Analytical algorithms of compressive bending capacity of bolted circumferential joint in metro shield tunnels

Abstract

The integrity and bearing capacity of segment joints in shield tunnels are associated closely with the mechanical properties of the joints. This study focuses on the mechanical characteristics and mechanism of a bolted circumferential joint during the entire bearing process. Simplified analytical algorithms for four stress stages are established to describe the bearing behaviors of the joint under a compressive bending load. A height adjustment coefficient, α, for the outer concrete compression zone is introduced into a simplified analytical model. Factors affecting α are determined, and the degree of influence of these factors is investigated via orthogonal numerical simulations. The numerical results show that α can be specified as approximately 0.2 for most metro shield tunnels in China. Subsequently, a case study is performed to verify the rationality of the simplified theoretical analysis for the segment joint via numerical simulations and experiments. Using the proposed simplified analytical algorithms, a parametric investigation is conducted to discuss the factors affecting the ultimate compressive bending capacity of the joint. The method for optimizing the joint flexural stiffness is clarified. The results of this study can provide a theoretical basis for optimizing the design and prediciting the damage of bolted segment joints in shield tunnels.

Graphical abstract

Keywords

shield tunnel / segment joint / joint structural model / failure mechanism

1 Introduction

Shield tunneling is widely used in urban tunnel construction owing to its safety, high speed, high efficiency, and insignificant construction impact [1,2]. Scholars have investigated the stability of shield tunnels primarily based on two aspects: the soil layer and structural characteristics. The soil layer stability of a shield tunnel is affected by the stress variation on the tunnel lining structure and the excavation surface caused by stratum disturbance during tunnel construction [3–10]. As the bearing component of the tunnel, the bearing of the shield tunnel lining structure directly determines its stability [11–15].

The shield tunnel lining is a prefabricated assembly structure comprising several arc-shaped segments connected using bolts. The present engineering practice and relevant studies indicate that segment joints are weak links of the tunnel lining. The bearing capacity and safety of a tunnel structure are directly determined by the mechanical performance of its joints [16–21].

The mechanical properties of segment joints are primarily investigated via compressive bending tests, whereas further analytical derivations and numerical simulation studies are occasionally supplemented. Ding et al. [22] conducted a full-scale test on the segment joint of a water tunnel and proposed a mechanical model that considered the elastoplastic properties of the concrete and the pre-tightening force of the bolt. Li et al. [23,24] investigated the mechanical behavior of prototype segment joints via failure tests and numerical simulations. The effects of the axial force and pre-tightening bolt force on the joint opening were analyzed, and a progressive model was proposed to predict the mechanical behavior of the joints. Jin et al. [25] performed tests and finite element simulations to investigate the joint deformation and flexural rigidity of the joint section of a complex water tunnel prototype under positive and negative bending moments. By conducting full-scale tests on segment joints fabricated using steel fiber concrete and conventional reinforced concrete, Gong et al. [26] showed that steel fiber concrete joints can be used instead of conventional reinforced concrete joints to maintain the bearing capacity and control cracks. Caratelli et al. [27] investigated the performance of bolt-free flat circular joints by perfomring full-scale tests on fiber-reinforced concrete members without conventional reinforcing bars. Feng et al. [28] conducted a full-scale segment joint test, established a corresponding three-dimensional finite element model, and analyzed the failure process of a segment joint. Although the simulation results were consistent with the test results, the bolt was simulated using a beam element, and the contact between the bolt and segment concrete was omitted. Liu et al. [29–31] investigated the failure mechanism and bearing performance of the circumferential joint of a shield tunnel segment under different operating conditions via a full-scale experimental test and then deduced the stress model of the circumferential joint.

In general, most of the studies above were conducted to design the stiffness of joints and focused on the change in the joint flexural stiffness under an external force. However, the related theoretical models involve many parameters, and the derivation process is relatively complicated. Consequently, using these models for field use is time consuming.

Hence, a simplified analytical model of the compressive bending capacity of a bolted circumferential joint during the entire bearing process is established in this study. Using this model, the mechanical characteristics of the circumferential segment joint are determined by considering a typical metro segment under a compressive bending load; additionally, the bearing performance and failure mechanism of the segment joint are clarified. Furthermore, the relationship between various influencing factors and the flexural stiffness of segment joints is analyzed. The results obtained can provide a theoretical basis for optimizing the design of segment joints in shield tunnels as well as for predicting and alleviating joint damages.

