A calculation method for the deformation between the rings of a newly built jacking pipe passing through an existing shield tunnel

Abstract

The present invention discloses a method for calculating the deformation of the inter-ring of a newly built top pipe passing through an existing shield tunnel, including establishing a longitudinal ring-spring-ring shield tunnel model, calculating the additional stress caused by the excavation and unloading of the newly built comprehensive pipe gallery tunnel, simplifying the existing shield tunnel into the longitudinal ring-spring-ring shield tunnel model on the modified Winkler foundation; according to the functional relationship between the state vectors on the cross-sections of the starting and end ends of a single pipe ring segment of the existing shield tunnel and the functional relationship between the state vectors on the cross-sections of the left and right adjacent pipe ring segments of the inter-ring joint segment, the transfer relationship between the state vectors of the starting cross-section of the entire existing shield tunnel and the state vectors of the end cross-section is obtained, and the opening amount and the misalignment amount at any inter-ring of the existing shield tunnel are derived. The present invention can more truly reflect the actual deformation characteristics of the inter-ring of the newly built top pipe passing through the existing shield tunnel.

Description

Technical Field

[0001] The invention belongs to the technical field of subway shield tunnels, and in particular relates to a method for calculating the deformation between rings of a newly built top pipe passing through an existing shield tunnel. Background Art

[0002] The rapid development of urban underground space has led to the construction of a large number of subway tunnels, which will inevitably lead to the construction of new pipe-jacking tunnels over existing subway tunnels. Subway tunnels are almost all constructed using the shield method. Since the construction of new tunnels over the existing subway tunnels will cause deformation of the underlying soil, the subsequent soil-tunnel interaction will cause deformation of the existing subway tunnels. If the induced deformation exceeds the design limit of the existing subway tunnel structure, the structural integrity of the existing shield tunnel will be seriously affected. Subway tunnel lines are extremely important lifelines for large cities, and any potential risks in the tunnel structure may threaten the lives of millions of people. Therefore, reasonable construction control measures are extremely important to ensure the maintainability and integrity of subway tunnels.

[0003] At present, the calculation method for the deformation (opening and misalignment) of the inter-ring of the existing shield tunnel under the construction of the new top pipe is mostly to regard the shield tunnel as a continuous long beam on a linear or nonlinear foundation, and to give the beam longitudinal stiffness in an equivalent way. However, the equivalent continuous beam model ignores the difference in stiffness between the shield tunnel segments and the inter-ring joints, which may lead to incorrect calculation of the deformation of the inter-ring of the existing shield tunnel under the construction of the new top pipe. The Chinese invention patent with application number 202210344699 discloses a method for analyzing the longitudinal deformation of the shield tunnel under the asymmetric thrust of the shield tail, which calculates the longitudinal deformation of the shield tunnel caused by the asymmetric thrust of the shield tail during the excavation or correction along the curve. The external load acting on the shield tunnel is generated by its own construction. However, this deformation analysis method cannot be applied to the deformation analysis of the existing shield tunnel under the construction of the new top pipe. In fact, the shield tunnel uses bolts to connect the rings, which makes the stiffness of the joints lower than that of the segments, resulting in the inter-ring joints being the weak position of the shield tunnel, especially under the construction of the new top pipe. The impact on the existing shield tunnel is more complicated. Summary of the invention

[0004] The purpose of the present invention is to provide a method for calculating the deformation between the rings of a newly built jacking pipe passing through an existing shield tunnel, so as to solve the above technical problems.

[0005] The present invention provides a method for calculating the deformation between rings of a newly built top pipe passing through an existing shield tunnel, comprising the following steps:

[0006] Step (1), decomposing the existing shield tunnel longitudinally into pipe ring segments and inter-ring joint segments connected to each other at intervals, and establishing a longitudinal ring-spring-ring shield tunnel model;

[0007] Step (ii), simplifying the existing shield tunnel into a longitudinal ring-spring-ring shield tunnel model on a modified Winkler foundation model, treating the beginning and end of the tunnel as free boundaries, and calculating the additional stress on the longitudinal axis of the existing shield tunnel caused by the unloading of the top pipe excavation of the new comprehensive pipe gallery based on the Mindlin solution of elastic mechanics;

[0008] Step (iii): Based on the functional relationship between the state vectors on the cross-sections at the beginning and end of a single pipe ring segment of an existing shield tunnel and the functional relationship between the state vectors on the cross-sections of the left and right adjacent pipe ring segments of the inter-ring joint segment, the transfer relationship between the state vectors of the cross-section at the beginning and end of the entire shield tunnel is obtained, and the opening amount and misalignment amount between any rings of the shield tunnel are derived.

