.2.1. This confirms the stage characteristics of bending failure. After the load reached the cracking load, the concrete on the inner arc surface cracked, and the strain in the outer arc surface rapidly increased owing to concrete compression. When the segment entered the ultimate stage, the cracks expanded rapidly, and the strain exhibited a rapid growth trend, with the concrete extrusion degree on the outer arc surface increasing significantly.

The W01 and W04 positions on the outer arc surface of the same cross-section were selected, and their strain variation curves are shown in Fig.6(b). The trends of the two curves’ conformed to the four stages of the bending test. For the third stage, the difference between the curves increased. At its maximum, the strain of W01 was 63% higher than that of W04. As W01 is near the load cushion, the external load leads to stress concentration near the load cushion, and thus, the strain is higher than in rest of the external arc surface.

(2) Concrete strain of inner arc surface

In the test, the inner arc concrete cracked first, and the crack width was positively related to the load. When the crack width is large, the strain gauge at this position is pulled off, leading to the failure of subsequent strain data collection. Therefore, the strain in the inner arc surface corresponds to the data before cracking.

Monitored points N10, N11, and N12, located on the inner arc surface in the same longitudinal section were selected to show their strain variation curves in Fig.7(a). At the cracking load, the strain at N11 first mutated, and the strains at N10 and N12 mutated almost simultaneously, indicating that the concrete at N11 cracked earlier than that at N10 and N12, and the crack developed from N11 to N10 and N12 simultaneously. When N10, N11, and N12 cracked, a through crack was formed on the vertical section. With the increase in load, the strain magnitude and change rules of the three locations tended to be consistent; the width of each location of the crack expanded at the same rate.

As shown in Fig.7(b), the strain at N07 mutated first, and the value was more significant than that at N08 and N09 in the whole process, indicating that N07 cracked first. After cracking, the strain at the three locations decreased from inside to the outside, indicating that internal concrete cracking was greater than external concrete cracking. As internal grouting holes weaken the crack resistance of concrete near the opening, they differed from the strain rules at each position on the right vertical section.

Positions N08 and N12 on the inner arc surface of the same cross-section were selected to have their strain variation curves shown in Fig.8. The strain at N12 mutated first, and the value was greater than that at N08 after the mutation, which shows that N12 cracked first, and the degree of cracking was more significant than that at N08 in the entire process. The longitudinal section where N12 is located is within the position of the load cushion and adjacent to it. As the first crack appeared on the longitudinal section of N12, it can be proven that the first crack of the segment is near the force application position of the load-distributing beam.

(3) Concrete strain on the side

As shown in Fig.9, the concrete at C05 near the inner arc surface was in tension throughout the process, and the strain level changed abruptly after reaching the cracking load. Meanwhile, the concrete at C03 near the outer arc surface was in compression during the entire process, and the strain increased sharply after reaching the ultimate load, indicating that the concrete on the outer arc surface was fully extruded. With an increase in the load, cracks on the inner arc surface developed into the outer arc surface, and the height of the concrete compression zone on the side of the segment continued to shrink. The concrete at C04 was first compressed and then subjected to tension.

3 Numerical analysis

3.1 Numerical model

A three-dimensional model was constructed based on the test segment. The finite element model consisted of a concrete segment, reinforcement framework, and load cushions. The concrete segment was modeled using solid elements, and the boundary conditions were set to limit the vertical displacement at both ends of the segment. The load cushion was modeled as a rigid body and attached to the segment. A vertical load was applied to the segment in the form of uniformly distributed force through the load cushions. The finite element model was meshed with hexahedral elements, as shown in Fig.10(a). The reinforcement framework was modeled using truss elements, and only the longitudinal reinforcement and stirrups were considered. The tensile longitudinal reinforcement consisted of eight ϕ20 rebars and the compressive longitudinal reinforcement consisted of four ϕ16 and four ϕ20 rebars. The rebars were embedded in the concrete segment. This method does not consider if the structure at the embedded position of the rebars is empty or whether there is relative slip between the rebars and segment. Therefore, this method can ensure that the rebars and concrete bear the external forces together and jointly improve the bearing capacity of the segment. The reinforcement framework is illustrated in Fig.10(b).

3.2 Numerical parameters

The ideal elastic–plastic constitutive model was selected to describe the mechanical behavior of the reinforcement, and the elastoplastic damage constitutive model was applied to concrete [25–27]. The reinforcement grade in the segment was Q235 steel, with a yield strength of 235 MPa, elastic modulus of 206 GPa, and Poisson’s ratio of 0.3. According to the concrete constitutive relationship in the Code for Design of Concrete Structures GB 50010-2010 [28], the stress–strain relation models of concrete under uniaxial compression and tensile stress are shown in Fig.11.

