Simplified nonlinear model and 3D printing model experiment verification of longitudinal joints of shield tunnel segments

The segment joints represent vulnerable and core-bearing areas within shield tunnel structures. Their flexural stiffness, starting from the contact surface of longitudinal joints to the bolt hole, exhibits continuous changes under bending loads, typically demonstrating nonlinear mechanical characteristics. This study investigates the influence of joint contact surfaces on flexural stiffness, extending from the joint contact surface to damaged concrete in the compressed zone. A new simplified model for the variable nonlinear stiffness of longitudinal segment joints in shield tunnels is proposed, considering the flexural stiffness of the joint contact surface and the width of the damaged concrete zone in the segment ring while simulating the flexural stiffness decay induced by the opening of segment joints. Based on the minimum potential energy principle, a design calculation method for joint flexural stiffness is derived, establishing the nonlinear bending stiffness coupling relationship between the segment and the joint. Through similarity theory, 3D printing technology, and indoor loading tests, the theoretical model's accuracy regarding tunnel segment joints is verified. This theoretical model provides a basis for designing and assessing the performance of tunnel-segment joints.


Introduction
Shield tunnel structures usually consist of segments connected by bolts.The segment joint, created by the bolted connection, is a critical and vulnerable part of the tunnel structure, directly affecting its load-bearing safety.The flexural stiffness of the segment, from the joint contact surface to the bolt hole, undergoes continuous change under bending loads and typically displays nonlinear mechanical behavior.Thus, accurately describing the nonlinear flexural stiffness of segment joints is crucial for determining the load-bearing capacity of shield tunnel structures.
Numerous scholars have explored diverse methodologies to investigate the flexural stiffness of shield tunnel segment joints [1].Liu et al. [2] conducted full-scale tests, model experiments, numerical simulations, and theoretical analyses.Numerical and theoretical analyses serve as common approaches for evaluating the flexural stiffness of segment joints.For example, Xia et al. [3] obtained a reasonable bending stiffness by using a fixed-point iterative method and multiple iterations based on the formulas for the internal force of each longitudinal joint and then established a nonlinear equation for the flexural stiffness of segment joints.Feng et al. [4] employed a finite element model with one complete ring and two half-rings to simulate the segment lining.Additionally, he devised an iterative algorithm to delineate the three-dimensional surface of flexural stiffness within the segment joint and established a reasonable convergence criterion.Huang et al. [5] developed an algorithm for calculating the effective ratio of the lateral flexural stiffness of segments based on the flexural stiffness of longitudinal joints and defined the flexural stiffness of the modified uniform stiffness ring model.In subsequent studies, Huang et al. [6] considered factors such as the elastic modulus of the segment ring material, segment diameter, segment ring width, and number of segment rings.By means of theoretical analysis, they derived an analytical algorithm for determining the longitudinal flexural stiffness of shield tunnels.While numerical and theoretical analyses offer convenience, simulating the nonlinear characteristics of segment joints proves challenging.Undoubtedly, full-scale tests represent the most direct and effective approach for investigating the mechanical performance of segment joints.Feng et al. [7] conducted a series of full-scale tests on segment joints and compared the results with those of numerical simulations to analyze the occurrence and development of joint failure.Zhang et al. [8] studied and summarized the failure processes of two types of segment lining structures, namely straight joint assembly and staggered joint assembly, used in the Shiziyang Tunnel through prototype tests.Zhou et al. [9] investigated the shear behavior of a novel circumferential joint with a sleeve-bolt combination in large-diameter shield tunneling, analyzing the displacement, bolt stress, and joint damage modes under shear loading.Li et al. [10,11] conducted full-scale continuous loading tests on the longitudinal joints of a subway tunnel until complete failure.Subsequently, they proposed a progressive model for predicting the flexural performance of segment joints, which was based on the observed joint failure mechanism.The high cost, long-term preparation, and high-loading device requirements make it less feasible as a common method for studying the performance of segment joints.
A reduced-scale model test is typically adopted to obtain the performance changes and stress states of the tunnel structure under various working conditions, while ensuring the accuracy and precision of the data.Shi et al. [12] proposed a time-dependent constitutive model for EPDM rubber pads used in segment joints.Their focus was on elucidating the degradation of material properties over time and its consequent impact on joint stiffness.Wang et al. [13] presented a method for measuring the joint opening between segments in reduced-scale model tests.Ground stone glass was utilized to fabricate a shield tunnel model, enabling the accurate measurement of joint openings between segments.In scenarios involving relatively large-scale model tests, the implementation of bolted connections between model segments proves feasible for practical operations.Nevertheless, various factors contribute to the flexural stiffness of model tunnel segment joints, including the material composition of the segments, the properties of connecting bolts, elastic pads, hydrophilic sealing gaskets, and the pre-tension force applied to longitudinal bolts.Achieving similarity between the flexural stiffness of the model joint and its prototype counterpart necessitated meticulous attention to ensuring adequate similarity across all influential variables.Ye et al. [14] used polymethylmethacrylate and aluminum wires to simulate segment and joint bolts, respectively, and created three models of segment rings for model experiments.Multistage loading tests were conducted utilizing a custom-made loading device to investigate the effective ratio of lateral flexural stiffness within the joints.Certain researchers have suggested techniques involving both internal and external grooving to emulate the reduction in bending stiffness observed in segment joints.Wang et al. [15,16] used the grooving method to investigate the impact of cracks of different lengths, quantities, and positions on the damage evolution of shield tunnel segments, considering subway tunnels and large-diameter underwater linings as engineering objects.Model experiments offer a more comprehensive understanding of the variation patterns exhibited by segment joints; however, they often lack a universally applicable theoretical model.Consequently, accurately describing the nonlinear mechanical characteristics of typical segment joints proves challenging.
To address the limitations inherent in the aforementioned methodologies, this study introduces a novel simplified model for the variable nonlinear stiffness of longitudinal segment joints within shield tunnels.This model takes into account the flexural stiffness of the joint contact surface and the extent of damage within the concrete zone of the segment ring while also simulating the decrease in flexural stiffness resulting from the opening of segment joints.Subsequently, three physical models of shield tunnels were constructed to assess the viability of the proposed design theory, utilizing 3D printing technology.Internal structural forces and deformations were scrutinized through digital image correlation (DIC) loading tests, thereby ascertaining the feasibility and practicality of the proposed nonlinear variable stiffness model for longitudinal segment joints.

