An equivalent flexural stiffness calculation method for synchronous grouting reinforcement of shield tunnels

Abstract

The present invention discloses a calculation method for the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting, including: 1) establishing a finite element calculation model of the shield tunnel considering the effect of synchronous grouting; 2) conducting stress-strain analysis at the segment joints of the shield tunnel and establishing a stress-strain equilibrium differential equation; 3) deriving the theoretical solution of the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting based on the principle of similar triangles and the definition of the equivalent flexural stiffness of the shield tunnel; 4) quantifying the synchronous grouting sensitive parameters and the equivalent flexural stiffness as a relationship matrix through calibration β , α , Z , and analyzing the influence of the thickness of the synchronous grouting layer, the elastic modulus of the grout, and the curing time on the equivalent flexural stiffness of the shield tunnel. The method of the present invention can quickly calculate the equivalent flexural stiffness of the shield tunnel reinforced by synchronous grouting while ensuring the accuracy of the calculation results, and can be widely applied to the reinforcement projects of shield tunnels, providing guidance for the design of shield tunnels reinforced by synchronous grouting.

Description

Technical Field

[0001] The 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 shell is larger than that 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 time, it may cause large ground loss, resulting in excessive deformation of the newly-built shield tunnel structure, excessive opening of the segment joints and other disease problems. 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 the material mechanical properties, grouting diffusion mechanism, grouting effect and grouting detection of synchronous grouting, and no relevant theoretical research on the equivalent flexural stiffness of the tunnel by synchronous grouting has been found. The equivalent flexural stiffness of the tunnel is one of the extremely important key parameters for calculating 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:

[0004] (1) Theoretical research: The current theory mainly focuses on the research of the equivalent flexural stiffness of un-reinforced shield tunnels, and there is no theoretical research on shield tunnels reinforced by synchronous grouting at present, so that the influence of synchronous grouting on the equivalent flexural stiffness of shield tunnels cannot be analyzed quickly and accurately.

[0005] (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 performance of synchronous grouting slurry and the filling rate of synchronous grouting. However, the above research is based on the conclusions obtained 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 test cost and time cost. 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

[0006] 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 segments. 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.

[0007] To achieve the above object, the present invention adopts the following technical solutions.

[0008] A calculation method for the equivalent flexural stiffness of a shield tunnel reinforced by synchronous grouting, comprising the following steps:

[0009] 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;

[0010] 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;

[0011] 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;

[0012] 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 synchronous grouting line stiffness; 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.

[0013] Specifically, the basic assumptions of the model in step S1 include:

[0014] 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;

[0015] Assumption 2: The longitudinal deformation of the bolts obeys elastic deformation, and the stress-strain of the bolts obeys linear deformation;

[0016] Assumption 3: The longitudinal bolt stiffness is equivalent to a uniformly distributed stiffness along the center line of the tunnel segment; k l ;

[0017] 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.

[0018] Specifically, the stress-strain equilibrium differential equation expression under the action of synchronous grouting is established at the joints of the shield tunnel segments in step S2 as follows:

[0019] The longitudinal axial force balance equation at the joints of the shield tunnel segments under the action of synchronous grouting is:

[0020] ;

[0021] The bending moment balance equation at the joints of the shield tunnel segments under the action of synchronous grouting is:

[0022] ;

[0023] ;

[0024] ;

[0025] 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; hz is the thickness of the synchronous grouting; r z is the grouting radius, i.e., 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 line; E z is the elastic modulus of the synchronous grouting.

[0026] 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:

[0027] Step S31: Establish the stress equilibrium equation of the local longitudinal bolts and the concrete of the tunnel segment;

[0028] ;

[0029] 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:

[0030] ;

[0031] 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.