The expression of the stress–strain relationship model of concrete under uniaxial compression is as follows:

(1)\sigma =\left(1-d_{\mathrm{c}}\right)E_{\mathrm{c}}\epsilon ,

where \sigma is the stress of C50 concrete; d_{\mathrm{c}} is the uniaxial compressive damage evolution parameter of concrete; E_{\mathrm{c}} represents the elastic modulus for C50 concrete, which is 34.5 GPa; \epsilon is the strain of C50 concrete.

(2)d_{\text{c}}=\{\begin{array}{ll}1-{\displaystyle \frac{{\rho }_{\text{c}}n}{n-1+{\left(\epsilon /{\epsilon }_{\text{c}}\right)}^n}},& \epsilon /{\epsilon }_{\text{c}}\le 1,\\ 1-{\displaystyle \frac{{\rho }_{\text{c}}}{{\alpha }_{\text{c}}{(\epsilon /{\epsilon }_{\text{c}}-1)}^2+\epsilon /{\epsilon }_{\text{c}}}},& \epsilon /{\epsilon }_{\text{c}}>1,\end{array}

where {\epsilon }_{\mathrm{c}} represents the peak compressive and the value can be taken as 1.68 × 10⁻³; {\alpha }_{\mathrm{c}} is the parameter value of the descending section of the uniaxial compressive stress–strain curve.

(3){\rho }_{\mathrm{c}}=\frac{f_{\mathrm{c}}^{\mathrm{s}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{c}}},n=\frac{E_{\mathrm{c}}{\epsilon }_{\mathrm{c}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{c}}-f_{\mathrm{c}}^{\mathrm{s}}},

where f_{\mathrm{c}}^{\mathrm{s}} represents the uniaxial compressive of the concrete and the value for C50 concrete can be taken as 32.4 MPa.

The expression for the stress−strain relationship model of concrete under uniaxial tensile stress is as follows:

(4)\sigma \text{=}\left(1-d_{\mathrm{t}}\right)E_{\mathrm{c}}\epsilon ,

where d_{\mathrm{t}} is the tensile damage evolution parameter of concrete.

(5)d_{\mathrm{t}}=\{\begin{array}{ll}1-{\rho }_{\mathrm{t}}\left[1.2-0.2{\left(\epsilon /{\epsilon }_{\mathrm{t}}\right)}^5\right],& \epsilon /{\epsilon }_{\mathrm{t}}\le 1,\\ 1-{\displaystyle \frac{{\rho }_{\mathrm{t}}}{{\alpha }_{\mathrm{t}}{(\epsilon /{\epsilon }_{\mathrm{t}}-1)}^{1.7}+\epsilon /{\epsilon }_{\mathrm{t}}}},& \epsilon /{\epsilon }_{\mathrm{t}}>1,\end{array}

where {\epsilon }_{\mathrm{t}} represents the peak tensile strain of concrete and the value can be taken as 7.50 × 10⁻⁵; {\alpha }_{\mathrm{t}} is the parameter value of the descending section of the tensile stress–strain curve.

(6){\rho }_t=\frac{f_{\mathrm{t}}^{\mathrm{s}}}{E_{\mathrm{c}}{\epsilon }_{\mathrm{t}}},

where f_{\mathrm{t}}^{\mathrm{s}} represents the uniaxial tensile strength of the concrete and the value of C50 concrete can be taken as 2.64 MPa.

3.3 Comparison of simulation and test results

3.3.1 Segment deflection

As shown in Fig.12, the relationships between the load and deflection in the simulation and test are the same, and all the load−deflection variation curves can be divided into four stages.

The deflection difference between the simulation and test is minimal in stages I, II, and III, but a significant difference appears in stage IV. This is because the finite element model of the segment existed as a continuum throughout the process, whereas the segment structure in the test showed strong discontinuity owing to the development of cracks. The width of the cracks in stages I to III is diminutive; therefore, the error between the continuum and non-continuum is not apparent. After entering stage IV, the crack propagates rapidly, and the error between the simulation and the test is larger and larger.

Although there is a significant deflection difference between the simulation and test curves in stage IV, the segment reaches the bearing capacity limit state at the beginning of this stage; hence, the subsequent failure characteristics have little reference significance for the analysis of the bending bearing capacity. Therefore, the numerical model can accurately simulate the bending performance of a segment.

