A method for analyzing the impact of shield tunneling on existing tunnels based on analytical solution

Abstract

The present invention belongs to the technical field of shield construction, and specifically discloses a method for analyzing the impact of shield underpass on existing tunnels based on analytical solutions. The method includes using a Pasternak two-parameter foundation model to simulate the interaction between the tunnel structure and the foundation. In the first stage, the additional stress formula of the construction load and soil loss is derived, and the additional stress at the axis of the existing tunnel caused by the additional thrust of the cutter head, the friction of the shield shell, the synchronous grouting pressure and the soil loss is comprehensively considered; in the second stage, the deformation stiffness equation of the beam is established to calculate the deformation and bending moment of the existing tunnel, analyze the impact of different influencing factors on the settlement of the existing tunnel, and analyze the laws of the impact of the foundation elastic modulus, clear distance and stratum loss rate on the existing tunnel. The present invention can quickly and accurately evaluate the impact of the existing tunnel under the condition of shield close construction.

Description

Technical Field

[0001] The present invention belongs to the technical field of shield construction, and more specifically, relates to a method for analyzing the influence of shield underpass on an existing tunnel based on an analytical solution. Background Art

[0002] The adjacent construction of a tunnel will release the original stress of the soil, causing additional stress in the existing tunnel, which will have a certain impact on the structural safety of the existing tunnel. Many literatures have analyzed the effects of adjacent excavation on the existing tunnel structure, including field tests, centrifuge model tests, numerical analysis and semi-analytical methods. Compared with field tests, centrifuge tests and complex finite element modeling that require a lot of manpower and material resources, analytical methods are a simple and low-cost effective method to evaluate the response of existing tunnels to adjacent shield construction in the initial design stage. At present, the two-stage analysis method is mostly used to study the impact of adjacent construction on adjacent structures. The mechanical model and model parameters of the soil and structure will have a significant impact on the calculation results. Summary of the invention

[0003] In view of the above defects or improvement needs of the prior art, the present invention provides a method for analyzing the impact of shield tunneling on existing tunnels based on analytical solutions, which can consider the impact of the main shield construction load on the deformation of the existing tunnel during the tunneling process. At the same time, the present invention analyzes the influence of the deformation of the existing tunnel structure and the internal force caused by important influencing factors such as the elastic modulus of the underlying layer, the tunnel spacing, and the soil loss rate. The method of the present invention can accurately evaluate the impact of existing tunnels under the condition of shield close construction.

[0004] To achieve the above object, the present invention proposes a method for analyzing the impact of shield tunneling on an existing tunnel based on an analytical solution, which is characterized by comprising the following steps:

[0005] S1 uses Mindlin formula to derive the additional stress formula caused by shield construction, and calculates the additional stress at the action point through double Gauss-legendre numerical integration;

[0006] S2 uses the mirror method to calculate the additional stress caused by soil loss, and obtains the additional stress at the point of action through triple Gauss-legendre numerical integration;

[0007] S3 regards the existing tunnel as a Euler-Bernoulli beam on a Pasternak two-parameter foundation, applies the additional stress obtained in steps S1 and S2 to the existing tunnel axis action point, uses the Pasternak two-parameter foundation beam theory to establish the deformation control differential equation, and uses the finite difference format to calculate the deformation and bending moment of the existing tunnel under the additional stress.

[0008] As further preferred, step S1 includes the following steps:

[0009] S11 The additional pressure q of the cutterhead acts evenly on the excavation surface, and the additional stress under the additional thrust q of the cutterhead is calculated;

[0010] S12 The shield friction force f acts evenly on the shield in the opposite direction of the shield machine excavation. Calculate the additional stress under the action of the shield friction force f.

[0011] The S13 synchronous grouting pressure p is evenly distributed along the entire radial section of the two ring segments at the shield tail, and the additional stress under the action of the synchronous grouting pressure p at the shield tail is calculated.

[0012] As a further preferred embodiment, in step S11, the additional stress σ under the additional thrust q of the cutter head is z-q for:

[0013]

[0014] Among them, R s is the radius of the cutter disc;

[0015] In step S12, the additional stress σ under the friction force f of the shield z-f for:

[0016]

[0017] Wherein, L is the length of the shield machine;

[0018] In step S13, the additional stress under the synchronous grouting pressure p of the shield tail should include the additional stress of the horizontal component and the additional stress of the vertical component. for:

[0019]

[0020] The additional stress of the vertical component for:

[0021]

[0022] Wherein, m is the width of the synchronous grouting pressure action surface.

[0023] As a further preferred embodiment, in step S2, only the stratum loss caused by the shield tail gap is considered, and it is assumed that the shield machine movement mode is an elliptical non-equivalent radial soil movement mode. The additional stress at the point (x, y, z) caused by the shield tail gap soil loss is:

[0024]

[0025] Among them, σ zlossis the additional stress at the point (x, y, z) caused by the loss of soil in the shield tail gap, dσ zloss is the additional stress caused by soil loss, R is the excavation radius of the shield machine, r is the radius of the shield machine, and L is the length of the shield machine.

[0026] As a further preferred embodiment, in step S3, the additional stress obtained in step S1 and step S2 is applied to the existing tunnel axis action point. At this time, the calculation formula for the additional stress of the existing tunnel caused by shield tunneling is:

[0027] q(x)=σ(x)=σ zq +σ zf +σ zpv +σ zph +σ zloss (11)

[0028] Where q(x) is the total additional stress at the axis of the existing tunnel; σ zq represents the additional vertical stress at the axis of the existing tunnel caused by the soil pressure; σ zf is the additional vertical stress on the tunnel axis caused by the friction between the shield and the soil; the synchronous grouting pressure is evenly distributed along the radial direction of the lining, and the components in the horizontal and vertical directions are p h and p v , p h and p v The vertical additional stresses induced are zph and σ zpv .

