Introduction

To solve the issue of regional water resource imbalance, inter-basin water delivery becomes a major engineering measure. This promotes the application of tunnel boring machine (TBM) technology in the construction of long-distance hydraulic tunnels1,2,3,4. Some successful water conservancy projects in China include the Water Diversion Project from Yangtze River to Han River5, the Songhua River Water Diversion Project6, the Water Resources Allocation Project of Pearl River Delta7, the Central Area Water Diversion Project of Yunnan Province8. Another is the Egypt Suez Canal Water Conveyance Tunnel Project9. In the construction of shield tunnels, many uncertain risks are always faced because of the hidden and complex nature in geological condition, the size of tunnel, excavation method and supporting patterns10,11. Once these risks evolve into accidents, an enormous economic and even loss of life may be caused12,13. This highlights the significance to ensure the safety of the segment lining of shield tunnel. Therefore, it is crucial to clarify the stress state of segments if defects exist in segment lining during installation process.

To solve this problem, monitoring the loading effects with strain or deformation sensors is a direct method, however it is expensive14. Full-scale test on single segment can simulate TBM thrust on different zones of segment under various support conditions, accounting for the frequent situation of imperfect contact between the segments of two next rings and highlighting crack patterns associated with different loads15. Besides, the finite element analysis is an effective approach for complex non-rod structures16, although a semi-analytical solution for the ultimate bearing capacity of directly jointed lining of a tunnel was proposed to analyze the stress and deformation of the lining under different vertical loads17. In fact, the analytical results of finite element model (FEM) have been used for design of many concrete structures under multi static and dynamic actions, such as large-scale silo18, hollow-section bridge slab19, invert siphon20 and prestressed cylinder concrete pipe21. This encourages the application of finite element method in the design and the mechanical property analysis of shield tunnels. Combined with an in-situ test of the shield tunnel with a single-layer segment lining with internal diameter of 5.4 m for the China Water Resources Allocation Project of Pearl River Delta, a numerical analysis was conducted using a three-dimensional (3D) FEM with surrounding rock and segment lining, results showed that the segment lining structure exhibits a transverse ellipse with outward expansion, the opening of joints and the stress of locating pins increase with the internal water pressure, the capacity of sharing internal water pressure by surrounding rock can be enhanced with the improvement of mechanical properties of surrounding rock22. This provides a reference for the design of segment lining in condition that a rational percent of internal water pressure is shared by surrounding rock. A similar study was conducted to analyze the effect of surrounding rock on the load-bearing performance of segment lining for a deeply embedded hydraulic tunnel with high external water pressure, results showed that the external water pressure is a main factor affecting the design of segment lining23. A rock-grouting-segment integrated model was proposed for a shield tunnel supported using parallelogram segments with internal diameter of 4.6 m, results indicated that the stress and deformation of segment lining are smaller during the construction period, except for the stress concentration mainly occurs near the joints because of the driving force of jacks; the lining segments in a ring independently bear internal water pressure with a discontinuous displacement and tensile stress during operation period, which are connected only by locating pins24,25. Meanwhile, a parametric numerical study was developed for evaluating the improvement effectiveness of the performance in combination with high strength reinforced concrete and high-performance fiber reinforced concretes, aiming to reducing the lining thickness or the amount of reinforcement, and modifying the geometric configuration of longitudinal joints26. A study was carried out on the size effect of segmental tunnel linings on the overall structure stiffness and the damage evolution mechanism with an exploration of the number and location of the yielded bolts, results showed that the large-diameter lining rings have a relatively greater increasing rate of convergence deformation, fewer plastic hinges emerged in large-scale lining rings before the structures reached their ultimate strength, leading to more brittle failure characteristic of large-diameter than that of small-diameter one and a prone to become unstable geometry of large-size structures27. Moreover, a mechanical model of the segment under the constraints of elastic support was built considering the influence of joint displacements, results showed that the displacements of the segment joints cause an unloading effect of the corresponding internal force of the joints, leading to internal force redistribution of each segment28. Another study was concerned about building a completely 3D FEM of double shield TBM tunnel considering the complex interaction between the rock mass, the tunnel machine, its system components, and the tunnel support, to make design analyses of tunnels with spalling hazard or squeezing of weak surrounding rocks29.