2 Proposed model

2.1 Problem description

Based on the geometric structure and caulking on the inner and outer sides of the circumferential joint of a segment, the joint surface can be categorized into outer-edge concrete, core concrete, bolt, and inner-edge concrete. The specific structural diagram of a circumferential joint between shield tunnel segments is shown in Fig.1 and Fig.2. The pressure bearing of a segment joint is a continuous mechanical process. By determining the circumferential joint stress, a segmented simplified analysis model describing the entire process of joint pressure bearing can be established to clarify the mechanical state of the joints in each stage, thus providing theoretical guidance for the prediction of joint safety performance and design theories.

To analyze the entire bearing process of the joint, the section model must be simplified. Therefore, the following settings and ssumptions were adopted.

1) A lining joint was adopted to replace the arc joint of an actual segment. The joint section was assumed to be flat during the bearing process, and the deformation of the joint was primarily bolt deformation and concrete compression. The rotation angle of the joint is defined as:

(1)\theta =\frac{{\delta }_1+{\delta }_2}{H},

where {\delta }_1 and {\delta }_2 represent the internal and external openings of the joint, respectively, and H is the height of the joint, as shown in Fig.3.

2) The waterproof gasket, positioning hole, and positioning rod contributed negligibly to the flexural stiffness of the joint, and the effect of bolt pre-stress was not considered. Bolt pre-stress is generally regarded as the safety redundancy for the tunnel structure. The effect of the joint bolt pre-stress was not considered in this study; hence, conservative calculation results were obtained.

3) The segment joint deformation directly affects the deformation characteristics of the tunnel structure, and as an underground structure, the tunnel is allowed only an insignificant amount of deformation. Tunnel segment joints in the elastic stage are typically used. Therefore, to simplify the theoretical derivation process, the concrete and bolts of the segment joint were assumed to be linear elastic bodies. Additionally, to determine the ultimate bending stiffness of the joint, concrete was assumed to be crushed when its compressive stress exceeded its ultimate compressive strength.

4) Based on the bearing characteristics of a reinforced concrete structure, when the entire section of the joint concrete was compressed, the section stress distribution exhibited a standard quadratic parabola and rectangle. After the joint presented an opening angle, considering the strong nonlinearity of concrete in the compression zone, the distribution of the compressive stress in the compression zone was assumed to be a standard quadratic parabola, and the apex of the parabola was at the edge of the compression zone.

5) The concrete compressive deformation {\delta }_{\mathrm{c}} at the edge of the joint compression zone was calculated as follows [32]:

(2){\delta }_{\mathrm{c}}=\frac{{\sigma }_{\mathrm{c}}}{E}l,

where E is the deformation modulus of concrete; l is the influence depth of the compressive strain, which is also regarded as the height of the compression zone.

2.2 Analytical solutions

The mechanical state of the circumferential joint can be categorized into four main stages: (a) stage I, when the core concrete in the entire section is compressed; (b) stage II, when the core concrete below the bolt is compressed; (c) stage III, when the core concrete above the bolt is compressed; (d) stage IV, when the outer-edge concrete and the core concrete are both compressed. The stress forms at each stage are shown in Fig.4.

At the initial bearing stage (stage I), when the core concrete in the full section is compressed, the height of the concrete compression zone y is equal to the height of the core concrete h. Additionally, the bolt is not stressed, i.e., T_{\mathrm{b}}=0, and the core concrete is in the elastic stage. The stress distribution diagram under an axial force N and bending moment M is shown in Fig.4(a). As the external load increased, the stress state of the joint evolved gradually from stage I to stage II when the concrete in the compression zone exhibitsed significant nonlinearity. The stress analysis results are presented in Fig.4(b). At this stage, the height range of the concrete compression zone is d≤ y < h, and the bolt is not stressed, i.e., T_{\mathrm{b}}=0.

