A method for calculating the response of a shield tunnel considering misaligned joint based on state space method

Abstract

The application discloses a kind of based on state space method's considering staggered joint assembling shield tunnel opening response calculation method, including shield tunnel, open a rectangle on shield tunnel, opening tunnel segment response calculation method includes: using the mode of virtual joint, tunnel segment is divided into multiple sections along longitudinal joint, state equation is used to transfer in section, joint equation is used to transfer between section;Considering the shear deformation of section, the mechanical behavior of lining segment is simulated using Timoshenko beam theory;The mechanical behavior of longitudinal joint is simulated using radial, tangential and rotating three-way spring, and the mechanical behavior of inter-ring joint is simulated using two-way shear spring;The interaction between tunnel and stratum is simulated using Winkler soil spring.The method can conveniently calculate the mechanical response of tunnel segment after considering staggered joint assembling shield tunnel opening, and various complex working conditions can be calculated by flexibly adjusting joint position, number, opening size and other parameters according to actual engineering.

Description

Technical Field

[0001] This invention relates to the field of shield tunnels, and in particular to a method for calculating the opening response of a shield tunnel considering staggered joint assembly based on the state-space method. Background Technology

[0002] Due to their convenient, fast, and reliable construction, shield tunnels have become the most important carrier for urban rail transit. With the rapid development of shield tunneling technology in recent years, large-diameter shield tunnels have also played a vital role in highway transportation. To achieve interconnectivity in underground space networks, numerous underground connecting structures are needed. For example, subway stations built using shield tunnels are connected to the stations through openings. Furthermore, connecting passages between subway and highway tunnels are created by constructing cross passages to connect two parallel shield tunnels.

[0003] From a functional perspective, to achieve interconnectivity in underground spatial networks, shield tunnel construction often involves openings. For example, transverse connecting passages between subway and highway tunnels require openings to connect two parallel tunnels. The method of tunneling subway stations using shield tunnels also necessitates openings to connect the shield tunnel and the station. Furthermore, highway shield tunnels also require long openings to connect with ramps. From a structural stress perspective, while prefabrication and assembly of tunnel segments greatly facilitate shield tunnel construction, they inevitably weaken the longitudinal and transverse stiffness and integrity of the shield tunnel, considered a weak point in the tunnel structure. Furthermore, openings in shield tunnels significantly compromise the tunnel's integrity, weakening the tunnel ring stiffness at the opening location and impacting the tunnel's safety and normal performance. Therefore, it is necessary to study the mechanical response characteristics of shield tunnels after openings to assess the safety of the opening condition and implement corresponding reinforcement measures to avoid engineering risks. There has been a great deal of research on this problem, but it has mainly focused on numerical calculations and model experiments. Analytical calculation methods for this problem have not yet been reported. However, analytical calculation methods are widely needed in engineering design practice because they are simple to calculate and easy to modify.

[0004] The state-space method employs the duality theory and symplectic state space based on the Hamiltonian system, and uses internal forces and displacements—two types of variables with energy duality—as unknowns, avoiding the higher-order nature of differential equations and making the solution more flexible. In recent years, the state-space method has been successfully applied in civil engineering to the static response studies of segmented linings in shield tunnels and the static response studies of large-diameter pile foundations under horizontal loads. These applications provide a reference for the application of the state-space method in calculating the opening response of shield tunnels considering staggered joint assembly. Summary of the Invention

[0005] The technical problem to be solved by this invention is to provide a method for calculating the opening response of staggered-joint shield tunnels based on the state-space method. This method facilitates the calculation of the mechanical response of each ring after the opening of a staggered-joint shield tunnel, avoiding the problems of complex finite element modeling and difficulty in modification, and providing a basis for optimization and reinforcement measures for the opening segments and their vicinity under the opening condition.

[0006] To address this, this invention provides a method for calculating the opening response of a staggered-joint shield tunnel based on the state-space method. The shield tunnel is assembled from multiple tunnel segments and longitudinal and interlocking joints. A rectangular opening is made in the tunnel. The specific steps of the tunnel segment response calculation method are as follows: A virtual joint is used to divide the tunnel segments into multiple segments along the longitudinal joints. State equations are used for propagation within each segment, and joint equations are used for propagation between segments. Considering the shear deformation of the cross-section, the mechanical behavior of the lining segments is simulated using Timoshenko beam theory. The mechanical behavior of the longitudinal joints is simulated using triaxial springs (radial, tangential, and rotational), and the mechanical behavior of the interlocking joints is simulated using biaxial shear springs. The interaction between the tunnel and the strata is simulated using Winkler soil springs. The specific steps of the method for calculating the opening response of a staggered-joint shield tunnel are as follows:

[0007] A state-space method-based calculation method for the opening response of a shield tunnel considering staggered joint assembly includes a shield tunnel, which is assembled from multiple tunnel segments, ring joints, and longitudinal joints. The shield tunnel has a rectangular opening. The shield tunnel opening response calculation method includes the following steps:

[0008] (1) Consider a single lining segment as a curved beam with a rectangular cross section;

[0009] In the segment curved beam, the radius of the curve is denoted as R. Coordinate axes zs are established along the radial and tangential directions, where z represents the radial direction and s represents the tangential direction. The radial displacement is denoted as w, and the tangential displacement as u. The segment cross-section rotation angle is also considered and denoted as θ. The shear force, axial force and bending moment on the segment section are denoted as Q, N and M, respectively. A segment curved beam is taken as the analysis object to establish a mechanical model of the shield tunnel segment.

