Equivalent flexural rigidity calculation method for synchronous grouting reinforcement shield tunnel

Abstract

The invention discloses an equivalent flexural rigidity calculation method for reinforcing a shield tunnel through synchronous grouting. The equivalent flexural rigidity calculation method comprises the steps that (1) a shield tunnel finite element calculation model considering the synchronous grouting effect is established; (2) stress-strain analysis is conducted on a shield tunnel segment joint, and a stress-strain balance differential equation is established; 3) based on a triangle similarity principle and shield tunnel equivalent flexural rigidity definition, deducing to obtain a shield tunnel equivalent flexural rigidity theoretical solution under the synchronous grouting effect; and 4) quantifying the synchronous grouting sensitive parameters and the equivalent flexural rigidity into a relation matrix [# imgabs0 #, beta, alpha, Z] through calibration, and analyzing the influence of the synchronous grouting layer thickness, the grout elasticity modulus and the maintenance time on the equivalent flexural rigidity of the shield tunnel. According to the method, the equivalent flexural rigidity of the synchronous grouting reinforcement shield tunnel is rapidly calculated, meanwhile, the precision of a calculation result is ensured, the method can be widely applied to reinforcement engineering of the shield tunnel, and guidance is provided for design of the synchronous grouting reinforcement shield tunnel.

Description

Technical Field

[0001] The present invention belongs to the technical field of shield tunnel reinforcement, and particularly relates to a calculation method for the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting. Background Art

[0002] During the construction of a newly built shield tunnel, since the diameter of the shield machine housing is larger than the outer diameter of the tunnel segment, there is a certain gap between the tunnel segment lining and the soil. If synchronous grouting is not carried out in a timely manner, it may cause relatively large ground loss, resulting in diseases such as excessive deformation of the newly built shield tunnel structure and excessive opening of the segment joints. To ensure the smooth construction of the newly built shield tunnel and reduce ground soil loss, synchronous grouting is generally an essential option and is particularly important. Therefore, synchronous grouting is widely used in tunnel reinforcement projects. In actual shield construction, it generally includes two construction processes: tunnel segment installation and synchronous grouting. However, current research mainly focuses on aspects such as the material mechanics properties of synchronous grouting, grouting diffusion mechanism, grouting effect, and grouting detection, and no relevant theoretical research on the equivalent flexural stiffness of the tunnel by synchronous grouting has been found. And the equivalent flexural stiffness of the tunnel is one of the extremely important key parameters for calculating the tunnel deformation. Therefore, to make up for the influence of synchronous grouting on the equivalent flexural stiffness of the shield tunnel, relevant theoretical research needs to be carried out, and it has become an urgent problem to be solved in the tunnel reinforcement project to quickly evaluate the influence of synchronous grouting on the mechanical properties of the shield tunnel.

[0003] Existing research on the mechanical properties of shield tunnels reinforced by synchronous grouting has certain limitations: (1) Theoretical research: The current theory mainly focuses on the research of the equivalent flexural stiffness of unreinforced shield tunnels, and there is no theoretical research on shield tunnels reinforced by synchronous grouting at the present stage, so that the influence of synchronous grouting on the equivalent flexural stiffness of shield tunnels cannot be analyzed quickly and accurately.

[0004] (2) Model test: Researchers have carried out model tests on shield tunnels reinforced by synchronous grouting and found that synchronous grouting can significantly improve the equivalent flexural stiffness of the tunnel, and mainly focus on the research of the properties of synchronous grouting slurry and the filling rate of synchronous grouting. However, the above research is all based on the conclusions drawn from model tests, and analyzing the influence of different synchronous grouting parameters on the equivalent flexural stiffness of the tunnel requires a large amount of experimental cost and time cost, while theoretical research has the advantage of quickly analyzing the influence of synchronous grouting on the equivalent flexural stiffness of shield tunnels compared with model tests. Summary of the Invention