[0032] Step S32: Establish the deformation coordination equation of the bolts and the concrete of the tunnel segment, which is expressed as:

[0033] ;

[0034] ;

[0035] In the above formula, θ refers to the bending rotation angle of the tunnel segment; l t is the width of the tunnel segment;

[0036] Step S33: Obtain the relationship between the tensile strain and compressive strain of the tunnel segment concrete based on the equations established in Step S31 and Step S32;

[0037] ;

[0038] Step S34: Further obtain the expression of the synchronous grouting strain and the compressive strain of the concrete segment based on the principle of similar triangles:

[0039] ;

[0040] ;

[0041] Step S35: Based on the definition of the equivalent flexural stiffness of the shield tunnel, combined with the equations in Step S2, Step S31 to Step S34, jointly solve the expression of the equivalent flexural stiffness equation of the shield tunnel under the action of synchronous grouting:

[0042] ;

[0043] ;

[0044] In the above formula, is the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting.

[0045] 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 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. The specific process is as follows:

[0046] Step S41: Through 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 as a relationship matrix , β , α , Z , and the expression of the relationship matrix is as follows:

[0047] ;

[0048] 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, calibrated through the equilibrium differential equation; β is the influence index of the tunnel structure design parameters, calibrated through calculation ; α is the grouting radius rz Ratio to the tunnel radius R 0 That is, the comprehensive influence index of the synchronous grouting thickness and the tunnel geometric parameters 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;

[0049] Step S42. According to the calibrated quantization relation matrix , β , α , Z , analyze the influence characteristics of the synchronous grouting layer thickness, the slurry elastic modulus, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel.

[0050] Furthermore, in step S42, analyzing the influence characteristics of the synchronous grouting layer thickness, the slurry elastic modulus, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel includes:

[0051] The equivalent flexural stiffness of the shield tunnel is linearly and positively correlated with the synchronous grouting layer thickness and the slurry elastic modulus; the equivalent flexural stiffness of the shield tunnel is non-linearly and 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 flexural stiffness of the shield tunnel, reduce the deformation of the shield tunnel and enhance the stability of the shield tunnel; when synchronously grouting and reinforcing shield tunnels with different structural designs, due to the different influence indexes of the tunnel structural design parameters, the reinforcement effects are also different.

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] 1. The method of the present invention establishes a stress-strain balance differential equation under synchronous grouting, and derives a theoretical solution of the equivalent flexural stiffness of the shield tunnel under synchronous grouting through integral transformation. The theoretical solution of the equivalent flexural stiffness of the shield tunnel under synchronous grouting is generated into a function and with the help of technical means such as computer programming, while realizing the rapid calculation of the equivalent flexural stiffness of the shield tunnel reinforced by synchronous grouting, ensuring the accuracy of the calculation results, and being widely applicable to the reinforcement projects of shield tunnels.

[0054] 2. The method of the present invention calibrates and quantifies the relationship between the grouting layer thickness, the slurry elastic modulus, the grouting curing time and the equivalent flexural stiffness of the shield tunnel into a relationship matrix. Based on the relationship matrix, the influence characteristics of the synchronous grouting layer thickness, the slurry elastic modulus, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel can be analyzed, providing guidance for the design of the shield tunnel reinforced by synchronous grouting. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] To more clearly illustrate the technical solutions of the embodiments of the present disclosure, the following briefly introduces the accompanying drawings required for 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, without creative efforts, other related drawings can also be obtained based on these drawings.

[0056] 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;

[0057] Figure 2 is a schematic diagram of stress-strain analysis at the joint of segment linings in an embodiment of the present invention;

[0058] Figure 3 is a schematic diagram of equivalent stiffness of longitudinal bolts as a uniform distribution stiffness along the center line of tunnel segment linings in an embodiment of the present invention;

[0059] Figure 4 is a schematic diagram of comparison and verification between the theoretical calculation results and numerical simulation results of the present invention;

[0060] In the figure: 1, bolt; 2, tunnel segment lining; 3, synchronous grouting layer. Specific embodiments

[0061] To facilitate the understanding and implementation of the present invention by those of ordinary skill in the art, the following details 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.