Establishment of theoretical del for longitudinal segment joint
Stiffness reduction near the joint within the tunnel model was conducted to accommodate varying degrees of stiffness attenuation resulting from the opening of joint contact surfaces and concrete damage within the compressed zone.This methodology serves to substitute bolted connections, thereby streamlining the intricate procedures associated with modeling bolts and joints.
To elucidate the design methodology concerning the nonlinear stiffness of the longitudinal segment joint, it was assumed that the diameter of the model segment ring mirrored that of the actual segment ring.Subsequently, an analysis was conducted to establish the equivalence between the nonlinear stiffness of the longitudinal segment joint model and that of the bolted segment joint.Assuming that the pure bending moment M acts at both ends of the stiffness reduction section of the stiffness damage model joint, the same pure bending moment M is applied to both ends of the corresponding segmental curved beam with the same center angles and bolted connections.The lengths of both bending beams (referred to as the stiffness damage beam and joint beam) were calculated using the arc length of their neutral axis when bending, respectively L and L1, as shown in Figure 1.In this scenario, the equivalent theoretical assumption for the nonlinear stiffness of the longitudinal segment joint model and the segment joint connected by bolts is that the relative rotation angles at both ends of the two beams under the action of a pure bending moment M are equal.Assuming that the model joint adopts a stiffness damage method (as shown in Figure 1), the bending stiffness of the joint contact surface is first reduced to EI1, and the stiffness reduction extends along the joint contact surface towards the bolt holes on both sides with the same width of damage on both sides.The bending neutral axis of the nonlinear stiffness within the longitudinal segment joint model remained consistent both prior to and following stiffness damage.Additionally, the length of the damaged stiffness beam corresponded to that of the joint beam.
The variable stiffness beam is jointly constrained by concentrated force P and axial force N at the support end at where is the deflection of the beam at the origin that satisfies the geometric constraints.The displacement was divided as follows: Assuming the stiffness variation of the section is The strain energy of the beam is When the undetermined coefficient becomes c  , the strain energy becomes: IOP Publishing doi:10.1088/1755-1315/1333/1/012054 The work done by external forces is: The work done by external forces is divided into According to the principle of minimum potential energy, namely: By substituting Equation (6) and Equation (8) into Equation ( 9), the following is obtained.
Due to the arbitrary nature of variational c  : According to Equation ( 11), the constant c can be obtained: By substituting Equation (12) into Equation ( 2), the vertical displacement function of a variable stiffness beam can be expressed as The relative rotation angles at both ends of the variable stiffness beam are The relative rotation angle of the cross-section at both ends of the curved beam is caused by two parts under the pure bending moment M: one is caused by the opening of the joint at the bolted connection area, and the other is caused by the bending of the curved beam.Set the stiffness of the segment joint to Kθ.
The θ2, where is the opening of the segment joint of the bolted connections, is given by: We can obtain the θ3 caused by the bending of the joint curved beam: The relative rotation angle θ1 of the cross-section at both ends of the joint curved beam can be expressed as Therefore, the equivalent theoretical assumption between the nonlinear stiffness of the longitudinal segment joint model and the bolted segment joint can be described as follows: Substituting Equation (17) and Equation ( 18) into a formula of I′ obtained according to Equation ( 14): Suppose that the thickness of the damaged section of the joint contact surface of the nonlinear stiffness of the longitudinal segment joint model is h, and the moment of inertia of the section is By considering Equation ( 19) and Equation ( 20), the thickness h can be given by By substituting Equation (21) into Equation ( 4), the stiffness damage function of the joint can be obtained as follows: In this model, the impact of segment joint openings on the attenuation of bending stiffness was simulated by decreasing the bending stiffness of the joint contact surface.Additionally, the stiffness reduction was implemented by considering the hand holes on both sides, using the stiffness of the joint contact surface as the boundary condition.It's noteworthy that the stiffness of the lining is not solely contingent upon the simulated joint stiffness but is also influenced by the width of the concrete damage area within the segment ring.