As the joint opening expanded further, the height of the compression zone spanned the bolt length, resulting in tension in the bolt (stage III). At this time, the height of the compression zone is 0 < y < d, and the associated stress is as shown in Fig.4(c). As the external load continued to increase, the outer arc surface of the joint continued to be compressed, and the outer-edge concrete began to borne the compressive stress simultanouesly with the core concrete (stage IV). At this stage, the height of the compressive zone in the core concrete decreased further, whereas the compressive zone height of the outer concrete increased gradually. The limit state of this stage was reached when the outer-edge concrete exceeded its ultimate bearing capacity. This stage reflects the effect of the joint structure on its bearing characteristics and revealed several influencing factors via mechanical analysis. Furthermore, this stage is key to determining the safety performance of the joint because it reflects the ultimate bearing state. The mechanical analysis results above show that the force mode of the joint at each stage is the same; additionally, because the outer concrete participates in the bearing at the final stage, many influencing factors are involved in the mechanical analysis, and the derivation process is relatively complex. Therefore, by establishing an analytical model for determining the joint mechanical properties in the final stage, the mechanical properties of the other stages can be determined using the established model.

This study focuses on analyzing the critical mechanical properties of a joint when its outer-edge concrete is crushed at stage IV. An adjustment coefficient α is introduced for the mathematical expression of the compression zone height in the outer-edge concrete. Thus, the height of the compression zone in the outer edge when the concrete is crushed can be expressed by the outer-edge concrete height multiplied by α. The stress analysis for this state is shown in Fig.4(d). Using the stress diagram, the corresponding mechanical balance equation can be established as follows:

(3)N_4+T_{\mathrm{b}}-C_1-C_2=0,

(4)\begin{array}{rl}{M}_4-& T_{\mathrm{b}}(d-h_1)+C_1\left(h_1-\frac{3}{8}y\right)\\ +& C_2\left[h_1+h_3+\left(1-\frac{3}{8}\alpha \right)h_2\right]=0,\end{array}

(5)C_1=\frac{2}{3}{\sigma }_{\mathrm{c}}by,

(6)C_2=\frac{2}{3}{\sigma }_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}b \alpha h_2,

where M₄ and N₄ are the bending moment and axial force when the concrete on the outer arc surface of the joint is crushed, respectively; C₁ and C₂ are the combined compressive stress of the core concrete and the outer-edge concrete, respectively; h₁, h₂, and h₃ are the distance from the bottom edge of the sealing gasket to the neutral axis of the segment, concrete thickness of outer edge and caulking width of sealing gasket, respectively; y represents the compression zone height of segment; α represents the height adjustment coefficient of outer concrete compression zone; d represents the distance from the lower edge of the sealing gasket to the bolt; b represents the segment thickness; σ_c and σ_cmax are the compressive stress on the upper edge of the segment core concrete and the maximum compressive stress of the outer-edge concrete, respectively; T_b represents the bolt tension; k represents the bolt stiffness.

Based on the assumption and concrete deformation {\delta }_{\text{c}}^{\prime } and {\delta }_{\text{c}} relationship, the compression deformation of the concrete in the outer edge and core region is expressed as follows, respectively:

(7){\delta }_{\text{c}}^{\prime }=( h_2+ h_3+ y)\theta -\mathrm{\Delta }= 2\frac{{\sigma }_{\text{cmax}}}{E}\alpha h_2,

(8){\delta }_{\mathrm{c}}=y\theta =2\frac{{\sigma }_{\mathrm{c}}}{E}y.

where θ represents the joint corner; Δ represents the segment joint gap.

Similarly, the opening amount {\delta }_{\mathrm{b}} of the joint at the bolt is expressed as:

(9){\delta }_{\mathrm{b}}=(d-y)\theta .

Based on the physical relationship of bolt deformation,

(10)k{\delta }_{\mathrm{b}}=T_{\mathrm{b}}.

Using the formula above and setting \beta =2{\sigma }_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}\alpha h_2+E∆, we obtain:

(11){\sigma }_{\mathrm{c}}=\frac{\theta E}{2},

(12)T_{\mathrm{b}}=k\theta (d-y),

(13)\theta =\frac{2{\sigma }_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}\alpha h_2+E\mathrm{\Delta }}{E(h_2+h_3+y)},

(14)y=\frac{3\beta kd-E(h_2+h_3)(2{\sigma }_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}b\alpha h_2-3N)}{E(2{\sigma }_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}b\alpha h_2+\beta b-3N)+3\beta k}.