As is known, the initial defects of segment lining are inevitable because of errors created in fabrication and assembly. Even if these defects are not harmful to the construction safety, an adverse effect may be produced on the safely operation of the shield tunnel, due to the changes of ground condition, surrounding excavation, or crossing by other tunnels30,31. By investigating the segment damages and FEM analyzing the loading properties of a tunnel, the main causes of the concrete cracks of segments include the high thrust of TBM jacks, the uneven segment ring surface, the eccentric compression on segment caused by jack gripper, the mismatching of TBM attitude to the curve section of segment lining, and the over pressure of grouting32. The crack distribution on 3D FEM segments were analyzed under conditions of jacking forces and relatively twist between connected segments in construction stage and those under different external forces in service stage, results indicated that the probability of crack and breakage can be effectively reduced by improving the crack resistance of concrete near hand holes and bolt holes33. A study of the effect of TBM attitude on inner force and dislocation of segments indicated that, the attitude deflection causes biggish dislocation between segments, the key segment is the most affected and weakest part in same ring which causes irregular displacement and dislocation in whole tunnel structure, the circumferential joint is in general much more affected than longitudinal joint under squeezing action of TBM, the curved bolt may be extruded bolt hole and the crack or breakage frequently concentrates in key segment and adjacent segment under the squeezing action if the segment dislocation exceeds the limit34. Comparatively, more FEM numerical analyses have been carried out on the segment construction under TBM thrust action, results indicated that some segments in certain conditions of TBM thrust may be cracked due to the tensile stress was over than the tensile strength of concrete, which required a countermeasure to strengthen the cracking resistance of segment itself35,36,37. This raises a topic to effectively improve the cracking and tensile resistance of segments using steel fiber reinforced concrete37,38,39.

However, except for the segment damage caused by above reasons, the misaligned defect of segments is common that induces tunnel diseases to influence the health of tunnels40. The statistical results obtained by Sharghi et al.41 and Cavalaro et al.42 concluded that the deviation between the lining ring assembly before the completion of tunnel construction is a main cause of tunnel damage. Another study believed that during the installation of tunnel lining, the main issues are misalignments and gaps between the segments43. The cracks and staggered displacement of tunnel segments come from the non-filling in time with peastone grouting after the installation of the segment lining, which led to outward or misalignment displacement of the lateral segment lining44. The studies on the structural bearing performance of segment lining with different eccentricity defects showed that the transverse horizontal elongation with misalignments is the most unfavorable working condition of segment lining, the initial misalignment significantly impacts on the segment stress, while the internal force of segment lining increases with the misalignment45,46. Meanwhile, a misalignment displacement caused by the buoyancy of synchronous grouting slurry exhibits a first upwards and then downwards after the segment detached from the shield machine47.

In summary, the 3D FEM is an effective method for the research of load-bearing performances of segment lining, considering the integrity of surrounding rock, grouting and segment lining. The load-bearing performances of segment lining have been studied in condition without defect or with some defects, and the causes and distribution of defects in segment lining of shield tunnel have been preliminarily identified. However, the impact of segment misalignment on the load-bearing performance of segment lining has not attach enough importance, especially no research concerns about the changes of the grouting caused by the segment misaligned defects. This may be more important for the shield tunnels with weak surrounding rocks29. Therefore, a refined 3D FEM composited with the segment lining, peastone grouting and surrounding rock was established in this study to completely simulate the conditions of misaligned segment lining in multi rings. To recognize the impact degree of the misalignment defect on the load-bearing performance of segment lining, an analysis is conducted combined a shield tunnel for the Anyang City's West Route of South-to-North Water Diversion Project of China10. The geological condition with weak surrounding rock is set as V-class rock. Two kinds of misaligned defects are considered, one is the misalignment elongating transverse horizontal axis of segment lining, another is the misalignment between adjacent segment rings. The circumferential stress of segments, the compressive contact stress on joint surface, and the tensile stress of locating pins are presented. The results are valuable for the guidance of quality acceptance of the shield tunnel in construction stage to ensure its reliable operation.