Model fabrication 2.2.1. 3D printing model production method.
3D Printing, also referred to as "additive manufacturing," is a rapid prototyping technology.This technique, based on the laminated slice technology of digital models, utilizes adhesive materials such as powdered metal or plastic to construct objects by printing and stacking them layer by layer.Generally, three-dimensional (3D) printing technology is categorized based on the forming method of printing materials.Table 1 provides an overview of commonly used non-metallic printing materials and their corresponding forming methods.This experiment employed a fused deposition modeling (FDM) 3D printer, functioning as follows: Initially, filamentous PLA (a thermoplastic polylactic acid polymer material derived from corn and potato starch) was heated and melted.Subsequently, a computer-controlled 3D nozzle was utilized to apply the material layer by layer onto the working platform, adhering to the designed cross-sectional profile information.Upon cooling, each layer formed a cross-section with a reduced thickness, and this process iterated until the entire solid model was printed.The FDM 3D printer and PLA material utilized in this experiment are detailed in Table 1, boasting a layering thickness ranging from 0.05 to 0.50 mm, a printing accuracy spanning from 0.1 to 0.2 mm, and a material diameter of 1.75 mm.The production of 3D printing models can be broadly divided into the following steps: (1) Model establishment: First, a 3D digital model (shown in Figure 3) was established using a 3D modeling software.The 3D digital model was exported as a stereolithography file in a simple format that could describe 3D information.
(2) Slicing processing: After creating a 3D model, it is sliced to represent it in a layered manner.When slicing, printing parameters such as fill rate and layer thickness should be set.
(3) 3D model and post-processing: The 3D model was imported into the printer, and the 3D printer created the model by layer printing according to relevant operations.

Similarity ratio of model test segments.
Establishing the similarity ratio parameters between the prototype and model tunnels constitutes a pivotal step in guaranteeing the precision of model testing.This concept of similarity entails that when both the model and prototype tunnels demonstrate akin mechanical behaviors, the parameters encompassing geometric dimensions, material attributes, external loads, and static responses of the segment also mirror each other.Consequently, when subjected to corresponding loads, the flexural performance of the segment emerges as a crucial metric for gauging structural deformation in both the prototype and model configurations.Therefore, the segment parameters of the similarity relationship include the geometric dimensions (outer diameter and width of the segment ring and thickness of the segment), material characteristics (elastic modulus of the segments and bolts, unit weight of the segment, and Poisson's ratio), load conditions (concentrated force, uniformly distributed load, and bending moment), and stress and strain (internal stress and strain in the segment).Given that the dead weight of the segment ring significantly pales in comparison to the pressure exerted by the upper soil layer, it is commonly disregarded in model tests, a stance upheld in this paper as well.The experimental design undertaken in this study solely concentrated on replicating concentrated force loading, omitting consideration of uniformly distributed loads and bending moments.Drawing upon the principles outlined in the three theorems of similarity (namely, the similarity positive theorem, π theorem, and similarity inverse theorem) and taking into account the printing precision and platform dimensions of FDM3D printers, the preliminary determination of the similarity relationship between the model and the prototype tunnel is presented in Table 2.