[0005] The object of the present invention is to provide a calculation method for the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting in view of the deficiencies of the prior art. First, a stress-strain equilibrium differential equation under the action of synchronous grouting is established at the joint of the tunnel segment. According to the definition of the equivalent flexural stiffness of the tunnel, the theoretical solution of the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting is derived. While realizing the rapid calculation of the equivalent flexural stiffness of the shield tunnel reinforced by synchronous grouting, the accuracy of the calculation result is ensured, and it can be widely applied to the reinforcement projects of shield tunnels, providing guidance for the design of shield tunnels reinforced by synchronous grouting.

[0006] To achieve the above object, the present invention adopts the following technical solutions.

[0007] A calculation method for the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting, comprising the following steps: Step S1: Establish a finite element calculation model of the shield tunnel considering the action of synchronous grouting, obtain the basic design parameters of the shield tunnel foundation and set the basic assumptions of the model; Step S2: According to the established finite element calculation model of the shield tunnel, conduct stress-strain analysis on the joints of the shield tunnel segments, and establish a stress-strain equilibrium differential equation under the action of synchronous grouting at the joints between the shield tunnel segments based on the stress-strain analysis results; Step S3: According to the stress-strain equilibrium differential equation established in Step S2, combined with the principle of similar triangles and the definition of the equivalent flexural stiffness of the shield tunnel, derive the theoretical solution of the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting; Step S4: Quantify the relationships among the grouting layer thickness, the elastic modulus of the grout, the grouting curing time and the equivalent flexural stiffness of the shield tunnel into a relationship matrix , β , α , Z , in the relationship matrix: is the angle between the neutral axis under the action of synchronous grouting and the geometric center line of the tunnel, β is the influence index of the tunnel structure design parameters, α is the grouting radius r z and the tunnel radius R 0 ratio, Z is the linear stiffness of synchronous grouting; according to the obtained relationship matrix , β , α , Z analyze the influence characteristics of the grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel.

[0008] Specifically, the basic assumptions of the model in Step S1 include: Assumption 1: The cross-section of the tunnel is assumed to be a circular plane, and the deformation of each point on the cross-section is proportional to the distance from the neutral axis. Assumption 2: The longitudinal deformation of the bolts obeys elastic deformation, and the stress-strain of the bolts obeys linear deformation. Assumption 3: The stiffness of the longitudinal bolts is equivalent to a uniformly distributed stiffness along the center line of the tunnel segments. k l ; Assumption 4: The thickness of the synchronous grouting layer is uniformly distributed along the outer diameter of the tunnel, and the stiffness of the synchronous grouting slurry is equivalent to a uniformly distributed stiffness.

[0009] Specifically, in step S2, the stress-strain equilibrium differential equation expression under the action of synchronous grouting at the joints between the segments of the shield tunnel is as follows: The longitudinal axial force equilibrium equation at the joints between the segments of the shield tunnel under the action of synchronous grouting is: ; The bending moment equilibrium equation at the joints between the segments of the shield tunnel under the action of synchronous grouting is: ; ; ; In the above formula, M is the bending moment; ε tt and ε tc are the maximum tensile strain and maximum compressive strain of the concrete segment respectively; ε zt and ε zc are the maximum tensile strain and maximum compressive strain of the synchronous grouting slurry respectively; α is the grouting radius r z and the tunnel radius R 0 ratio; t is the thickness of the tunnel segment; R 0 is the radius from the center line of the tunnel segment to the center point of the circle; h z is the thickness of the synchronous grouting; r z is the grouting radius, that is, the radius from the center line of the synchronous grouting to the center point of the circle; E c is the elastic modulus of the concrete segment; is the angle between the neutral axis under the action of synchronous grouting and the geometric center line of the tunnel; Z is the line stiffness of the synchronous grouting; E z is the elastic modulus of the synchronous grouting.