3D FEM for numerical analysis

Composition of segment lining

This study focuses on a tunnel of the Anyang City's West Route of South-to-North Water Diversion Project of China. According to the design document of this project, TBM technology is applied for the excavation passed through Class III, IV and V surrounding rocks. Peastone grouting is adopted after the installation of segment lining. The segments of shield tunnel are made of C50W8F100 concrete. Figure 1 shows the segment lining structure in a sectional view and an assembly diagram. The thickness of a single segment is 0.3 m. The segment lining is 3.5 m of inner diameter and 4.1 m of outer diameter. Each ring of the segment lining consists of four hexagonal honeycomb-shaped segments, and adjacent segments are connected by two locating pins in the ring joints. No protrusions are used on the joint surfaces of the ring and longitudinal joints of the segments. Figure 2 shows the assembly diagram of the segments. The segment lining uses a staggered assembly method, with each segment in width of 1.2 m. The longitudinal joints are connected using slots without locating pins.

Figure 1
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Segment structure diagram.

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Figure 2
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Segment assembly diagram.

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The 3D FEM of segment lining

With the aid of the finite element software of ANSYS 17.0 (ANSYS Inc. USA, www.ansys.com), the coordinates of the 3D FEM of segment lining were set as the x-axis for transverse horizontal direction of the tunnel section, the y-axis for transverse vertical direction of the tunnel section, and the z-axis for longitudinal direction along the tunnel. The boundary was determined with a distance from the lining no less than 8–10 times the tunnel diameter48. The entire 3D FEM was built with the original coordinate at the sectional center of the tunnel. The left and right vertical boundary and the bottom boundary were 44 m that was far from the center. The top boundary was the ground surface, which was selected at a poor geological condition with V-class rock, and the average burial depth is 87 m for the tunnel.

To build the 3D FEM, the mesh of surrounding rock was first divided to simulate the initial excavation to achieve a stress equilibrium. Then the peastone grouting, the segment and the locating pins were successively established. The normal constraints were set on the vertical boundaries, and the fixed constraints were set in all three directions at the bottom boundary. Due to the misalignment segment led to the misalignment displacement of adjacent segment, seven rings of segments were built for the 3D FEM, in which the middle ring was aimed to give out the analytical results. The segments, peastone grouting, and surrounding rocks were simulated using SOLID45 elements, while the locating pins were simulated using BEAM188 elements. To ensure the accuracy of analysis in complex geometries, all FEM meshes were divided using mapped hexahedral elements. The surrounding rock within a 4 m radius of the tunnel was more finely meshed. The misalignment defects of the segments are simulated by applying enforced displacement to the segments. The gaps generated by the enforced displacement are managed using element management options, ensuring the gaps filled with peastone grouting. The contact is established using TARGE170 target elements which describe the 3D target surfaces for contact with CONTA173 elements. CONTA173 contact elements are 4-node surface-to-surface contact elements which describe the contact and sliding conditions between the TARGE170 target elements and the deformable surfaces defined by these elements49. The built 3D FEM is shown in Fig. 3.

Figure 3
figure 3

Built 3D FEM for numerical analysis of shield tunnel.

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Focused on the impact of misalignment defects on segment lining, and considered the computational efficiency with a model complexity, the rebars in segment were uniformly smeared to increase the modulus of elasticity of concrete18,49. This can indirectly increase the tensile resistance of segment concrete. The connected grooves between segments in a ring are simplified to planar contacts with an applied friction coefficient to simulate the behavior of the contact surfaces. This helps to identify the mechanical response on the contact interfaces. The locating pins between segment rings are modeled using rod elements coupled with the segment elements to capture the contribution of the pins to overall structural stability. These simplifications were made based on a thorough consideration of the research objectives and key influencing factors.