[0010] (2) Under the mechanical model of shield tunnel segments, the shear deformation of the segment cross section is considered. The Timoshenko beam theory is used to simulate the mechanical behavior of the lining. The mechanical behavior of the inter-ring joints between rings is simulated by the radial shear spring coefficient k. fz and tangential shear spring coefficient k fs The normalized radial shear spring coefficient is obtained after normalization. and tangential shear spring coefficient

[0011] Set the radial soil spring coefficient k zand tangential soil spring coefficient k s Winkler soil springs were used to simulate the interaction between the shield tunnel and the strata, and the normalized radial soil spring coefficient was obtained after normalization. and tangential soil spring coefficient

[0012] Obtain the radial load q borne by the tunnel segment z and tangential load q s The normalized radial load is obtained after normalization. and tangential load

[0013] Establish the state equations for the tunnel segments;

[0014] The standard solution is obtained from the segment state equation, yielding the transfer equation and transfer matrix within a segment. A segment refers to a circumferential segment; all segments within the lining ring located in this region fall under the category of a segment.

[0015] (3) Set the radial spring coefficient k of the joint w , Joint tangential spring coefficient k u , Joint rotation spring coefficient The mechanical behavior of longitudinal joints connecting shield tunnel segments along the circumferential direction is simulated using radial, tangential, and rotational three-way joint springs. Considering the staggered assembly of shield tunnel segments, the principle of continuous internal force but discontinuous displacement is followed at longitudinal joints where there are circumferential joints; virtual joints are set at locations where there are no joints, following the principle of continuous internal force and displacement. The equations for longitudinal joints connecting shield tunnel segments along the circumferential direction are established. The equations are transformed to obtain the transfer equations between the end of one segment and the beginning of the next segment, and then rearranged into a matrix equation form to obtain the joint transfer matrix.

[0016] (4) Divide the longitudinal joints connecting the tunnel segments along the circumferential direction into multiple segments (considering the angles of the longitudinal joints and openings in the segment circumferential connection). Following the sequence of multiple segments, alternately use the transfer equations obtained in step (2) for the internal segments and the transfer equations obtained in step (3) between the end of one segment and the beginning of the next segment, until the circumferential direction is completed. This establishes the initial complete ring matrix equation. The left side of the equation is the initial coefficient matrix multiplied by the state vector composed of the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of all segment starting and ending sections. The right side of the equation is a vector composed of the load integral vector and the zero vector. Due to the presence of openings, the initial coefficient matrix will have fewer rows than columns, therefore additional conditions are needed for solving.

[0017] (5) The length of the rectangular opening along the axial direction of the shield tunnel is an integer multiple of the width of the tunnel segment. Free boundary conditions are used to simulate the rectangular opening, with internal forces of 0, which is the supplementary condition. A corresponding supplementary coefficient matrix is ​​constructed, and the internal forces corresponding to the opening location in the state vector are selected from the supplementary coefficient matrix and set to their corresponding zero vectors. The supplementary opening equation is then obtained.

[0018] The selection of supplementary coefficients should be such that the supplementary coefficient matrix, when multiplied on the right by the state vector composed of the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of all segment starting and ending sections, equals the internal force corresponding to the opening position in the state vector.

[0019] (6) Integrate the initial integral ring matrix equation obtained in step (4) with the supplementary opening equation obtained in step (5) to form the final integral ring matrix equation. Solve the final integral ring matrix equation to obtain the state vector composed of normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of all segment starting and ending sections. Then, based on the transfer equation inside the segment obtained in step (2), obtain the state vector composed of normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of any section in the circumferential direction of all segments. By inverse normalizing the obtained state vector composed of normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of each section, the opening response of the shield tunnel considering staggered joint assembly can be obtained.

[0020] Based on the obtained response data from the opening of a staggered-joint shield tunnel, shield tunnel designers can design the reinforcement of the tunnel segments and the strengthening measures for the area near the opening. This ensures the safe implementation of the opening operation and avoids engineering accidents such as excessive deformation or even segment damage in staggered-joint shield tunnels due to the opening. Simultaneously, parameters can be adjusted to calculate different opening locations, different joint arrangements, and different radial spring coefficients (k) for various joints. w , Joint tangential spring coefficient k u , Joint rotation spring coefficient Radial shear spring coefficient k fz and tangential shear spring coefficient k fs The corresponding shield tunnel opening response is analyzed, and the design scheme of the shield tunnel opening is optimized to minimize risks and save costs while ensuring safety.

[0021] In step (2), the state equation for the tunnel segment is:

[0022]

[0023] In the formula, This is the state vector composed of normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment. θ is the normalized tangential coordinate. The normalized system matrix includes beam cross-sectional information, material information, shear spring information between rings, and soil spring information. This is the normalized load vector.

[0024] The standard solution to the state equation is the transfer equation:

[0025]

[0026] In the transfer equation The state vector representing the normalized radial displacement, tangential displacement, section rotation, shear force, axial force, and bending moment of the cross section with normalized tangential coordinate θ is given in the transfer equation. The state vector represents the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of the section with normalized tangential coordinate θ0.

