Influence of the Diameter Size on the Deformation and Failure Mechanism of Shield Precast Segmental Tunnel Lining under the Same Burial Depth

: With the development of large-diameter shield tunnels, how to realize effective security and stability control of shield tunnel lining has become a significant research topic. This paper investigates the deformation and failure mechanism of lining large diameter shield tunnels in depth and discusses the deformation characteristics and influencing factors of the lining of the shield tunnel with various diameters through the software of finite element analysis ABACUS. A set of models with varying diameters is built under identical stress conditions in order to maintain control over the variable. The utilization of the elastic–plastic model is observed in the application of bolts and rebar. The utilization of the Concrete Damage Plasticity model has been taken into account for the concrete lining. For the sake of comparison, the crown displacement of the shield tunnel, strain in tension and compressive zones, bolt stress and strain, deformation and intemal force distribution around the shield tunnel, and cracks in the tension zone, are carefully studied. An in-depth analysis is conducted to elucidate the variations in damage evolution mechanisms across linings of different sizes, within the framework of plastic hinge theory. The results indicate that the convergence deformation of large-diameter tunnel lining increases significantly during loading compared with that of small-diameter tunnel. Moreover, the probability of brittle failure is higher in big-diameter shield tunnels compared to small-diameter tunnels, indicating that these larger tunnel structures are more prone to suffering geometric instability.