[0009] As a further preference of the above scheme:

[0010] The derivation method of the functional relationship between the state vectors at the beginning and end sections of a single ring section of the existing shield in step (iii) is:

[0011] Step (1), calculate the additional stress q(y) on the longitudinal axis of the existing shield tunnel caused by the construction of the new top pipe:

[0012]

[0013] Where: p is the unloading load caused by the excavation of the pipe jacking tunnel; H n H is the vertical distance from the bottom of the jacking tunnel to the ground; t is the vertical distance between the longitudinal centerline of the existing shield tunnel and the ground; L1 and L2 are the straight-line distances between the starting shaft and the top pipe excavation surface and the intersection of the new and old tunnel planes, respectively; L e is the width of the excavation area of ​​the pipe jacking machine; v is the Poisson's ratio of the soil layer; R1 and R2 are parameters;

[0014] The unloading load p generated by excavation of a newly built pipe-jacking tunnel can be expressed as:

[0015]

[0016] Where: γ s , γ t and γ n are the unit weights of the soil excavated by the pipe jacking machine, the pipe section of the pipe jacking machine, and the drag-reducing mud around the pipe wall; h e and L are the height of the excavation area of ​​the pipe jacking machine and the width of the pipe jacking section respectively; h and t c are the height and thickness of the newly built jacking pipe section respectively;

[0017] Step (2): When the existing shield tunnel undergoes longitudinal deformation w, the Winkler foundation model provides an external load kwD to the tunnel.t , the equations for the bending moment, shear force and rotation angle of the pipe ring are expressed as:

[0018]

[0019]

[0020]

[0021]

[0022] Where: M is the tunnel bending moment; Q is the tunnel shear force; EI is the tunnel bending stiffness; is the tunnel section rotation angle; y is the tunnel section longitudinal position; D t is the tunnel diameter; k is the foundation reaction coefficient; q(y) is the additional stress;

[0023] Step (3), the displacement and internal force equations of a single ring section of the existing shield tunnel are converted into a matrix form and dimensionless, and the resulting state equation is:

[0024]

[0025] Where: is a dimensionless state vector, including unknown quantities such as section displacement, rotation angle, bending moment and shear force;

[0026]

[0027] in, is the dimensionless external load state vector, is the dimensionless system matrix, which can be expressed as:

[0028]

[0029]

[0030] The dimensionless quantities are expressed as:

[0031]

[0032] Where: L is the calculated total length of the tunnel, E is the elastic modulus of the tunnel tube ring, A is the cross-sectional area of ​​the tunnel, and I is the moment of inertia of the tunnel section;

[0033] Step (4), the standard solution of the state equation is the functional relationship between the state vector on any cross section of a single tunnel ring segment and the state vector of the cross section at its starting end:

[0034]

[0035] Where: is the longitudinal position of the cross section at the beginning of a single tunnel ring segment, is the state vector of the section at the beginning of a single tunnel ring segment, is the state vector of any cross section of a single tunnel ring segment, is the transfer matrix, is the vector integral of the dimensionless external load;

[0036] Step (5), assuming that the longitudinal position of the end section of a single pipe ring segment is y1, take y=y1 to obtain the state vector of the beginning section of a single pipe ring segment of the tunnel: and the end section state vector The matrix transfer relationship between them is:

[0037]

[0038] As a further preference of the above scheme:

[0039] The parameters R1 and R2 are:

[0040]

[0041]

[0042] As a further preference of the above scheme:

[0043] The derivation method of the functional relationship between the state vectors on the cross-sections of the left and right adjacent pipe ring sections of the existing shield tunnel inter-ring joint section is:

[0044] Step (1), according to the continuity conditions of shear force and bending moment of the joint segment, the expression of the internal force at the jth joint is obtained as follows:

[0045]

[0046]

[0047] Where: M0 (j+1) 、M1 (j) are the bending moments at the right and left ends of the pipe ring of the jth joint respectively; Q0 (j+1) 、Q1 (j) are the shear forces at the right and left pipe ring ends of the jth joint, respectively; are the rotation angles of the right and left pipe ring end sections of the jth joint respectively; w0 (j+1) 、w1 (j) are the displacements of the right and left pipe ring ends of the jth joint, respectively;

[0048] Step (2), convert the expression of step (1) into a dimensionless matrix form, that is, the functional relationship between the state vectors on the cross-sections of the left and right adjacent pipe ring sections of the tunnel ring joint section. The dimensionless form of the joint rotation and shear stiffness is:

[0049]

[0050] As a further preference of the above scheme:

[0051] The method for deriving the opening amount and the misalignment amount at any ring interval of the existing shield tunnel in step (iii) is:

[0052] Step (1), based on the obtained functional relationship between the cross-section state vectors at the beginning and end of a single pipe ring segment and the functional relationship between the cross-section state vectors of the adjacent pipe ring segments on the left and right of the joint segment, the transfer relationship between the cross-section state vectors at the beginning and end of the entire shield tunnel including n+1 pipe ring segments and n annular joint segments is obtained as follows:

[0053]

[0054] Where: is the state vector of the end section of the n+1th tunnel ring segment; is the state vector of the end section of the n+1th tunnel ring segment; is the state vector of the end section of the nth tunnel ring segment; is the state vector of the section at the beginning of the first ring section of the tunnel; is the transfer matrix on the n+1th ring section of the tunnel; is the vector integral of the external load on the n+1th ring section of the tunnel; is the total transfer matrix, is the total external load vector integral, which can be expressed as:

[0055]

[0056]

[0057] Step (2), setting the boundary conditions of the existing shield tunnel as follows:

[0058]

[0059] Where: Q0 1 、Q1 n+1 are the shear forces on the first ring start section and the n+1th ring end section of the existing shield tunnel respectively; M0 1 、M1 n+1 are the bending moments on the sections at the beginning of the first ring and the end of the n+1th ring of the existing shield tunnel respectively;