[0062] 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:

[0063] 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;

[0064] 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 segment linings, and establish a stress-strain equilibrium differential equation under the action of synchronous grouting at the joints between the shield tunnel segment linings based on the stress-strain analysis results;

[0065] Step S3: Based on 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, the theoretical solution of the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting is derived;

[0066] 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 , where in the relationship matrix: is the angle between the neutral axis and the geometric center line of the tunnel under the action of synchronous grouting, β is the influence index of the tunnel structure design parameters, α is the grouting radius r z ratio of R 0 to the tunnel radius Z is the linear stiffness of synchronous grouting; according to the obtained relationship matrix , β , α , Z , 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.

[0067] Specifically, the basic assumptions in Step S1 include:

[0068] 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;

[0069] Assumption 2: The longitudinal deformation of the bolts obeys elastic deformation, and the bolt stress-strain obeys linear deformation;

[0070] Assumption 3: The longitudinal bolt stiffness is equivalent to a uniformly distributed stiffness along the center line of the tunnel segment k l ;

[0071] Assumption 4: The synchronous grouting layer thickness is uniformly distributed along the outer diameter of the tunnel, and the synchronous grouting slurry stiffness is equivalent to a uniformly distributed stiffness.

[0072] As Figure 2 shown is the stress-strain analysis schematic diagram of the segment joint in this embodiment. According to the stress-strain analysis of the segment joint, the stress-strain equilibrium differential equation expression under the action of synchronous grouting at the segment joint of the shield tunnel in Step S2 is as follows:

[0073] The longitudinal axial force balance equation at the segment joint of the shield tunnel under the action of synchronous grouting is:

[0074] ;

[0075] The bending moment balance equation at the segment joint of the shield tunnel under the action of synchronous grouting is as follows:

[0076] ;

[0077] ;

[0078] ;

[0079] 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.

[0080] 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:

[0081] Step S31: Establish the stress balance equation of the local longitudinal bolt and the tunnel segment concrete;

[0082] ;

[0083] In the above formula, l l is the length of the longitudinal bolt; δ f is the bolt strain; kl is the linear stiffness of longitudinal bolts evenly distributed. As Figure 3 shown, the stiffness of longitudinal bolts is equivalent to the linear stiffness evenly distributed along the center line of the tunnel segment. k l It is expressed as:

[0084] ;

[0085] 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.

[0086] Step S32: Establish the deformation coordination equation between the bolt and the tunnel segment concrete, which is expressed as:

[0087] ;

[0088] ;

[0089] In the above formula, θ refers to the bending rotation angle of the tunnel segment; l t is the width of the tunnel segment;

[0090] Step S33: Based on the equations established in Step S31 and Step S32, obtain the relationship between the tensile strain and compressive strain of the tunnel segment concrete;

[0091] ;

[0092] 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:

[0093] ;

[0094] ;

[0095] Step S35: Based on the definition of the equivalent flexural stiffness of the shield tunnel, combined with the equations in Step S2, Step S31 to Step S34, jointly solve the expression of the equivalent flexural stiffness equation of the shield tunnel under the action of synchronous grouting:

[0096] ;

[0097] ;

[0098] In the above formula, is the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting.

[0099] 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:

[0100] 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 into a relationship matrix through calibration , β , α , Z , and the expression of the relationship matrix is as follows:

[0101] ;

[0102] In the above formula, is the angle between the neutral axis under synchronous grouting and the geometric center line of the tunnel, 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;

[0103] Step S42: According to the calibrated quantitative relationship matrix , β , α , Z , 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.

[0104] 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:

[0105] 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 tunnel structure design parameters, the reinforcement effects are also different:

[0106] Next, through specific numerical examples, the method of the present invention will be compared 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.

[0107] Example 1: Comparison with the traditional classical equivalent flexural stiffness theoretical calculation method;

[0108] 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:

[0109] ;

[0110] ;

[0111] ;

[0112] 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;

[0113] 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.

[0114] Table 1. Design parameters of the shield tunnel segments and bolts

[0115] ;

[0116] Table 2. Differences in calculation results

[0117] ;

[0118] As can be seen from Table 2 above, the differences between the calculation results of the theoretical solution proposed by the method of the present invention and the calculation results of the classical theory are all less than 1%, thus confirming the reliability of the calculation method of the present invention.

