Calculation method of longitudinal mechanical response of shield tunnel based on weak connection effect between rings

Abstract

The invention discloses a method for calculating the longitudinal mechanical response of a shield tunnel based on the weak connection effect between rings, including a shield tunnel, wherein the shield tunnel is composed of a plurality of segments and inter-ring joints spliced ​​and assembled, and the inter-ring joints use waterproof cushions and / or bolts as connection materials, and the specific steps of the longitudinal mechanical response analysis method are as follows: the inter-ring joints divide the tunnel into short beams in the longitudinal direction, and the lining between two adjacent ring joints is regarded as a tubular hollow continuous short beam; considering the shear deformation of the cross section, the Timoshenko beam theory is used to simulate the mechanical behavior of the lining, and the cross section before and after the deformation conforms to the flat section assumption; the mechanical behavior of the inter-ring joints is simulated by shear and bending two-way joint springs; the interaction between the tunnel and the stratum is simulated by Winkler soil springs. This method is not only applicable to the currently widely used equivalent homogenized continuous tunnel model, but also provides a convenient and fast analysis method for the analysis of the longitudinal mechanical characteristics of the tunnel.

Description

Technical Field

[0001] The invention patent relates to the field of shield tunnels, and in particular to a method for calculating the longitudinal mechanical response of shield tunnels based on the weak connection effect between rings. Background Art

[0002] In the past two decades, shield tunnels have been widely used in China's urban rail transit construction and are an important means to relieve urban traffic congestion. The prefabrication and assembly technology of the segments has greatly facilitated the construction of shield tunnels, but it also inevitably brings about the deformation control and waterproofing problems of the joints between the segments and the lining rings. The existence of joints weakens the longitudinal and transverse stiffness of the shield tunnel and is considered to be the weak link of the tunnel structure. The water and soil pressure around the tunnel forms a "tightening" effect on the transverse section of the tunnel. As long as the segments do not have joint cracks that affect waterproofing, the lateral stability and safety of the lining are guaranteed. Compared with the influence of lateral deformation, the shield tunnel structure is more sensitive to longitudinal displacement. The additional longitudinal displacement of the tunnel lining structure caused by uneven strata, surface overload, the influence of new tunnels, excavation in adjacent areas, etc. is one of the main causes of a series of diseases such as misalignment, excessive opening of the ring joints, leakage or tensile cracks on the surface of the segments, which affects or even endangers the safety and normal use of the tunnel.

[0003] The study of the longitudinal mechanical response of shield tunnels to changes in the surrounding environment has attracted the attention of many scholars. In terms of theoretical analysis, the two-stage idea is usually used to study the impact of changes in surrounding environmental factors such as nearby excavation, ground overload, and new tunnels. That is, first, the existence of existing tunnels is not considered, and the displacement or stress changes at the elevation of the existing tunnel are obtained through elastic theory or empirical formulas, which are used as active loads to apply to the longitudinal mechanical model of the tunnel. As for the mechanical model of the tunnel body, there are mainly two types: longitudinal beam-spring model and equivalent continuation model. The former considers the role of joints by applying a reduction factor to the tunnel stiffness, while the latter simplifies the tunnel in the longitudinal direction into a short beam connected by joint spring units. In current research, the former is the most widely used, and the Euler beam model is usually used to simulate the longitudinal mechanical behavior of the lining. However, when the tunnel is displaced longitudinally, there is not only obvious opening deformation at the inter-ring joint, but also shear misalignment, which is the main component of the longitudinal differential deformation of the tunnel. However, both cannot avoid the problem of the value of the tunnel equivalent stiffness. Taking half of the lining ring on both sides of the joint as the analysis unit, assuming that when the joint is opened, the tension is borne by the bolts in the tension zone and the pressure is completely borne by the segments in the compression zone, the equivalent bending stiffness of the lining is derived based on the force balance relationship and the deformation equivalence principle. Based on similar assumptions, the lateral stiffness effective rate is introduced to consider the influence of the change in cross-sectional shape caused by the lateral deformation of the tunnel on the equivalent bending stiffness of the tunnel. It is believed that the shear force between the rings is mainly borne by the bolts, and the longitudinal equivalent shear stiffness of the tunnel is derived. The Euler beam and Timoshenko beam analytical results of the existing tunnel response problem under the influence of the new tunnel are compared. It is found that the joint opening obtained by the Euler beam model is smaller than that of the Timoshenko beam, while the shear misalignment between the rings is greater than that of the Timoshenko beam. The above research provides a theoretical means for the value of equivalent bending stiffness.