[0060] Step (3), the relationship between the neutral axis angle Ψ and the rotation angle θ of the tunnel section is:

[0061]

[0062]

[0063] Where: E c is the elastic modulus of a single segment; I c is the longitudinal inertia moment of the tunnel; n is the number of bolts at the joint; k b is the average linear stiffness of the joint bolts, k b =E b A b / l b , E b is the elastic modulus of the bolt, A b is the cross-sectional area of ​​the bolt, l b is the bolt length; l s is the ring width; A c is the cross-sectional area of ​​the tunnel segment, r is the position from the segment axis to the bolt;

[0064] The rotational stiffness k of the tunnel annular joint is obtained as θ The calculation formula is:

[0065]

[0066] Where: η is the rotation stiffness influence coefficient;

[0067] Step (4), according to the joint angle, the tunnel ring opening is obtained as follows:

[0068]

[0069] Tunnel annular joint shear stiffness k s The calculation formula is:

[0070]

[0071] Where: K b and K c are the Timoshenko shear coefficients of bolts and segment rings respectively; G b and G c are the shear stiffness of bolts and segments respectively;

[0072] Step (5), based on the functional relationship between the state vectors on the cross sections at the beginning and end of the existing single pipe ring segment, and the functional relationship between the state vectors on the cross sections of the adjacent pipe ring segments on the left and right of the tunnel ring joint segment, the state vector at any cross section of the tunnel can be obtained. Finally, the dimensionless quantity is converted into a real quantity according to the following formula to obtain the opening amount and misalignment amount between any rings of the tunnel:

[0073]

[0074] Beneficial effects of the present invention: a longitudinal ring-spring-ring shield tunnel model capable of simultaneously considering the inter-ring opening and shear misalignment deformation is proposed, and then based on the longitudinal ring-spring-ring shield tunnel model and the modified Winkler foundation model, a two-stage analysis method and a state space method are used to derive an analytical solution to the longitudinal inter-ring deformation of a newly built top pipe passing through an existing shield tunnel, as follows:

[0075] 1. Compared with the traditional shield tunnel ring deformation calculation method, it can more realistically reflect the actual deformation characteristics of the existing shield tunnel under the construction of the new top pipe;

[0076] 2. It can truly calculate the longitudinal displacement, ring opening and misalignment of the existing shield tunnel under the construction of the newly built top pipe;

[0077] 3. It can accurately calculate the most dangerous parts of the deformation between the rings of the existing shield tunnel under the construction of the new jacking pipe, thus providing a reference for the rapid and safe protection of the tunnel. BRIEF DESCRIPTION OF THE DRAWINGS

[0078] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0079] Figure 1 It is a longitudinal ring-spring-ring shield tunnel model in an embodiment of the present invention;

[0080] Figure 2 is a cross-sectional view of a calculation model in an embodiment of the present invention;

[0081] Figure 3 is a plan view of a calculation model in an embodiment of the present invention;

[0082] Figure 4 It is the force analysis of the micro-element section of the existing shield tunnel in the embodiment of the present invention;

[0083] Figure 5 The longitudinal joint section of the existing shield tunnel in the embodiment of the present invention is deformed;

[0084] Figure 6 It is the existing shield tunnel state vector transmission route in the embodiment of the present invention;

[0085] Figure 7 It is a plan view of the general situation of the Wuhan Huangxiaohe comprehensive pipe gallery passing through Metro Line 3 in the comparative example of the present invention;

[0086] Figure 8 It is a cross-sectional view of the general situation of the Wuhan Huangxiaohe comprehensive pipe gallery passing through Metro Line 3 in the comparative example of the present invention;

[0087] Fig. 9 It is another cross-sectional view of the engineering overview of Wuhan Huangxiaohe comprehensive pipe gallery passing through Metro Line 3 in the comparative example of the present invention;

[0088] Fig.10 It is the uplift value of the right tunnel in the comparison between the uplift value of the No. 3 line tunnel calculated by the method of the present invention and the on-site monitoring and equivalent continuous beam model calculation results;

[0089] Fig.11 It is the uplift value of the left tunnel in the comparison between the uplift value of the No. 3 line tunnel calculated by the method of the present invention and the results of the on-site monitoring and the calculation of the equivalent continuous beam model;

[0090] Fig.12 The comparison between the right-line tunnel ring opening calculated by the method of the present invention and the calculation result of the equivalent continuous beam model;

[0091] Fig.13 The inter-ring stagger of the right-line tunnel calculated by the method of the present invention is compared with the calculation result of the equivalent continuous beam model; DETAILED DESCRIPTION

[0092] The present invention is further described below in conjunction with specific embodiments. These embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. The structural features of the present invention are now described in detail in conjunction with the accompanying drawings of the specification.