Mechanical properties of printing materials.
The mechanical properties of PLA materials were analyzed through unconfined compression strength (UCS) and direct tensile strength (DTS) tests.Two sets of standard specimens were prepared using each of the three PLA materials, as depicted in Fig. 4. Following the standard test method, the standard specimen underwent uniaxial tensile and compressive tests.Its tensile and compressive elastic moduli were obtained, and its stress-strain curve was drawn.In this study, the filling rate of the model samples used in the experiment was set to 100%, and the height of each layer was 0.15 mm.
Firstly, the UCS test was performed by the "MTS pressure testing machine."The dimensions of the cylindrical PLA specimen were 35 mm × 70 mm, as shown in Fig. 4 (a).(three-dimensional strain collection instrument) was used to record the axial and lateral expansion displacements with loading rates controlled at 2 mm/min.Then, the DTS test was performed by the "MTS tensile testing machine."The thickness of the PLA tensile specimen was 2 mm, and its dimensions are shown in Fig. 4 (b).Strain gauges were applied to the front and back surfaces of the tensile specimen to monitor axial extension.All samples were subjected to controlled pulling at a rate of 2 mm/min until failure occurred.Consequently, the axial strain was measured, enabling the derivation of Young's modulus and peak tensile strength.The compressive strength and Young's modulus of the 3D printed specimen were determined through the UCS test, as illustrated in Figure 5(a).The stress-strain curves of the three PLA materials demonstrated similar trends.Initially, the stress of the material increased rapidly and linearly before reaching the peak stress.Subsequently, upon reaching the peak stress, the load could no longer be applied, leading to continued plastic deformation of the material until failure occurred.Notably, the compression stress-strain curve indicated distinct elastic-plastic characteristics in the PLA test.The compression specimen exhibited plastic expansion when it failed, and the surface of the specimen did not crack, as shown in Figure 6.(a).
The tensile strength and Young's modulus of the 3D printed specimen were acquired through the DTS test, depicted in Figure 5 (b).The stress-strain curves of the three PLA materials exhibited consistent variations.Initially, during the initial loading phase, the stress-strain curve displayed an approximately linear elastic behavior.Subsequently, as the load continued to increase, the material transitioned into ductile behavior, with the stress continuously rising; however, the rate of increase gradually diminished.When the load reached the tensile failure strength of the material, brittle fracture occurred, and all three specimens broke in the middle, as shown in Figure 6   Table 3 presents the compressive and tensile elastic moduli of three PLA materials.It is evident from the table that the ratio of the elastic modulus of material PLA a to C50 concrete is 10.29, which closely aligns with the similarity constant CE=10 established for the elastic modulus.Consequently, the material PLA was ultimately chosen as the 3D printing material for the segment.This study employed 3D printing technology to fabricate a scaled and refined segment structure, encompassing three distinct types of segment structure models: a homogeneous circular ring model, a nonlinear variable stiffness model, and a refined scaled model.Additionally, the mechanical characteristics of these three tunnel models were investigated.The production processes of the three model tunnels are outlined as follows: 1. Homogeneous circular ring model tunnel (1) A 3D modeling software was used to establish a 3D homogeneous circular digital model (Figure 7. (a)).
(2) Slice the model with a height of 0.15 mm for each layer and a filling rate of 100%.
(3) Import the 3D model into the printer and print the solid model tunnel directly (Figure 7).(b)).(c) Taking into account the flexural stiffness EI of the segment in the undamaged area as the boundary condition at the opposite end, the reduction in stiffness of the structure within the joint damage area was accomplished by decreasing the material's elastic modulus within the affected stiffness region and altering the cross-sectional dimensions of the structure.In this particular experiment, the stiffness reduction was achieved through the adjustment of the cross-sectional size.
(2) Fabrication of nonlinear variable stiffness model tunnel (a) The stiffness reduction coefficient of the joint contact surface was determined to be 0.008, indicating a reduction in the width of the section by 0.2 times its original value.Subsequently, utilizing the outcomes of the numerical simulation, the stiffness reduction zone of the model was delineated as the region between the two bolt holes of the longitudinal joint.To visualize this, a 3D modeling software was employed to construct a homogeneous circular digital model, as illustrated in Figure 8