Introduction
The tunnel shield machine has gained significant popularity in long tunnel construction due to its versatility and ability to ensure safety in various construction situations.The shield tunnel plays a crucial role in accommodating many types of pressures during the building phase, including soil pressure from the surrounding environment, water pressure, and potential local stresses resulting from future activities in the operational stage.The utilization of segmented prefabricated lining in tunnel excavation is prevalent due to its convenient assembly throughout the building process and effective quality control measures.Nevertheless, the segmented tunnel also presents numerous drawbacks.In practical engineering, the connection of the lining section may encounter challenges like as water leakage and stress concentration.The fragility of the junction in the segmented lining renders it crucial in determining the overall strength of the lining, so establishing a significant relationship with the bolts of the node.To address the issue of overcrowded urban areas in certain Chinese cities, there has been a gradual increase in the dimensions of tunnels, hence maximizing the utilization of subterranean space [1][2][3].
Studying the deformation and stress on shield segments during shield compaction is scientifically important and has practical relevance due to the following reasons: (1) Studying the deformation and stress of shield lining and panels provides a theoretical foundation for shield tunneling construction, improving understanding.This framework guides the design, implementation, and quality control of shield tunneling construction during the construction phase to ensure the safety, effectiveness, and stability of shield tunneling operations.(2) Improving tunnel engineering quality: Studying the deformation and stress of shield lining and panels helps enhance tunnel engineering quality by optimizing design, improving construction processes, and selecting more resilient materials.Studying the deformation and stress of shield lining and panels helps reduce project costs and improve project results.Studying the deformation and stress of shield lining and panels is an important area of research in tunnel technology, helping to advance the field.This investigation has the potential to enhance the development and creativity of tunnel technology, providing future tunnel projects with advanced technologies and approaches [4][5][6].
Numerical simulation, model testing, and theoretical analysis are three widely employed research methodologies.In summary, numerical simulation, model testing, and theoretical analysis methodologies each possess distinct characteristics.Numerical simulation is capable of replicating the behavior of physical structures in diverse operational scenarios, albeit necessitating computational resources.The model test has the capability of replicating and evaluating the physical structure; however, it is vital to construct the matching model for this purpose.The utilization of the theoretical analysis method enables the acquisition of precise performance indices for the structure.However, this approach necessitates a profound understanding of mathematics and mechanical theories.In the context of practical engineering applications, it is common for all three methodologies to be mutually supportive and employed in conjunction.
(1) Numerical simulation: Li et al. (2023) [7] introduced an innovative precast segmental lining (PSL) for tunnel engineering, including both physical and numerical modeling techniques.The study incorporates three segmentation principles and constructs a threedimensional finite element model.The study examines the impact of joint thickness and stiffener on the performance of Parallel Strand Lumber (PSL).The findings indicate that cracks primarily manifest at the interfaces between blocks, and the use of the maximum moment position principle approach is deemed ideal for minimizing deformation and bending moments.The study conducted by Huang et al. (2020) [8] focused on the assessment of shield tunnel lining deterioration in Tianjin Metro Line 1 as a result of soil-water influx.The occurrence was a consequence of inadequate waterproofing during the drilling process, leading to the formation of a vulnerable area susceptible to seepage.Despite concerted efforts to stabilize the tunnel, the shifting of the soil resulted in significant damage.This study employs numerical simulation and analysis of field monitoring data to derive insights that can inform future shield tunneling projects conducted under comparable engineering conditions.Li et al. (2022) [9] employed a three-dimensional numerical model to examine the mechanical characteristics of a shield tunnel as it traverses an active ground fissure in Xi'an, China.The lining construction of the tunnel was subjected to both tension and compression, resulting in deformations and separation of the surrounding soil.The tunnel undergoes significant shear force, segment displacement, and fissures in the lining.The study proposes that the use of structural reinforcing measures, such as the incorporation of specialized segments and the use of bolts, is necessary in order to effectively reduce the associated risk.Li et al. (2023) [10] used an enhanced finite element model to examine the damage characteristics of tunnels that were reinforced with steel plates.The experimental evaluation of the model was performed on a stagger-jointed shield tunnel, and further simulations were undertaken to analyze the effects of deformation on the post-reinforcing lining.The findings of the study indicate that the primary source of bond damage in the vicinity of the crown is predominantly attributed to tension and shear forces, but in the vicinity of the waist, the primary cause of bond damage is primarily attributed to shear forces.Xu et al. (2019) [11] examined the tunnel connecting Shangteng and Dadao stations on Fuzhou metro line one.This tunnel traverses two-layered composite strata.The findings of the study indicate that longitudinal and corner fractures are the primary forms of cracking observed, and their patterns are influenced by variations in jacking thrusts.Cracks have an impact on the distribution of internal forces along the perimeter of a tunnel, exhibiting three distinct levels of influence: significant, moderate, and minimal.The failure process of the lining can be divided into four distinct stages, characterized by the development of pre-existing cracks in the longitudinal direction.
(2) Model testing: Zhang et al. (2022) [12] examined instances of urban ground collapses and tunnel failures resulting from the joint leakage of shield tunnel liners.The prediction of sand layer settlement and water and sand loss is accomplished by the utilization of model testing and particle flow simulations.The study, moreover, formulates a seepage tracer apparatus and utilizes a numerical model based on particle flow to comprehend the intricate consequences of sand particle migration and depletion at a small scale.Zhang et al. (2023) [13] examined the impact of loading and unloading on the coupling effect of segmental shield tunnel linings in densely populated urban regions.The results of physical model experiments indicate that the reaction of tunnels is considerably increased when subjected to coupling situations.The proposed coupling impact index serves as a means to assess the degree of coupling between various disturbances, with a notable emphasis on the loading phase preceding the unloading phase.Gou et al. (2023) [14] examined the viability of utilizing a shield tunnel as a means of traversing ground fissures in Xi'an, China.The findings from the model test and numerical simulation demonstrate