[0027] matrix That is, the normalized transfer matrix from θ0 to θ. This is the normalized load integral vector from θ0 to θ.

[0028]

[0029] In equation (30), e is the natural constant.

[0030] In step (3), the transfer equation between the end of one segment and the beginning of the next segment is as follows:

[0031]

[0032] in Let be the normalized state vector at the starting end of the (j+1)th segment. Let be the normalized state vector at the end of the j-th segment. Let be the normalized connector transfer matrix for the j-th connector.

[0033] In step (4), the initial integral ring matrix equation is:

[0034]

[0035] In the formula The initial coefficient matrix is ​​obtained by alternately integrating the transfer equation inside the segment obtained in step (2) and the transfer equation between the end of one segment and the beginning of the next segment obtained in step (3). It is a state vector consisting of normalized radial displacement, tangential displacement, section rotation, shear force, axial force, and bending moment for all segments at the beginning and end sections. A vector consisting of the load integral vector and the zero vector.

[0036] In step (5), the opening equation is supplemented:

[0037]

[0038] In the formula This is to supplement the coefficient matrix. This is a state vector consisting of normalized radial displacement, tangential displacement, section rotation, shear force, axial force, and bending moment at the starting and ending sections of all tunnel segments. (Supplementary coefficient matrix) The number of columns and the initial coefficient matrix Same, equal to the state vector The number of rows. Supplementary coefficient matrix. The number of rows is determined by the ratio of the length of the opening along the axial direction of the shield tunnel to the width of the segment. The number of rows of the zero vector on the right-hand side of the supplementary opening equation is equal to the number of rows in the supplementary coefficient matrix. number of rows.

[0039] In step (6), the final integral ring matrix equation is:

[0040]

[0041] in:

[0042]

[0043] Solving the final complete loop matrix equation yields the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of all segment starting and ending sections. Then, based on the transfer equations within the segments obtained in step (2), the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of any section in the circumferential direction of all segments are obtained as state vectors. By inverse normalizing the obtained state vectors of normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment for each section, the opening response of the shield tunnel considering staggered joint assembly can be obtained.

[0044] The above derivation process did not include the soil spring stiffness coefficient (k) within each ring. z ,k s ), the form of load on the tunnel (q) z ,q s ) and the stiffness (k) of each longitudinal joint w ,k u ,k ψ ) and ring joint stiffness (kfz ,k fs Assumptions are made regarding the value of ), and the soil spring stiffness of each segment ring, the joint stiffness between rings of each joint, and the load on the tunnel are assigned values ​​according to actual needs, in order to consider the non-uniformity of the strata along the longitudinal direction of the tunnel, the differences in the stiffness values ​​of joints at various locations, and the influence of arbitrary load forms.

[0045] Compared with the prior art, the present invention has the following advantages:

[0046] This invention presents a method for calculating the opening response of staggered-joint shield tunnels based on the state-space method. The correctness of the proposed solution is verified by comparing the results with those obtained from a finite element model built using ABAQUS and existing analytical solutions. This method can conveniently calculate the mechanical response of each ring after the opening of a staggered-joint shield tunnel, avoiding the problems of complex finite element modeling and difficulty in modifying the model. It provides a basis for optimization and reinforcement measures for the opening segments and their vicinity under opening conditions. Attached Figure Description

[0047] Figure 1 Construct a route diagram for the state equation;

[0048] Figure 2 This is a schematic diagram of the load.

[0049] Figure 3 This is a schematic diagram of the calculation model for a three-ring curved beam.

[0050] Figure 4 The diagram shows a longitudinal joint inside the ring, where (a) is a schematic diagram of a three-way spring joint and (b) is a schematic diagram of a virtual joint.

[0051] Figure 5 A schematic diagram of the calculation model for the opening condition of a shield tunnel with staggered joints;

[0052] Figure 6 The result of the finite element modeling for Example 1;

[0053] Figure 7 Figure showing the mechanical response of the 8th ring lining under the condition of staggered joint assembly shield tunnel opening. Detailed Implementation

[0054] The technical solutions in the embodiments of this patent will be clearly and completely described below with reference to the present invention. Obviously, the described embodiments are only a part of the embodiments of this patent, and not all of them. Based on the embodiments of this patent, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this patent.

[0055] Reference Figure 1The present invention provides a method for calculating the opening response of a shield tunnel based on the state-space method, considering staggered joint assembly, including a shield tunnel, such as... Figure 5 As shown, the shield tunnel is assembled from multiple segments, ring joints, and longitudinal joints. Considering the existence of an opening in the tunnel, the specific steps of the analytical method for analyzing the mechanical response of the staggered-joint shield tunnel opening are as follows:

[0056] (1) Consider a single lining segment as a curved beam with a rectangular cross section;

[0057] A mechanical model of a shield tunnel segment is established, with coordinate axes zs along the radial and tangential directions. The z-direction is radial, and the s-direction is tangential. The radial displacement is denoted as w, and the tangential displacement is denoted as u. The section rotation angle is also considered and denoted as θ. The shear force, axial force, and bending moment on the cross section are denoted as Q, N, and M, respectively. A segment curved beam is taken as the analysis object.