3.3.2 Final failure state

Fig.13(a) shows the final failure states of the inner arc surface. The number and distribution of the through cracks in the test and simulation were the same. There was one crack in the middle of the segment with a slight degree of cracking, and three on each side. The four cracks near the middle exhibited the most significant degree of cracking.

The final failure states of the side are shown in Fig.13(b), and the depths of the cracks in the test and simulation were the same. The maximum depths of the cracks were located around the sealing ring, and the four cracks on the inner side have a more considerable cracking degree.

Based on the above analysis, the final failure states of the test and simulation are similar, indicating that the numerical model can accurately simulate the bending failure characteristics of the segment.

3.3.3 Characteristic load

During the test, the initial load of each stage was defined as the characteristic load. In the simulation, the characteristic load of each stage should be determined as an important parameter that reflects the bending capacity of the segment.

(1) Cracking load

In the test, the cracking of the concrete on the inner arc surface indicated that the segment had reached the cracking load, and the deflection started to increase. In the numerical simulation, the cracking of the segment was determined based on the magnitude of tensile strain in the concrete. The ultimate tensile strain of the C50 concrete model was 7.5 × 10⁻⁵. When the segment deflection started to increase and the equivalent plastic tensile strain was greater than 7.5 × 10⁻⁵. Therefore, it can be considered that the segment reached the cracking load.

(2) Failure load

The segment reached the failure load when all the through cracks of the inner arc surface were formed, and the deflection increased rapidly. Based on the load−deflection curves shown in Fig.12 and the equivalent plastic strain nephogram, the failure load value of the finite element model can be obtained. From the inversion of the equivalent plastic strain results corresponding to the failure load value, it is concluded that when the equivalent plastic tensile strain is greater than 5.0 × 10⁻³ in the numerical simulation, the segment reached the failure load.

(3) Ultimate load

The test results showed that when the segment reached the ultimate load, the maximum crack width of the inner arc concrete was more than 2 mm, and the deflection started to increase sharply. From the load−deflection curve shown in Fig.12 and the equivalent plastic strain nephogram, the ultimate load value of the numerical model can be obtained. Based on the inversion of the equivalent plastic strain results corresponding to the ultimate load value, it is concluded that when the equivalent plastic tensile strain is greater than 1.1 × 10⁻² in the numerical simulation, the segment reached the ultimate load.

Based on the test and simulation results, a method for determining the characteristic loads in a numerical simulation was proposed. The method errors were validated and were all within 8.5%, as shown in Tab.3.

4 Parametric analysis

The arrangement of the segment reinforcement is an essential factor that affects the bending bearing capacity of the segment [14,29]. Shield segments generally have material waste generated by an excessive reinforcement ratio. Meanwhile, if the proportion of tensile longitudinal reinforcement in the principal reinforcement is too large, the final failure form of the segment becomes an over-reinforced failure, which makes the segment unable to fully exert the design strength [3,13]. Therefore, it is necessary to clarify the influence of the reinforcement ratio and distribution of the tensile–compressive longitudinal reinforcement on the bearing capacity.

To approach the evaluation criteria of the actual design to the maximum extent, an equivalent plastic strain nephogram was used to determine the crack location and relative cracking degree in the simulation. First, the judgment standard of the characteristic load value was determined according to the deflection curves and equivalent plastic strain nephogram, as described in Subsubsection 3.3.3. Subsequently, the characteristic load values under different working conditions were determined by the criterion, and the influence of different reinforcement layout schemes was evaluated by the characteristic load values [30,31]. Therefore, in the numerical simulation, the characteristic load was another manifestation of the crack width.

To perform the parametric analysis better, only the principal reinforcement (tensile and compressive longitudinal reinforcement) was considered. The reinforcement ratio adopted is the ratio of the principal reinforcement area to the cross-sectional area. The proportion of the tensile and compressive longitudinal reinforcements represents the proportional relationship between the principal reinforcements. Because the sum was 100%, the proportion of the tensile longitudinal reinforcement was selected as the variable of the working conditions.

In summary, the research objectives of the parametric analysis are the reinforcement ratio and proportion of tensile longitudinal reinforcement. The reinforcement ratio of the test segment was 1.1%, and the proportion of tensile longitudinal reinforcement was 55%.

4.1 Reinforcement ratio

An investigation of shield segments in many cities revealed that the reinforcement ratio was typically between 1.0% and 1.6% [28,32,33]. A control variable method was adopted to explore the influence of the reinforcement ratio on the bearing capacity. The other parameters were consistent with the test, and the reinforcement ratio was set as a variable for the 12 working conditions. The reinforcement designs for the different working conditions are listed in Tab.4.