The equations above show that at the final stage, when the outer-edge concrete is crushed, the height of the compressive region y in the core concrete can be obtained once the adjustment coefficient α is determined. Subsequently, \theta, T_{\mathrm{b}}, and {\sigma }_{\mathrm{c}} can be solved using Eqs. (11)–(13). Thus, the mechanical properties of the joint in the ultimate load state are clarified.

3 Determining α via numerical simulation

3.1 Factors affecting α

When performing a theoretical analysis of the entire joint bearing process, one must clarify the height adjustment coefficient α at stage IV, i.e., when the concrete in the outer edge is crushed (i.e., the compressive stress is borne by the outer-edge concrete and core concrete simultaneously). Subsequently, the mechanical properties of the joint at this stage can be determined. However, α is affected by many factors. To determine the value of α under different conditions, a three-dimensional solid model of the circumferential joint under a compressive bending load was established using the finite element software ABAQUS. Orthogonal analysis was performed on the factors affecting α using a five-factor four-level orthogonal table L16 (4⁵). The selected influencing factors and their levels are listed in Tab.1.

The value of each level of the influencing factors was determined based on the following.

1) To ensure that the outer-edge concrete reaches the crushing state under an external load, the minimum value of M/N was specified as 0.32 m, based on the joint mechanical analysis model deduced in this study. The step was set as 0.08 m to fully reflect the effect of the external load on the mechanical properties of the joint.

2) A change in the sealing liner caulking width directly affects the thickness of the core concrete, which consequently affects the mechanical properties of the joint. Based on current practices in subway shield tunnel engineering, the sealing liner caulking width is 35–50 mm.

3) The position of the connecting bolts affects the restraint of the bolts during joint expansion. Studies show that when the distance between the bolt and inner edge of the segment is within 1/3 to 1/2 of the segment thickness, the bolt significantly restrains the joint expansion under positive and negative bending moments.

4) The joint gap affects the time node when the outer-edge concrete supports the load; nonetheless, the joint gap should not be excessively large for waterproofing. Therefore, based on the relevant code and the current practices in subway shield tunnel engineering, the joint gap was set to 2–8 mm.

3.2 Numerical model

The calculation model was established based on typical subway shield tunnel forms. The joints were connected using 5.8 grade M30 straight bolts, and the concrete grade was C50. The outer diameter, inner diameter, and width of the lining ring were 6400, 5800, and 1200 mm, respectively. The effects of the positioning groove, positioning rod, and elastic gaskets were disregarded. Because the segment was symmetric, it was modeled as a semi-structure in the width direction of the segment to simplify calculations, and a symmetry plane was established for analysis. The calculation model is shown in Fig.5, where the global size of the model grid was 20 mm. The grids of the model were locally densified in the critical analysis region (i.e., the core concrete and outer-edge concrete), and the seed size was specified as 5 mm. The specific geometric parameters of the segment model are showen in Tab.2.

Both the concrete and bolt were simulated using the elastic constitutive model, and their elastic modulus was 34.5 and 210 kN/mm², respectively. Friction contact was set between the bolts and bolt holes via a friction coefficient of 0.15, whereas it was specified between the segments via a friction coefficient of 0.55 [33]. The front and back planes of the model were symmetric with a set of symmetric boundaries, and the constraints at the bottom of the left and right sides were simply supported. In this model, the bending moment M was set for the joint, which was specified as 600 kN·m under different operating conditions. The left and right sides of the model were subjected to an axial force N in the form of a surface load, and the value of N was determined based on the M/N ratio under different operating conditions. The model calculation was performed in two loading steps. First, the axial force of the segment was uniformly applied to the design value. Subsequently, a bending moment was applied to the segment with a constant axial force until the segment disintegrated.

3.3 Derivation of α

A five-factor four-level orthogonal table L16 (4⁵) was selected for the orthogonal numerical test. The specific parameters and calculated α are shown in Tab.3. The blank column represents the error column, which contains the random errors of the test and interaction. The results can be analyzed via range analysis to quantitatively analyze the influence degree of each factor on the simulation result.