that the tunnel experiences many forms of deformation, such as axial bending, localized horizontal torsion, and axial tension-compression.The lining undergoes biased pressure and exhibits cracking in the vicinity of fasteners.Deformation and failure have significant implications for the hanging wall and footwall rings, necessitating the implementation of reinforcement measures.The study additionally presents potential strategies and recommendations aimed at alleviating the adverse consequences associated with ground cracks.Zhu et al. (2019) [15] performed prototype structural loading experiments on a shield tunnel lining with a unique shape.The results of their investigation demonstrated that as the burial depth grew, there was a corresponding increase in both internal forces and deformation.The locations of the highest negative bending moments in the lining were identified to be near the shoulder portions.The design characteristics, namely the effective transverse rigidity ratio and rotational stiffness, underwent evolution as a result of loading.Cracks formed at various stages of stress, exhibiting varying diameters throughout time.The cracks were primarily localized on the segmental bodies within the rings that were near the longitudinal joints.Liang et al. (2023) [16] employed a plastic hinge-based methodology to investigate the failure causes associated with underwater shield tunnel lining structures.The definition of joint and segmental plastic hinges is established by analyzing failure patterns, while their placements are determined by displacement requirements.The results obtained from comprehensive testing conducted on two distinct types of structures demonstrate that the manner in which the components are assembled has a substantial impact on the observed failure patterns.This study highlights the significance of incorporating assembly methodology and the selection of plastic hinge type in the design of underwater shield tunnel lining structures.Allawi et al. (2019) [17] investigated the structural behavior of precast concrete segmental bridges using post-tensioned segmental girders with different joint configurations.The results showed slight differences in behavior across the four types of girders.Cracks were observed in concrete adjacent to the epoxy mortar, attributed to the high tensile strength of the epoxy compared to concrete.Ibrahim et al. (2023) [18] investigated the flexural performance of reinforced concrete specimens with encased GFRP beams, focusing on the effect of shear studs on composite interaction.Results show that the embedded GFRP beam enhances peak loads by 65% and 51%, respectively.A non-linear finite element model was developed to validate these findings, showing increased peak loads for different compressive strengths.
(3) Theoretical analysis: Yan et al. (2023) [19] put up a novel approach to detect structural deterioration in shield tunnel lining.This method involves the utilization of back analysis and Partial Least Squares Regression.The proposed methodology aims to establish a correlation between mechanical factors and structural response by employing a finite element model.The methodology utilizes monitoring data to ascertain the location of load and damage, so effectively addressing concerns related to randomization.The accuracy of the approach is showcased across various damage situations, wherein circumferential strain exhibits superior sensitivity compared to radial displacement.Zhang et al. (2021) [20] introduced a mathematical solution that is analytically derived to estimate the settlements caused by ground consolidation due to train loading.This solution takes into account the effects of lining leakage and the rheological properties of the medium and is based on the Terzaghi-Rendulic theory.The proposed solution utilizes semi-permeable boundary conditions, triangular cyclic loading, and the Boltzmann viscoelastic model in order to accurately replicate the geological properties of interest.The dissipation solution for the excess pore water pressure and ground reactions in viscoelastic porous clays is obtained by the utilization of conformal mapping techniques.The validity of the analytical solutions is assessed by the examination of their agreement with both in situ observable data and numerical simulation findings.The present study additionally examines the impact of several parameters on surface consolidation settlements.Huang and Zhang (2016) [21] introduced a conceptual framework aimed at evaluating the resilience of shield tunnels.The primary objective of their model was to analyze the robustness of performance and the speed of recovery.The validity of the model is demonstrated through its application to a real-life scenario involving an extreme surcharge event on the Shanghai metro tunnel.During this event, the performance of the tunnel was significantly interrupted, with a reduction of 70% to 80% in normal functioning.However, the recovery rate observed was only 1%.
The deformation and force distribution laws of shield tunnel lining slabs are significantly influenced by the diameter size [22][23][24][25].The following are a few key factors that exert influence.The radial distribution of a pipe slab becomes more uniform as its diameter increases.This implies that the pipe slab experiences more consistent pressures and torques along its radial axis, leading to a reduction in deformation and an enhancement in its stability.The regularity of axial distribution in a pipe slab is significantly influenced by the diameter of the slab.In instances when the diameter of a pipe slab is very large, it is possible for the axial distribution to exhibit non-uniformity, leading to significant deformations in regions experiencing higher axial forces over the operational lifespan of the pipe slab.The connection between pipe slabs plays a crucial role in determining the distribution of deformation and forces within the pipe slab.Overall, the diameter of the pipe sheet plays a crucial role in influencing the deformation and force distribution characteristics of the shield bunker lining pipe sheet.In order to enhance the stability and safety of shield construction, it is imperative to employ a reasonable approach in selecting the diameter of the pipe sheet based on the unique engineering requirements.Additionally, it is crucial to implement appropriate design and construction procedures accordingly.
In general, there is a significant difference in deformation and force distribution regularities between large-diameter shield tunneling lining and small-diameter shield tunneling lining.However, previous similar studies have only looked at tunnels of specific diameters (whether small or large).This paper investigates the deformation and failure mechanism of lining large-diameter shield tunnels in depth, and discusses the deformation characteristics and influencing factors of the lining of the shield tunnel with various diameters through finite element analysis using ABACUS software.A set of models with varying diameters is built under identical stress conditions in order to maintain control over the variable.The utilization of the elastic-plastic model is observed in the application of bolts and rebar.The utilization of the Concrete Damage Plasticity model has been taken into account for the concrete lining.For the sake of comparison, the crown displacement of the shield tunnel, strain in tension and compressive zones, bolt stress and strain, deformation and intemal force distribution around the shield tunnel, and cracks in the tension zone, are carefully studied.An in-depth analysis is conducted to elucidate the variations in damage evolution mechanisms across linings of different sizes, within the framework of plastic hinge theory.