[0004] Although the equivalent homogenization model is simple in concept, easy to calculate, and widely used, it cannot directly reflect the discontinuity of displacement at the joint compared to the longitudinal beam-spring model. The longitudinal beam-spring model directly uses joint springs to simulate the mechanical behavior of the joint, which increases the complexity of the problem and has few related analytical studies. The current analytical solutions are mainly aimed at specific working conditions, and the strata considered are mainly homogeneous or several discontinuous strata, which are too ideal. In actual engineering, the spatial variation characteristics of strata are not only manifested between different strata, but also include the non-uniformity of soil within the same stratum. The current theoretical research on the longitudinal mechanical characteristics of shield tunnels has not fully considered its impact.

[0005] Theoretical analysis of mechanical models ultimately boils down to solving control equations. At present, the means of solving control equations of tunnel longitudinal mechanical models is to solve high-order equations of single variables after elimination. There are many coefficients to be determined, the process is too cumbersome, and the applicable boundary conditions are too simple. It is difficult to deal with the problem of satisfying continuous and discontinuous deformation conditions of many control nodes caused by the longitudinal changes of model properties such as multiple joint nodes and spatial variability of strata.

[0006] Invention patent content

[0007] The technical problem to be solved by the present invention is to provide a method for calculating the longitudinal mechanical response of a shield tunnel based on the weak connection effect between rings. This method is convenient for handling the longitudinal non-uniform characteristics of the system model caused by problems such as discontinuous deformation of multiple longitudinal joint nodes of the tunnel, local joint weakening, and spatial variability of the strata that are difficult to handle with the equivalent homogenized model, providing a convenient and fast analysis method for the analysis of the longitudinal mechanical characteristics of the tunnel.

[0008] To this end, the present invention provides a method for calculating the longitudinal mechanical response of a shield tunnel based on the weak connection effect between rings, including a shield tunnel, wherein the shield tunnel is composed of a plurality of segments and inter-ring joints, and the inter-ring joints are connected by waterproof cushions and / or bolts. The specific steps of the longitudinal mechanical response analysis method are as follows:

[0009] S1. The inter-ring joints divide the tunnel into short beams longitudinally. The lining between two adjacent ring joints is regarded as a tubular hollow continuous short beam.

[0010] A longitudinal mechanical model of a shield tunnel is established, wherein the coordinate axis of the mechanical model is along the longitudinal direction of the tunnel, the z direction is vertically upward, the x direction is horizontal, the vertical displacement is recorded as w, and a short beam between rings is taken as the analysis object;

[0011] S2. The shear deformation of the cross section is simulated by Timoshenko beam theory, and the cross section before and after deformation conforms to the plane cross section assumption;

[0012] S3. The mechanical behavior of the inter-ring joint is simulated by using shear and bending two-way joint springs;

[0013] S4. The interaction between the tunnel and the stratum is simulated by Winkler soil springs.

[0014] Furthermore, in the above step S2, the axial displacement u of any point on the segment cross section is z It can be expressed as:

[0015]

[0016] Among them, ψ is the cross-sectional rotation angle, which consists of two parts:

[0017]

[0018] and

[0019]

[0020] Among them, κ is the shear shape factor, which is generally taken as 0.5 for a circular ring section. Q is the cross-sectional shear force, G is the cross-sectional shear modulus, and A is the cross-sectional area;

[0021] Then the strain at any point on the tunnel cross section is

[0022]

[0023] The bending moment M is expressed as

[0024]

[0025] Where, E is the section elastic modulus, I is the section moment of inertia;

[0026] From equation (1), the energy density of a Timoshenko beam on an elastic foundation subjected to an arbitrary load is written as