Constitutive relationships

Given that the primary objective of this research is to analyze the structural response during the construction phase, and the stress levels are below the failure limit of the concrete, the linear elastic models of segment concrete and peastone grouting can provide a sufficient accuracy, while the constitutive relationship of locating pins was set as the elastoplastic. The main parameters are summarized in Table 1. The Mohr–Coulomb constitutive model was used for the surrounding rock with the main parameters shown in Table 2. The interfaces between segments and peastone grouting, and those between segments were set as a face-to-face contact, which was complied with the Coulomb's law of friction along the tangent direction. When the tangential stress reached a critical value, a slipping was created with a friction coefficient of 0.520,49. Meanwhile, a hard contact was applied to the normal direction of the interfaces, which was allowed to separate. The contact settings are shown in Fig. 4.

Table 1 Main physical and mechanical parameters of segment lining materials.
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Table 2 Main physical and mechanical parameters of surrounding rock.
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Figure 4
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Structural contact diagram.

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Working conditions

By consulting engineers and reviewing relevant quality acceptance reports23,41,46, two kinds of segment misalignment are commonly presented during the installation process, as shown in Fig. 5. The left one is the misalignment that elongates the transverse horizontal diameter of segment lining, the right one is the misalignment between adjacent rings of segments, due to the compression of top surrounding rock and the slurry-induced uplift.

Figure 5
figure 5

Defect types of the misalignment of segment lining.

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In the acceptance reports for these two types of segment misalignment, the maximum misalignment was found to reach 3% the tunnel diameter (D) and 30 mm, respectively. Due to the complexity and unpredictability in the operation of the tunnel, the segment misalignments may exceed these values. Therefore, to explore the stress variations in misaligned segment structures, the maximum segment misalignments are set as 4%D and 40 mm in this study. The surrounding rock of the shield tunnel was V-class. To consider the influence of different amounts of segment misalignment, the transverse horizontal eccentricity of segments was set to 0.5%D, 1%D, 2%D, 3%D, or 4%D. The continuous misalignment between adjacent rings was set to a total misalignment of 5 mm, 10 mm, 20 mm, 30 mm, or 40 mm. Meanwhile, a 3D FEM was built for the shield tunnel without segment defect, which is used for a basis of analysis. Therefore, eleven working conditions were considered as exhibited in Table 3.

Table 3 Working conditions of the shield tunnel with segment misaligned defects.
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This tunnel is a non-pressure water deliver tunnel with an operation water of 1.3 m deep. After comparison for the numerical results of the tunnel with or without water, the load-bearing behavior is worse for the tunnel without water. Therefore, the result analysis is concentrated on those conditions of the tunnel in maintenance period without water.

Analytical locations of the segment lining

When the misalignments of segments exist in the segment lining, the failure of segment lining will occur if the compressive stress is over the compressive strength of segment concrete, or the misalignment enlarges to break away from the contact of segments. Except that, cracks may be appeared due to the tensile stress is over the tensile strength of segment concrete16. Because the shield tunnel mainly bears the vertical load of surrounding rock, in order to reduce the influence of boundary effects on the analytical results, the middle ring of segment lining along the longitudinal direction of the 3D FEM is selected as the analytical object after the misaligned defects are exerted. The middle cross-section of the middle ring segment is selected to analyze the circumferential stress, in which 360 pairs of circumferential stresses are extracted from the middle cross-section at different angle. The joint surfaces of the misaligned segment are selected to analyze the contact performance, which are numbered 1 to 4 as shown in Fig. 6a). Meanwhile, the locating pins numbered 1 to 8 as shown in Fig. 6b), are selected to analyze the bearing performance of locating pins, which are pins for the locating of the middle segment ring of 3D FEM in this study.

Figure 6
figure 6

Analytical location of the 3D FEM.

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Results analysis and discussion

Segment lining with misaligned defects elongating transverse horizontal diameter

Circumferential stress of segment

The circumferential stresses in both outer and inner layers are selected at the midsection of the misaligned segment lining. The direction towards the center of the segment lining is defined as positive, while the direction away from the center is considered negative. Tensile stress is denoted as positive, and compressive stress as negative.