3D Finite Element Model for the Shield Tunnel Lining Structure
The model studied in this paper refers to the common structural dimensions of subway tunnel lining in Shenzhen, China, and uses ABACUS software (2021 edition) for finite element modeling.Figure 1 shows the cross-section diagram of the shield precast segmental tunnel lining.Table 1 is a summary of the general structural dimensions of subway tunnel lining, including thickness, diameter, block, bolt number, etc.A set of models with varying diameters is built under identical stress conditions in order to maintain control over the variable.
Buildings 2024, 14, x FOR PEER REVIEW 5 of 19 with varying diameters is built under identical stress conditions in order to maintain control over the variable.The utilization of the elastic-plastic model is observed in the application of bolts and rebar.The utilization of the Concrete Damage Plasticity model has been taken into account for the concrete lining.For the sake of comparison, the crown displacement of the shield tunnel, strain in tension and compressive zones, bolt stress and strain, deformation and intemal force distribution around the shield tunnel, and cracks in the tension zone, are carefully studied.An in-depth analysis is conducted to elucidate the variations in damage evolution mechanisms across linings of different sizes, within the framework of plastic hinge theory.

3D Finite Element Model for the Shield Tunnel Lining Structure
The model studied in this paper refers to the common structural dimensions of subway tunnel lining in Shenzhen, China, and uses ABACUS software (2021 edition) for finite element modeling.Figure 1 shows the cross-section diagram of the shield precast segmental tunnel lining.Table 1 is a summary of the general structural dimensions of subway tunnel lining, including thickness, diameter, block, bolt number, etc.A set of models with varying diameters is built under identical stress conditions in order to maintain control over the variable.

Model Parameters and Boundary Conditions
The structure of the reinforced liner comprises distinct segments, steel reinforcement bars, and interconnecting bolts.Hence, it is imperative to establish definitions for three

Model Parameters and Boundary Conditions
The structure of the reinforced liner comprises distinct segments, steel reinforcement bars, and interconnecting bolts.Hence, it is imperative to establish definitions for three distinct types of material.The elastoplastic constitutive model was utilized for the reinforcement and bolts, whilst the Concrete Damaged Plasticity model, which effectively characterizes the irreversible damage in the concrete segment, was employed for the segmental lining.The simulated portions of the lining are composed of C55 concrete, and their behavior in terms of compressive damage and tensile damage may be shown in Table 2.The utilization of HPB335, a material possessing both high strength and favorable plasticity, as a reinforcing component in tunnel linings is a common practice.This material exhibits a yield strength of 335 MPa.The connecting bolts utilized in this system are M30 high-strength prestressed bolts, which possess the ability to experience deformation when subjected to tension.The segment has a thickness of 350 mm and a width of 1200 mm.The nuts and screws of the bolts have sizes of 50 mm and 30 mm, respectively, and lengths of 30 mm and 400 mm, respectively.To simplify the modeling process, we exclude intricate features such as sealing grooves and plugging grooves, as they have minimal impact on the response characteristics of the lining structure.Table 3 provides the specifications of steel bars and bolts utilized in the calculations.These specifications are determined based on prior publications and the standards established in China.Figure 2 shows the loading system, and the loading system is set to be consistent in all working conditions.The change of load is divided into two stages.The first stage is the earth stress stage, which simulates the stress state completely dominated by soil pressure.The second stage is the additional load stage, which mainly simulates the additional load that appears in the future service stage.Two loading scenarios were hypothesized in order to replicate the ground pressure during Stage I and the overloading condition during Stage II, as depicted in Figure 2b.Corresponding to the actual state of the project, Stage I represents the stress state under normal buried depth load, and Stage II renders the response of internal force and deformation under overload conditions after the load gradually increases.In the subsequent analysis, it is observed that the loading process for Stage I of shield segments with varying diameters remains consistent.This implies that the tunnel center, with variable diameters as examined in this study, is situated at an identical burial depth (H = 18.175 m).There are three types of loads that are taken into account in both stages.Additionally, a total of 24 loads are evenly distributed around the outside surface of segments, with each load being placed at 15 • intervals.In order to mitigate the concentration of stress, the concentrated loads applied at the loading sites are distributed to the nearest surfaces of the two columns by the continuum-distributing coupling mechanism.This mechanism alone facilitates the transmission of the load without any further effects.This paper focuses on the influence of diameter size on the deformation and failure mechanism of shield tunnel lining under the same burial depth, as shown in Figure 3.The relationship of the three concentrated forces can be represented by ( ) ( ) . When the tunnel crown buried depth (H) is the same: where the superscript represents the concentrated force when the radius R is of different radii.This paper focuses on the influence of diameter size on the deformation and failure mechanism of shield tunnel lining under the same burial depth, as shown in Figure 3.The relationship of the three concentrated forces can be represented by When the tunnel crown buried depth (H) is the same:    Figure 4 displays the crown displacement of the shield tunnel with various diameters along with the increase in load P1.Table 4 renders the vertical and lateral loads for the shield tunnel with various diameters (kN).
gradually decreases with the increase in diameter.
In general, under this buried depth condition, the strain at the inflection point of th curve of the tunnel with a large diameter (D > 6 m) is larger than that at the inflection poi of the curve of the tunnel with a small diameter (D = 6 m).Moreover, the slope (stiffnes of the first stage (c.f.Stage I in Figure 2) on the curve of the tunnel with a large diamet (D > 6 m) is smaller than that of the curve of the tunnel with a small diameter (D = 6 m) Specifically, when the tunnel diameter increases from 6 m to 14 m, the strain at th inflection point on the curve gradually increases, and the slope (stiffness) of the first stag on the curve gradually decreases.When the tunnel diameter increases from 14 m to 18 m the strain at the inflection point on the curve gradually decreases, and the slope (stiffnes of the first stage on the curve gradually increases.The failure process of a single-ring structure can be divided into three distinct stages: elastic stage, elastoplastic stage, and plastic stage.In the elastic stage, bolts, steel bars, and concrete are all in an elastic state, there is no plastic deformation in the structure, and the convergence deformation of the structure increases linearly.Until the concrete at the joint reaches the peak stress and the structure enters the elastoplastic stage, the overall stiffness of the structure begins to decline and the development of convergence and deformation accelerates.Then, the structure formed the first plastic hinge at the first structural damage position and entered the plastic stage.Finally, there are four plastic hinges in the structure, which cause the mechanism to lose stability, or the brittle failure of the joint causes the structural system to change, and the structure reaches the ultimate bearing capacity state.Since then, the bearing capacity of the structure begins to decline and the deformation develops unsteadily. Under the condition of the same buried depth at the top of the tunnel, the larger the tunnel diameter D, the larger the value of P 2 .Although the ultimate load P 1 of a large-diameter tunnel (D > 6 m) is larger than that of a small-diameter tunnel (D = 6 m) under this buried depth condition, the difference between P 1 and P 2 -of vertical and lateral loads gradually decreases with the increase in diameter.
In general, under this buried depth condition, the strain at the inflection point of the curve of the tunnel with a large diameter (D > 6 m) is larger than that at the inflection point of the curve of the tunnel with a small diameter (D = 6 m).Moreover, the slope (stiffness) of the first stage (c.f.Stage I in Figure 2) on the curve of the tunnel with a large diameter (D > 6 m) is smaller than that of the curve of the tunnel with a small diameter (D = 6 m).
Specifically, when the tunnel diameter increases from 6 m to 14 m, the strain at the inflection point on the curve gradually increases, and the slope (stiffness) of the first stage on the curve gradually decreases.When the tunnel diameter increases from 14 m to 18 m, the strain at the inflection point on the curve gradually decreases, and the slope (stiffness) of the first stage on the curve gradually increases.