[0029] As a further preferred embodiment, in step S3, the shield tunnel structure deformation control equation under the additional stress is as follows:

[0030]

[0031] Where EI is the equivalent longitudinal stiffness of the existing tunnel, w(x) is the deflection at the tunnel axis x, x is the horizontal coordinate of the tunnel, k is the foundation coefficient, D is the tunnel diameter, G is c is the shear stiffness of the shear layer, and q(x) is the total additional stress at the axis of the existing tunnel.

[0032] As further preferred, step S3 specifically includes the following steps:

[0033] The shield tunnel structure is discretized into n+5 units of length l, and the differential expressions of the second-order, third-order and fourth-order derivatives of the unit deformation are:

[0034]

[0035]

[0036] In the formula, w i is the vertical displacement of the i-th point, x i is the coordinate of the i-th point, l is the selected tunnel length, (EI) eq is the equivalent bending stiffness, k is the soil base coefficient, D is the diameter of the tunnel, G c is the stiffness of the shear layer, q is the additional stress;

[0037] Two virtual units are set at both ends of the foundation beam, and the deformation control equation of the shield tunnel structure under the action of additional stress is as follows:

[0038] [K t ]{w}+[K s ]{w}-[G]{w}={Q} (15)

[0039] In the formula, [K t ],[K s ] and [G] are the stiffness matrices of the bending beam, foundation and shear layer respectively, {w} and {Q} are the column vectors of the longitudinal settlement and additional stress of the existing tunnel structure;

[0040] If both ends of the foundation beam are regarded as free, the shear force Q and bending moment M at both ends of the foundation beam can be calculated by the following formula:

[0041]

[0042] Where M o is the bending moment at point 0, M n is the bending moment at the nth point, w(x) is the deflection at the tunnel axis x, Q0 is the shear force at both ends of the foundation beam, Q n is the shear force at the nth point of the foundation beam;

[0043] Write formula (11) in finite difference format:

[0044]

[0045]

[0046] In the formula, w -1 、w -2 、w n+1 and w n+2 are the virtual points at both ends of the foundation beam;

[0047] By combining equations (11) and (13), we can obtain the virtual points w at both ends of the foundation beam: -1 , w -2 , w n+1 and w n+2 The linear equations are:

[0048]

[0049] Substituting equation (14) into equation (8), eliminating w -1 , w -2 , w n+1 and w n+2 , respectively obtain the stiffness matrix of the foundation beam and the stiffness matrix of the shear layer;

[0050] Let [K] = [K t ]+[K s ]-[G], multiply both sides by [K] -1 , then the solution of equation (10) is:

[0051] {w}=[K] -1 {Q} (23)

[0052] Where, [K] -1 is the inverse of the matrix [K];

[0053] The bending moment on the foundation beam is:

[0054]

[0055] In the formula, [K M ] is the bending stiffness of the foundation beam, where

[0056] As a further preferred embodiment, the calculation formula of the longitudinal equivalent stiffness EI of the existing tunnel is as follows:

[0057]

[0058]

[0059] k b =E b A b / l b (27)

[0060] Among them, k b is the stiffness coefficient of the longitudinal connecting bolt, E b is the Young's modulus of the bolt; A b is the cross-sectional area of ​​the bolt, r b is the radius of the bolt; n is the number of longitudinal bolts; l is the width of the segment; E c is the Young’s modulus of the lining segment; I is the inclination angle of the central axis; c A is the moment of inertia of the longitudinal section of the segment; c is the cross-sectional area of ​​the tunnel segment, l b is the length of the bolt.

[0061] As a further preferred method, the calculation formula of the underlying foundation modulus k is as follows:

[0062]

[0063] Among them, E s is the elastic modulus of soil, B is the width of the foundation beam, EI is the flexural stiffness of the foundation beam, and μ is the Poisson's ratio of soil.

[0064] As a further preferred embodiment, the calculation formula of the shear layer shear stiffness is as follows:

[0065]

[0066] Among them, h t is the thickness of the shear layer, μ is the Poisson's ratio of the soil, E s is the elastic modulus of soil.

[0067] In general, the above technical solution conceived by the present invention has the following technical advantages compared with the prior art:

[0068] 1. The present invention adopts the Parsternak two-parameter foundation model to consider the interaction between adjacent Winkler springs, calculates the additional stress caused by the main factors according to the Mindlin formula and the mirror method, regards the existing tunnel as the Euler-Bernoulli beam on the Parsternak two-parameter foundation, and modifies the three important model parameters of equivalent longitudinal bending stiffness, bed coefficient and shear stiffness. Then, the finite difference method is used to establish the deformation stiffness equation of the beam, and the influence of the shield construction load on the deformation of the existing tunnel is analyzed.

[0069] 2. The present invention adopts a two-stage analysis method to further correct the calculation formula of the base bed coefficient, and introduces the Parsternak dual-parameter foundation model to simulate the interaction between soil and tunnel structure, which can accurately estimate the response of the existing tunnel caused by shield excavation, which is consistent with the actual monitoring results.