[0058] (2) Considering the shear deformation of the segment cross section, the mechanical behavior of the lining is simulated using Timoshenko beam theory;

[0059] (3) The mechanical behavior of the longitudinal joint of the single-ring inner segment is simulated by radial, tangential and rotational joint springs (k w ,k u , The mechanical behavior of the inter-ring joints between rings is simulated using radial and tangential shear springs (k). fz ,k fs );

[0060] (4) The interaction between the tunnel and the strata was simulated using Winkler soil springs (k z ,k s );

[0061] (5) The tunnel has a rectangular opening, and the free boundary conditions (internal forces are 0) are used to simulate the opening;

[0062] (6) The segments bear arbitrary radial and tangential loads (q) z ,q s ).

[0063] like Figure 3 As shown, we first consider the case of a curved beam system composed of three adjacent rings. The subscript i is used to distinguish the physical quantities of the three rings. The value of i is 1, 2, 3. Its characteristic is: in the above step (2), the displacement u at any point on the cross section along coordinates z and s zi (s,z) and u si (s,z) is:

[0064]

[0065] In equation (1), w i (s),u i (s), These represent the radial displacement, tangential displacement, and rotation angle of the section with circumferential coordinate s of the i-th ring lining. s and z are the coordinates on the s-axis and z-axis, respectively. The normal strain ε at any point on the section with circumferential coordinate s of the i-th ring lining is... i and shear strain γ i They are respectively

[0066]

[0067] In equation (2), z is the coordinate on the z-axis, R is the radius of the curve in the lining, and the rest are defined in equation (3):

[0068]

[0069] In equation (3), w i ,u i , These represent the radial displacement, tangential displacement, and rotation angle of the section with circumferential coordinate s in the i-th ring lining. R is the radius of the curve in the lining.

[0070] According to Hooke's law, the normal stress σ and shear stress τ at any point on the cross section with circumferential coordinate s of the i-th ring lining are respectively

[0071]

[0072] In equation (4), E and G are Young's modulus and shear modulus, respectively, κ is the section shear correction factor, z is the coordinate on the z-axis, and R is the radius of the curve in the lining. Therefore, the internal forces on the section are:

[0073]

[0074] In equations (5)-(7), N i M i Q i Let w represent the axial force, bending moment, and shear force at the section with circumferential coordinate s of the i-th ring lining. i ,u i , These represent the radial displacement, tangential displacement, and section rotation angle of the i-th ring lining with circumferential coordinate s. R and A are the curve radius and cross-sectional area of ​​the lining, respectively. E and G are the Young's modulus and shear modulus of the lining material, respectively. κ is the section shear correction factor. Parameter β is a dimensionless quantity related to the cross-sectional shape, defined as:

[0075]

[0076] In equation (8), R and A are the curve radius and cross-sectional area in the lining, respectively. z is the coordinate on the z-axis.

[0077] The total potential energy of the system can be written as the sum of the potential energies π1, π2, and π3 of the three segments (each segment's potential energy includes the elastic potential energy of the segment, the potential energy of the soil spring, and the work done by the external force) and the potential energies π4 and π5 of the connection between the two rings, that is:

[0078] π = π1 + π2 + π3 + π4 + π5 (9)

[0079] in:

[0080]

[0081] In equations (10)-(12), N i M i Q i Let w represent the axial force, bending moment, and shear force of the i-th ring lining, respectively. i ,u i , These represent the radial displacement, tangential displacement, and section rotation angle of the i-th ring lining, respectively. zi q si Let represent the radial and tangential loads of the i-th ring lining, respectively. Let b be the width of the segment and h be the thickness of the segment. z ,k s represents the stiffness coefficients of the radial and tangential soil springs, respectively. fz ,k fs These represent the radial and tangential stiffness coefficients of the interring shear spring, respectively.

[0082] By performing variational analysis on the system's potential energy and applying the principle of minimum potential energy, the equilibrium equations that the internal forces must satisfy can be obtained as follows:

[0083]

[0084] In equation (13), N i M i Q i Let N1, M2, and Q2 represent the axial force, bending moment, and shear force of the i-th ring lining, respectively. Let N2, M2, and Q2 represent the axial force, bending moment, and shear force of the second ring lining, respectively. i ,u i q represents the radial and tangential displacements of the i-th ring lining, respectively. w1 and u1 represent the radial and tangential displacements of the first ring lining, respectively. w2 and u2 represent the radial and tangential displacements of the second ring lining, respectively. w3 and u3 represent the radial and tangential displacements of the third ring lining, respectively. zi ,q si Let q represent the radial and tangential loads of the i-th ring lining, respectively. z2 ,q s2 These represent the radial and tangential loads of the second ring lining, respectively. b is the width of the segment, and h is the thickness of the segment.z ,k s represents the stiffness coefficients of the radial and tangential soil springs, respectively. fz ,k fs These represent the radial and tangential stiffness coefficients of the interring shear spring, respectively.