Strain in Tension and Compressive Zones
Figure 5 shows the strain in the tension zone of the shield tunnel with various diameters.When the lining bears the load distribution with a large vertical load, the concrete on the inside of 180 • will be pulled, leading to cracks, and the steel bar will bear the stress.Because the reinforcement rate of the lining can generally meet the requirements, it is difficult for the reinforcement to reach the yield state, so the analysis only focuses on the tensile damage of concrete.
Under the same burial depth of the tunnel crown, the concrete first enters the elastic area and the strain increases linearly, then gradually yields, and the strain increases nonlinearly, and, finally, reaches the limit state.It can be seen that the 6 m diameter lining has a more intuitive yield phase and a relatively slow mutation.In this working condition, the vertical load is unchanged, and the lateral load change can affect the stability of the lining from two aspects.First, when the vertical and lateral loads are close, the lateral pressure can provide support for the lining and helping to help maintain the stability of the lining.On the other hand, when the lateral load is too large or too small, it will cause a burden on the structure.
Moreover, when the diameter of the tunnel is small, the strain difference of different positions is obvious.When the diameter of the tunnel is large, the strain difference at different positions is not obvious, that is, the circle part in the figure .Strain in the compressive zone of the shield tunnel with various diameters is indicated in Figure 6.Those places are located at the junction of each section of the lining, which is the weak point of the whole ring tunnel lining.The crushing of concrete leads causes a significant decrease in the stiffness of the overall structure and a rapid increase in the stress of the bolts.Moreover, the slope of the elastic phase gradually increases, which means that the deformation at the plasticity is less and the destructive form is more brittle.Strain in the compressive zone of the shield tunnel with various diameters is indicated in Figure 6.Those places are located at the junction of each section of the lining, which is the weak point of the whole ring tunnel lining.The crushing of concrete leads causes a significant decrease in the stiffness of the overall structure and a rapid increase in the stress of the bolts.Moreover, the slope of the elastic phase gradually increases, which means that the deformation at the plasticity is less and the destructive form is more brittle.