Based on the calculated values presented in the orthogonal table, a range analysis was performed to determine the optimal level of each influencing factor, the optimal level combination, and the priority order. The results of the range analysis are presented in Tab.4. Here, K is the average value of the sum of the simulation results at different levels of each factor, which is used to determine the optimal level of each factor and the optimal combination for the simulations. R is the extreme difference in each factor, which reflects the variation range of the simulation results. The larger the R value, the greater is the influence of this factor on the simulation index. In other words, this factor is the main influencing factor.

Based on the results shown in Tab.4, the joint gap exerts the most significant influence on α, followed by M/N, the distance between the bolt and the inner edge of the segment, and the caulking width of the sealing gasket. Moreover, the optimum α was achieved under the following conditions: M/N = 0.32; sealing liner caulking width = 50 mm; distance between the bolt and inner edge = 110 mm; joint gap = 4 mm.

Considering that range analysis cannot distinguish the source controibuting to the difference in the simulation results for each factor level, a variance analysis was performed to quantitatively estimate the influence of each factor on α. The analysis results are presented in Tab.5. The degree of freedom (DOF) for each factor listed in the orthogonal table was three. The blank column represents the error column; thus, the degree of freedom of the error is three. According to the F distribution table, F_0.01(3,3) = 29.46, F_0.05(3,3) = 9.28, and F_0.10(3,3) = 5.36. The F value of M/N is F_A = 1.96 < F_0.10(3,3); the F value of the caulking width of the sealing gasket is F_B = 0.24 < F_0.10(3,3); the F value of the distance between the bolt and the inner edge of the segment is F_C = 0.62 < F_0.10(3,3); the F value of the joint gap is F_D = 20.33 > F_0.05(3,3). The results of the variance analysis show that the joint gap significantly affected α. By contrast, the caulking width of the sealing gasket and the distance between the bolt and inner edge did not impose significant effects.

Therefore, by comprehensively considering the range and variance analysis results, when analyzing the effects of various factors on α, the effects of M/N, the caulking width of the sealing gasket, and the distance between the bolt and inner edge are negilgible. In other words, the effect of the joint gap should be considered.

To further clarify the relationship between α and the joint gap, the following optimized values were used: M/N = 0.32; caulking width of the sealing gasket = 50 mm; distance between the bolt and inner edge = 110 mm. Based on current engineering data, the range and gradient of the joint gap is 1–10 and 1 mm, rsepectively. The basic setting of the model was the same as that of the orthogonal analysis model. A numerical simulation model was established and calculations were performed; the results are shown in Fig.6.

The analysis above shows that α is associated closely with only to the joint gap under different operating conditions. After performing a substantial amount of numerical simulations, the relationship between α and the joint gap ∆ of conventional subway segments can be preliminarily established as follows:

(15)\alpha \text{=}-\text{0}\text{.0003}{\mathrm{\Delta }}^3 +0.0034{\mathrm{\Delta }}^2-0.0087\mathrm{\Delta }+\mathrm{0.2066.}

In practice, for most urban metro shield tunnels, the joint gap of segments is typically set between 2 and 8 mm. According to the derivation formula, α can be set as approximately 0.2.

4 Case study

A three-dimensional numerical model of the segment joint was developed to verify the rationality of the simplified theoretical joint force analysis. The design parameters were determined based on a subway shield tunnel segment in Shanghai, China. The analytical and numerical solutions of the ultimate bearing capacity under different axial forces were analyzed and verified based on the test results of Liu et al. [30].

The geometric dimensions of the joint surface of the calculation model are shown in Fig.7, and the settings of the model calculation parameters were consistent with those described in Subsection 3.2. During the compressive bending process of the joint, an external load is provided by the axial force and bending moment. The bending moment promotes joint rotation, and the axial force compresses the joint, thereby limiting the development of joint rotation. Therefore, the axial force contributes positively to the flexural rigidity of the joint within a specific range. To clarify the relationship between the axial force and joint bearing capacity, the axial force was varied gradually from 0 to 2000 kN. The ultimate bearing capacity of the joints under different axial forces can be calculated using the deduced analytical solution and the corresponding numerical model. The changes in the ultimate bending moment, joint angle, and ultimate flexural stiffness with the axial force are shown in Fig.8-Fig.10, respectively. The joint stiffness was modeled using a twofold line model, and the ultimate bending stiffness of the joint was calculated as follows:

(16)k_{\text{θ}}=\frac{M_4-M_1}{\theta },

where M₄ and θ are the bending moment and rotation angle when the concrete on the outer arc surface of the joint is crushed, respectively; M₁ is the bending moment when the joint is opened.