[0027]

[0028] According to the principle of minimum potential energy, we get the following equilibrium equation:

[0029] M'=Q,Q'+kw-q z =0 (7)

[0030] Here k is the soil reaction spring stiffness coefficient, q z is the longitudinal load acting on the tunnel;

[0031] Writing equations (1), (2), (3), (5) and (7) in matrix form gives

[0032]

[0033] The physical quantities in the above formula are normalized as follows

[0034]

[0035] The equation of state for the Timoshenko beam on an elastic foundation can be obtained:

[0036]

[0037] in, is the state vector, which consists of the displacement and internal force of the cross section, is the system matrix, which stores the physical and geometric information of the cross section. is the external load vector, which is composed of the vertical external load on the tunnel; their specific forms are shown in the following formula:

[0038]

[0039] The standard solution of formula (10) is

[0040]

[0041] Formula (12) is the coordinates on the tunnel and Two-point state vector and The transitive relationship between them. is the transfer matrix of the system, are the load integral vectors of the system, and their expressions are as follows:

[0042]

[0043] Coefficient g i With the system matrix The formula (13) is applicable to any load. Once the form of the external load is determined, the external load vector can be obtained according to the above formulas.

[0044] Substituting the coordinates of the two ends of the short beam into equation (12), the transfer relationship between the physical quantities at the two ends of the short beam is obtained as follows:

[0045]

[0046] in, and are the state vectors of the cross sections at both ends of the j-th short beam, and

[0047] Furthermore, in step S3, the inter-ring joint can transmit shear force and bending moment, and the magnitude of the internal force is determined by the relative displacement at the joint, satisfying the continuity condition of the force at the joint and the discontinuity condition of the displacement. The tunnel range length is L, and there are n segments in it, so there are n-1 inter-ring joints in total. According to the constitutive relationship of the two-way spring, the physical quantity transfer relationship on both sides of the inter-ring joint is

[0048]

[0049] The above formula can be written in the following matrix form after normalization:

[0050]

[0051] in,

[0052]

[0053]

[0054] The physical quantity transfer relationship at both ends of the short beam and the physical quantity transfer relationship at the inter-ring joint are obtained. Equations (14) and (17) are used for the short beam and the inter-ring joint along the longitudinal direction of the tunnel, and finally the transfer relationship between the state vectors at both ends of the tunnel is obtained as follows:

[0055]

[0056] in,

[0057]

[0058] If the distance between the two ends of the tunnel is far, then both ends are considered free, that is,

[0059]

[0060] The above formula can be written in matrix form as

[0061]

[0062] Modify the boundary conditions at both ends according to actual needs.

[0063] Combining formula (19), we can get the final algebraic equation:

[0064]

[0065] Furthermore, in the above derivation process, no assumptions are made about the soil spring stiffness coefficients within each ring, the load form of the tunnel, and the values ​​of the joint stiffness between each ring. According to actual needs, the soil spring stiffness of each segment ring, the joint stiffness between each joint, and the load on the tunnel are assigned values ​​to take into account the longitudinal heterogeneity of the stratum along the tunnel, the differences in the joint stiffness values ​​at various locations, and the influence of any load form. When the joint between the rings degenerates into a continuous concrete section.

[0066] Furthermore, in the above step S4, as shown in the parameters in Table 1, the longitudinal response of the shield tunnel affected by the stratum heterogeneity under the uniformly distributed load is analyzed, wherein the shield tunnel model is made of polyethylene pipe, and the interaction between the tunnel and the stratum is simulated by two springs with different stiffnesses. The stiffnesses of the springs are 419.70 kN / m and 63.55 kN / m, respectively, and the converted reaction coefficients are 5.08×10 4 kN / m 3 and 7.70×10 3 kN / m 3 .