[0093] Example

[0094] This embodiment provides a method for calculating the deformation of the inter-ring of a newly built top pipe passing through an existing shield tunnel, including establishing a longitudinal ring-spring-ring shield tunnel model, establishing a calculation model, calculating the additional stress caused by the unloading of the newly built tunnel excavation, and deriving the longitudinal inter-ring deformation of the existing shield tunnel, specifically including the following steps:

[0095] Step 1: Establish a longitudinal ring-spring-ring shield tunnel model

[0096] The shield tunnel is decomposed longitudinally into tube ring segments and inter-ring joint segments connected to each other, as a longitudinal ring-spring-ring shield tunnel model, as shown in Figure 1 As shown in the figure, each ring segment is regarded as an Euler-Bernoulli short beam, whose length is consistent with the length of the segment; the inter-ring joint segment is composed of a linear rotation spring and a shear spring. Different from the equivalent continuous beam model in the prior art, this model simultaneously considers the inter-ring opening and shear misalignment deformation of the shield tunnel.

[0097] Among them, the displacement and internal force equations of each tunnel ring segment are expressed as:

[0098]

[0099]

[0100]

[0101] Where: is the tunnel section rotation angle; w is the tunnel longitudinal displacement; M and Q are the tunnel bending moment and shear force respectively; EI is the tunnel bending stiffness; q(y) is the external load acting on the tunnel, and y is the tunnel longitudinal position.

[0102] Among them, the bending moment and shear force of each tunnel joint section are expressed as:

[0103]

[0104] Q j =k s Δw=k s (w r -w l ) (2b)

[0105] Where: M j , Q j are the bending moment and shear force at the tunnel joint, respectively; k θ , k s are the rotational stiffness and shear stiffness of the tunnel joint, respectively; Δw are the relative rotation angle and shear misalignment of the tunnel sections at both ends of the joint; are the tunnel section angles at the right and left ends of the joint respectively; w r 、w l are the displacements of the tunnel sections at the right and left ends of the joint, respectively.

[0106] Step 2: Establish a calculation model

[0107] The existing shield tunnel under the new jacking pipe is regarded as a longitudinal ring-spring-ring shield tunnel model resting on the modified Winkler foundation model, as shown in Figure 2 and Figure 3 As shown in the figure, the beginning and end of the existing shield tunnel are regarded as free constraints, that is, the shear force and bending moment at both ends of the existing shield tunnel are 0, that is:

[0108]

[0109] Where: Q0 1 、Q1 n+1 are the shear forces on the first ring start section and the n+1th ring end section of the existing shield tunnel respectively; M0 1 、M1 n+1 They are the bending moments on the starting section of the 1st ring and the ending section of the n+1th ring of the existing shield tunnel respectively.

[0110] Step 3: Calculate the additional stress caused by unloading the new tunnel excavation

[0111] According to the calculation model established in step 2, the new pipe jacking tunnel will cause additional stress in the longitudinal direction of the existing shield tunnel below due to the unloading effect of excavation. The stratum is equivalent to a homogeneous elastic half space, and the Mindlin stress solution is used to solve the additional stress. In order to consider the arbitrary crossing angle α between the new tunnel and the existing shield tunnel, the global coordinate system o-xyz and the local coordinate system o′-x′y′z′ are established, as follows: Figure 2 and Figure 3 The relationship between the global coordinate system o-xyz and the local coordinate system o′-x′y′z′ can be expressed as:

[0112]

[0113] Where: m is the vertical distance between the x' axis in the local coordinate system and the x axis in the global coordinate system; n is the vertical distance between the y' axis in the local coordinate system and the y axis in the global coordinate system.

[0114] According to Mindlin stress solution and formula (4), the additional stress q(y) in the longitudinal direction of the existing shield tunnel below caused by the excavation unloading effect of the newly built jacking tunnel can be obtained as follows:

[0115]

[0116] Where: p is the unloading load caused by the excavation of the pipe jacking tunnel; H n H is the vertical distance from the bottom of the jacking tunnel to the ground; t is the vertical distance between the longitudinal centerline of the existing shield tunnel and the ground; L1 and L2 are the straight-line distances between the starting shaft and the top pipe excavation surface and the intersection of the new and old tunnel planes, respectively; L e is the width of the excavation area of ​​the pipe jacking machine; v is the Poisson's ratio of the soil layer; parameters R1 and R2 are expressed as:

[0117]

[0118]

[0119] The unloading load p generated by excavation of a newly built pipe-jacking tunnel can be expressed as:

[0120]

[0121] Where: γ s , γ t and γ n are the unit weights of the soil excavated by the pipe jacking machine, the pipe section of the pipe jacking machine, and the drag-reducing mud around the pipe wall; h eand L are the height of the excavation area of ​​the pipe jacking machine and the width of the pipe jacking section respectively; h and t c They are the height and thickness of the newly built jacking pipe section respectively.

[0122] Step 4: Derivation of the longitudinal inter-ring deformation of the existing shield tunnel. The derivation method is:

[0123] (1) Take Figure 2 The stress analysis of the existing shield tunnel micro-element segment is carried out, such as Figure 4 As shown, according to Figure 4 , the existing shield tunnel ring segment shear equilibrium equation, that is, equation 1(c), can be further expressed as

[0124]

[0125] Where: D t is the existing shield tunnel diameter; k is the foundation reaction coefficient, expressed as

[0126]

[0127] Where: E s is the elastic modulus of the soil layer; ω is the influence coefficient of the tunnel depth, ω=1+1 / (1.7H t / D t ), formula (9) needs to satisfy H t / D t >0.5 condition.