As illustrated in Fig. 7, the observed distribution of circumferential stress of segment without misaligned defect is consistent to the finding of previous studies22,25. This demonstrates the reliability of the FEM analytical results. With a relatively low strength of V-class surrounding rock, the pressure release due to excavation makes the surrounding rock into a post-peak softening stage with the compressive stress redistribution. This leads a concentrated pressure through the peastone grouting on the segment lining along vertical direction. As a result, the ring of segment lining has a tendency to shorten in vertical axis and elongate in horizontal axis. Therefore, the compressive stress is more pronounced at the outer layer of crown and bottom, and at the inner layer of hance, while the largest value at the bottom. Correspondingly, the tensile stress is pronounced at the inner layer of crown and bottom, and at the outer layer of hance. This creates a maximum circumferential tensile stress of 1.05 MPa, and a maximum circumferential compressive stress of 0.79 MPa.

Figure 7
figure 7

Circumferential stress analysis.

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From Fig. 7a), with the increase of segment misalignment that elongates the transverse horizontal axis of segment lining from 0.5%D to 3%D, the circumferential tensile stress of the inner layer and the circumferential compressive stress of the outer layer apparently increases at the bottom and the hance, and the variation are more pronounced at the bottom. This creates an increase of 8%, 12%, 20% and 23% in maximum tensile stress, and 14%, 11%, 33% and 34% in maximum compressive stress. A maximum circumferential tensile stress is 1.87 MPa, and a maximum circumferential compressive stress is 1.78 MPa, as shown in Fig. 7b), when the segment misalignment reaches 3%D. Generally, the stress variation is normal with the increase of segment misalignment for the segment lining under the pressure of surrounding rock. This reflects an entire loading bearing behavior of the ring of segment lining. In the worst case of segment misalignment at 3%D, the maximum tensile stress reaches 76% of the standard tensile strength of C50 concrete, which increases a risk of segment cracking. Therefore, with a much higher capacity to withstand compressive stress of segment C50 concrete, tensile stress becomes a crucial index affecting structural safety when a segment misaligned defect occurs.

However, when the segment misalignment reaches 4%D, a sharp increase of circumferential stress appears at the bottom segment, while the hances remain at a compression status. This shows that the transforming of load-bearing behavior from entire ring of segment lining to a single segment at bottom, because the entirety of segment lining is distracted by the large misalignment. In this case, the circumferential tensile and compressive stresses at the midpoint of bottom segment are 2.28 MPa and 2.77 MPa in maximum, respectively. This creates a risk of segment cracking on the bottom segment as shown in Fig. 8, due to the maximum tensile stress reaches 86% of the standard tensile strength of C50 concrete. Because the tensile stress increases with the magnitude of misalignment which elongates the transverse horizontal axis of segment ring, the inner surface of the bottom segment is identified as the weakest zone of the segment structure. This will lead to the cracks at this location, or even structural failure of the tunnel. Therefore, it is worth noting that under the 4%D misaligned defect, the local segment at bottom should be especially concerned about to prevent an abrupt fracture of segment lining.

Figure 8
figure 8

Segment failure tendency in condition of misalignment elongating transverse horizontal axis.

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Segment contact stress

Since the assembly of segment lining and the misaligned defect are symmetrical with the vertical axis, the contact stress on the middle ring joints of segment lining exhibits a same symmetry under different working conditions of 1 to 6. Therefore, the contact stresses on the joint surface 1 and 2 marked in Fig. 6a) are shown in Figs. 9 and 10, respectively.

Figure 9
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Contact stress on joint surface 1 under different working conditions.

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Figure 10
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Contact stress on joint surface 2 under different working conditions.

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In the working condition of segment without misaligned defect, a vertical load induced by V-class surrounding rock is exerted on the segment lining, which is transferred mainly in compression on the normally contacted segments. As a result, the joint surface 2 bears the pressure transferred from the joint surface 1. This makes the joint surfaces 1 and 2 primarily experience compression, with the compressive stress on joint surface 2 exceeding that on joint surface 1. The maximum compressive stresses on the joint surfaces 1 and 2 are 0.62 MPa and 0.78 MPa, respectively.