[0067] In the present invention patent, a general solution to the longitudinal beam-spring model of the shield tunnel under arbitrary load is given, which is conveniently degenerated into the solution of the equivalent continuous model, thus achieving the unification of the two. The correctness of the solution in this paper is verified by comparing with the current indoor test results and the existing analytical solutions. In addition, according to the same equivalent principle, the conversion formula between the equivalent continuous model and the beam-spring model is given. The case study shows that when the equivalent principle is consistent, the results of the two types of models are basically the same, which confirms the effectiveness of the equivalent continuous model from the side. The method in this paper is not only applicable to the currently widely used equivalent homogeneous continuous tunnel model, but also can easily deal with the longitudinal non-uniform characteristics of the system model caused by problems such as discontinuous deformation of multiple longitudinal joint nodes of the tunnel, local joint weakening, and spatial variability of the strata, which are difficult to deal with by the equivalent homogenized model, providing a convenient and fast analysis method for the analysis of the longitudinal mechanical properties of the tunnel. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 Schematic diagram of the tunnel's longitudinal stress-deformation mode and its simplified mechanical model;

[0069] Figure 2 Schematic diagram of the stress and deformation of the short beam element;

[0070] Figure 3 Schematic diagram of the longitudinal transfer of physical quantities based on the state space method;

[0071] Figure 4 Schematic diagram of the vertical settlement of hard-soft alternating strata;

[0072] Figure 5 Schematic diagram of vertical settlement in hard-soft-hard alternating strata. DETAILED DESCRIPTION

[0073] The following will be combined with the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0074] Reference Figure 1-Figure 5 As shown, the present invention provides a method for calculating the longitudinal mechanical response of a shield tunnel based on the weak connection effect between rings, including a shield tunnel, wherein the shield tunnel is composed of a plurality of segments and inter-ring joints, and the inter-ring joints are connected by waterproof cushions and / or bolts. The specific steps of the longitudinal mechanical response analysis method are as follows:

[0075] S1. The inter-ring joints divide the tunnel into short beams longitudinally. The lining between two adjacent ring joints is regarded as a tubular hollow continuous short beam.

[0076] A longitudinal mechanical model of a shield tunnel is established, wherein the coordinate axis of the mechanical model is along the longitudinal direction of the tunnel, the z direction is vertically upward, the x direction is horizontal, the vertical displacement is recorded as w, and a short beam between rings is taken as the analysis object;

[0077] S2. The shear deformation of the cross section is simulated by Timoshenko beam theory, and the cross section before and after deformation conforms to the plane cross section assumption;

[0078] S3. The mechanical behavior of the inter-ring joint is simulated by using shear and bending two-way joint springs;

[0079] The interaction between the S4 tunnel and the stratum is simulated using Winkler soil springs.

[0080] In the above step S2, the axial displacement u of any point on the segment cross section is z It can be expressed as:

[0081]

[0082] Among them, ψ is the cross-sectional rotation angle, which consists of two parts:

[0083]

[0084] and

[0085]

[0086] Among them, κ is the shear shape factor, which is generally taken as 0.5 for a circular ring section. Q is the cross-sectional shear force, G is the cross-sectional shear modulus, and A is the cross-sectional area;

[0087] Then the strain at any point on the tunnel cross section is

[0088]

[0089] The bending moment M is expressed as

[0090]

[0091] Where, E is the section elastic modulus, I is the section moment of inertia;

[0092] From equation (1), the energy density of a Timoshenko beam on an elastic foundation subjected to an arbitrary load is written as

[0093]

[0094] According to the principle of minimum potential energy, we get the following equilibrium equation:

[0095] M'=Q,Q'+kw-q z =0 (7)

[0096] Here k is the soil reaction spring stiffness coefficient, q z is the longitudinal load acting on the tunnel;

[0097] Writing equations (1), (2), (3), (5) and (7) in matrix form gives

[0098]

[0099] The physical quantities in the above formula are normalized as follows

[0100]

[0101] The equation of state for the Timoshenko beam on an elastic foundation can be obtained:

[0102]

[0103] in, is the state vector, which consists of the displacement and internal force of the cross section; It is the system matrix, which stores the physical and geometric information of the cross section; is the external load vector, which is composed of the vertical external load on the tunnel; their specific forms are shown in the following formula:

[0104]

[0105] The standard solution of formula (10) is

[0106]

[0107] Formula (12) is the coordinates on the tunnel and Two-point state vector and The transitive relationship between them. is the transfer matrix of the system, are the load integral vectors of the system, and their expressions are as follows:

[0108]

[0109] Coefficient g i With the system matrix The formula (13) is applicable to any load. Once the form of the external load is determined, the external load vector can be obtained according to the above formulas.