[0070] 3. The present invention regards the existing tunnel as a Euler-Bernoulli beam on the Parsternak foundation, and adopts a two-stage method to analyze the influence of shield parameters and soil loss on the settlement of the existing tunnel. 2m and 10m before the shield machine cutter head passes through the intersection, the soil loss contributes the most to the settlement. The additional thrust of the cutter head, the friction of the shield shell and the synchronous grouting pressure of the shield tail will cause the existing tunnel to bulge; among them, the friction of the shield shell contributes the most to the uplift of the existing tunnel, and the synchronous grouting pressure has little effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1The present invention relates to a flow chart of a method for analyzing the impact of shield tunneling on an existing tunnel based on an analytical solution;

[0072] Figure 2 It is a structural schematic diagram of a shield tunneling model involved in the present invention;

[0073] Figure 3 It is a schematic diagram of additional stress in an existing tunnel caused by shield excavation involved in the present invention;

[0074] Figure 4 (a) in (a) is the deformation of the existing tunnel before crossing 10m away from the intersection. Figure 4 (b) shows the deformation of the existing tunnel before crossing 2m from the intersection;

[0075] Figure 5 (a) is the influence of foundation elastic modulus and flexural stiffness on the maximum settlement of the existing tunnel. Figure 5 (b) shows the influence of foundation elastic modulus and flexural stiffness on the bending moment of the existing tunnel;

[0076] Figure 6 (a) is the effect of foundation elastic modulus on the settlement of existing tunnel. Figure 6 (b) shows the influence of foundation elastic modulus on the bending moment of the existing tunnel;

[0077] Figure 7 (a) shows the influence of different tunnel spacing S on the settlement of existing tunnels. Figure 7 (b) shows the influence of different tunnel spacing S on the bending moment of the existing tunnel;

[0078] Figure 8 (a) shows the influence of different stratum loss rates on the settlement of existing tunnels. Figure 8 (b) shows the influence of different stratum loss rates on the bending moment of the existing tunnel;

[0079] Fig. 9 This is a tunnel settlement cloud map based on FLAC3D shield tunneling according to an embodiment of the present invention. DETAILED DESCRIPTION

[0080] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0081] like Figure 1As shown, an embodiment of the present invention provides a method for analyzing the impact of shield tunneling on existing tunnels based on an analytical solution. The Parsternak two-parameter foundation model is used to consider the interaction between adjacent Winkler springs. The additional stress caused by the main factors is obtained according to the Mindlin formula and the mirror method. The existing tunnel is regarded as an Euler-Bernoulli beam on the Parsternak two-parameter foundation, and the three important model parameters of equivalent longitudinal bending stiffness, bed coefficient and shear stiffness are corrected. Then, the finite difference method is used to establish the deformation stiffness equation of the beam, and the influence of the shield construction load on the deformation of the existing tunnel is analyzed.

[0082] In order to overcome the defects of the existing analytical solutions, this paper proposes a more accurate analytical method for evaluating the response calculation of existing tunnels under shield proximity construction conditions. This method uses the Pasternak two-parameter foundation model to simulate the interaction between the tunnel structure and the foundation. The Pasternak two-parameter model can take into account the interaction between adjacent springs. The formula is as follows:

[0083]

[0084] In the formula, G c is the shear stiffness of the shear layer, k is the base coefficient, unit is MPa / m, w(x) is the deflection at the tunnel axis x, and x is the horizontal coordinate of the foundation beam.

[0085] The proposed analytical solution method adopts a two-stage analysis method, and the construction process is shown in Figure 1 . In the first stage, the additional thrust q of the cutter head, the friction force f of the shield, and the synchronous grouting pressure p are obtained by using the Mindlin formula to obtain the additional stress formula, and the discrete value of the additional stress at the point of action is obtained by double Gauss-legendre numerical integration. The additional stress formula caused by soil loss is derived based on the mechanical principle of the mirror image method, and the discrete value of the additional stress at the point of action is obtained by triple Gauss-legendre numerical integration; in the second stage, various additional stresses are applied to the existing tunnel axis point of action, and the deformation control differential equation is established using the Pasternak two-parameter foundation beam theory. The deformation and bending moment of the existing tunnel under the action of additional stress are calculated using the finite difference format.

[0086] The analytical solution method proposed in the present invention can take into account the influence of the main shield construction load on the deformation of the existing tunnel during the penetration process. At the same time, the present invention analyzes the influence law of the deformation and internal force of the existing tunnel structure caused by important influencing factors such as the elastic modulus of the underlying layer, the tunnel spacing, and the soil loss rate.

[0087] First, the additional stress formula of construction load and soil loss is derived, and the additional stress at the axis of the existing tunnel caused by the additional thrust of the cutter head, the friction of the shield, the synchronous grouting pressure and the soil loss is comprehensively considered. The details are as follows:

[0088] 1. Derivation of the additional stress formula caused by shield construction using Mindlin formula

[0089] The Mindlin formula is based on the Boussinesq solution, and is a theoretical solution to the stress and strain at any point in the semi-infinite space caused by vertical or horizontal loads acting on the interior of a semi-infinite elastic body. Therefore, the Mindlin formula must satisfy the following assumptions: the soil is a homogeneous, isotropic, linear elastic body in a semi-infinite space. The additional pressure q of the cutterhead acts evenly on the excavation surface, and the friction force f of the shield acts evenly on the shield in the opposite direction along the tunneling of the shield machine; the synchronous grouting pressure p is evenly distributed along the entire radial section of the two ring segments at the tail of the shield. Figure 2 As shown, the position of the concentrated force point is (0,0,c), where c=Z0, Among them, (x, y, z) is an arbitrary point in the xyz coordinate system, and Z0 is the z-coordinate of the point where the concentrated force acts.