[0128] (2) To facilitate the solution of subsequent equations, the displacement and internal force equations (1a), (1b) and (8) of a single tunnel ring segment are transformed into the following matrix form:

[0129]

[0130] Where: X is the tunnel section state vector; Q is the external load state vector; A is the system matrix; respectively expressed as

[0131]

[0132] Q=[0 0 -qD t 0] (12)

[0133]

[0134] In order to ensure the stability of the calculation of formula (10), formula (10) is further dimensionless as formula (14):

[0135]

[0136] Where: is the dimensionless tunnel cross-section state vector; is the dimensionless external load state vector; is the dimensionless system matrix; respectively expressed as

[0137]

[0138]

[0139]

[0140] The dimensionless physical quantities in equations (15) to (17) are

[0141]

[0142] Where: L t is the total length of the calculated tunnel; E is the Young's modulus of the segment; A is the cross-sectional area of ​​the existing tunnel; I is the moment of inertia of the cross-sectional area of ​​the existing tunnel.

[0143] According to the state space method, the solution of equation (14) can be obtained, that is, the functional relationship between the state vector on any cross section of a single tunnel ring segment and the state vector of its starting cross section is:

[0144]

[0145] Where: is the longitudinal position of the cross section at the beginning of a single tunnel ring segment, is the state vector of the section at the beginning of a single tunnel ring segment, is the state vector of any cross section of a single tunnel ring segment, is the transfer matrix, is the vector integral of the dimensionless external load, expressed as

[0146]

[0147]

[0148] When there are m concentrated loads P acting on the tunnel interval y0~y, equation (21) can be further expressed as

[0149]

[0150] Where: is the location where the ith concentrated load acts, expressed as is a dimensionless load, expressed as l i is the acting length of the ith concentrated load along the y-axis.

[0151] (3) If the longitudinal position of the end section of a single pipe ring segment is y1, y = y1 is taken for equation (19) to obtain the state vector of the section at the beginning of a single pipe ring segment of the tunnel: and the end section state vector The matrix transfer relationship between them is

[0152]

[0153] (4) After obtaining the functional relationship between the state vectors at the beginning and end of a single pipe ring segment, it is necessary to derive the functional relationship between the state vectors at the left and right adjacent pipe ring segments of the tunnel inter-ring joint segment, such as Figure 5 As shown in Figure 2, according to the continuity conditions of shear force and bending moment in the joint segment and combined with formula (2), the expression of the internal force at the jth joint is obtained as follows:

[0154]

[0155]

[0156] Where: M0 (j+1) 、M1 (j) are the bending moments at the right and left ends of the pipe ring of the jth joint respectively; Q0 (j+1) 、Q1 (j) are the shear forces at the right and left pipe ring ends of the jth joint, respectively; are the rotation angles of the right and left pipe ring end sections of the jth joint respectively; w0 (j+1) 、w1 (j) are the displacements of the right and left pipe ring ends of the jth joint, respectively.

[0157] To facilitate the solution of the subsequent equations, equation (24) is converted into the following dimensionless matrix form, that is, the functional relationship between the state vectors on the cross-sections of the left and right adjacent pipe ring sections of the tunnel ring junction section:

[0158]

[0159] Where: is the state vector of the section at the beginning of the j+1th tunnel ring segment; is the state vector of the end section of the jth tunnel ring segment, Transfer matrix for joint segment

[0160]

[0161] The dimensionless form of the joint rotation and shear stiffness is

[0162]

[0163] (5), reference Figure 6As shown in Figure 2, after obtaining the functional relationship between the state vectors of the cross sections at the beginning and end of a single pipe ring segment (Equation (23)) and the functional relationship between the state vectors of the cross sections of the adjacent pipe ring segments on the left and right of the joint segment (Equation (25)), the transfer relationship between the state vectors of the cross sections at the beginning and end of the entire shield tunnel containing n+1 pipe ring segments and n annular joint segments is obtained by combining equations (23) and (25) as follows:

[0164]

[0165] Where: is the state vector of the end section of the n+1th tunnel ring segment; is the state vector of the end section of the n+1th tunnel ring segment; is the state vector of the end section of the nth tunnel ring segment; is the state vector of the section at the beginning of the first ring section of the tunnel; is the transfer matrix on the n+1th ring section of the tunnel; is the vector integral of the external load on the n+1th ring section of the tunnel; is the total transfer matrix, is the total external load vector integral, which can be expressed as:

[0166]

[0167]

[0168] (6) Calculation formula for spring stiffness of the inter-ring joint section: When the tunnel is subjected to end thrust, without considering the preload of the bolts, the following relationship is established for the neutral axis angle Ψ and the rotation angle θ of the tunnel section:

[0169]

[0170]

[0171] Where: E c is the elastic modulus of a single segment; I c is the longitudinal inertia moment of the tunnel; n is the number of bolts at the joint; k b is the average linear stiffness of the joint bolts, k b =E b A b / l b , E b is the elastic modulus of the bolt, A b is the cross-sectional area of ​​the bolt, l b is the bolt length; l s is the ring width; A c is the cross-sectional area of ​​the tunnel segment, and r is the position from the segment axis to the bolt.

[0172] According to formula (31), the rotation stiffness k of the tunnel annular joint can be obtained as θ The calculation formula is

[0173]

[0174] Where: η is the rotation stiffness influence coefficient, which is 1 when the joint reinforcement is not considered.