[0110] Substituting the coordinates of the two ends of the short beam into equation (12), the transfer relationship between the physical quantities at the two ends of the short beam is obtained as follows:

[0111]

[0112] in, and are the state vectors of the cross sections at both ends of the j-th short beam, and

[0113] The expression of gi above, the expression of the characteristic root of the system matrix is ​​as follows:

[0114] The expression of the characteristic root of the system matrix A is as follows,

[0115]

[0116] For tunnel projects in soft soil areas, generally speaking, At this time the matrix There are two pairs of conjugate roots. Then the conjugate root can be expressed as,

[0117]

[0118] in,

[0119]

[0120] in To be related to the matrix characteristic roots, the expression is as follows,

[0121]

[0122]

[0123]

[0124]

[0125] In the above step S3, the inter-ring joint can transmit shear force and bending moment, and the magnitude of the internal force is determined by the relative displacement at the joint, satisfying the continuity condition of the force at the joint and the discontinuity condition of the displacement. The tunnel range length is L, and there are n segments in it, so there are n-1 inter-ring joints in total. According to the constitutive relationship of the two-way spring, the physical quantity transfer relationship on both sides of the inter-ring joint is

[0126]

[0127] The above formula can be written in the following matrix form after normalization:

[0128]

[0129] in,

[0130]

[0131]

[0132] refer to Figure 3 As shown in the figure, the physical quantity transfer relationship at both ends of the short beam and the physical quantity transfer relationship at the inter-ring joint are obtained. Equations (14) and (17) are used for the short beam and the inter-ring joint respectively along the longitudinal direction of the tunnel, and finally the transfer relationship between the state vectors at both ends of the tunnel is obtained as follows:

[0133]

[0134] in,

[0135]

[0136] If the distance between the two ends of the tunnel is far, then both ends are considered free, that is,

[0137]

[0138] The above formula can be written in matrix form as

[0139]

[0140] The boundary conditions at both ends can be modified accordingly according to actual needs.

[0141] Combining formula (19), the final algebraic equation is:

[0142]

[0143] In the above derivation process, no assumptions are made about the soil spring stiffness coefficients within each ring, the load form of the tunnel, and the values ​​of the joint stiffness between each ring. According to actual needs, the soil spring stiffness of each segment ring, the joint stiffness between each joint, and the load on the tunnel are conveniently assigned to take into account the longitudinal heterogeneity of the stratum along the tunnel, the differences in the joint stiffness values ​​at various locations, and the influence of any load form. When the ring joint is degenerated into a continuous concrete section, the proposed method can be easily degenerated into an equivalent continuous model solution.

[0144] In the above step S4, as shown in the parameters in Table 1, the longitudinal response of the shield tunnel affected by the stratum heterogeneity under the uniform load is analyzed, where the shield tunnel model is made of polyethylene pipe, and the interaction between the tunnel and the stratum is simulated by two springs with different stiffnesses. The stiffness of the springs is 419.70 kN / m and 63.55 kN / m, respectively, and the converted reaction coefficients are 5.08×10 4 kN / m 3 and 7.70×10 3 kN / m 3 .

[0145] refer to Figure 4 and Figure 5 As shown in the figure, the settlement values ​​calculated by the analytical method in this paper are compared with the actual results of the model test and the numerical calculation results of Huang et al. in the two formations of hard-soft alternation and hard-soft-hard alternation. It can be seen that the results of the three methods are in good agreement in terms of trend and value. This proves the effectiveness of the method in this paper, and also shows that the analytical method in this paper can easily consider the influence of formation discontinuity.

[0146] Table 1: Numerical parameters of the model test of Huang et al.