[0090] (1) Calculation of additional stress under the additional thrust q of the cutter head

[0091] The micro-element area of ​​any point on the surface of the cutter head additional thrust q is dA=rdrdθ, and the corresponding concentrated force dP h =qrdrdθ, its coordinate system is x′y′z′, c=z0-rsinθ, dP h The additional stress at any point (x, y, z) in the xyz coordinate system under the action is:

[0092]

[0093]

[0094] Where r is the radius of the microelement area, θ is the angle between the line of action of the additional thrust q of the cutter head and the thrust direction of the shield machine, P h is the concentrated force, q is the additional thrust of the cutter head, y is the coordinate value of the concentrated force along the thrust direction of the shield machine, υ is the Poisson's ratio of the soil, z is the height of the concentrated force from the ground, z0 is the z-coordinate of the position of the concentrated force action point, R is the excavation radius of the shield machine, σ z-q For additional stress.

[0095] (2) Calculation of additional stress under the action of shield friction force f

[0096] For any infinitesimal element dA = R s dsdθ, the corresponding concentrated force on the microelement is dPh =fR s dsdθ, whose spatial coordinates are (x′, y′, z′), dP h The additional stress at any point (x, y, z) in the xyz coordinate system under the action:

[0097]

[0098]

[0099] Among them, R s is the radius of the cutterhead, (x′, y′, z′) is the spatial coordinate of any infinitesimal element, υ is the Poisson’s ratio of the soil, and z0 is the z-coordinate of the point where the concentrated force acts.

[0100] (3) Calculation of additional stress under the action of synchronous grouting pressure p at the shield tail

[0101] The micro-element area of ​​any point on the shield tail synchronous grouting pressure p action surface dA=R A dsdθ, the corresponding concentrated force dp=pR s dsdθ, its coordinate system is x′y′z′, c=z0-R s sinθ, decomposes dp into the horizontal component dp h =pR s cosθdsdθ and vertical component dp v =pR s sinθdsdθ,dp v The additional stress at any point (x, y, z) in the xyz coordinate system under the action:

[0102]

[0103]

[0104] dp h The additional stress at any point (x, y, z) in the xyz coordinate system under the action:

[0105]

[0106]

[0107] Among them, p is the synchronous grouting pressure of the shield tail, v is the Poisson's ratio of the soil, z0 is the z-coordinate of the position of the concentrated force action point, and m is the action width of the synchronous grouting pressure of the shield tail.

[0108] 2. Calculation of additional stress caused by soil loss using the mirror method

[0109] In order to simplify the calculation of additional stress caused by soil loss, only the stratum loss caused by the shield tail gap is considered, and it is assumed that the shield machine movement mode is an elliptical non-equivalent radial soil movement mode. For any infinitesimal element dV=rdrdθdl in space, the coordinates of the small circle are (r cosθ,l,z0-rsinθ), and the coordinates of the large circle are (r cosθ,l,z0-(R s -r)-rs inθ), the additional stresses caused by all the microelements in the shield tail gap are superimposed to obtain the total additional stress. Since this paper only needs to find the vertical additional stress at the axis of the existing tunnel, the additional stress at the point shown in the figure is dσ zloss , under the effect of all shield tail gaps σ zloss .

[0110]

[0111] Where: zloss is the additional stress at the point (x, y, z) caused by the loss of soil in the shield tail gap, dσ zloss is the additional stress caused by soil loss, R is the excavation radius of the shield machine, r is the radius of the shield machine, and L is the length of the shield machine.

[0112] 3. Additional vertical stress on the existing tunnel axis caused by shield excavation

[0113] The vertical additional stress at the center axis of the existing tunnel caused by shield excavation includes the shield soil compartment pressure, the friction between the shield shell and the soil, the synchronous grouting pressure and the soil loss. The formula for the additional stress of the existing tunnel caused by shield excavation is:

[0114] q(x)=σ(x)=σ zq +σ zf +σ xpv +σ zph +σ zloss (11)

[0115] Where: q(x) is the total additional stress at the existing tunnel axis; σ zq represents the additional vertical stress at the axis of the existing tunnel caused by the soil pressure; σ zf is the additional vertical stress on the tunnel axis caused by the friction between the shield and the soil; the synchronous grouting pressure is evenly distributed along the radial direction of the lining, and the components in the horizontal and vertical directions are p h and p v The vertical additional stresses caused are σ zph and σ zpv After obtaining the total vertical additional stress of each factor, the differential equation of the tunnel deflection curve is established according to the foundation beam theory.

[0116] Secondly, the deformation stiffness equation of the beam is established to calculate the deformation and bending moment of the existing tunnel, analyze the influence of different influencing factors on the settlement of the existing tunnel, and analyze the influence of foundation elastic modulus, clear distance and stratum loss rate on the existing tunnel.

[0117] The additional stress obtained in the first stage is applied to the axis of the existing tunnel. The deformation control differential equation is established according to the microelement equilibrium of the beam. The finite difference method is used to convert the continuity equation into a finite difference format to obtain the deformation stiffness equation of the beam, thereby calculating the deformation and bending moment of the beam.

[0118] 1. Deformation control differential equation of existing tunnels

[0119] In order to overcome the shortcoming of Winkler foundation beam that does not consider the foundation's resistance to shear deformation, the Parsternak two-parameter foundation model is used to describe the interaction between soil and tunnel structure, and the existing tunnel structure is regarded as an Euler-Bernoulli infinite length beam, see Figure 3 This method assumes that the tunnel structure and the soil are always in close contact, and does not consider the slip between the structure and the soil; moreover, the foundation is regarded as an isotropic linear elastic material, and the plastic mechanical behavior of the rock and soil is not considered. The deformation control equation of the shield tunnel structure under the action of additional stress is as follows:

[0120]

[0121] Where EI is the equivalent longitudinal stiffness of the existing tunnel, w(x) is the deflection at the tunnel axis x, x is the horizontal coordinate of the tunnel, k is the foundation coefficient, D is the tunnel diameter, G is c is the shear stiffness of the shear layer, and q(x) is the total additional stress at the axis of the existing tunnel. If the shear layer parameter G c is 0, the stiffness matrix formula of the foundation beam will degenerate into the governing equation of the Euler-Bernoullli infinite length beam on the commonly used Winkler foundation model.