[0175] According to the joint angle, the opening between the existing shield tunnel rings can be obtained as

[0176]

[0177] Tunnel annular joint shear stiffness k s The calculation formula is

[0178]

[0179] Where: K b and K c are the Timoshenko shear coefficients of bolts and segment rings respectively; G b and G c are the shear stiffness of the bolt and the segment respectively, both of which are related to their respective elastic moduli; λ is the shear stiffness influence coefficient, which takes the value of 1 when the joint reinforcement is not considered.

[0180] (7) Equation solution: Combining the boundary conditions of equations (3) and (28), the state vector (i.e., displacement, rotation, bending moment, and shear force) of the entire tunnel start section can be obtained. Then, according to equations (19) and (25), the state vector at any section of the tunnel can be obtained. Then, according to equation (18), the dimensionless quantity is converted into a real quantity to obtain the displacement and internal force at any section of the tunnel. Finally, according to equations (2b) and (33), the misalignment and opening between any adjacent rings of the existing shield tunnel can be obtained.

[0181] Comparative Example

[0182] In order to verify the applicability and rationality of the method proposed in this application in calculating the opening and misalignment between the rings of the existing shield tunnel when a new jacking pipe is passing through and under it, this comparative example provides the monitoring results of the Wuhan Huangxiaohe Integrated Pipeline Corridor passing through Metro Line 3 and the calculation results of two commonly used equivalent continuous beam models (Euler-Bernoulli beam and Timoshenko beam) for comparative analysis.

[0183] Project Overview: The Huangxiaohe integrated pipeline corridor passes through the double shield tunnel of Metro Line 3. The integrated pipeline corridor is excavated using an earth pressure balance (EPB) pipe jacking machine of 8.32 meters (width) × 5.27 meters (height). Figure 7The relative position of the integrated pipe gallery and the subway line 3 tunnel and the deformation monitoring layout of the line 3 tunnel are given. The integrated pipe gallery is 138 meters long and has a total of 92 prefabricated reinforced concrete pipe segments. Each segment has a ring length of 1.5 meters, a thickness of 0.5 meters, a width of 8.3 meters, and a height of 5.25 meters. The integrated pipe gallery crosses the line 3 tunnel at an angle of 70.4°. The integrated pipe gallery and the line 3 tunnel are buried at depths of 8m and 18.47m below the surface, respectively. The horizontal distance between the left and right tunnels of Line 3 is 8.4m. The minimum clear distance between the line 3 tunnel and the pipe gallery is 4.72 meters. The structural parameters of the subway line 3 tunnel are listed in Table 1. The soil parameters related to this project are listed in Table 2. The pipe jacking machine was started from the starting well on March 10, 2021 and arrived at the receiving well on April 30, 2021. The total construction time is 51 days.

[0184] The basic parameters set by the above three methods (the method proposed in this application, Euler-Bernoulli (EB) and Timoshenko (TM) equivalent continuous beam methods) are: the longitudinal length of the existing left and right line shield tunnels of Line 3 is 200 rings, the width of each ring segment is 1.2m, and the total length L=240m; the burial depth of the existing left and right line tunnels H t =18.47m, the bottom burial depth of the integrated pipe gallery is H n =10.65m; the structural parameters of the pipe segment and joint bolts of the integrated pipe gallery are shown in Table 1; the foundation reaction coefficient k = 5800kN / m 3 , Poisson's ratio v = 0.35; the rotation stiffness k of the existing left and right shield tunnel joints is calculated by equations (32) and (34): θ and shear stiffness k s 6.5×10 7 kN·m / rad and 2.2×10 6 kN / m. Equivalent bending stiffness (EI) of existing left and right line shield tunnels eq and equivalent shear stiffness (GA) eq They are 7.8×10 7 kN·m 2 and 2.1×10 6 kN / m; for the right-line tunnel, the excavation lengths of the integrated pipeline gallery L1 and L2 are 84m and 54m respectively; for the left-line tunnel, L1 and L2 are 100m and 38m respectively.

[0185] Table 1 Structural parameters of Line 3 tunnel

[0186]

[0187] Table 2 Physical parameters of soil layers

[0188]

[0189] Fig.10 and Fig.11 The uplift value of Line 3 tunnel calculated by the method of the present invention is compared with the results of on-site monitoring and equivalent continuous beam model calculation. Fig.10 It can be seen that the uplift values ​​of the right tunnel of Line 3 calculated by the method of the present invention and the TM continuous beam model are in good agreement with the measured results, while the EB continuous beam model slightly underestimates the uplift value of the right tunnel. Fig.11 The uplift values ​​of the left tunnel of Line 3 calculated by the method of the present invention and the TM continuous beam model slightly overestimate the uplift values ​​of the left tunnel, but in general, the three methods can better predict the longitudinal uplift deformation of the existing shield tunnel under the construction of the integrated pipeline corridor. Figure 7 Further observations show that the uplift value curve of Line 3 Tunnel calculated by the method of the present invention is composed of a series of short straight lines with inclined angles, and there are obvious jumps at the joints, while the uplift value curve of Line 3 Tunnel calculated by the TM and EB continuous beam models is smooth and continuous. This is because the method of the present invention takes into account the difference in stiffness at the joints and pipe sections of the actual shield tunnel. The joint section is more easily deformed than the pipe ring section, while the TM and EB continuous beam models regard the shield tunnel as a continuous long beam and ignore the difference in stiffness at the tunnel joints and pipe sections. Since the on-site monitoring points cannot be densely arranged, this phenomenon cannot be monitored. The above results show that the method of the present invention can more realistically reflect the longitudinal deformation characteristics of the actual shield tunnel than the traditional equivalent continuous beam model.