When the segment lining is arranged with a misaligned defect that varies from 0.5%D to 4.0%D, the adjacent segments are not aligned, resulting in uneven force on the joint surfaces. With a filling of peastone grouting to the outside misalignment part, a pressure is subjected by the outside part of upper segment, while it is subjected by the inside part of lower segment at misaligned joint surface 1. This leads to an incurve for the crown and bottom segments, and a necked-out for the segments on left and right sides. Therefore, the compressive stress on outside part of joint surface 1 while that on inside part of joint surface 2 increases with the amount of misaligned defect. At the misaligned defect of 4%D, the maximum compressive stress on joint surface 1 is 0.85 MPa, and that on joint surface 2 is 1.10 MPa. Because the compressive stress on the joint surfaces is far less than the compressive strength of C50 concrete, the misaligned defect within 4%D has a less influence on the bearing property of joint surfaces between segments.

Stress of locating pins

Figure 11 illustrates the tensile stresses of the eight locating pins on the middle ring segment lining under the six working conditions. Because the vertical pressure of surrounding rock is transmitted from the top segment to bottom segment through both sides of the segment, the maximum deformation happens on the crown segment, resulting in a maximum tensile stress of corresponded locating pins. However, the tensile stress is small in value that is only 1.10 MPa. With the increase of misaligned defect from 0.5%D to 4%D, the position of the maximum tensile stress of locating pins gradually shifts from the crown to the bottom. This coincides to the greater deformation of bottom segment due to the eccentric compression on the segment ring with the misaligned defect. As a result, the maximum tensile stress of 10.8 MPa happens on the locating pains 5 and 6 for the bottom segment. Since the tensile stress is far less than the tensile strength of steel pin, slight influence exists on the safety of locating pins.

Figure 11
figure 11

Stress of locating pins under different working conditions.

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Segment lining with misaligned defects between adjacent rings

Circumferential stress of segment

The maximum circumferential stresses of the outer and inner layer of the middle section of segment lining under V-class surrounding rock at working conditions 1 and 7–11 are shown in Fig. 12. With the increase of misalignment between adjacent rings of segments, the convex-in segments at arch crown subject to the upper pressure of surrounding rock with a maximum sedimental deformation from excavation. This is pronounced with a stress variation of the crown segments. Meanwhile, the stress variation of the bottom segments also becomes obvious. The maximum tensile stress of segments at different working conditions has an increase of 5%, 7%, 20%, 35%, and 30%. Simultaneously, the maximum compressive stress of segments has an increase of 33%, 12%, 36%, 38%, and 33%. When the misalignment reaches 40 mm between adjacent rings of segments, the segment lining experiences maximum stress. The maximum tensile stress on the segment lining reaches 2.49 MPa, while the maximum compressive stress reaches 2.95 MPa. This makes the tensile stress reach to 94% of the standard tensile strength of C50 concrete, which may create cracks on the crown segments.

Figure 12
figure 12

Circumferential stress analysis.

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Extracting the circumferential stresses of segment rings at middle points of crown along the axial of shield tunnel, a stress path analysis is made to exhibit the impact of continuous misalignment between segment rings. The circumferential stress distribution of both the outer and inner layers of the crown segments is depicted in Fig. 13. With the accumulated misalignment between adjacent rings, the fourth ring of segment lining, that is the middle ring in the 3D FEM, has the largest defect among all the segment rings, which subjects a greatest pressure of surrounding rock compared with its adjacent segment ring. Consequently, it experiences the maximum stress. With a 30 mm misaligned defect, the tensile stress on the middle segment ring is close to the standard tensile strength of C50 concrete. with a 40 mm misaligned defect, the tensile stresses on both middle segment ring and two adjacent segment rings are close to the standard tensile strength of C50 concrete. This indicates that a potential risk of cracking on inner surface of crown segments raises with the increase of misaligned defect between adjacent rings. As shown in Fig. 14, as the tensile stress increases with the magnitude of misalignment between adjacent rings, the middle zone of inner surface of the crown segment is identified to be a weakest location which will be cracked even fracture in the segment structure.

Figure 13
figure 13

Stress path analysis of the arch crown segments.

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Figure 14
figure 14

Segment failure tendency in condition of misalignment between adjacent rings.

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It can be observed that compared with the misaligned defect elongating transverse horizontal axis of segment lining, the misaligned defect between adjacent rings of segments has a greater impact on the circumferential stress of segment lining. Notably, for the two kinds of misaligned defects, tensile stress remains a prominent indicator that has apparent impact on the safety of segment lining.