[0122] 2. Establish the stiffness matrix equation

[0123] In order to simplify the calculation process, the deformation control equation of the beam (Equation (12)) is a fourth-order ordinary differential equation, which is generally difficult to obtain an analytical solution. The tunnel structure is discretized into n+5 units of length l, and the differential expressions of the second-order, third-order and fourth-order derivatives of the deformation are (13); two virtual units are set at both ends of the foundation beam. Therefore, the finite difference format of equation (9) can be written as (14):

[0124]

[0125]

[0126] In the formula, w i is the vertical displacement of the i-th point, x i is the coordinate of the i-th point, l is the selected tunnel length, (EI) eq is the equivalent bending stiffness, k is the soil base coefficient, D is the diameter of the tunnel, G c is the stiffness of the shear layer, and q is the additional stress.

[0127] Equation (14) can be further written in the following matrix form:

[0128] [K t ]{w}+[K s ]{w}-[G]{w}={Q} (15)

[0129] In the formula, [K t ],[K s ] and [G] are the stiffness matrices of the bending beam, foundation and shear layer respectively, {w} is the longitudinal settlement of the existing tunnel structure, and {Q} is the column vector of the additional stress of the existing tunnel structure. Let {w} = {w0,w1,…,w i ,w i+1 ,…,w n} T and {Q}={q(x0),q(x1)…q(x i )…,q(x n )} T D, where {Q} can be obtained by the formula.

[0130] Stiffness matrix K s for:

[0131]

[0132] Assuming that both ends of the foundation beam are free, the shear force Q and bending moment M at both ends can be calculated by the following formula:

[0133]

[0134] Among them, M o is the bending moment at point 0, M n is the bending moment at the nth point, w(x) is the displacement at x, Q0 is the shear force at both ends of the foundation beam, Q n is the shear force at the nth point of the foundation beam.

[0135] Formula (17) can be written in finite difference format:

[0136]

[0137]

[0138] By combining equations (18) and (19), we can obtain the virtual points w at both ends of the foundation beam: -1 , w -2 , w n+1 and w n+2 The linear equations are:

[0139]

[0140] Substituting equation (20) into equation (14), eliminating w -1 , w -2 , w n+1 and w n+2 , the stiffness matrix of the foundation beam and the stiffness matrix of the shear layer can be obtained respectively:

[0141]

[0142]

[0143] Let [K] = [K t ]+[K s ]-[G], multiply both sides by [K] -1 , then the solution of equation (15) is:

[0144] {w}=[K] -1 {Q} (23)

[0145] Where, [K] -1 is the inverse of the matrix [K].

[0146] The bending moment on the foundation beam is:

[0147]

[0148] In the formula, [K M ] is the bending stiffness of the foundation beam, where

[0149] In the present invention, the longitudinal equivalent stiffness EI of the existing tunnel, the modulus k of the underlying foundation and the elastic modulus E of the surrounding soil layer are s It has a significant impact on the elastic foundation beam under additional stress.

[0150] 1. Determination of equivalent bending stiffness of shield segments

[0151] EI is a key parameter that reflects the deformation resistance of existing tunnels. In fact, tunnel segments are not continuous tubular structures, but are connected by longitudinal and circumferential bolts. Therefore, it is necessary to reduce the longitudinal stiffness of the tunnel to a certain extent. Theoretical analysis and model tests are widely used methods to calculate the longitudinal equivalent stiffness. The theoretical formula for the longitudinal equivalent bending stiffness is:

[0152]

[0153]

[0154] k b =E b A b / l b (27)

[0155] Where: k b is the stiffness coefficient of the longitudinal connecting bolt, E b is the Young's modulus of the bolt; A b is the cross-sectional area of ​​the bolt, r b is the radius of the bolt; n is the number of longitudinal bolts; l is the width of the segment; E c is the Young’s modulus of the lining segment; I is the inclination angle of the central axis; c A is the moment of inertia of the longitudinal section of the segment; c is the cross-sectional area of ​​the tunnel segment, l b is the length of the bolt.

[0156] 2. Determination of foundation base coefficient k

[0157] The underlying bed coefficient k is another important parameter that reflects the interaction between soil and tunnel structure. Many scholars have proposed various formulas for estimating the bed coefficient k. Vesic derived an empirical formula for estimating the underlying bed coefficient k by assuming that an infinitely long foundation beam is placed on the elastic surface:

[0158]

[0159] Where: E s is the elastic modulus of the soil, B is the width of the foundation beam, EI is the flexural stiffness of the foundation beam, and μ is the Poisson's ratio of the soil.

[0160] Since the buried depth of urban subway tunnels is generally 10m to 40m below the surface, in order to consider the influence of the embedded depth of foundation beams on the base coefficient, the reduction factor η is introduced:

[0161]

[0162] In the formula, k h is the base coefficient corresponding to the buried depth h, B is the width of the foundation beam, k ∞ It is the base bed coefficient corresponding to the infinite burial depth.