[0085] Rewriting equations (5)-(7) and (13) in matrix form, we have:

[0086]

[0087] in:

[0088]

[0089] q = [000-q z1 bq s1 b 0000-q z2 bq s2 b 0000-q z3 bq s3 b 0] T (16)

[0090]

[0091]

[0092] In equations (15)-(20), N1, M1, and Q1 represent the axial force, bending moment, and shear force of the first ring lining, respectively. N2, M2, and Q2 represent the axial force, bending moment, and shear force of the second ring lining, respectively. N3, M3, and Q3 represent the axial force, bending moment, and shear force of the third ring lining, respectively. w1, u1, These represent the radial displacement, tangential displacement, and section rotation angle of the first ring lining, respectively. w2,u2, These represent the radial displacement, tangential displacement, and section rotation of the second ring lining, respectively. w3,u3, These are the radial displacement, tangential displacement, and section rotation angle of the third ring lining, respectively. z1 ,q s1 q represents the radial and tangential loads of the first ring lining, respectively. z2 ,q s2 q represents the radial and tangential loads of the second ring lining, respectively. z3 ,q s3 These represent the radial and tangential loads of the third ring lining, respectively. b is the width of the segment, and h is the thickness of the segment. k z ,k s represents the stiffness coefficients of the radial and tangential soil springs, respectively. fz ,k fsrepresents the radial and tangential stiffness coefficients of the inter-ring shear spring, respectively. R and A are the curve radius and cross-sectional area in the lining, respectively. E and G are the Young's modulus and shear modulus of the lining material, respectively. κ is the section shear correction coefficient. The parameter β is a dimensionless quantity related to the cross-sectional shape, as shown in equation (8).

[0093] The physical quantities in the above equation are normalized as follows:

[0094]

[0095] In equation (21), R and A are the radius of curvature and cross-sectional area of ​​the lining, respectively. E is the Young's modulus of the lining material. b is the width of the segment, and h is the thickness of the segment. N, M, and Q are the axial force, bending moment, and shear force of the lining, respectively. w, u, These are the radial displacement, tangential displacement, and section rotation of the lining, respectively. z ,k s represents the stiffness coefficients of the radial and tangential soil springs, respectively. fz ,k fs These represent the radial and tangential stiffness coefficients of the inter-ring shear spring, respectively. 's' is the coordinate on the s-axis. Quantities marked with underlines above are normalized versions of the original physical quantities; θ is the normalized version of 's'.

[0096] The state equations for the tunnel segments can be obtained:

[0097]

[0098] In equations (23)-(28) These are the normalized axial force, bending moment, and shear force of the first ring lining, respectively. These are the normalized axial force, bending moment, and shear force of the second ring lining, respectively. These are the normalized axial force, bending moment, and shear force of the third ring lining, respectively. These are the normalized radial displacement, tangential displacement, and section rotation of the first ring lining, respectively. These are the normalized radial displacement, tangential displacement, and section rotation of the second ring lining, respectively. These are the normalized radial displacement, tangential displacement, and section rotation of the third ring lining, respectively. These represent the normalized radial and tangential loads of the first ring lining, respectively. These represent the normalized radial and tangential loads of the second ring lining, respectively. These represent the normalized radial and tangential loads of the third ring lining, respectively. These represent the normalized radial and tangential soil spring stiffness coefficients, respectively. represents the normalized radial and tangential stiffness coefficients of the inter-ring shear springs, respectively. R and A are the curve radius and cross-sectional area in the lining, respectively. E and G are the Young's modulus and shear modulus of the lining material, respectively. κ is the section shear correction coefficient. The parameter β is a dimensionless quantity related to the cross-sectional shape, as shown in equation (8).

[0099] The standard solution to equation (22) is

[0100]

[0101] Equation (29) represents the state vector of two points with coordinates θ and θ0 on the tunnel. and The transitive relationship between them. Among them, The transfer matrix of the system. Let the load integral vectors of the system be given, and their expressions are as follows:

[0102]

[0103] In equation (30), e is the natural constant.

[0104] From the coefficient matrix of the three rings above, the coefficient matrix for the multi-ring case can be derived.

[0105] In step (3), such as Figure 4 As shown in (a), the longitudinal joint can transmit shear force, axial 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 force and the discontinuity condition of displacement at the joint. According to the constitutive relation of the triaxial spring, the transmission relationship of physical quantities on both sides of the joint with loop is as follows:

[0106]

[0107] In equation (31), the subscript i represents the lining ring number, 0 in the subscript represents the starting end of the segment, 1 represents the ending end of the segment, and the superscripts j and j+1 represent the segment number and the joint number. w ,k u , These represent the radial, circumferential, and rotational spring stiffness of the longitudinal joint within the ring, respectively. These represent the radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment at the end of the j-th segment of the i-th ring, respectively. These represent the radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment at the beginning of the (j+1)th segment of the i-th ring, respectively.

[0108] Considering that shield tunnels mostly use staggered joint assembly, such as Figure 4 As shown in (b), a virtual joint is introduced to unify joint treatment. The internal forces and displacements on both sides of the virtual joint remain continuous, i.e.:

[0109]

[0110] In formula (32), the subscript i represents the lining ring number, 0 in the subscript represents the beginning end of the segment, 1 represents the end end of the segment, and the superscript j and j+1 represent the segment number and the joint number.

[0111] These represent the radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment at the end of the j-th segment of the i-th ring, respectively. These represent the radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment at the beginning of the (j+1)th segment of the i-th ring, respectively.