As shown in Fig.8, the change in the ultimate bending moment with the joint axial force obtained analytically and numerically is similar. As the axial force increased, the analytical and numerical values of the joint bending moment showed consistency. Furthermore, those values slightly exceeded the experimental values obtained by Liu et al. [30]. Based on Fig.9, the rotation angle of the joint did not change significantly with the axial force, and the error rate between the analytical and numerical values remained within 10%. Because the concrete and bolts were assumed to be linear elastic bodies, the analytical and numerical values of the joint rotation angle exceeded the experimental value. As shown in Fig.10, the flexural stiffness of the joint exhibits a linear positive relationship with the axial force. The experimental value is between the analytical and numerical values, and the analytical value is similar to the experimental value. Therefore, the mechanical properties of the joint can be estimated analytically.

Liu et al. [30] performed an experiment and discovered that at a bending moment of 90 kN·m, the bolts were stretched and began to support the force. As the bending moment increased, the joint bolts yielded when the bending moment was 180 kN·m. The outer-edge concrete at the both side of joint contacted with each other when the bending moment reached 203 kN·m and then crushed when the bending moment increased to 240 kN·m. In Liu’s test, the joint bolts yielded before the outer-edge concrete touched each other, thus resulting in joint stiffness dissipation and a rapid development of joint rotation. By contrast, in the present study, the joint bolt was assumed to be in the elastic stage throughout the entire process. Therefore, both the analytical and numerical values of the ultimate bending moment exceeded the experimental values. Additionally, in the numerical simulation, the axial force of the joint was applied vertically to the end of the segment as the surface load. During the compressive bending deformation of the joint, the end of the segment rotated, which caused the axial force to generate a vertical stress component, thus resulting in an additional bending moment on the joint. However, this additional bending moment was not considered in the analytical solution; hence, the ultimate bending moment obtained analytically exceeded that obtained numerically. As the axial force increased, the opening angle and rotation degree of the segment decreased gradually, thus diminishing the effect of the additional bending moment in the numerical simulation. Consequently, the difference between the analytical and numerical values decreased gradually.

The ultimate flexural stiffness of the joint increased linearly with the axial force of the joint, which reflects the restraint of the axial force on the development of the rotation angle. Based on this law, the bending resistance of the joint can be improved by applying a pre-tightening force to the joint bolts without changing the joint characteristics. However, the appropriate waterproof measures for shield tunnels below the water level will provide a uniform confining pressure to the structure, thus rendering the segment assembly more compact as well as increasing the axial force and flexural rigidity of the joints. This implies that the underwater tunnel performs better than its counterpart over the water level. Therefore, for an underwater shield tunnel, the joint stiffness need not be improved, whereas the waterproofing of the tunnel should be prioritized.

Based on a comprehensive analysis of the mechanical properties of the segment joints, the theoretical solution presents a slight error in deriving the bending capacity of the segment joints, whereas the theoretical solution can reflect the bearing performance of the segment joints. In summary, the proposed simplified theoretical model can reasonably estimate the mechanical properties of compressive bending joints and provide a theoretical basis for their design and damage prediction/treatment.

5 Parametric investigation

Segment joint stiffness is vital to the general mechanical characteristics of a tunnel. Therefore, to provide a basis for the design and optimization of segment joints, factors affecting the ultimate flexural stiffness and their laws should be clarified. Based on the proposed simplified analytical model, the relationship between the influencing factors and the ultimate flexural stiffness of the joint is analyzed systematically in this section based on the section geometric characteristics and material properties of the segment joint. A method for optimizing the joint flexural stiffness is presented, which can provide a reference for the design of segment joints.