[0163] Combining formulas (28) and (29), we can get the foundation coefficient k h :

[0164]

[0165] 3. Shear layer shear stiffness

[0166] The shear layer parameter is crucial in the Parsternak two-parameter model. c The empirical formula is:

[0167]

[0168] In formula (28), h t is the thickness of the shear layer, μ is the Poisson's ratio of the soil, E s is the elastic modulus of the soil, generally, h t =2.5D.

[0169] Example 1

[0170] This example is based on the Wuhan Metro Line 2 passing through the Metro Line 4 at the Zhongnan Road Transfer Station of the Rail Transit. Line 2 and Line 4 are upper and lower cross tunnels, and the angle of the plane projection is approximately 90°. The diameter of the shield machine is D = 6.2m, the length is 7.5m, and the outer diameter of the existing tunnel segment is also 6.2m. The soil layer where the existing tunnel is located is soft clay, the Poisson's ratio is 0.28, the soil elastic modulus is 24.5Mpa, and the soil gravity γ = 20KN / m 3 , the thickness of the lining structure is 0.3m, the internal friction angle Cohesion c = 25.2 kpa, the burial depth of the new shield tunnel is 30 m, the burial depth of the existing tunnel is z0 = 18 m, the lining structure concrete of the existing tunnel is C50, and the elastic modulus is E c =34.5Gpa.

[0171] The additional stress is taken according to the engineering construction data. The front additional thrust is q = 295 kPa, the shield friction is f = 180 kPa, the shield tail synchronous grouting additional pressure is p = 236 kPa, and the longitudinal overall equivalent bending stiffness (EI) of the existing tunnel is eq =57.5GPa. According to formula (25), the base coefficient of the existing tunnel can be obtained as k = 1.2 × 10 4 kN / m 3 .

[0172] This paper analyzes the deformation and bending moment of the existing tunnel caused by the additional thrust of the cutter head, the friction of the shield shell, the grouting pressure of the shield tail, and the soil loss under three working conditions: working condition 1 is that the distance between the shield cutter head and the intersection is 10m before the shield tunneling; working condition 2 is that the distance between the shield cutter head and the intersection is 2m before the shield tunneling; working condition 3 is that the distance between the shield cutter head and the intersection is 10m after the shield tunneling. Let the position of the existing tunnel action point x∈[-100m,100m] be divided into 200 units with a unit length of 1m. According to the above additional stress calculation formula, the 6-node Gauss-legrende integral formula is used to calculate the additional stress value {Q} at each discrete point on the existing tunnel. w and M are calculated by formulas (23) and (24). Figure 4 (a) and (b) show the vertical deformation and bending moment of the existing tunnel when the cutterhead is 10m and 2m away from the intersection point before it passes through, respectively.

[0173] from Figure 4 It can be seen that before the shield machine cutterhead passes through, soil loss will cause the existing tunnel to settle, and the three shield construction parameters will cause the existing tunnel to bulge. The amount of uplift of the existing tunnel caused by the construction parameters before the shield machine passes through is: shield friction f> cutterhead additional thrust q> shield tail synchronous grouting pressure p, the uplift of the existing tunnel caused by the shield friction f is 2.2mm, the cutterhead additional thrust will also cause a certain degree of uplift, and the influence of the full-section synchronous grouting pressure is very small. Therefore, in the crossing construction, the two important parameters of the jack thrust and the cutterhead additional thrust should be reasonably set, not too small or too large, to ensure the safety of the underpass construction.

[0174] For the third working condition, when the shield machine passes the intersection, the shield construction parameters of the cutterhead additional thrust q, the shield shell friction f and the synchronous grouting pressure p of the shield tail all cause tensile stress in the soil behind the cutterhead, but the soil cannot withstand large tensile stress. Therefore, it is not reasonable to use the above analytical method to calculate the deformation law of the existing tunnel caused by the shield construction parameters of working condition three.

[0175] In order to further analyze the influence of different important influencing factors on the deformation and bending moment of existing tunnels, this paper will conduct parametric research on several important parameters such as the elastic modulus of the underlying foundation, tunnel spacing, and soil loss rate.

[0176] 1. Influence of foundation elastic modulus

[0177] The construction of shield tunnels close to the existing lines will affect the operational safety of the existing lines. In engineering practice, certain preventive measures are usually taken, such as strengthening the intermediate strata and strengthening the longitudinal bending stiffness of the existing tunnels. eq Set to (EI) respectively eq 、10(EI) eq 、100(EI) eqThree situations are analyzed to analyze the changes of deformation and bending moment of existing tunnels with the elastic modulus of the foundation.

[0178] from Figure 5 It can be seen that the maximum settlement and bending moment of the existing tunnel increase with the increase of the elastic modulus of the foundation; with the increase of the longitudinal bending stiffness, the slope of the maximum settlement curve increases. The stiffness ratio of the existing tunnel to the soil is the key factor causing the settlement and internal force variation of the existing tunnel, and a dimensionless standardized parameter (EI) can be defined eq / EsD 4 To measure the relative stiffness ratio. With (EI) eq / EsD 4 The maximum settlement and maximum bending moment of the existing tunnel gradually increase, and the maximum settlement and maximum bending moment of the existing tunnel gradually decrease. In engineering, steel sections can be used to reinforce the sections of the existing tunnel that are more affected by the underpass, so as to increase the relative stiffness of the existing tunnel and the soil and reduce deformation.

[0179] Figure 6 Describes the existing tunnel longitudinal bending stiffness of 100 (EI) eq The settlement and bending moment curves of the existing tunnel under different underlying foundation elastic modulus conditions are shown in Figure 2. It can be found that the maximum settlement and maximum negative bending moment of the existing tunnel increase with the increase of foundation modulus, and the peak value of the settlement trough curve of the existing tunnel increases and the curve shape becomes steeper with the increase of foundation modulus.