[0112] Taking the three-ring ABA-type staggered assembly as an example, the above relationship can be written in matrix form:

[0113]

[0114] in Let be the state vector of the (j+1)th lining segment's starting end. Let G be the state vector at the end of the j-th lining segment. j The connector transfer matrix I and H for the j-th connector are shown in the following equation:

[0115]

[0116] In equation (34), k w ,k u , Let represent the radial, circumferential, and rotational spring stiffnesses of the longitudinal joint within the ring, respectively. Similarly, by dimensionless transformation of the transfer equations, we have:

[0117]

[0118] Corresponding:

[0119]

[0120] in:

[0121]

[0122] In equation (37), k w ,k u , These represent the radial, circumferential, and rotational spring stiffnesses of the longitudinal joint within the ring, respectively. R and A represent the curve radius and cross-sectional area in the lining, respectively. E is the Young's modulus of the lining material. These are the normalized stiffness of the joint spring.

[0123] Based on the above theory, the transmission can be extended from the three-ring case to multiple rings. The entire ring lining is divided into n blocks along the longitudinal joint (considering the angle of the longitudinal joint and the angle of the opening in the division), and then the matrix equation is listed. Each ring is spliced ​​from m segments, so n is not less than m. When the joint is continuous, n = m, and when the joint is staggered, n > m. The physical quantity transmission relationship at both ends of the curved beam and the physical quantity transmission relationship at the joint between rings are obtained. Using equations (29) and (35) along the tunnel circumference, we have:

[0124]

[0125] Writing:

[0126]

[0127] In the coefficient matrix of equation (38), odd-numbered rows represent the transmission within each segment, i.e., equation (29), and even-numbered rows represent the transmission between the end of the previous segment and the beginning of the next segment, i.e., equation (35). Note: Segments refer to circumferential divisions, such as the segment from 30 degrees to 90 degrees. A segment contains all rings, and a ring refers to a tunnel ring, which also contains all segments. In the state vector, The superscript indicates the number of the tube segment, starting from the lowest point (0 degrees) and numbering clockwise. The subscript indicates the beginning (0) or the end (1). The number of rows is determined by the number of rings involved in the calculation. Because of the presence of openings, the number of rows in the state vector of the pipe segment within the opening range will be less than that of other segments, so special handling is required when transferring between open and closed segments. Equation (38) shows the case when the p-th segment is an open segment and only this segment is an open segment (when multiple segments are located within the opening angle, the dimensions of the transfer equation and the joint equation will be less, and this method is also applicable). When transferring from the end of the (p-1)-th segment to the beginning of the p-th segment, the joint matrix It's not a square matrix, but a matrix with fewer rows than columns. When passing data from the beginning to the end of the p-th segment, the transfer matrix... The dimension of this joint is lower than that of the unopened segment's transfer matrix. At the end of the p-th segment, the transfer direction on this joint is opposite to that of the other joints, transferring from the beginning of the (p+1)-th segment to the end of the p-th segment. The corresponding joint matrix... It needs to be adjusted according to equation (31), resulting in a matrix with fewer rows than columns. It can be observed that this special treatment will lead to an increase in the coefficient matrix. Since the number of rows is less than the number of columns, boundary conditions at the opening need to be added to obtain a solution to the equation. Specifically, this means constructing a boundary condition matrix to find the internal forces (Q, N, M) corresponding to the open loop at the end of the segment before the opening segment (p-1) and the beginning of the segment after the opening segment (p+1), setting them to 0, and writing them as:

[0128]

[0129] Finally, we obtain the equation:

[0130]

[0131] in:

[0132]

[0133] Solving equation (41) yields the normalized state vectors at the beginning and end of each segment. Then, according to equation (29), the normalized state vectors of any loop at any angle can be obtained. Finally, the results are inversely normalized to obtain the response after the tunnel opening.

[0134] The above derivation process did not include the soil spring stiffness coefficient (k) within each ring. z ,k s ), the form of load on the tunnel (q) z ,q s ) and the stiffness (k) of each longitudinal joint w ,k u ,k ψ ) and ring joint stiffness (k fz ,k fs Assumptions are made regarding the value of ), and the soil spring stiffness of each segment ring, the joint stiffness between rings of each joint, and the load on the tunnel are assigned values ​​according to actual needs, in order to consider the non-uniformity of the strata along the longitudinal direction of the tunnel, the differences in the stiffness values ​​of joints at various locations, and the influence of arbitrary load forms.

[0135] like Figure 7 As shown, the mechanical response results of a staggered shield tunnel opening are presented. The analytical calculation results of this invention are compared with the ABAQUS numerical calculation results (numerical model as shown). Figure 6 The results are shown in the figure. The comparison reveals that the two results agree very well. This demonstrates the effectiveness of the method of the present invention.

[0136] Table 1 Calculation parameters for Example 1

[0137]

[0138] Table 2 Calculation parameters of the finite element model

[0139]

[0140] Note:

[0141] 1. The soil spring parameters and inter-ring spring parameters in the finite element model in Table 2 differ from the corresponding parameters in Table 1 because the springs are uniformly distributed in the analytical model, while the springs in the numerical model can only be point springs. Therefore, a conversion is required.