[0010] Furthermore, the derivation process of the theoretical solution for the equivalent flexural stiffness of the shield tunnel under synchronous grouting in step S3 is as follows: Step S31: Establish the stress equilibrium equation of the local longitudinal bolts and the tunnel segment concrete; ; In the above formula, l l is the length of the longitudinal bolt; δ f is the bolt strain; k l is the uniformly distributed linear stiffness of the longitudinal bolts. The stiffness of the longitudinal bolts is equivalent to the uniformly distributed linear stiffness along the center line of the tunnel segment k l It is expressed as: ; In the above formula, n refers to the number of bolts; A b refers to the cross-sectional area of the bolt; E b refers to the elastic modulus of the bolt.

[0011] Step S32: Establish the deformation coordination equation of the bolts and the tunnel segment concrete, which is expressed as: ; ; In the above formula, θ refers to the bending rotation angle of the tunnel segment; l t is the width of the tunnel segment; Step S33: Based on the equations established in step S31 and step S32, obtain the relationship between the tensile strain and the compressive strain of the tunnel segment concrete; ; Step S34: Based on the principle of similar triangles, further obtain the expressions of the synchronous grouting strain and the compressive strain of the concrete segment: ; ; Step S35: Based on the definition of the equivalent flexural stiffness of the shield tunnel, combine the equations in step S2, step S31 to step S34 to jointly solve the expression of the equivalent flexural stiffness equation of the shield tunnel under synchronous grouting: ; ; In the above formula, The equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting.

[0012] Specifically, in step S4, the relationship between the grouting layer thickness, the elastic modulus of the grout, the grouting curing time and the equivalent flexural stiffness of the shield tunnel is quantified into a relationship matrix , β , α , Z , and the influence characteristics of the synchronous grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel are analyzed. The specific process is as follows: Step S41: By calibration, the relationship between the grouting layer thickness, the elastic modulus of the grout, the grouting curing time and the equivalent flexural stiffness of the shield tunnel is quantified into a relationship matrix , β , α , Z , and the expression of the relationship matrix is as follows: ; In the above formula, is the angle between the neutral axis and the geometric center line of the tunnel under the action of synchronous grouting, which is calibrated by the equilibrium differential equation; β is the influence index of the tunnel structure design parameters, which is calibrated by calculation ; α is the grouting radius r z and the tunnel radius R 0 ratio, that is, the comprehensive influence index of the synchronous grouting thickness and the tunnel geometric parameters, which is calibrated by calculating the synchronous grouting thickness, the segment thickness and the tunnel radius; Z is the synchronous grouting line stiffness, which is calibrated by calculating the synchronous grouting thickness and the elastic modulus; Step S42: According to the calibrated quantitative relationship matrix , β , α , Z , the influence characteristics of the synchronous grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel are analyzed.

[0013] Furthermore, in step S42, analyzing the influence characteristics of the synchronous grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel includes: The equivalent flexural stiffness of a shield tunnel is linearly and positively correlated with the thickness of the synchronous grouting layer and the elastic modulus of the grout; the equivalent flexural stiffness of a shield tunnel has a non-linear positive correlation with the grouting curing time; increasing the synchronous grouting thickness, grouting elastic modulus, and grouting curing time can significantly improve the equivalent flexural stiffness of a shield tunnel, reduce the deformation of the shield tunnel, and enhance the stability of the shield tunnel; when using synchronous grouting to reinforce shield tunnels with different structural designs, due to the different influence indices of tunnel structural design parameters, the reinforcement effects are also different.

[0014] Compared with the prior art, the present invention has the following beneficial effects: 1. By establishing a stress-strain equilibrium differential equation under synchronous grouting and deriving the theoretical solution of the equivalent flexural stiffness of a shield tunnel under synchronous grouting through integral transformation, the present invention generates a function of the theoretical solution of the equivalent flexural stiffness of a shield tunnel under synchronous grouting and, with the help of technical means such as computer programming, while achieving rapid calculation of the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting, ensures the accuracy of the calculation results and can be widely applied to the reinforcement projects of shield tunnels.