The refined scaled model tunnel (1) Segmentation production
According to the geometric parameters of the model tunnel, the slices underwent processing before being imported into a 3D printer.This printer was utilized to print each segment separately.Due to the model tunnel's small volume and the challenges associated with assembly, two minor modifications were implemented to the bolt hole of one segment.These adjustments were made without compromising the structural integrity or causing deformation.Firstly, the arc surface of the bolt hole was transformed into an inclined surface to allow for easier nut installation while retaining the original position and form of the bolt hole.Additionally, the diameter of the bolt hole was suitably increased to facilitate bolt installation.Furthermore, appropriate material was selected for crafting the bolt; however, neither the form nor the length of the bolt underwent alterations.
The selection of the model bolts must satisfy the assembly requirements of the segment, and the geometric dimensions and elastic moduli must be similar.Based on the geometric similarity ratio CL=31 and elastic modulus similarity CE=10, aluminum welding wires were selected for secondary processing to produce bending bolts, combined with the research results of YE and ZHENG.The elastic modulus of the bolt was 3.3 GPa, with a diameter of 2 mm and length of 20 mm, and it was connected to 12 longitudinal bolts.
Initially, the aluminum welding wire was cut into bars measuring 20 mm in length.Subsequently, a 2 mm precision screw mold was employed to manually create threads at both ends of the bolt.Following this, a bender was utilized to curve the bolt to the specified curvature, after which nuts and washers were affixed to both ends.It is noteworthy that nuts and washers compatible with a 2 mm diameter bolt model are commercially available and can be acquired directly, obviating the need for manual production.
(3) Assembly of model The model segments and bolts were connected sequentially for assembly into rings.The main process of model-making is shown in Figure 9.

Test loading and monitoring plan.
When designing model experiments to investigate transverse structural deformation in tunnels, scholars commonly employ methods such as sandbox-layered sand filling and weight-graded loading.Additionally, displacement gauges and strain gauges are typically utilized to monitor structure displacement and strain.However, in order to meticulously examine the deformation characteristics of segments under various joint treatment methods, this experiment employed an MTS pressure testing machine to apply load at the arch bottom of the tunnel.Furthermore, three-dimensional digital image correlation (3D-DIC) technology was employed to dynamically monitor the deformation and strain of the entire cross-section of the model tunnel in realtime.Figure 10 illustrates the loading and monitoring methods employed.Additionally, an additional homogeneous circular ring model tunnel was prepared for a pre-experiment to determine load application.The experimental loading plan was as follows: (1) Initially, a homogeneous circular ring-model tunnel was subjected to loading, utilizing displacement control recording.The tunnel was loaded until its destruction, with a loading rate set at 2 mm/min.
(2) Determine the appropriate load value based on the load-displacement curve of the preexperimental model.
(3) The three model tunnels undergo loading using the force control method.Based on preexperimental results, a load corresponding to structural deformation approximately within the linear elastic stage was chosen for loading.In this experiment, the maximum load was 180N, and the control recording rate was 1N/S.(1) Matte spray paint was used to make speckles in the sections of the three tunnel models, and the speckles were distributed randomly, as shown in Figure 11.
(2) A DIC test system was set up, and the height of the equipment was adjusted to ensure that it was horizontally aligned with the sample to be monitored, as shown in Figure 12.
(3) Calibrate of the camera.The accurate distance between two pairs of identification points on the signboard was set as the scale, and two cameras were used to capture the image data in different directions on the signboard.Subsequently, the three-dimensional coordinates of the identification points were identified, and an algorithm was used to calculate the internal and external parameters of the camera.
(4) Position the experimental tunnel model onto the loading platform of the MTS pressuretesting machine, adhere to the predetermined loading plan for data collection, and synchronize the timing of image capture with the commencement of loading.
(5) Post-processing and data analysis