Bolt Stress and Strain
Figure 7 displays the bolt strain of the shield tunnel with various diameters.Under this working condition, the 6 m lining is gradually surrendered due to the small lateral load constraint, and the structural instability is rapidly reached when it reaches the additional load stage.Other larger diameters fail quickly in the second stage, and the yield phase is very short.It can be found that the strain curve of the large diameter tunnel lining will be slightly protruding, with a sharp point to the left before the bolt strain changes.

Bolt Stress and Strain
Figure 7 displays the bolt strain of the shield tunnel with various diameters.Under this working condition, the 6 m lining is gradually surrendered due to the small lateral load constraint, and the structural instability is rapidly reached when it reaches the additional load stage.Other larger diameters fail quickly in the second stage, and the yield phase is very short.It can be found that the strain curve of the large diameter tunnel lining will be slightly protruding, with a sharp point to the left before the bolt strain changes.This may correspond to the transient stress transfer caused by the sudden failure of other weak points in the structure.
Figure 7 displays the bolt strain of the shield tunnel with various diameters.Under this working condition, the 6 m lining is gradually surrendered due to the small lateral load constraint, and the structural instability is rapidly reached when it reaches the additional load stage.Other larger diameters fail quickly in the second stage, and the yield phase is very short.It can be found that the strain curve of the large diameter tunnel lining will be slightly protruding, with a sharp point to the left before the bolt strain changes.This may correspond to the transient stress transfer caused by the sudden failure of other weak points in the structure.Figure 8 shows the bolt stress cloud diagram of the shield tunnel with various diameters.In the segmented tunnel lining, the bolt, as the key component of each section lining, is the weak point in the structure, and also the place where the plastic hinge is easy to appear.The stress cloud map of bolts at 8° and 73° and three deformations (magnifactor 50) are summarized.As for the deformation of the bolt, it is often accompanied by the rotation of the contact surface between the lining, which will also bring about various engineering problems during the future service period: water leakage, displacement, etc.Under the same burial depth of the tunnel crown, the bolt at 8° of 6 m diameter of the lining has fully reached the ultimate bearing capacity of 400 Mpa, but only the upper half of the bolt at 73° has reached the yield, but it has been destroyed at this time.As the diameter increases, it can be seen that the entire area of the bolt at 73° has yielded and shows a large deformation near the nut.  Figure 8 shows the bolt stress cloud diagram of the shield tunnel with various diameters.In the segmented tunnel lining, the bolt, as the key component of each section lining, is the weak point in the structure, and also the place where the plastic hinge is easy to appear.The stress cloud map of bolts at 8 • and 73 • and three deformations (magnifactor 50) are summarized.As for the deformation of the bolt, it is often accompanied by the rotation of the contact surface between the lining, which will also bring about various engineering problems during the future service period: water leakage, displacement, etc.Under the same burial depth of the tunnel crown, the bolt at 8 • of 6 m diameter of the lining has fully reached the ultimate bearing capacity of 400 Mpa, but only the upper half of the bolt at 73 • has reached the yield, but it has been destroyed at this time.As the diameter increases, it can be seen that the entire area of the bolt at 73 • has yielded and shows a large deformation near the nut.

Deformation and Internal Force Distribution Around the Shield Tunnel
(e) Figure 8 shows the bolt stress cloud diagram of the shield tunnel with various diameters.In the segmented tunnel lining, the bolt, as the key component of each section lining, is the weak point in the structure, and also the place where the plastic hinge is easy to appear.The stress cloud map of bolts at 8° and 73° and three deformations (magnifactor 50) are summarized.As for the deformation of the bolt, it is often accompanied by the rotation of the contact surface between the lining, which will also bring about various engineering problems during the future service period: water leakage, displacement, etc.Under the same burial depth of the tunnel crown, the bolt at 8° of 6 m diameter of the lining has fully reached the ultimate bearing capacity of 400 Mpa, but only the upper half of the bolt at 73° has reached the yield, but it has been destroyed at this time.As the diameter increases, it can be seen that the entire area of the bolt at 73° has yielded and shows a large deformation near the nut.