[0142] 2. The load parameters of the finite element model in Table 2 are twice the corresponding parameters in Table 1 because the load applied in the finite element model is a line load, and the influence of the tunnel width needs to be considered manually, so it is multiplied by the tunnel width b.

[0143] Under ground loads, the opening of a shield tunnel experiences significant displacement due to reduced stiffness. The impact of the opening is transmitted to adjacent rings via inter-ring shear springs. Simultaneously, the opening segment resembles a cantilever beam, with the cantilever's endpoint being the nearest longitudinal joint within the ring. The length of this "cantilever" directly affects the mechanical response caused by the opening. Therefore, the mechanical response modeling for the opening of an assembled shield tunnel, as provided in this invention, must begin with the segment body model, the force transmission from the longitudinal joints within the rings and the inter-ring shear joints, and the interaction between the soil and the tunnel.

[0144] The method for calculating the opening response of staggered-joint shield tunnels provided in this invention is an analytical calculation method based on the curved beam-spring model and state-space method, and fully considers the shield tunnel assembly effect. Comparison with the results of the ABAQUS finite element model and the results of this invention proves the effectiveness of the proposed method. This method can conveniently calculate the mechanical response of each ring after the opening of a staggered-joint shield tunnel, avoiding the problems of complex finite element modeling and difficult modification, and providing a basis for optimization and reinforcement measures for the opening segments and their vicinity under the opening condition.

[0145] The state-space method uses both energy-dual forces and displacements as solution quantities, avoiding the problem of determining undetermined coefficients when solving higher-order equations. It can conveniently obtain the physical quantity transfer relationships between tunnel sections under arbitrary loads. This method introduces the state-space method and presents a calculation method for the opening response of staggered-joint shield tunnels. The correctness of the solution is verified by comparing the results with those of the finite element model. The proposed method can conveniently calculate the opening response of staggered-joint shield tunnels under arbitrary diameter, geological conditions, joint conditions, opening angle, and number of opening rings, providing a convenient and rapid calculation method for the mechanical response calculation of shield tunnel opening conditions.

[0146] The above description is merely a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principle of the present invention should also be considered within the scope of protection of the present invention.

Claims

1. A method for calculating the opening response of a shield tunnel based on the state-space method, considering staggered joint assembly, comprising a shield tunnel, wherein the shield tunnel is assembled from multiple segments and interlocking joints and longitudinal joints, and the shield tunnel is provided with a rectangular opening, characterized in that, The shield tunnel opening response calculation method comprises the following steps: (1) regarding a single lining segment as a curved beam with rectangular cross section; The curve radius in the segmental curved beam is denoted as R, the coordinate axes z-s are established along the radial and tangential directions, the z direction is the radial direction, the s direction is the tangential direction, the radial displacement is denoted as w, the tangential displacement is denoted as u, and the segmental cross section rotation angle is denoted as The shear force, axial force and bending moment on the segmental cross section are denoted as Q, N and M respectively, a segmental curved beam is taken as an analysis object, and a shield tunnel segmental mechanical model is established. (2) Considering the shear deformation of the segment cross-section under the mechanical model of the shield tunnel segment, the Timoshenko beam theory is used to simulate the mechanical behavior of the lining, and the mechanical behavior of the inter-segment joint between the rings is simulated by using the radial shear spring coefficient k fz and the tangential shear spring coefficient k fs , and the normalized radial shear spring coefficient and the tangential shear spring coefficient Setting radial soil spring coefficient k z and tangential soil spring coefficient k s , Winkler soil spring is used to simulate the interaction between shield tunnel and stratum, and the normalized radial soil spring coefficient and tangential soil spring coefficient q is the radial load borne by the segment z q is the tangential load borne by the segment s q is the radial load borne by the segment q is the tangential load borne by the segment establishing a segment state equation; obtaining a standard solution according to the segment state equation, and obtaining a segment internal transfer equation and a segment transfer matrix; (3) Set joint radial spring coefficient k w , joint tangential spring coefficient k u , joint rotational spring coefficient The radial, tangential and rotational joint spring is used to simulate the mechanical behavior of the longitudinal joint of the shield tunnel along the pipe piece ring. considering the misaligned joint assembling condition of the shield tunnel segment, at the longitudinal joint along the segment ring connection where the segment ring connection exists, the principle of internal force continuity and displacement discontinuity is followed; at the place where the joint does not exist, a virtual joint is set, the principle of internal force and displacement continuity is followed, a longitudinal joint equation of the shield tunnel along the segment ring connection is established, the equation is transformed to obtain a transfer equation between the end of a segment and the start of the next segment, and the transfer equation is arranged into a matrix equation form to obtain a joint transfer matrix; (4) dividing the longitudinal joint of the shield tunnel along the segment ring connection into multiple segments, and according to the order of the multiple segments, the segment internal transfer equation obtained in step (2) and the transfer equation between the end of a segment and the start of the next segment obtained in step (3) are alternately used along the tunnel ring until the ring turns back to one round, that is, an initial whole ring matrix equation is established, the left side of the equation is an initial coefficient matrix multiplied by a state vector composed of normalized radial displacement, tangential displacement, cross section rotation angle, shear force, axial force and bending moment of all segment start ends and end sections, and the right side of the equation is a vector composed of a load integral vector and a zero vector; the initial whole ring matrix equation will lead to that the number of rows of the initial coefficient matrix is less than the number of columns due to the existence of the opening, and therefore a supplementary condition is needed for solving; (5) the length of the rectangular opening along the axial direction of the shield tunnel is an integer multiple of the width of the segment, a free boundary condition is used to simulate the rectangular opening, the internal force is 0, that is, the supplementary condition, a corresponding supplementary coefficient matrix is constructed, the internal force corresponding to the opening position in the state vector is screened out through the supplementary coefficient matrix, and the internal force is set to be a corresponding zero vector to obtain a supplementary opening equation; (6) the initial whole ring matrix equation obtained in step (4) and the supplementary opening equation obtained in step (5) are integrated to form a final whole ring matrix equation, the final whole ring matrix equation is solved to obtain a state vector composed of normalized radial displacement, tangential displacement, cross section rotation angle, shear force, axial force and bending moment of all segment start ends and end sections; and then according to the segment internal transfer equation obtained in step (2), a state vector composed of normalized radial displacement, tangential displacement, cross section rotation angle, shear force, axial force and bending moment of all segment ring sections is obtained, and the state vectors composed of normalized radial displacement, tangential displacement, cross section rotation angle, shear force, axial force and bending moment of all sections are denormalized to obtain the opening response of the misaligned joint assembling shield tunnel.