Radial deformation analysis of the model tunnels
Following the construction of the model tunnels, loading tests were conducted on three distinct types: the homogeneous circular ring model, the nonlinear variable stiffness model, and the finely detailed model.The monitoring outcomes of these experimental trials are synthesized and subsequently deliberated upon.
Under the same experimental conditions, the radial deformation results of the three model tunnels at the end of the loading, when the load reached 180N, are shown in Fig. 13.It can be seen that: (1) Similar deformation characteristics: The deformation characteristics of the three model tunnels were essentially identical, exhibiting symmetrical deformations along the vertical axis.The maximum deformation occurred at the arch bottom, followed by the positions along the horizontal diameter, whereas the four points at 45°, 135°, 225°, and 315°showed relatively smaller deformations.
(2) Increasing radial deformation with load: As the load continued to escalate, the radial deformation of all three tunnel models exhibited a steady rise.Under equivalent loading circumstances, the deformation of the homogeneous circular ring model tunnel proved the most minimal, followed by the nonlinear variable stiffness model tunnel, with the finely detailed model tunnel displaying the greatest deformation.
(3) Growing discrepancy in radial deformation: As the load continued to increase, the numerical disparity in radial deformation became increasingly pronounced among the homogeneous circular ring, nonlinear variable stiffness, and finely detailed model tunnels.When subjected to a load of 180N, the tunnel models exhibited varying degrees of deformation, with the homogeneous circular ring model experiencing the least, followed by the nonlinear variable stiffness model and the finely detailed model registering the highest deformation.This discrepancy in deformation among the three models continued to widen as the load increased.Analysis of radial deformation revealed a significant decrease in flexural stiffness in the finely detailed model tunnel due to the presence of longitudinal joints compared to the homogeneous circular ring model, highlighting a notable limitation in utilizing the latter to replicate real shield tunnels.The deformation of the nonlinear variable stiffness model tunnel was approximately 50% greater than that of the homogeneous circular ring model, approaching but slightly less than that of the finely detailed model tunnel, suggesting comparable flexural performance between the nonlinear variable stiffness model and the finely detailed model tunnels.

Convergence deformation analysis of the model tunnels
In the experiment, the convergence value of the deformation measured inside the tunnel was negative, and that outside the tunnel was positive.According to the general similarity law of model tests, the test results of the model tunnel deformation are converted into corresponding shield tunnel prototypes using conversion formulas such as Equation (23) and Equation (24).(1) The vertical convergence values for all three tunnel models exhibited a gradual decrease as the concentrated load increased, while the horizontal convergence values showed a gradual increase, signifying distinct elliptical deformation under the load.The deformation characteristics of both vertical and horizontal convergences across the three models were largely consistent during the loading stage of 0-250kN, demonstrating an approximately linear pattern of change.The primary discrepancy among the three curves lay in the rate of change.
(2) After surpassing a load of 250 kN, the convergence value of the homogeneous circular ring model maintained an approximately linear trend.Conversely, both horizontal and vertical convergence values of the nonlinear variable stiffness and finely detailed model tunnels experienced rapid escalation.This observation underscores the substantial impact of joints on the flexural bearing capacity of the lining structure, resulting in swift structural deformation.
(3) The Pearson correlation coefficients between the two models were obtained by calculating the mean and variance of the curves of the nonlinear variable stiffness model and the finely detailed model.The correlation coefficient for vertical convergence was 0.95, while that for horizontal convergence was 0.98.These findings denote a strong positive correlation between the two models.Conversely, the data from the homogeneous circular ring model exhibited noticeable deviation, indicating a larger margin of error.
As per the designated control parameter for maximum convergence deformation in shield tunnels, the prototype tunnel's maximum convergence deformation value was recorded at 37.2 mm, selected as 0.6% of the design value.As depicted in Figure 15, under a load of 500kN, the maximum vertical and horizontal convergence deformations of the prototype tunnel, as per the nonlinear variable stiffness model and the finely detailed model, were measured at 46.05 mm and 46.29 mm, respectively.At this juncture, the convergence deformation of the prototype tunnel no longer adheres to the specified requirements.

Analysis of the failure characteristics of model tunnels
To further study and analyze the deformation and failure characteristics of the model tunnel, after the proposed loading stage was completed (loading to 180 N), a further load was applied until the structure failed.From the model tunnels designed with different joint treatment modes in Figure 15, it can be observed that the failure modes of the tunnel structures have different characteristics under concentrated loads at the two points.
In the case of the homogeneous circular ring model (representing the tunnel), both vertical and horizontal convergence values remained relatively modest until structural failure ensued.During the loading process, tunnel deformation exhibited a steady increase proportionate to the applied load, following a linear growth pattern.With ongoing load augmentation, the stress in the tensile zone of the structure at the load application point steadily escalated until it reached the material's ultimate tensile strength.Subsequently, the lining structure experienced sudden fracture, characterized by the emergence of a single fracture crack at the load application point.This failure mode was typified by brittle failure.
In the case of the nonlinear variable stiffness model, both vertical and horizontal convergences of the tunnel steadily escalated with augmented loads, resulting in a noticeable elliptical deformation of the tunnel structure.As loading progressed, notable structural deformation transpired at the junctions between adjacent and standard blocks, attributed to the gradual increase in deformation owing to joint stiffness reduction.Upon reaching a certain load threshold, structural deformation persisted in its ascent, culminating in an unstable failure of the lining structure: In the finely detailed model tunnel, a clear elliptical deformation was observed in the tunnel structure.Throughout the loading process, segment joints experienced inward opening under positive bending moment loads and outward opening under negative bending moment loads.Analogous to the tunnel modeled with nonlinear variable stiffness, the most pronounced deformation of segment joints in the finely detailed model occurred at connections between adjacent and standard blocks.As the load steadily increased, joint angles augmented, and structural stabilization proved elusive, signifying that joint failure under negative bending moments ultimately precipitated the overall unstable failure of the lining structure.
It is evident that the nonlinear variable stiffness model tunnel and the finely detailed model tunnel exhibit similar characteristics in terms of deformation, convergence, and failure.By contrast, the homogeneous circular ring model cannot effectively reflect the mechanical response of the shield tunnel structure.This suggests that the nonlinear reduction joint model that considers the stiffness reduction in the joint contact surface and concrete damage zone can better simulate the flexural performance of the joints.