Cracks in Tension Zone
Figure 11 renders the cracks in the tension zone of the shield tunnel with various diameters.
Due to the material characteristics of the concrete, the tensile damage is easy to occur in the tunnel lining structure.The CDP constitutive model used in this study can better simulate the tensile damage of concrete.Except for the 6 m diameter tunnel lining, the compression damage positions of other diameter lining are concentrated on the inside of 73°, and the results shown in the cloud map are relatively consistent, so only the tensile damage of concrete is analyzed.The tensile damage in the simulation can be simply considered that the concrete produces cracks and cannot withstand more loads, and then the stress will be transferred to the internal reinforcement.
Under the same burial depth of the tunnel crown, the tensile damage is distributed on the inside of the upper and lower segments and the outside of the left and right segments.The obvious trend is that the lateral tensile damage area will become larger with the increase in diameter.It is worth noting that the damage to the concrete tensile area has nothing to do with the ultimate bearing capacity.At this time, the damage to the concrete will only lead to the transfer of stress to the steel bar, and the general reinforcement of the  Due to the material characteristics of the concrete, the tensile damage is easy to occur in the tunnel lining structure.The CDP constitutive model used in this study can better simulate the tensile damage of concrete.Except for the 6 m diameter tunnel lining, the compression damage positions of other diameter lining are concentrated on the inside of 73 • , and the results shown in the cloud map are relatively consistent, so only the tensile damage of concrete is analyzed.The tensile damage in the simulation can be simply considered that the concrete produces cracks and cannot withstand more loads, and then the stress will be transferred to the internal reinforcement.

Cracks in Tension Zone
Under the same burial depth of the tunnel crown, the tensile damage is distributed on the inside of the upper and lower segments and the outside of the left and right segments.The obvious trend is that the lateral tensile damage area will become larger with the increase in diameter.It is worth noting that the damage to the concrete tensile area has nothing to do with the ultimate bearing capacity.At this time, the damage to the concrete will only lead to the transfer of stress to the steel bar, and the general reinforcement of the steel bar can meet the actual engineering needs.Therefore, the analysis focuses on where the cracks are created, rather than focusing on finding the size of the tensile damage.
x FOR PEER REVIEW 17 of 19 steel bar can meet the actual engineering needs.Therefore, the analysis focuses on where the cracks are created, rather than focusing on finding the size of the tensile damage.

Conclusions
This paper investigates the deformation and failure mechanism of lining large-diameter shield tunnels in depth, and discusses the deformation characteristics and influencing factors of the lining of the shield tunnel with various diameters through a finite element analysis using ABACUS software (2021 edition).A set of models with varying diameters is built under identical stress conditions in order to maintain control over the variable.The crown displacement of the shield tunnel, strain in tension and compressive zones, bolt stress and strain, deformation and intemal force distribution around the shield tunnel, and cracks in the tension zone are carefully studied.Based on the findings, some conclusions can be drawn as follows: (1) Under this buried depth condition, the strain at the inflection point of the curve of the tunnel with a large diameter (D > 6 m) is larger than that at the inflection point of the curve of the tunnel with a small diameter (D = 6 m).Moreover, the slope (stiffness) of the first stage (c.f.Stage I in Figure 2) on the curve of the tunnel with a large diameter (D > 6 m) is smaller than that of the curve of the tunnel with a small diameter (D = 6 m).(2) When the tunnel diameter increases from 6 m to 14 m, the strain at the inflection point on the curve gradually increases, and the slope (stiffness) of the first stage on the curve gradually decreases.When the tunnel diameter increases from 14 m to 18 m, the strain at the inflection point on the curve gradually decreases, and the slope (stiffness) of the first stage on the curve gradually increases.(3) The results indicate that the convergence deformation of large-diameter tunnel lining increases significantly during loading compared with that of a small-diameter tunnel.Moreover, the probability of brittle failure is higher in big-diameter shield tun-