[0147]

[0148]

[0149] Under the action of longitudinal load, the shield tunnel is displaced, and the load is transmitted to the far end through the deformation of the lining ring itself and the differential deformation between the rings. The deformation of the lining is related to the inherent stiffness of the structure, and the differential deformation between the lining rings is determined by the local structure of the connection between the rings. The joints between the lining rings use waterproof cushion layers, bolts, etc. as connecting materials, and their mechanical behavior shows obvious flexible characteristics. They will experience misalignment and rotation deformation after being subjected to force. The surrounding strata constrain the deformation of the flexible lining and act on the lining structure in the form of reaction force. Therefore, the longitudinal mechanical modeling of the shield tunnel in the method provided by the present invention must start from the lining body model, joint force transmission, and the interaction between the surrounding rock and the structure.

[0150] The method for calculating the longitudinal mechanical response of a shield tunnel based on the inter-ring weak connection effect provided by the present invention is based on the longitudinal beam-spring model and the state space method, and provides the parameter conversion relationship between two types of models, namely the longitudinal beam-spring model and the equivalent continuous model. The comparison with the existing model test results and analytical results proves the effectiveness of the method in this paper. The method can be conveniently degenerated into an equivalent continuous model solution, and by comparing the calculation results of the two types of models, it is found that the calculation results of the two types of models are basically the same when the equivalent principle is consistent. The method in this paper is applicable to any form of longitudinal load action, that is, it can not only be used for several existing commonly used working conditions: simple cases such as single uniform strata and non-continuous strata, but also for the analysis of complex situations such as the longitudinal non-uniform changes in joint stiffness and the longitudinal non-uniform characteristics of the interaction between the tunnel and the stratum caused by the spatial variation characteristics of the stratum.

[0151] The state-space method uses the force and displacement of the energy duality as the solution quantity at the same time, avoiding the problem of determining the undetermined coefficients when solving high-order equations, and can easily obtain the transfer relationship of physical quantities between tunnel sections under arbitrary loads. The analysis and study of the transverse mechanical characteristics of shield tunnels show that this transfer characteristic can effectively deal with the problem of discontinuous deformation of multi-joint nodes such as tunnel linings. This method introduces the state-space method and gives a general solution to the longitudinal beam-spring model of shield tunnels under arbitrary loads, which can be easily degenerated into the solution of the equivalent continuous model, thus achieving the unification of the two. The correctness of the solution in this paper is verified by comparing with the current indoor test results and the existing analytical solutions. In addition, according to the same equivalence principle, the conversion formula between the equivalent continuous model and the beam-spring model is given. The example study shows that when the equivalence principle is consistent, the results of the two models are basically consistent, which indirectly confirms the effectiveness of the equivalent continuous model. The method proposed in this paper is not only applicable to the currently widely used equivalent homogenized continuous tunnel model, but can also easily deal with the longitudinal non-uniform characteristics of the system model caused by problems that are difficult to deal with by the equivalent homogenized model, such as discontinuous deformation of multiple longitudinal joints in the tunnel, local joint weakening, and spatial variability of the strata, providing a convenient and fast analysis method for the analysis of the longitudinal mechanical characteristics of the tunnel.