[0015] 2. By calibrating, the present invention quantifies the relationship between the grouting layer thickness, grout elastic modulus, grouting curing time and the equivalent flexural stiffness of a shield tunnel into a relationship matrix. Based on the relationship matrix, the influence characteristics of the synchronous grouting layer thickness, grout elastic modulus, and grouting curing time on the equivalent flexural stiffness of a shield tunnel can be analyzed, providing guidance for the design of synchronous grouting to reinforce shield tunnels. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] In order to more clearly illustrate the technical solutions of the embodiments of the present disclosure, the following will briefly introduce the drawings required for use in the embodiments. It should be understood that the following drawings only show some embodiments of the present disclosure and should not be regarded as limiting the scope. For those of ordinary skill in the art, other related drawings can be obtained based on these drawings without creative efforts.

[0017] Figure 1 is a flowchart of a method for calculating the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting according to the present invention; Figure 2 is a schematic diagram of stress-strain analysis at the joint of the segment in the embodiment of the present invention; Figure 3 is a schematic diagram of the equivalent stiffness of longitudinal bolts being equivalent to a uniform distribution stiffness along the center line of the tunnel segment in the embodiment of the present invention; Figure 4 is a schematic diagram for comparing and verifying the theoretical calculation results and numerical simulation calculation results of the present invention; In the figure: 1. Bolt; 2. Tunnel segment; 3. Synchronous grouting layer. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0018] For the convenience of those of ordinary skill in the art to understand and implement the present invention, the following provides a detailed description of each step of the method proposed by the present invention. It should be understood that these embodiments are only used to illustrate the present invention and not to limit the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art can make various changes or modifications to the present invention, and these equivalent forms also fall within the scope defined by the appended claims of this application.

[0019] As Figure 1 shown, the present invention discloses a method for calculating the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting, including the following steps: Step S1: Establish a finite element calculation model of the shield tunnel considering the effect of synchronous grouting, obtain the basic design parameters of the shield tunnel foundation and set the basic assumptions of the model; Step S2: According to the established finite element calculation model of the shield tunnel, conduct stress-strain analysis at the segment joints of the shield tunnel, and establish a stress-strain equilibrium differential equation under the action of synchronous grouting at the segment joints of the shield tunnel based on the stress-strain analysis results; Step S3: According to the stress-strain equilibrium differential equation established in Step S2, combined with the principle of similar triangles and the definition of the equivalent flexural stiffness of the shield tunnel, derive the theoretical solution of the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting; Step S4: Quantify the relationships between the grouting layer thickness, the elastic modulus of the grout, the grouting curing time and the equivalent flexural stiffness of the shield tunnel into a relationship matrix , β , α , Z , in the relationship matrix: is the angle between the neutral axis under the action of synchronous grouting and the geometric center line of the tunnel, β is the influence index of the tunnel structure design parameters, α is the grouting radius r z and the tunnel radius R 0 ratio, Z is the linear stiffness of synchronous grouting; according to the obtained relationship matrix , β , α , Z analyze the influence characteristics of the grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel.

[0020] Specifically, the basic assumptions of the model in Step S1 include: Assumption 1: Assume that the tunnel cross-section is a circular plane, and the deformation of each point on the cross-section is proportional to the distance from the neutral axis; Assumption 2: The longitudinal deformation of the bolt obeys elastic deformation, and the stress-strain of the bolt obeys linear deformation; Assumption 3: The stiffness of the longitudinal bolts is equivalent to a uniformly distributed stiffness along the center line of the tunnel segment. k l ; Assumption 4: The thickness of the synchronous grouting layer is uniformly distributed along the outer diameter of the tunnel, and the stiffness of the synchronous grouting slurry is equivalent to a uniformly distributed stiffness.