Segment contact stress

The contact stress distributions on joint surface 1 and 2 of middle segment ring with misaligned defect between adjacent rings under different working conditions 1 and 7–11 are illustrated in Figs. 15 and 16.

Figure 15
figure 15

Contact stress on joint surface 1 under different working conditions.

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Figure 16
figure 16

Contact stress on joint surface 2 under different working conditions.

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In the presence of continuous misalignment between adjacent segment rings, the crown of the misaligned rings bears a larger pressure of surrounding rock. With the pressure transmission through the joint surfaces, the compressive stress occurs on the joint surfaces. Compared figures between Figs. 11 and 12, the pressure on joint surface 1 is more pronounced than that on joint surface 2 with the increase of misaligned defect. With the maximum misaligned defect, the maximum compressive stress on joint surface 1 is 1.00 MPa, while that on joint surface 2 is 0.91 MPa. This is unlike the maximum compressive stress appeared on joint surface 2 with a misaligned defect elongating transverse horizontal axis. Whatever, under all working conditions in this study, the contact compressive stress is less than the compressive strength of C50 concrete.

Stress of locating pins

Figure 17 exhibits the tensile stresses of the eight locating pins of the middle segment ring with different misaligned defects between adjacent segment rings. Corresponded to a larger pressure that is subjected by the crown segment, a higher tensile stress occurs on the locating pins of this segment. This indicates a larger deformation between the middle segment ring and the adjacent segment rings. With the increase of misaligned defect between adjacent segment rings, the thickness of the peastone grouting at the crown gradually decreases, resulting that the crown segment becomes the primary element bearing the rock pressure. With a 40 mm misaligned defect, the maximum stress of the locating pins occurs at positions 1 and 2, reaching to 13.9 MPa.

Figure 17
figure 17

Stress of locating pins under different working conditions.

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Conclusions

A refined three-dimensional finite element model composited with the segment lining, peastone grouting and surrounding rock was established in this study to completely simulate the conditions of misaligned segment lining in multi rings for a hydraulic project. Based on the numerical analysis of the load-bearing behaviors of segment lining with misaligned defects elongating the transverse horizontal axis or occurring between adjacent rings for a hydraulic shield tunnel, conclusions can be drawn as follows:

  • The misaligned defects elongating the transverse horizontal axis creates an obvious impact on the circumferential stress of bottom segment, while the misaligned defect between adjacent rings of segments creates a pronounced impact on the circumferential stress of crown segment. With a 3%D misaligned defect elongating the transverse horizontal axis, the tensile stress on the inner layer of the bottom segment reaches 76% of the standard tensile strength of C50 concrete. With a 40 mm misaligned defect between adjacent rings of segments, the tensile stress on the inner side of the crown segment reaches 94% of the standard tensile strength of C50 concrete. This raises a similar risk of internal surface cracking of segment in the opposite location at bottom to crown. It is worth noting that when the segment misalignment elongating the transverse horizontal axis reaches 4%D, a sharp increase of circumferential stress appears at the bottom segment with a single incurve with a circumferential tensile stress of 2.28 MP.

  • The two kinds of misaligned defects of segments induce the increase of contact stress on the joint surface between adjacent segments. The misaligned defect elongating the transverse horizontal axis creates a greater contact stress on the lower joint surface, while the misaligned defect between adjacent rings produces a greater contact stress on the upper joint surface. However, the contact stresses are smaller compared to the strength of C50 concrete. This raises a slight impact on the safety of segment lining.

  • Corresponded to the increase of circumferential tensile stress of segments with the two kinds of misaligned defects, the locating pins for segments at bottom and crown separately bear an increased tensile stress. However, the tensile stresses are smaller than the tensile strength of steel. This leads a slight impact on the safety of segment lining.

  • This study recognized the crown or bottom segment failure tendency with concrete cracking from two misalignment defects during segment installation stage. The failure risk of segment lining raised by other misalignment should be further studied to predict the biggish influencing factor to ensure a safe operation of the tunnel.