[0180] The distance S between the new tunnel and the existing tunnel is the key factor in the proximity division in the construction of the upper and lower crossing tunnels. Figure 7 As shown in (a) and (b), it characterizes the law of the change of settlement and bending moment of the existing tunnel with the spacing when the buried depth of the new tunnel is 30m; it can be seen that when the spacing between the two tunnels increases from 9m to 24m, the settlement and bending moment of the existing tunnel decrease, the maximum settlement decreases by 22%, and the maximum negative bending moment (absolute value) decreases by 57%. The reduction in the maximum bending moment is significantly greater than the maximum settlement value, indicating that the additional internal force of the existing tunnel is sensitive to deformation, so the deformation of the existing tunnel should be strictly controlled during construction to ensure that the existing structure is not damaged.

[0181] In this embodiment, the formation loss rate is set to 0.2%, 0.8%, 1.4% and 2% to analyze the distribution law of the settlement and bending moment of the existing tunnel along the tunnel axis.

[0182] from Figure 8It can be seen that with the increase of the stratum loss rate, the maximum settlement and maximum bending moment both increase linearly. This is because the greater the stratum loss rate, the greater the disturbance to the soil around the existing tunnel, resulting in greater stress release, which increases the settlement and bending moment of the existing tunnel accordingly. Therefore, during the shield tunneling construction, the advancement speed should be strictly controlled, and the secondary grouting should be timely increased to increase the grouting filling rate to reduce the stratum loss. In particular, attention should be paid to the excavation posture control of the shield machine to avoid serpentine and knocking.

[0183] In order to verify the effectiveness of the proposed analytical method, FLAC 3D is used to simulate the entire process of the shield tunnel passing through the existing tunnel. Fig. 9 The settlement of existing tunnels calculated by FLAC3D, the Winkler-based method and the theoretical formula method proposed in this paper are compared. It can be seen that the Winkler-based method overestimates the settlement of the existing tunnel compared with the results of numerical simulation. This is because the Vesic's base bed coefficient used in the method based on the Winkler foundation model underestimates the relative stiffness of the existing tunnel. The maximum settlement calculated by the theoretical formula method proposed in this invention is only slightly larger than the FLAC3D simulation result, and the settlement curve has a similar variation pattern. The results show that the proposed two-stage analysis method further corrects the calculation formula of the base bed coefficient, and introduces the Parsternak two-parameter foundation model to simulate the interaction between soil and tunnel structure. According to the above analysis, the proposed theoretical formula can reasonably calculate the response of the existing tunnel caused by shield excavation.

[0184] In general, the two-stage analysis method proposed in this paper further corrects the calculation formula of the base coefficient, and introduces the Parsternak two-parameter foundation model to simulate the interaction between soil and tunnel structure. The results show that the proposed analytical solution method can more accurately estimate the response of the existing tunnel caused by shield excavation, which is consistent with the actual monitoring results. The existing tunnel is regarded as a Euler-Bernoulli beam on the Parsternak foundation, and the two-stage method is used to analyze the influence of shield parameters and soil loss on the settlement of the existing tunnel. The results show that at 2m and 10m before the shield machine cutterhead crosses the intersection, the soil loss contributes the most to the settlement, and the additional thrust of the cutterhead, the friction of the shield shell and the synchronous grouting pressure of the shield tail will cause the existing tunnel to bulge; among them, the friction of the shield shell contributes the most to the bulge of the existing tunnel, and the synchronous grouting pressure has little effect. With the increase of foundation elastic modulus, the relative stiffness ratio of the existing tunnel decreases, and the maximum settlement, maximum longitudinal positive bending moment and maximum longitudinal negative bending moment (absolute value) of the existing tunnel increase accordingly. In engineering, the deformation of the existing tunnel can be reduced by increasing the stiffness ratio of the tunnel to the soil. Under the same burial depth of the new tunnel, the peak settlement, longitudinal positive bending moment and longitudinal negative bending moment of the existing tunnel gradually decrease with the increase of the distance between the two tunnels. The peak settlement, longitudinal bending moment and longitudinal negative bending moment (absolute value) of the existing tunnel increase linearly with the increase of the stratum loss rate. Measures should be taken to reduce stratum loss during shield tunneling construction.

[0185] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for analyzing the impact of shield tunneling on existing tunnels based on analytical solutions, characterized in that: The following steps are involved: S1 uses Mindlin formula to derive the additional stress formula caused by shield construction, and calculates the additional stress at the action point through double Gauss-legendre numerical integration; Step S1 includes the following steps: S11 The additional pressure q of the cutterhead acts evenly on the excavation surface, and the additional stress under the additional thrust q of the cutterhead is calculated; S12 The shield friction force f acts evenly on the shield in the opposite direction of the shield machine excavation. Calculate the additional stress under the action of the shield friction force f. S13 The synchronous grouting pressure p is evenly distributed along the radial cross-section of the two ring segments at the tail of the shield, and the additional stress under the synchronous grouting pressure p at the tail of the shield is calculated; S2 uses the mirror method to calculate the additional stress caused by soil loss, and obtains the additional stress at the point of action through triple Gauss-legendre numerical integration; In step S2, only the stratum loss caused by the shield tail gap is considered, and it is assumed that the shield machine movement mode is an elliptical non-equivalent radial soil movement mode. The additional stress at the point (x, y, z) caused by the shield tail gap soil loss is: Among them, σ zloss is the additional stress at the point (x, y, z) caused by the loss of soil in the shield tail gap, dσ zloss is the additional stress caused by soil microelement loss, R is the excavation radius of the shield machine, r is the radius of the shield machine, and L is the length of the shield machine; S3 regards the existing tunnel as a Euler-Bernoulli beam on a Pasternak two-parameter foundation, applies the additional stress obtained in steps S1 and S2 to the existing tunnel axis action point, uses the Pasternak two-parameter foundation beam theory to establish the deformation control differential equation, and uses the finite difference format to calculate the deformation and bending moment of the existing tunnel under the additional stress.