[0021] As Figure 2 shown in the stress-strain analysis schematic diagram of the segment connection in this embodiment, according to the stress-strain analysis of the segment connection, the stress-strain equilibrium differential equation expression under the action of synchronous grouting at the joint of the shield tunnel segments in step S2 is as follows: The longitudinal axial force balance equation at the joint of the shield tunnel segments under the action of synchronous grouting is: ; The bending moment balance equation at the joint of the shield tunnel segments under the action of synchronous grouting is: ; ; ; In the above formula, M is the bending moment; ε tt and ε tc are the maximum tensile strain and maximum compressive strain of the concrete segment respectively; ε zt and ε zc are the maximum tensile strain and maximum compressive strain of the synchronous grouting slurry respectively; α is the grouting radius r z and the tunnel radius R 0 ratio; t is the thickness of the tunnel segment; R 0 is the radius from the center line of the tunnel segment to the center point of the circle; h z is the thickness of the synchronous grouting; r z is the grouting radius, that is, the radius from the center line of the synchronous grouting to the center point of the circle; E c is the elastic modulus of the concrete segment; is the angle between the neutral axis under the action of synchronous grouting and the geometric center line of the tunnel; Z is the linear stiffness of the synchronous grouting; E z is the elastic modulus of the synchronous grouting.

[0022] Furthermore, the derivation process of the theoretical solution of the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting in step S3 is as follows: Step S31: Establish the stress equilibrium equation of the local longitudinal bolts and the tunnel segment concrete; ; In the above formula, l l is the length of the longitudinal bolt; δ f is the bolt strain; k l is the uniformly distributed linear stiffness of the longitudinal bolts. As shown in Figure 3 , the stiffness of the longitudinal bolts is equivalent to the uniformly distributed linear stiffness along the center line of the tunnel segment k l is expressed as: ; In the above formula, n refers to the number of bolts; A b refers to the cross-sectional area of the bolt; E b refers to the elastic modulus of the bolt.

[0023] Step S32: Establish the deformation coordination equation of the bolts and the tunnel segment concrete, which is expressed as: ; ; In the above formula, θ refers to the bending rotation angle of the tunnel segment; l t is the width of the tunnel segment; Step S33: Based on the equations established in step S31 and step S32, obtain the relationship between the tensile strain and the compressive strain of the tunnel segment concrete; ; Step S34: Based on the principle of similar triangles, further obtain the expressions of the synchronous grouting strain and the compressive strain of the concrete segment: ; ; Step S35: Based on the definition of the equivalent flexural stiffness of the shield tunnel, combined with the equations in step S2 and steps S31 to S34, jointly solve the expression of the equivalent flexural stiffness equation of the shield tunnel under the action of synchronous grouting: ; ; In the above formula, is the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting.

[0024] Specifically, in step S4, the relationship between the grouting layer thickness, the elastic modulus of the grout, the grouting curing time and the equivalent flexural stiffness of the shield tunnel is quantified as a relationship matrix , β , α , Z , and the influence characteristics of the synchronous grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel are analyzed. The specific process is as follows: Step S41: Quantify the relationship between the grouting layer thickness, the elastic modulus of the grout, the grouting curing time and the equivalent flexural stiffness of the shield tunnel as a relationship matrix , β , α , Z through calibration. The expression of the relationship matrix is as follows: ; In the above formula, is the angle between the neutral axis and the geometric center line of the tunnel under the action of synchronous grouting, which is calibrated through the equilibrium differential equation; β is the influence index of the tunnel structure design parameters, which is calibrated through calculation ; α is the grouting radius r z and the tunnel radius R 0 ratio, that is, the comprehensive influence index of the synchronous grouting thickness and the tunnel geometric parameters, which is calibrated through the synchronous grouting thickness, the segment thickness and the tunnel radius; Z is the synchronous grouting line stiffness, which is calibrated through the synchronous grouting thickness and the elastic modulus; Step S42: Analyze the influence characteristics of the synchronous grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel according to the calibrated quantitative relationship matrix , β , α , Z .