Conclusion
The joint of the tunnel segments is a vulnerable and crucial load-bearing area in a shield tunnel, directly impacting the structural bearing safety.To further investigate the nonlinear mechanical behavior of the segment joint, this study proposes a simplified nonlinear model for the longitudinal joint and verifies its accuracy using model experiments.The main conclusions are as follows: 1.This study considers the influence of concrete damage in the joint contact surface and compression zone on flexural stiffness and proposes a new nonlinear variable stiffness model for longitudinal segment joints of shield tunnels.By considering and deducing the width of the damaged concrete zone in the segment ring, a method for calculating the flexural stiffness of the joint was derived based on the principle of minimum potential energy and a nonlinear coupling relationship between the flexural stiffnesses of the segment and joint was established.
2. The theoretical model's accuracy and practical applicability for the longitudinal joint of a tunnel segment were confirmed through validation.Utilizing 3D printing technology, three models were produced: a homogeneous circular ring model, the nonlinear variable stiffness joint model proposed in this paper, and a finely detailed physical experimental model, which accounted for the specific structural characteristics of the joint.Comparative analysis of radial deformation, convergence deformation, and failure characteristics was conducted among the three models based on digital image correlation (DIC) loading tests.Compared to the homogeneous circular ring model, the variable stiffness joint model exhibited a nearly 50% increase in radial deformation, which was slightly less than that of the finely detailed model.In comparison to tunnel convergence deformation, the homogeneous circular ring model displayed notable discrepancies, whereas the convergence deformation of the variable stiffness joint model closely approximated that of the finely detailed model.The Pearson correlation coefficient for vertical convergence between the variable stiffness joint model and the finely detailed model was 0.95, with a coefficient of 0.98 for horizontal convergence, indicating a strong positive correlation.While the homogeneous circular ring model exhibited brittle failure under loading, both the variable stiffness joint model and the finely detailed model demonstrated an overall unstable failure of the lining structure due to joint failure at the connection between adjacent blocks and standard block segments, showcasing highly similar failure characteristics.
3. The experimental findings suggest that the nonlinear variable stiffness joint model proposed in this study more accurately replicates the stiffness variation between the segment and joint by accounting for flexural stiffness reduction in the joint contact surface and concrete damage zone.Additionally, the simplified theoretical model for the longitudinal joint outlined in this study offers a clear mechanical concept and requires fewer parameters.This model furnishes a quantitative measure of flexural stiffness in tunnel model joint design, streamlining experimental procedures, improving experimental precision, and concurrently offering a theoretical foundation for the design and assessment of tunnel segment joints.

Figure 1 .
Figure 1.Schematic diagram of joint damage and deformation.

1
  .The damage model of the joint area also satisfied the four basic assumptions described in the previous section.