The variation relationships between each characteristic load and reinforcement ratio based on the determination method of the characteristic loads of the simulation are shown in Fig.14.

4.1.1 Cracking load

Fig.14(a) shows the relationship between the cracking load and reinforcement ratio, with a positive correlation between them. However, the change in the load value was not apparent, indicating that an increase in the reinforcement ratio does not effectively enhance the crack resistance of the inner camber.

4.1.2 Failure load

When the segment reached the failure load, the failure form was such that the inner arc surface concrete was cracked under tension, and the outer arc surface had no apparent plastic failure. Therefore, the effect of the reinforcement ratio on the failure load was mainly reflected in the tensile performance of the inner arc surface. Fig.14(b) shows the variation relationship between the failure load and reinforcement ratio, and the percentage difference in the figure represents the change ratio of the load value in the latter working condition compared to that in the previous working condition. The variation curve of the failure load can be divided into two phases.

Phase one (0.7% ≤ ratio < 1.2%): The failure load increased sharply with a total load increase of 203 kN, and the difference percentage maintained a considerable value. The change in the reinforcement ratio significantly enhanced the tensile performance of the inner arc surface and rapidly increased the failure load of the segment.

Phase two (1.2% ≤ ratio ≤ 1.8%): The growth of the failure load slowed down, and the percentage difference fluctuated around 1%. At this stage, the tensile strength of the inner arc surface reached its limit, and the increase in the reinforcement ratio could no longer enhance the bearing capacity of the inner arc surface; thus, the failure load had almost no change.

4.1.3 Ultimate load

When the segment reached the ultimate load, the failure form showed that the inner arc surface had a significant degree of cracking, and a wide range of plasticity appeared on the outer arc surface; therefore, the influence of the reinforcement ratio was mainly reflected in the bearing performance of the inner and outer arc surfaces. As shown in Fig.14(c), the change curve of the ultimate load can be divided into three phases.

Phase one (0.7% ≤ ratio < 1.0%): The ultimate load of the segment increased rapidly with a total load increase of 105 kN, and the difference percentage maintained a considerable value. With an increase in the reinforcement ratio, the tensile and compressive properties of the inner and outer cambered surfaces were greatly enhanced, and the ultimate load rapidly increases.

Phase two (1.0% ≤ ratio < 1.2%): The growth of the ultimate load tends to be horizontal, indicating that the total bearing capacity of the inner and outer cambered surfaces was not enhanced at this stage, and the increase in the reinforcement ratio cannot effectively improve the ultimate load of the segment.

Phase three (1.2% ≤ ratio ≤ 1.8%): The ultimate load showed a rapid linear growth trend, while the difference percentage was generally smaller than that in the first stage, and the bearing capacity of the inner and outer arc surfaces gradually reached the limit.

In summary, the change in the reinforcement ratio had little effect on the cracking load but significantly affected the failure and ultimate loads. From 0.7% to 1.2%, the failure load of the segment increased sharply, with a total increase of 203 kN. From 0.7% to 1.0%, the ultimate load increased rapidly, with a total increase of 105 kN. For the shield segment of a subway, the normal service limit state is generally used as the criterion for judging the bearing performance. Therefore, it is recommended that the reinforcement ratio of the segment be selected within the range of 0.7%–1.2%.

4.2 Proportion of tensile longitudinal reinforcement

Statistics on numerous subway projects have showed that the proportion of tensile longitudinal reinforcement was typically between 50% and 60% [28,32]. To explore the influence of the proportion of the tensile longitudinal reinforcement on the bearing capacity, the control variable method was adopted. The other parameters were consistent with the test, and the proportion of tensile longitudinal reinforcement was set as a variable for the 13 working conditions. The reinforcement designs for the different working conditions are listed in Tab.5.

Based on the characteristic load determination method of the simulation, the variation relationships between each characteristic load and the proportion of the tensile longitudinal reinforcement are shown in Fig.15.

4.2.1 Cracking load

The relationship between the cracking load and proportion of the tensile longitudinal reinforcement is shown in Fig.15(a). The cracking load changed very minimal and was approximately a straight line, indicating that an increase in the proportion of tensile longitudinal reinforcement had little effect on the crack resistance of the inner arc surface.

4.2.2 Failure load

At the failure load, the influence of the proportion of the tensile longitudinal reinforcement on the bearing capacity is mainly reflected in the tensile properties of the inner arc surface. As shown in Fig.15(b), the change curve of the failure load can be divided into three phases.

Phase one (49% ≤ proportion < 51%): The failure load of the segment increased significantly, and the difference percentage remained at a high level. At this stage, the tensile performance of the inner arc surface could be substantially improved by adding tensile reinforcement.