[0190] Fig.12 The comparison between the inter-ring opening of the right-line tunnel calculated by the method of the present invention and the calculation results of the equivalent continuous beam model is given. As can be seen from the figure, the inter-ring opening of the right-line tunnel calculated by the three methods shows a "W" distribution along the longitudinal direction of the tunnel, and the location where the maximum inter-ring opening occurs is located directly below the central axis of the newly built comprehensive pipe corridor. The inter-ring opening of the right-line tunnel calculated by the method of the present invention is higher than the calculation result of the TM continuous beam and lower than the calculation result of the EB continuous beam. This is because the TM beam and the EB beam ignore the weakening effect of the shield tunnel joint, resulting in a miscalculation of the inter-ring opening of the tunnel. This shows that when predicting the inter-ring opening of an existing shield tunnel under pipe jacking construction, the TM continuous beam model may underestimate the tunnel inter-ring opening, while the EB continuous beam model may overestimate the tunnel inter-ring opening. Fig.13The inter-ring misalignment of the right-line tunnel calculated by the method of the present invention is compared with the calculation results of the equivalent continuous beam model. As can be seen from the figure, the maximum inter-ring misalignment of the right-line tunnel calculated by the method of the present invention and the TM continuous beam model is located at the tunnel 7.2m away from both sides of the central axis of the integrated pipe corridor. The inter-ring misalignment of the right-line tunnel calculated by the method of the present invention is lower than the calculation result of the TM continuous beam, and the inter-ring misalignment of the tunnel calculated by the EB continuous beam is 0. This is because the EB beam assumes that the shear stiffness of the tunnel is infinite and only bending deformation occurs, so the calculated misalignment is 0. The above results show that when predicting the inter-ring misalignment of the existing shield tunnel under the construction of the integrated pipe corridor, the EB continuous beam model cannot be used to calculate the inter-ring opening of the existing shield tunnel, and the TM continuous beam model may overestimate the inter-ring misalignment of the tunnel.

[0191] The above is only a preferred embodiment of the present invention, and 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 deformation of a newly built jacking pipe passing through an existing shield tunnel ring, characterized in that: The following steps are involved: Step (1), decomposing the existing shield tunnel longitudinally into pipe ring segments and inter-ring joint segments connected to each other at intervals, and establishing a longitudinal ring-spring-ring shield tunnel model; Step (ii), simplifying the existing shield tunnel into a longitudinal ring-spring-ring shield tunnel model on a modified Winkler foundation model, treating the beginning and end of the tunnel as free boundaries, and calculating the additional stress on the longitudinal axis of the existing shield tunnel caused by the unloading of the top pipe excavation of the new comprehensive pipe gallery based on the Mindlin solution of elastic mechanics; Step (iii), according to the functional relationship between the state vectors on the cross-sections of the start and end of a single pipe ring segment of the existing shield tunnel and the functional relationship between the state vectors on the cross-sections of the left and right adjacent pipe ring segments of the inter-ring joint segment, the transfer relationship between the state vectors of the cross-section of the start and end of the entire shield tunnel is obtained, and the opening amount and the misalignment amount between any rings of the shield tunnel are derived; The derivation method of the functional relationship between the state vectors at the beginning and end sections of a single ring section of the existing shield in step (iii) is: Step (1), calculate the additional stress q(y) on the longitudinal axis of the existing shield tunnel caused by the construction of the new top pipe: Where: p is the unloading load caused by the excavation of the pipe jacking tunnel; H n H is the vertical distance from the bottom of the jacking tunnel to the ground; t is the vertical distance between the longitudinal centerline of the existing shield tunnel and the ground; L1 and L2 are the straight-line distances between the starting shaft and the top pipe excavation surface and the intersection of the new and old tunnel planes, respectively; L e is the width of the excavation area of ​​the pipe jacking machine; v is the Poisson's ratio of the soil layer; R1 and R2 are parameters; The unloading load p generated by excavation of a newly built pipe-jacking tunnel is expressed as: Where: γ s , γ t and γ n are the unit weights of the soil excavated by the pipe jacking machine, the pipe section of the pipe jacking machine, and the drag-reducing mud around the pipe wall; h e and L are the height of the excavation area of ​​the pipe jacking machine and the width of the pipe jacking section respectively; h and t c are the height and thickness of the newly built jacking pipe section respectively; Step (2): When the existing shield tunnel undergoes longitudinal deformation w, the Winkler foundation model provides an external load kwD to the tunnel. t , the equations for the bending moment, shear force and rotation angle of the pipe ring are expressed as: Where: M is the tunnel bending moment; Q is the tunnel shear force; EI is the tunnel bending stiffness; is the tunnel section rotation angle; y is the tunnel section longitudinal position; D t is the tunnel diameter; k is the foundation reaction coefficient; q(y) is the additional stress; Step (3), the displacement and internal force equations of a single ring section of the existing shield tunnel are converted into a matrix form and dimensionless, and the resulting state equation is: Where: is a dimensionless state vector, including unknown quantities such as section displacement, rotation angle, bending moment and shear force; in, is the dimensionless external load state vector, is the dimensionless system matrix, which can be expressed as: The dimensionless quantities are expressed as: Where: L is the calculated total length of the tunnel, E is the elastic modulus of the tunnel tube ring, A is the cross-sectional area of ​​the tunnel, and I is the moment of inertia of the tunnel section; Step (4), the standard solution of the state equation is the functional relationship between the state vector on any cross section of a single tunnel ring segment and the state vector of the cross section at its starting end: Where: is the longitudinal position of the cross section at the beginning of a single tunnel ring segment, is the state vector of the section at the beginning of a single tunnel ring segment, is the state vector of any cross section of a single tunnel ring segment, is the transfer matrix, is the vector integral of the dimensionless external load; Step (5), assuming that the longitudinal position of the end section of a single pipe ring segment is y1, take y=y1 to obtain the state vector of the beginning section of a single pipe ring segment of the tunnel: and the end section state vector The matrix transfer relationship between them is: The derivation method of the functional relationship between the state vectors on the cross-sections of the left and right adjacent pipe ring sections of the existing shield tunnel inter-ring joint section is: Step (1), according to the continuity conditions of shear force and bending moment of the joint segment, the expression of the internal force at the jth joint is obtained as follows: Where: M0 (j+1) 、M1 (j) are the bending moments at the right and left ends of the pipe ring of the jth joint respectively; Q0 (j+1) 、Q1 (j) are the shear forces at the right and left pipe ring ends of the jth joint, respectively; are the rotation angles of the right and left pipe ring end sections of the jth joint respectively; w0 (j+1) 、w1 (j) are the displacements of the right and left pipe ring ends of the jth joint, respectively; Step (2), convert the expression of step (1) into dimensionless matrix form, that is, the functional relationship between the state vectors on the cross-sections of the adjacent pipe ring sections on the left and right sides of the tunnel ring joint section, Where: is the state vector of the section at the beginning of the j+1th tunnel ring segment; is the state vector of the end section of the j-th tunnel ring segment, Transfer matrix for joint segment The dimensionless form of the joint rotation and shear stiffness is:

2. The method for calculating the deformation of the inter-ring of a newly built jacking pipe passing through an existing shield tunnel according to claim 1 is characterized by: The parameters R1 and R2 are:

3. The method for calculating the deformation of the inter-ring of a newly built top pipe passing through an existing shield tunnel according to claim 1 is characterized by: The method for deriving the opening amount and the misalignment amount at any ring interval of the existing shield tunnel in step (iii) is: Step (1), based on the obtained functional relationship between the cross-section state vectors at the beginning and end of a single pipe ring segment and the functional relationship between the cross-section state vectors of the adjacent pipe ring segments on the left and right of the joint segment, the transfer relationship between the cross-section state vectors at the beginning and end of the entire shield tunnel including n+1 pipe ring segments and n annular joint segments is obtained as follows: Where: is the state vector of the end section of the n+1th tunnel ring segment; is the state vector of the section at the beginning of the n+1th tunnel ring segment; is the state vector of the end section of the nth tunnel ring segment; is the state vector of the section at the beginning of the first ring section of the tunnel; is the transfer matrix on the n+1th ring section of the tunnel; is the vector integral of the external load on the n+1th ring section of the tunnel; is the total transfer matrix, is the total external load vector integral, which can be expressed as: Step (2), setting the boundary conditions of the existing shield tunnel as follows: Where: are the shear forces on the first ring start section and the n+1 ring end section of the existing shield tunnel respectively; are the bending moments on the sections at the beginning of the first ring and the end of the n+1th ring of the existing shield tunnel respectively; Step (3), the relationship between the neutral axis angle Ψ and the rotation angle θ of the tunnel section is: Where: E c is the elastic modulus of a single segment; I c is the longitudinal inertia moment of the tunnel; n is the number of bolts at the joint; k b is the average linear stiffness of the joint bolts, k b =E b A b / l b , E b is the elastic modulus of the bolt, A b is the cross-sectional area of ​​the bolt, l b is the bolt length; l s is the ring width; A c is the cross-sectional area of ​​the tunnel segment, r is the position from the segment axis to the bolt; The rotational stiffness k of the tunnel annular joint is obtained as θ The calculation formula is: Where: η is the rotation stiffness influence coefficient; Step (4), according to the joint angle, the tunnel ring opening is obtained as follows: Tunnel annular joint shear stiffness k s The calculation formula is: Where: κ b and κ c are the Timoshenko shear coefficients of bolts and segment rings respectively; G b and G c are the shear stiffness of bolts and segments respectively; Step (5), based on the functional relationship between the state vectors on the cross sections at the beginning and end of the existing single pipe ring segment, and the functional relationship between the state vectors on the cross sections of the adjacent pipe ring segments on the left and right of the tunnel ring joint segment, the state vector at any cross section of the tunnel can be obtained. Finally, the dimensionless quantity is converted into a real quantity according to the following formula to obtain the opening amount and misalignment amount between any rings of the tunnel:

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