Figure 2 .
Figure 2. Simplified schematic diagram of nonlinear variable stiffness model.From the simple supported beam diagram, it can be observed that the geometric constraints
(a) Plastic failure diagram of compressed specimens (b) Fracture failure diagram of tensile specimen

Figure 7 .
Homogeneous circular ring model tunnel.2. The nonlinear variable stiffness model tunnel (1) Design of nonlinear variable stiffness model tunnel (a) Firstly, the stiffness reduction coefficient of the joint contact surface is determined by considering flexural stiffness η EI of the longitudinal segment joint contact surface as the boundary condition at the one end of the flexural stiffness attenuation area.(b) Determine joint damage areas.
(a).(b) Slice the model to a height of 0.15 mm for each layer and a filling rate of 100%.(c) Import the 3D model into the printer and print the solid model tunnel directly (Figure 8. (b)).
(a)Three-dimensional model (b)entity model

Figure 10 .
Figure 10.Loading and monitoring diagram.The specific test steps are as follows:(1) Matte spray paint was used to make speckles in the sections of the three tunnel models, and the speckles were distributed randomly, as shown in Figure11.(2)A DIC test system was set up, and the height of the equipment was adjusted to ensure that it was horizontally aligned with the sample to be monitored, as shown in Figure12.(3)Calibrate of the camera.The accurate distance between two pairs of identification points on the signboard was set as the scale, and two cameras were used to capture the image data in different directions on the signboard.Subsequently, the three-dimensional coordinates of the identification points were identified, and an algorithm was used to calculate the internal and external parameters of the camera.(4)Position the experimental tunnel model onto the loading platform of the MTS pressuretesting machine, adhere to the predetermined loading plan for data collection, and synchronize the timing of image capture with the commencement of loading.(5)Post-processing and data analysis

Figure 13 .
(a)Deformation of a homogeneous ring model tunnel under 180N Load (b)Deformation of nonlinear variable stiffness model tunnel model under 180N load (c)Deformation of refined scale model tunnel model under 180 N load (d)Radial deformation (load 180N) Model tunnel deformation situation.

F
where F C , E C ,and L C are the load similarity ratio, elastic modulus similarity ratio, and geometric similarity ratio, respectively.( ) are the concentrated loads of the prototype and model tunnels, respectively.The vertical and horizontal convergence deformations of the three model tunnels were analyzed.The results of this analysis are shown in Figure14.
(a)Vertical convergence (b)Horizontal convergence Figure 14.Horizontal and vertical convergence of model tunnels.

Figure 14
Figure14illustrates the variations in the vertical and horizontal convergences during the entire loading process for the three tunnel models.The observations were as follows:(1) The vertical convergence values for all three tunnel models exhibited a gradual decrease as the concentrated load increased, while the horizontal convergence values showed a gradual increase, signifying distinct elliptical deformation under the load.The deformation characteristics of both vertical and horizontal convergences across the three models were largely consistent during the loading stage of 0-250kN, demonstrating an approximately linear pattern of change.The primary discrepancy among the three curves lay in the rate of change.(2)After surpassing a load of 250 kN, the convergence value of the homogeneous circular ring model maintained an approximately linear trend.Conversely, both horizontal and vertical convergence values of the nonlinear variable stiffness and finely detailed model tunnels experienced rapid escalation.This observation underscores the substantial impact of joints on the flexural bearing capacity of the lining structure, resulting in swift structural deformation.(3)The Pearson correlation coefficients between the two models were obtained by calculating the mean and variance of the curves of the nonlinear variable stiffness model and the finely detailed model.The correlation coefficient for vertical convergence was 0.95, while that for horizontal convergence was 0.98.These findings denote a strong positive correlation between the two models.Conversely, the data from the homogeneous circular ring model exhibited noticeable deviation, indicating a larger margin of error.As per the designated control parameter for maximum convergence deformation in shield tunnels, the prototype tunnel's maximum convergence deformation value was recorded at 37.2 mm, selected as 0.6% of the design value.As depicted in Figure15, under a load of 500kN, the maximum vertical and horizontal convergence deformations of the prototype tunnel, as per the nonlinear variable stiffness model and the finely detailed model, were measured at 46.05 mm and 46.29 mm, respectively.At this juncture, the convergence deformation of the prototype tunnel no longer adheres to the specified requirements.
(a)Failure characteristics of homogeneous ring model tunnel and DIC monitoring deformation and failure cloud map ( b ) Nonlinear variable stiffness model tunnel failure characteristics and DIC monitoring deformation and failure cloud map ( c ) Fine-scaled model tunnel failure characteristics and DIC monitoring deformation and failure cloud map

Figure 15 .
Figure 15.Horizontal and vertical convergence of model tunnels.

Table 1 .
3D printing materials and their forming methods.

Table 2 .
Segment similarity relationship.test, selecting appropriate materials is a prerequisite for success.Based on the established similarity relationship, this study selected three different PLA materials and determined the appropriate segment-printing materials through mechanical performance testing experiments.

Table 3 .
Physical and mechanical parameters of PLA materials.The size parameters of the model tunnel were converted using similar constants, as listed in Table4.