2. The method for calculating the response of a shield tunnel opening considering misfit assembly based on the state space method according to claim 1, characterized in that, In step (2), the segment state equation is: wherein, is the state vector composed of normalized radial displacement, tangential displacement, cross-section rotation angle, shear force, axial force, and bending moment, and θ is the normalized tangential coordinate, is the normalized system matrix, including the cross-section information, material information, shear spring information between rings, and soil spring information of the beam, is the normalized load vector; The standard solution of the state equation is a transfer equation: the state vector consisting of the normalized radial displacement, the normalized tangential displacement, the cross-sectional rotation angle, the shear force, the axial force, and the bending moment on the cross section with the normalized tangential coordinate θ in the transfer equation the state vector consisting of the normalized radial displacement, the normalized tangential displacement, the cross-sectional rotation angle, the shear force, the axial force, and the bending moment on the cross section with the normalized tangential coordinate θ0 in the transfer equation the state vector consisting of the normalized radial displacement, the normalized tangential displacement, the cross-sectional rotation angle, the shear force, the axial force, and the bending moment on the cross section with the normalized tangential coordinate θ0 in the transfer equation matrix is the normalized transfer matrix from θ0to θ, is the normalized load integration vector from θ0to θ; In formula (30), e is a natural constant.

3. The method for calculating the response of a shield tunnel opening considering misfit assembly based on the state space method according to claim 1, characterized in that, In step (3), the transfer equation between the end of a segment and the start of the next segment is: wherein is the normalized state vector at the start of the j+1st segment of pipe, is the normalized state vector at the end of the jth segment of pipe, is the normalized joint transfer matrix for the jth pipe joint.

4. The method for calculating the response of a shield tunnel opening considering misfit assembly based on the state space method according to claim 2, characterized in that, In step (4), the initial whole ring matrix equation is: wherein is the initial coefficient matrix, which is obtained by alternately integrating the transfer equation of the segment of pipe segment and the transfer equation between the end of the segment of pipe segment and the start of the next segment of pipe segment, is the state vector composed of the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force, and bending moment of all the start and end sections of the segments of pipe segment, is the vector composed of the load integral vector and zero vector.

5. The method for calculating the response of a shield tunnel opening considering misfit assembly based on the state space method according to claim 4, characterized in that, In step (5), the supplementary opening equation is: wherein is the supplementary coefficient matrix, is the state vector composed of the normalized radial displacement, tangential displacement, section rotation, shear force, axial force and bending moment of all segment starting and ending sections, the supplementary coefficient matrix has the same column number as the initial coefficient matrix and the same row number as the state vector , the row number of the supplementary coefficient matrix is determined by the ratio of the length of the opening along the axial direction of the shield tunnel to the width of the segment, and the row number of the zero vector on the right side of the opening equation is equal to the row number of the supplementary coefficient matrix .

6. The state space method based response calculation method of considering misfit assembly of a shield tunnel opening according to claim 1, characterized in that, In step (5), the selection of the supplementary coefficient needs to make the state vector composed of the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force and bending moment of all the starting and ending sections of the segments multiplied by the supplementary coefficient matrix equal to the internal force corresponding to the opening position in the state vector.

7. The method for calculating the response of a shield tunnel opening considering misfit assembly based on the state space method according to claim 5, characterized in that, In step (6), the final ring matrix equation is: Wherein: Solving the final ring matrix equation (41), the state vector composed of the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force and bending moment of all the starting and ending sections of the segments is obtained, and then according to the transfer equation (29) of the segments obtained in step (2), the state vector composed of the normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force and bending moment of any section in the ring direction of all the segments is obtained. The normalized radial displacement, tangential displacement, section rotation angle, shear force, axial force and bending moment of each section are anti-normalized to obtain the response considering the staggered joint assembly of the shield tunnel opening.

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