5.1 Construction size of joint

The segment joint exhibits a specific structural form, and adjusting the joint size changes its mechanical properties. A parameter sensitivity analysis was conducted based on the current segment joint design. Using the control variable method, the caulking width of the sealing gasket, the bolt position, and the joint gap were considered to determine the relationship between the parameters and the ultimate flexural stiffness. The section sizes of the concrete segment joints are presented in Tab.2.

5.1.1 Caulking width of sealing gasket

In the joint section, the caulking width of the sealing gasket affets the height of the joint core concrete as well as the ultimate flexural joint stiffness. The relationship between the width of the sealing gasket and the ultimate flexural stiffness was analyzed using the control variable method.

Based on the derivation presented earlier, when the joint gap is 4 mm, α can be assumed to be 0.2. The axial force of the joint was set as 600 kN. The crushing of the outer-edge concrete was regarded as the limit state of the circumferential joint, and the bending joint stiffness in this state was considered as the ultimate flexural stiffness. The ultimate bending stiffness for different caulking widths of the sealing gasket can be calculated using the analytical solution deduced from the joint bending process. The results, as presented in Fig.11, show that the ultimate flexural stiffness increased gradually with the caulking width of the sealing gasket, which indicates a positive correlation. When the caulking width was changed from 35 to 50 mm, the change rate of the ultimate flexural stiffness was low (only 7.0%). Therefore, one may conclude that the caulking width of the sealing gasket does not significantly affect the flexural stiffness and that the caulking width of the sealing gasket should be designed based on the reasonable waterproof requirements of the joint.

5.1.2 Distance between bolt and inner edge of segment

The connecting bolt inhibits the development of joint rotation, whereas the position of the connecting bolt affects the development rate of joint rotation. The position of the joint connecting the bolt was changed to analyze its effect on the ultimate flexural stiffness. Calculations were performed using the proposed analytical model, and the results are shown in Fig.12. The ultimate bending stiffness decreased gradually as the distance between the joint bolt and the inner edge of the segment increased, and this relationship was linear. A greater distance between the bolt and the inner edge of the segment resulted in less time for the bolt to inhibit the development of the joint angle. Therefore, the joint angle was relatively large, and the ultimate bending joint stiffness decreased. When optimizing the circumferential joint section, the bolt position can be adjusted based on the linear relationship between the ultimate flexural stiffness and the distance between the bolt and the inner edge of the segment.

5.1.3 Segment joint gap

The segment joint gap changed the mechanical state of the compressive concrete at the outer edge of the joint and the ultimate flexural joint stiffness. A parameter sensitivity analysis was performed on the joint gap to clarify the effect of the joint gap on the ultimate flexural joint stiffness.

Based on the derivation presented earlier, when the joint gap is 2–8 mm, α can be set as 0.2. The relationship between the circumferential joint gap and ultimate flexural stiffness was determined based on the analytical solution deduced from the joint compressive bending process.

Fig.13 shows that the ultimate flexural stiffness exhibits a nonlinear relationship with the joint gap and that the change occurred in multiple stages. In particular, when the joint gap was 2–4 mm, the rate of change of the ultimate flexural stiffness was significant. By contrast, when the joint gap exceeded 4 mm, the change rate decreased gradually and then stabilized. Based on this trend, when optimizing the joint section, if the joint gap is large (4–8 mm), then the ultimate flexural stiffness will not change significantly by adjusting the joint gap. By contrast, if the joint gap is small (2–4 mm), then the ultimate flexural stiffness can be changed significantly by adjusting the joint gap.

5.2 Joint material properties

The load on the joint was borne by the concrete segment and connecting bolts. Therefore, changes in the material properties of the joint directly affect its bearing performance. A parameter sensitivity analysis was performed to investigate the relationship between each material attribute and the ultimate flexural capacity of the joint. The section sizes of the joints are presented in Tab.6.

5.2.1 Concrete grade

The ultimate flexural stiffness of the joint was obtained in the crushing state of the concrete at the outer edge; thus, the compressive strength of the concrete directly affects the flexural stiffness. The relationship between the concrete strength and ultimate flexural stiffness of the joint was analyzed based on the typically used concrete grade. The relationship between the concrete grade and the ultimate flexural stiffness can be determined by substituting the joint section parameters into the analytical solution, as shown in Fig.14.