[0025] Furthermore, in step S42, analyzing the influence characteristics of the synchronous grouting layer thickness, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel includes: The equivalent flexural stiffness of the shield tunnel is linearly and positively correlated with the thickness of the synchronous grouting layer and the elastic modulus of the grout; the equivalent flexural stiffness of the shield tunnel has a non-linear positive correlation with the grouting curing time; increasing the synchronous grouting thickness, grouting elastic modulus, and grouting curing time can significantly improve the equivalent flexural stiffness of the shield tunnel, reduce the deformation of the shield tunnel and enhance the stability of the shield tunnel; when using synchronous grouting to reinforce shield tunnels with different structural designs, due to the different influence indexes of the tunnel structural design parameters, the reinforcement effects are also different: The following is a specific example to compare the method of the present invention with the traditional theoretical calculation method and the finite element analysis method to verify the scientificity and reliability of the method of the present invention.

[0026] Example 1: Comparison with the traditional classical equivalent flexural stiffness theoretical calculation method; Since there is currently no theoretical calculation method for the equivalent flexural stiffness of a tunnel under the action of synchronous grouting reinforcement, in order to ensure the consistency of the calculation results, first, the formula derived in the present invention is degenerated into the theoretical solution of the equivalent flexural stiffness of a tunnel without synchronous grouting, that is, let Z = 0 (degenerate the theoretical solution without synchronous grouting); then introduce the equivalent flexural stiffness in the classical equivalent flexural stiffness theoretical calculation method, and its expression is: ; ; ; In the above formula: A c refers to the cross-sectional area of the shield tunnel; I c refers to the moment of inertia of the homogeneous shield tunnel, I c = π · t · R 0 3 , K b refers to the total tensile stiffness of the longitudinal bolts of the shield tunnel; Finally, substitute the shield tunnel design parameters in Table 1 below into the above formula and the special theoretical solution of synchronous grouting respectively, and list the two calculation results in Table 2 below.

[0027] Table 1. Design parameters of the shield tunnel segments and bolts ; Table 2. Differences in calculation results ; As can be seen from Table 2 above, the differences between the theoretical solution calculation results proposed by the method of the present invention and the classical theoretical calculation results are all less than 1%, thus verifying the reliability of the calculation method of the present invention.

[0028] Example 2: Comparison with numerical simulation calculation To further verify the reliability of the theoretical solution calculation model proposed by the present invention, the calculation results of the above-mentioned theoretical solution are compared and analyzed with the numerical simulation calculation results. There are numerical simulation analyses of the equivalent flexural stiffness of shield tunnels with and without synchronous grouting in existing literature. The design parameters of the shield tunnels in the simulation analysis are shown in Table 3 below. To ensure the consistency of the calculation results, the design parameters in Table 3 are first substituted into Z the theoretical solution with =0; secondly, the equivalent flexural stiffness of the tunnel with different numbers of bolts (the equivalent flexural stiffness of the tunnel without synchronous grouting) is calculated respectively; as Figure 4 shown; the comparison and verification diagram of the theoretical calculation results of the present invention and the numerical simulation calculation results is obtained.

[0029] Table 3. Design parameters of segment and bolt of shield tunnel ; It can be seen from Figure 4 that the theoretical calculation results of the present invention are in good agreement with the numerical simulation calculation results. The equivalent flexural stiffness of the shield tunnel has a linear positive correlation with the number of longitudinal bolts of the shield tunnel, and the error between the calculation results of the method of the present invention and the numerical simulation calculation results is 4.6% and the trends of the calculation results are highly consistent, further verifying the reliability of the theoretical solution calculation model proposed by the present invention.