Conclusions
This paper investigates the deformation and failure mechanism of lining large-diameter shield tunnels in depth, and discusses the deformation characteristics and influencing factors of the lining of the shield tunnel with various diameters through a finite element analysis using ABACUS software (2021 edition).A set of models with varying diameters is built under identical stress conditions in order to maintain control over the variable.The crown displacement of the shield tunnel, strain in tension and compressive zones, bolt stress and strain, deformation and intemal force distribution around the shield tunnel, and cracks in the tension zone are carefully studied.Based on the findings, some conclusions can be drawn as follows: (1) Under this buried depth condition, the strain at the inflection point of the curve of the tunnel with a large diameter (D > 6 m) is larger than that at the inflection point of the curve of the tunnel with a small diameter (D = 6 m).Moreover, the slope (stiffness) of the first stage (c.f.Stage I in Figure 2) on the curve of the tunnel with a large diameter (D > 6 m) is smaller than that of the curve of the tunnel with a small diameter (D = 6 m).(2) When the tunnel diameter increases from 6 m to 14 m, the strain at the inflection point on the curve gradually increases, and the slope (stiffness) of the first stage on the curve gradually decreases.When the tunnel diameter increases from 14 m to 18 m, the strain at the inflection point on the curve gradually decreases, and the slope (stiffness) of the first stage on the curve gradually increases.(3) The results indicate that the convergence deformation of large-diameter tunnel lining increases significantly during loading compared with that of a small-diameter tunnel.Moreover, the probability of brittle failure is higher in big-diameter shield tunnels compared to small-diameter tunnels, indicating that these larger tunnel structures are more prone to suffering geometric instability.

Figure 1 .
Figure 1.The cross-section diagram of the shield precast segmental tunnel lining.

Figure 1 .
Figure 1.The cross-section diagram of the shield precast segmental tunnel lining.

Figure 2 .
Figure 2. Boundary conditions and force processes: (a) boundary conditions; (b) evolution of force.

Figure 2 .
Figure 2. Boundary conditions and force processes: (a) boundary conditions; (b) evolution of force.

2 = 2 = 2 = 2 = 2 =Figure 2 .
Figure 2. Boundary conditions and force processes: (a) boundary conditions; (b) evolution of force represents the concentrated force when the radius R is of differen radii.

Figure 3 .
Figure 3. Shield tunnel with various diameters located at an identical burial depth (H = 18.175 m).

Figure 3 .
Figure 3. Shield tunnel with various diameters located at an identical burial depth (H = 18.175 m).

Figure 4 .
Figure 4. Crown displacement of the shield tunnel with various diameters (the red dot indicates the crown position).

Figure 4 .
Figure 4. Crown displacement of the shield tunnel with various diameters (the red dot indicates the crown position).

Figure 8 .
Figure 8. Bolt stress cloud diagram of the shield tunnel with various diameters.

Figures 9 7 .
Figures 9 and 10 indicate the deformation distribution around the shield tunnel, the axial force and bending moment distributions of the shield tunnel with various diameters.

Figure 8 .
Figure 8. Bolt stress cloud diagram of the shield tunnel with various diameters.

Figures 9 8 .
Figures 9 and 10 indicate the deformation distribution around the shield tunnel, the axial force and bending moment distributions of the shield tunnel with various diameters.

3. 4 .Figure 9 .
Figures 9 and 10 indicate the deformation distribution around the shield tunnel, the axial force and bending moment distributions of the shield tunnel with various diameters.

Figure 11
Figure 11  renders the cracks in the tension zone of the shield tunnel with various diameters.Due to the material characteristics of the concrete, the tensile damage is easy to occur in the tunnel lining structure.The CDP constitutive model used in this study can better simulate the tensile damage of concrete.Except for the 6 m diameter tunnel lining, the compression damage positions of other diameter lining are concentrated on the inside of 73 • , and the results shown in the cloud map are relatively consistent, so only the tensile damage of concrete is analyzed.The tensile damage in the simulation can be simply considered that the concrete produces cracks and cannot withstand more loads, and then the stress will be transferred to the internal reinforcement.Under the same burial depth of the tunnel crown, the tensile damage is distributed on the inside of the upper and lower segments and the outside of the left and right segments.The obvious trend is that the lateral tensile damage area will become larger with the increase in diameter.It is worth noting that the damage to the concrete tensile area has nothing to do with the ultimate bearing capacity.At this time, the damage to the concrete will only lead to the transfer of stress to the steel bar, and the general reinforcement of the steel bar

Figure 11 .
Figure 11.Cracks in tension zone of the shield tunnel with various diameters: (a) .

Figure 11 .
Figure 11.Cracks in tension zone of the shield tunnel with various diameters.

Table 1 .
Dimensions of tunnel lining and number of bolts.
Outer Diameter (D) Thickness (t) Number of Segments (n) Number of Bolts (nb)

Table 1 .
Dimensions of tunnel lining and number of bolts.
Outer Diameter (D) Thickness (t) Number of Segments (n) Number of Bolts (n b )

Table 2 .
Parameters of Concrete Damaged Plasticity model for the concrete C55.

Table 3 .
Main mechanical parameters of the rebars and bolts.

Table 4 .
Vertical and lateral loads for the shield tunnel with various diameters.