[0119] Example 2: Comparison with numerical simulation calculation

[0120] 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 calculation results of numerical simulation. 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 tunnels with different numbers of bolts (the equivalent flexural stiffness of tunnels without synchronous grouting) is calculated respectively; as Figure 4 shown; the comparison and verification schematic diagram of the theoretical calculation results of the present invention and the numerical simulation calculation results is obtained.

[0121] Table 3. Design parameters of segment and bolt of shield tunnel

[0122] ;

[0123] From Figure 4 it can be seen that the theoretical calculation results of the present invention are highly consistent 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.

[0124] The above is only a preferred embodiment of the present invention, and it is not intended to limit 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 with 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 fall within the protection scope of the technical solution of the present invention.

Claims

1. A calculation method for the equivalent flexural stiffness of synchronous grouting to reinforce shield tunnels, characterized in that It includes the following steps: Step S1: Establish a finite element calculation model of a shield tunnel considering the effect of synchronous grouting, obtain the basic design parameters of the shield tunnel foundation, and set up 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 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, deduce the theoretical solution of the equivalent flexural stiffness of the shield tunnel under the action of synchronous grouting; Step S4: 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 into a relationship matrix , β , α , Z . In the relationship matrix: is the angle between the neutral axis under 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 synchronous grouting line stiffness; According to the obtained relationship matrix , β , α , Z , 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.

2. The equivalent flexural stiffness calculation method for synchronous grouting reinforcement of shield tunnels according to claim 1, 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 bolts obeys elastic deformation, and the stress-strain of the bolts obeys linear deformation; Assumption 3: The stiffness of longitudinal bolts is equivalent to a uniform stiffness distribution along the center line of tunnel segments k l ; Assumption 4: The thickness of the synchronous grouting layer is evenly distributed along the outer diameter of the tunnel, and the stiffness of the synchronous grouting slurry is equivalent to a uniformly distributed stiffness.

3. The equivalent flexural stiffness calculation method for synchronous grouting reinforcement of shield tunnels according to claim 1, wherein, The expression of the stress-strain equilibrium differential equation under the action of synchronous grouting at the segment joints of the shield tunnel in Step S2 is as follows: The longitudinal axial force balance equation at the segment joints of the shield tunnel under the action of synchronous grouting is: ； The bending moment balance equation at the segment joints 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 simultaneous 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 simultaneous grouting; r z is the grouting radius, that is, the radius from the center line of the simultaneous 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 the simultaneous grouting and the geometric center line of the tunnel; Z is the line stiffness of the simultaneous grouting; E z is the elastic modulus of the simultaneous grouting.

4. The equivalent flexural stiffness calculation method for synchronous grouting reinforcement of shield tunnels according to claim 3, characterized in that 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 a stress equilibrium equation between the local longitudinal bolts and the concrete of the tunnel segments; ； 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 longitudinal bolt stiffness 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; Step S32: Establish a deformation coordination equation between the bolts and the concrete of the tunnel segments, which is expressed as: ； ； In the above formula, θ refers to the bending and 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 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 segments; ； ； 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.

5. The equivalent flexural stiffness calculation method for synchronous grouting reinforcement of shield tunnels according to claim 1, characterized in that 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 through calibration into a relationship matrix , β , α , Z , and the expression of the relationship matrix is as follows: ； In the above formula, is the included angle between the neutral axis under the action of synchronous grouting and the geometric center line of the tunnel, 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 quantization relation matrix , β , α , Z , analyze the influence characteristics of the thickness of the synchronous grouting layer, the elastic modulus of the grout, and the grouting curing time on the equivalent flexural stiffness of the shield tunnel.

6. The equivalent flexural stiffness calculation method for synchronous grouting reinforcement of shield tunnels according to claim 5, characterized in that, In Step S42, analyze the influence characteristics of the synchronous grouting layer thickness, slurry elastic modulus, and grouting curing time on the equivalent flexural stiffness of the shield tunnel, including: The equivalent flexural stiffness of the shield tunnel is linearly and positively correlated with the synchronous grouting layer thickness and slurry elastic modulus; the equivalent flexural stiffness of the shield tunnel is non-linearly and positively correlated 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.

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