[0152] The above is only the preferred implementation of the present invention. The protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A method for calculating the longitudinal mechanical response of a shield tunnel based on the weak connection effect between rings, comprising a shield tunnel, wherein the shield tunnel is composed of a plurality of segments and inter-ring joints, and the inter-ring joints are connected by waterproof cushions and / or bolts, and wherein: The specific steps of the longitudinal mechanical response analysis method are as follows: S1. The inter-ring joints divide the tunnel into short beams longitudinally. The lining between two adjacent ring joints is regarded as a tubular hollow continuous short beam. A longitudinal mechanical model of a shield tunnel is established, wherein the coordinate axis of the mechanical model is along the longitudinal direction of the tunnel, the z direction is vertically upward, the x direction is horizontal, the vertical displacement is recorded as w, and a short beam between rings is taken as the analysis object; S2. Considering the shear deformation of the short beam cross section, the Timoshenko beam theory is used to simulate the mechanical behavior of the lining. The cross section before and after deformation conforms to the plane section assumption. The axial displacement of any point on the segment cross section is u z It can be expressed as: (1) in, ψ is the cross-sectional angle, which consists of two parts: (2) and (3) in, κ is the shear shape factor, which is 0.5 for a circular ring section. Q is the cross-sectional shear force, G is the section shear modulus, A is the cross-sectional area; Then the strain at any point on the tunnel cross section is (4) Bending moment M Expressed as (5) in, E is the section elastic modulus, I is the section moment of inertia, From equation (1), the energy density of a Timoshenko beam on an elastic foundation subjected to an arbitrary load is written as (6) According to the principle of minimum potential energy, we get the following equilibrium equation: (7) Here k is the soil reaction spring stiffness coefficient, q z is the longitudinal load acting on the tunnel, Writing equations (1), (2), (3), (5) and (7) in matrix form gives (8) The physical quantities in the above formula are normalized as follows (9) The equation of state for the Timoshenko beam on an elastic foundation can be obtained: (10) in, is the state vector, which consists of the displacement and internal force of the cross section; It is the system matrix, which stores the physical and geometric information of the cross section; is the external load vector, which is composed of the vertical external loads on the tunnel; their specific forms are shown in the following formula: (11) The standard solution of formula (10) is (12) Formula (12) is the coordinates on the tunnel and Two-point state vector and The transitive relationship between them, is the transfer matrix of the system, are the load integral vectors of the system, and their expressions are as follows: (13) coefficient g i With the system matrix The formula (13) is related to the characteristic root of . It is applicable to any load. Once the form of the external load is determined, the external load vector can be obtained according to the above formulas. , Substituting the coordinates of the two ends of the short beam into equation (12), the transfer relationship between the physical quantities at the two ends of the short beam is obtained as follows: (14) in, and Respectively j The state vectors of the cross sections at both ends of the short beam are ; S3. The mechanical behavior of the inter-ring joint is simulated by using shear and bending two-way joint springs; S4. The interaction between the tunnel and the stratum is simulated by Winkler soil springs.

2. The method for calculating the longitudinal mechanical response of a shield tunnel based on the weak connection effect between rings according to claim 1 is characterized in that: In step S3, the inter-ring joint can transmit shear force and bending moment, and the magnitude of the internal force is determined by the relative displacement at the joint, satisfying the continuity condition of the force at the joint and the discontinuity condition of the displacement. The tunnel range length is L , which has a total of n pipe segments, then there are n -1 inter-ring joint, according to the constitutive relationship of the two-way spring, the physical quantity transfer relationship on both sides of the inter-ring joint is (16) The above formula can be written in the following matrix form after normalization: , (17) in, , (18a) , (18b) The physical quantity transfer relationship at both ends of the short beam and the physical quantity transfer relationship at the inter-ring joint are obtained. Equations (14) and (17) are used for the short beam and the inter-ring joint along the longitudinal direction of the tunnel, and finally the transfer relationship between the state vectors at both ends of the tunnel is obtained as follows: (19) in, , (20) If the distance between the two ends of the tunnel is far, then both ends are considered free, that is, (21) The above formula can be written in matrix form as (22) According to actual needs, the boundary conditions at both ends can be modified accordingly; Combining formula (19), we can get the final algebraic equation: (23) 。 3. The method for calculating the longitudinal mechanical response of a shield tunnel based on the inter-ring weak connection effect according to claim 1 or 2 is characterized in that: when When the joint between the rings degenerates into a continuous concrete section.

4. The method for calculating the longitudinal mechanical response of a shield tunnel based on the inter-ring weak connection effect according to claim 1 is characterized in that: In the above step S4, the longitudinal response of the shield tunnel affected by the stratum heterogeneity under the uniformly distributed load is analyzed, wherein the shield tunnel model is made of polyethylene pipe, and the interaction between the tunnel and the stratum is simulated by two springs with different stiffnesses. The stiffnesses of the springs are 419.70 kN / m and 63.55 kN / m, respectively, and the converted reaction coefficients are 5.08×10 4 kN / m 3 and 7.70×10 3 kN / m 3 .

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