Based on Fig.14, the ultimate flexural stiffness increased with the concrete grade, although the rate of increase was low. Specifically, when the concrete grade increased from C45 to C65, the ultimate flexural stiffness of the joint increased by only 1.97%. Therefore, the concrete grade did not significantly affect the ultimate flexural stiffness of the joint. Meanwhile, changing the concrete grade contributed negligibly to the ultimate flexural stiffness. In other words, the segment concrete grade can be specified based on the waterproof and deformation requirements of the tunnel segment instead of the joint stiffness.

5.2.2 Tensile line stiffness of bolt

The strength and stiffness of the bolt, which connects the segment joints, determine the mechanical properties of the circumferential joint and affect the general performance of the single ring. A reasonable selection of the connecting bolts can maximize the synergy between the concrete and bolts and improve the flexural stiffness of the joint. In this section, the crushing of the outer-edge concrete is regarded as the limit state of the circumferential joint, and the connecting bolt at this state is assumed to be “yielding” such that the effect of bolt stiffness can be quantitatively analyzed easily. The stiffness matching relationship between the connecting bolt and circumferential joint was calculated and analyzed using the proposed analytical model of the bolted circumferential joint in the entire compressive bending process.

Based on a typical metro shield tunnel project, the connecting bolts used for the circumferential joints were M24–M42, and their length was 400–700 mm. Therefore, the bolt tensile line stiffness of the circumferential joint can be assumed to be 100–800 MN/m. Based on the deduced analytical solution, the relationship between the bolt tensile line stiffness and the ultimate flexural stiffness of the circumferential joint was determined, as shown in Fig.15.

Based on Fig.15, the bolt tensile line stiffness shows a linear relationship with the ultimate flexural stiffness of the joint. This linear relationship is expressesd as shown in Eq. (17), which can be used a theoretical basis for the design and optimization of segment connecting bolts to improve the synergy between the joint concrete and bolts such that a reasonable flexural stiffness can be achieved between segment joints.

(17)k_{\text{θ}}=0.0078k+ \mathrm{3.0875.}

6 Conclusions

In this study, simplified analytical algorithms for bolted circumferential joints in metro shield tunnels were established. The rationality of the algorithms was verified via numerical simulations and model tests. Based on the proposed simplified analytical model, the relationship between the influencing factors and the ultimate flexural stiffness of the joint was systematically analyzed based on three aspects: the geometric size, external load, and material properties of the joint. The method for optimizing the flexural stiffness of the joint was clarified. The main conclusions are as follows.

1) A simplified analytical model for investigating the segmented stress behavior of a bolted circumferential joint under a compressive bending load was established. The results yielded by the simplified model were consistent with the numerical results and results from a previous test. It can be used to calculate the bearing capacity of a bolted circumferential joint under an arbitrary load, thereby providing a theoretical basis for optimizing the design and predicting the damage of the circumferential joint of shield tunnels.

2) The height of the compression zone at the outer edge was the most significantly affected by the joint gap. A calculation formula, which can yield the α of the compressive zone in the outer edge when concrete is crushed, was proposed based on a series of positive and back analyses conducted via numerical simulation and theoretical analysis. Notably, when the joint gap is 4–8 mm, the coefficient can be set as approximately 0.2.

3) Using the simplified analytical model of the bolted circumferential joint proposed herein, the relationship between the geometric characteristics of the joint and the ultimate flexural stiffness of the segment joint was analyzed. The ultimate flexural stiffness showed insignificant, negative linear, and segmented nonlinear relationships with the caulking width of the sealing gasket, the distance between the joint bolt and the inner edge of the segment, and the joint gap, respectively.

4) The ultimate flexural stiffness of the joint increased linearly with the joint axial force, which reflects the inhibitory effect of the axial force on the development of the joint angle. In practical engineering, applying preloading to the connecting bolt is recommended to increase the joint axial force and improve the joint performance.

5) The concrete grade of the segment contributed negligibly to the ultimate flexural stiffness of the joint. Meanwhile, the ultimate flexural stiffness of the joint exhibited a linear relationship with the tensile line stiffness of the bolt. The joint stiffness can be adjusted and optimized based on this law.

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