Phase two (51% ≤ proportion < 57%): The failure load showed a rapid growth trend. However, the difference percentage was generally smaller than that in the first phase, indicating that there was still ample space for improving the tensile performance of the inner arc surface, but the effect was not as significant as in the previous phase.

Phase three (57% ≤ proportion ≤ 65%): The growth of the failure load slowed down, indicating that the tensile properties of the inner arc surface had reached the limit, especially when the proportion of tensile longitudinal reinforcement reached 63%, and the tensile strength of the inner camber no longer increased.

4.2.3 Ultimate load

At the ultimate load, the effect of the proportion of the tensile longitudinal reinforcement was mainly reflected in the bearing properties of the inner and outer arc surfaces. The relationship between the ultimate load and proportion of tensile longitudinal reinforcement is shown in Fig.15(c), and the change curve of the ultimate load can be divided into four phases.

Phase one (49% ≤ proportion < 53%): The ultimate load of the segment increased slowly, and the difference percentage remained low. At this stage, the compressive performance of the outer camber concrete was firm, which limited the cracking of the inner camber concrete to a certain extent. Therefore, increasing the tensile reinforcement had little effect on the ultimate load.

Phase two (53% ≤ proportion < 55%): The variation in the tensile reinforcement sharply increased the ultimate load. Because the tensile and compressive reinforcement proportions were basically the same, the inner and outer arc surfaces bear the load cooperatively, significantly increasing the ultimate load.

Phase three (55% ≤ proportion < 57%): The ultimate load decreased significantly, indicating that an increase in the proportion of tensile longitudinal reinforcement had little effect on the improvement of the tensile capacity but dramatically reduced the compressive capacity of the outer arc surface.

Phase four (57% ≤ proportion ≤ 65%): The change in ultimate load tended to be stable, which showed that after the proportion of tensile longitudinal reinforcement increased to 57%, the compressive capacity of the outer arc surface was mainly provided by concrete, and the reduction in compressive reinforcement did not affect the compressive performance of the outer arc surface.

In summary, the change in the proportion of tensile longitudinal reinforcement had little effect on the cracking load but had a significant impact on the failure load and ultimate load. From 49% to 57%, the failure load of the segment increased rapidly by 11 kN for every 1% increase. From 53% to 55%, the ultimate load increased sharply by 48 kN for every 1% increase. However, when the proportion of tensile longitudinal reinforcement was more significant than 55%, the ultimate load tended to decrease. For the shield segment, the normal service limit state is typically used as the main criterion to judge the bearing performance, and the bearing capacity limit state is used as the secondary criterion. Therefore, it is suggested that the proportion of the tensile longitudinal reinforcement should be within the range of 49%–55%.

5 Conclusions

In this study, a full-scale bending failure test was conducted to investigate the bending failure performance of a shield tunnel segment. In addition, a series of numerical simulations was performed to conduct a comparative and parametric analysis, and the results can guide the optimization of the segment reinforcement design. The main conclusions are as follows.

1) The bending failure process of the segment can be divided into four stages: elastic stage, working stage with cracks, failure stage, and ultimate stage. The characteristic loads between each stage are the cracking load, failure load, and ultimate load.

2) The change in the reinforcement ratio had little effect on the cracking load but significantly affected the failure and ultimate loads. When the reinforcement ratio changed from 0.7% to 1.2%, the failure load increased sharply, with a total increase of 203 kN. When the reinforcement ratio changed from 0.7% to 1.0%, the ultimate load of the segment increased rapidly, with a total increase of 105 kN.

3) The change in the proportion of tensile longitudinal reinforcement had little influence on the cracking load but had a significant impact on the failure load and ultimate load. When the proportion of tensile longitudinal reinforcement varied from 49% to 57%, the failure load increased rapidly by 11 kN for every 1%. When the proportion of the tensile longitudinal reinforcement varied from 53% to 55%, the ultimate load of the segment increased sharply by 48 kN for every 1%. However, when the proportion of the tensile longitudinal reinforcement was more significant than 55%, the ultimate load started to decrease.

4) For the safety of subway operations, the normal service limit state (expressed by the failure load) of the segment is generally used as the main parameter to evaluate the bearing performance, and the bearing capacity limit state (expressed by the ultimate load) is used as the secondary parameter. Therefore, it is suggested that in the reinforcement design of the subway segment, the reinforcement ratio can be selected in the range of 0.7%–1.2%, and the proportion of tensile longitudinal reinforcement be selected in the range of 49%–55%, which allows the segment to effectively use the reinforcement and fully exert the design strength.

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