[0030] The above is only the preferred embodiment of the present invention, and it is not a limitation to the present invention in other forms. Any person skilled in the art may use the technical content disclosed above to make changes or modifications into equivalent embodiments of equivalent changes. However, any simple modification, equivalent change and modification made to the above embodiments based on the technical essence of the present invention without departing from the technical solution content of the present invention still belong to the protection scope of the technical solution of the present invention.

Claims

1. A method for calculating the equivalent bending stiffness of a synchronous grouting reinforced shield tunnel, characterized in that: The following steps are involved: Step S1, establishing a finite element calculation model of a shield tunnel considering the effect of synchronous grouting, obtaining the basic design parameters of the shield tunnel and establishing basic assumptions of the model; Step S2, according to the established shield tunnel finite element calculation model, perform stress-strain analysis on the shield tunnel segment joints, and establish a stress-strain equilibrium differential equation under the action of synchronous grouting at the shield tunnel segment joints based on the stress-strain analysis results; Step S3, according to the stress-strain equilibrium differential equation established in step S2, combined with the triangle similarity principle and the definition of equivalent bending stiffness of the shield tunnel, derive a theoretical solution of the equivalent bending stiffness of the shield tunnel under the action of synchronous grouting; Step S4, quantify the relationship between the grouting layer thickness, grouting elastic modulus, grouting curing time and shield tunnel equivalent bending stiffness into a relationship matrix [ , β , α , Z ], in the relationship matrix: is the angle between the neutral axis and the geometric center line of the tunnel under synchronous grouting, β is the influence index of tunnel structure design parameters, α is the grouting radius r z With tunnel radius R 0 The ratio, Z is the synchronous grouting line stiffness; according to the obtained relationship matrix [ , β , α , Z ]The influence of synchronous grouting layer thickness, slurry elastic modulus and grouting curing time on the equivalent flexural stiffness of shield tunnel is analyzed.

2. The method for calculating equivalent bending stiffness of a synchronous grouting reinforced shield tunnel according to claim 1 is characterized in that: The basic assumptions of the model in step S1 include: Assumption 1: The tunnel cross section is assumed to be a circular plane, and the deformation of each point on the cross section is proportional to the distance from the neutral axis; Assumption 2: The longitudinal deformation of the bolt obeys elastic deformation, and the stress-strain of the bolt obeys linear deformation; Assumption 3: The longitudinal bolt stiffness is equivalent to the uniformly distributed stiffness along the centerline of the tunnel segment k l ; Assumption 4: The thickness of the synchronous grouting layer is uniformly distributed along the outer diameter of the tunnel, and the stiffness of the synchronous grouting slurry is equivalent to the uniformly distributed stiffness.

3. The method for calculating equivalent bending stiffness of a synchronous grouting reinforced shield tunnel according to claim 1 is characterized in that: In step S2, the stress-strain equilibrium differential equation expression under the action of synchronous grouting is established at the joint between the shield tunnel segments as follows: The longitudinal axial force balance equation at the joint between shield tunnel segments under synchronous grouting is: ； The moment equilibrium equation at the joint between shield tunnel segments under synchronous grouting is: ； ； ； In the above formula, M is the bending moment; ε tt and ε tc are the maximum tensile strain and maximum compressive strain of the concrete segment, respectively; ε zt and ε zc are the maximum tensile strain and maximum compressive strain of synchronous grouting slurry, respectively; α is the grouting radius r z With tunnel radius R 0 ratio; t is the thickness of the tunnel segment; R 0 is the radius from the center line of the tunnel segment to the center point of the circle; h z is the thickness of synchronous grouting; r z is the grouting radius, i.e. the radius from the synchronous grouting centerline to the center point of the circle; E c is the elastic modulus of the concrete segment; It is the angle between the neutral axis and the geometric center line of the tunnel under the action of synchronous grouting; Z is the stiffness of synchronous grouting line; E z is the elastic modulus of synchronous grouting.