2. The method according to claim 1, characterized in that In step S11, the additional stress σ under the additional thrust q of the cutter head is z-q for: Among them, R s is the cutter head radius.

3. The method according to claim 2, characterized in that In step S12, the additional stress σ under the friction force f of the shield z-f for: Wherein, L is the length of the shield machine; 4. The method according to claim 2, characterized in that: In step S13, the additional stress under the synchronous grouting pressure p of the shield tail should include the additional stress of the horizontal component and the additional stress of the vertical component. for: The additional stress of the vertical component for: Where m is the width of the synchronous grouting pressure action surface.

5. The method according to claim 1, characterized in that In step S3, the additional stress obtained in step S1 and step S2 is applied to the existing tunnel axis action point. At this time, the calculation formula for the additional stress of the existing tunnel caused by shield tunneling is: q(x)=σ zq +s zf +s zpv +s zph +s zloss (11) Where q(x) is the total additional stress at the axis of the existing tunnel; σ zq represents the additional vertical stress at the axis of the existing tunnel caused by the soil compartment pressure; σ zf is the additional vertical stress on the tunnel axis caused by the friction between the shield and the soil; the synchronous grouting pressure is evenly distributed along the radial direction of the lining, and the components in the horizontal and vertical directions are p h and p v , p h and p v The additional vertical stresses induced are σ zph and σ zpv .

6. The method according to claim 1, characterized in that In step S3, the control equation of shield tunnel structure deformation under the additional stress is as follows: Where EI is the equivalent longitudinal stiffness of the existing tunnel, w(x) is the deflection at the tunnel axis x, x is the horizontal coordinate of the tunnel, k is the foundation coefficient, D is the tunnel diameter, G is c is the shear stiffness of the shear layer, and q(x) is the total additional stress at the axis of the existing tunnel.

7. The method according to claim 6, characterized in that Step S3 specifically includes the following steps: The shield tunnel structure is discretized into n+5 units of length l. The differential expressions of the second-order, third-order and fourth-order derivatives of the unit deformation are: In the formula, w i is the vertical displacement of the i-th point, x i is the coordinate of the i-th point, l is the selected tunnel length, (EI) eq is the equivalent bending stiffness, k is the soil base coefficient, D is the diameter of the tunnel, G c is the stiffness of the shear layer, q is the additional stress; Two virtual units are set at both ends of the foundation beam, and the deformation control equation of the shield tunnel structure under the action of additional stress is as follows: [K t ]{w}+[K s ]{w}-[G]{w}={Q} (15) In the formula, [K t ],[K s ] and [G] are the stiffness matrices of the bending beam, foundation and shear layer respectively, {w} and {Q} are the column vectors of the longitudinal settlement and additional stress of the existing tunnel structure; If both ends of the foundation beam are regarded as free, the shear force Q and bending moment M at both ends of the foundation beam can be calculated by the following formula: Where M o is the bending moment at point 0, M n is the bending moment at the nth point, Q0 is the shear force at both ends of the foundation beam, Q n is the shear force at the nth point of the foundation beam; Write formula (11) in finite difference format: In the formula, w -1 、w -2 、w n+1 and w n+2 are virtual points at both ends of the foundation beam; By combining equations (11) and (13), we can obtain the virtual points w at both ends of the foundation beam: -1 , w -2 , w n+1 and w n+2 The linear equations are: Substituting equation (14) into equation (8), eliminating w -1 , w -2 , w n+1 and w n+2 , respectively obtain the stiffness matrix of the foundation beam and the stiffness matrix of the shear layer; Let [K] = [K t ]+[K s ]-[G], multiply both sides by [K] -1 , then the solution of equation (10) is: {w}=[K] -1 {Q} (23) Where, [K] -1 is the inverse of the matrix [K]; The bending moment on the foundation beam is: In the formula, [K M ] is the bending stiffness of the foundation beam, where 8. The method according to claim 7, characterized in that: The calculation formula of the longitudinal equivalent stiffness EI of the existing tunnel is as follows: k b =E b A b / l b (27) Among them, k b is the stiffness coefficient of the longitudinal connecting bolt, E b is the Young's modulus of the bolt; A b is the cross-sectional area of ​​the bolt, r b is the radius of the bolt; n is the number of longitudinal bolts; l is the width of the segment; E c is the Young’s modulus of the lining segment; I is the inclination angle of the central axis; c A is the moment of inertia of the longitudinal section of the segment; c is the cross-sectional area of ​​the tunnel segment, l b is the length of the bolt.

9. The method according to claim 7, characterized in that: The calculation formula of the underlying foundation modulus k is as follows: Among them, E s is the elastic modulus of soil, B is the width of the foundation beam, EI is the flexural stiffness of the foundation beam, and μ is the Poisson's ratio of soil.

10. The method according to claim 7, characterized in that: The calculation formula of the shear layer shear stiffness is as follows: Among them, h t is the thickness of the shear layer, μ is the Poisson's ratio of the soil, E s is the elastic modulus of soil.

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