4. The method for calculating equivalent bending stiffness of a synchronous grouting reinforced shield tunnel according to claim 3 is characterized in that: The derivation process of the theoretical solution of the equivalent bending stiffness of the shield tunnel under the synchronous grouting in step S3 is as follows: Step S31, establishing a stress balance equation between local longitudinal bolts and tunnel segment concrete; ； In the above formula, l l is the longitudinal bolt length; δ f is the bolt strain; k l The longitudinal bolt stiffness is equivalent to the uniformly distributed linear stiffness along the center line of the tunnel segment. k l It is expressed as: ； In the above formula, n Refers to the number of bolts; A b Refers to the cross-sectional area of ​​the bolt; E b Refers to the elastic modulus of the bolt; Step S32: Establish the deformation coordination equation between the bolt and the tunnel segment concrete as follows: ； ； In the above formula, θ Refers to the bending rotation angle of the tunnel segment; l t is the tunnel segment width; Step S33, obtaining a relationship between the tensile strain and the compressive strain of the tunnel segment concrete based on the equations established in Step S31 and Step S32; ； Step S34: Based on the triangle similarity principle, the expression of synchronous grouting strain and concrete segment compressive strain is further obtained: ； ； Step S35, based on the definition of equivalent bending stiffness of the shield tunnel, the equations in step S2, step S31 to step S34 are combined to jointly solve the equivalent bending stiffness equation expression of the shield tunnel under the action of synchronous grouting: ； ； In the above formula, is the equivalent bending stiffness of the shield tunnel under the action of synchronous grouting.

5. The method for calculating equivalent bending stiffness of a synchronous grouting reinforced shield tunnel according to claim 1 is characterized in that: In step S4, the relationship between the grouting layer thickness, grout elastic modulus, grouting curing time and the equivalent bending stiffness of the shield tunnel is quantified as a relationship matrix [ , β , α , Z ], the influence of synchronous grouting layer thickness, grout elastic modulus, and grouting curing time on the equivalent bending stiffness of shield tunnels is analyzed. The specific process is as follows: Step S41: quantify the relationship between the grouting layer thickness, grout elastic modulus, grouting curing time and the equivalent bending stiffness of the shield tunnel into a relationship matrix [ , β , α , Z ], the expression of the relationship matrix is ​​as follows: ； In the above formula, It is the angle between the neutral axis and the geometric center line of the tunnel under the action of synchronous grouting, which is calibrated by the equilibrium differential equation; β The influence index of tunnel structure design parameters is calibrated by calculation ; α is the grouting radius r z With tunnel radius R 0 The ratio of the synchronous grouting thickness to the tunnel geometric parameters is the comprehensive influence index, which is calculated and calibrated by the synchronous grouting thickness, the segment thickness and the tunnel radius. Z The stiffness of the synchronous grouting line is calibrated by calculating the synchronous grouting thickness and elastic modulus; Step S42: According to the calibrated quantization relationship matrix [ , β , α , Z ], and analyze the influence of synchronous grouting layer thickness, grouting elastic modulus, and grouting curing time on the equivalent flexural stiffness of shield tunnels.

6. The method for calculating equivalent bending stiffness of a synchronous grouting reinforced shield tunnel according to claim 5, characterized in that: In step S42, the influence characteristics of the thickness of the synchronous grouting layer, the elastic modulus of the grouting liquid, and the grouting curing time on the equivalent bending stiffness of the shield tunnel are analyzed, including: The equivalent bending stiffness of the shield tunnel is linearly positively correlated with the thickness of the synchronous grouting layer and the elastic modulus of the slurry; the equivalent bending stiffness of the shield tunnel is nonlinearly positively correlated with the grouting curing time; increasing the synchronous grouting thickness, the grouting elastic modulus and the grouting curing time can significantly improve the equivalent bending stiffness of the shield tunnel, reduce the deformation of the shield tunnel and improve the stability of the shield tunnel.

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