Displacement deformation calculation method for tunnel with anti-floating anchor during excavation of overlapped tunnel

Abstract

The invention discloses a displacement calculation method for a tunnel with an anti-floating anchor during excavation of a stacked tunnel, and belongs to the technical field of underground engineering. The calculation method specifically comprises the steps that the position relation between the existing shield tunnel with the anti-floating anchor rods and a newly-built shield tunnel is established, the shield tunnel additional load with the anti-floating anchor rods caused by shield tunnel construction is calculated, the function of additional stress on the existing tunnel is calculated by considering a duct piece ring collaborative deformation model of an opening angle and a staggered slab, and the function of the additional stress on the existing tunnel is calculated. Through an energy variational method and a tunnel shearing staggered slab collaborative deformation model, a Fourier expansion and variational control equation is used to deduce a displacement prediction function which is in asymmetric distribution of a deformation displacement curve of the existing tunnel and is suitable for the overlapped tunnel. The shearing staggered slab collaborative deformation model provided by the prediction method is wider in application range, can estimate the condition that the tunnel displacement curve is symmetrically or asymmetrically distributed about the center at the same time, and is suitable for the working condition that a new tunnel upwards passes through or downwards passes through the existing tunnel at any angle.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of underground engineering, and particularly relates to a tunnel displacement calculation method with anti-floating anchor rods during the excavation of superimposed tunnels. BACKGROUND

[0002] In recent years, with the development of urban underground space engineering construction, the urban stratum space becomes more and more crowded, and the superimposed construction of a new tunnel above or below an existing tunnel in a city becomes more and more frequent. However, when a new tunnel is constructed above or below an existing tunnel, an unloading effect is generated on the soil around the existing tunnel, causing longitudinal uneven settlement deformation of the existing tunnel, which may cause deformation and damage of the shield tunnel structure and seriously threaten the safety of the tunnel.

[0003] At present, the control measures for the upward floating of a tunnel include the following: loading and weighting in the tunnel hole, setting anti-floating anchor rods, setting isolation piles, and grouting reinforcement. The anti-floating anchor rod utilizes the shear force between the anchor body composed of anchor rods and mortar and the rock-soil layer to resist the upward floating force borne by the underground structure. At present, the anti-floating anchor rod is mainly applied to basements, shallow-buried tunnels, water pools and other structures, and is rarely applied to shield tunnels. The effect of the anti-floating anchor rod on the upward floating of a shield tunnel still needs to be further researched. The main method of the existing research is finite element simulation, and the results have a great relationship with the restoration degree of the modeling to the actual working condition, and the accuracy of the results cannot be guaranteed. At present, there is no theoretical solution derivation method for the displacement deformation of an existing tunnel with anti-floating anchor rods caused by the excavation of a superimposed tunnel.

[0004] In summary, the research on the displacement of an existing tunnel with anti-floating anchor rods caused by the excavation of a superimposed tunnel is mostly concentrated in finite element numerical simulation, and it is difficult to ensure the accuracy. There is no theoretical solution derivation method, and relevant research is urgently needed. SUMMARY

[0005] The purpose of the present application is to provide a tunnel deformation calculation method with anti-floating anchor rods during the excavation of superimposed tunnels, which can estimate the symmetric or asymmetric distribution of the tunnel displacement curve about the center at the same time, and is suitable for the calculation of the tunnel displacement and deformation values caused by the additional load under the working condition that a new tunnel is constructed above or below an existing tunnel at an arbitrary angle.

[0006] To achieve the above purpose, the present application provides a tunnel displacement calculation method with anti-floating anchor rods during the excavation of superimposed tunnels in a first aspect, which comprises the following steps: S11, based on the related conditions of the excavation face of a new tunnel, a shield tunnel loading mechanical model with anti-floating anchor rods is established, the coordinate system and the shield tunneling direction are determined, and the excavation sequence of the uplink and downlink tunnels is determined; S12, based on the basic idea of Mindlin solution, the stress component caused by the additional load in the soil is calculated, the stress component of each load along the z axis at a specific position is obtained, and the additional stress component caused by the loss of soil is calculated, and then the total additional vertical stress is obtained; S13, based on energy variation method, calculate the longitudinal displacement of shield tunnel, consider the collaborative deformation model of segment ring, establish the function conservation equation, integrate the total additional vertical stress, ground resistance, inter-ring shear force and inter-ring tension along the longitudinal direction of the tunnel, introduce the influence of anti-floating anchor, construct the displacement function and solve the vertical displacement.

[0007] Further, in step S11, specifically comprising: S21, the anti-floating anchor is uniformly arranged on the specific ring number of the tunnel lining along the tunnel axis direction at a preset interval; S22, the ground surface above the tunnel excavation face is taken as the coordinate origin to establish the coordinate system, the xoy plane is located at the ground level, the x axis is parallel to the direction of the existing tunnel, and the z axis is vertically downward; S23, the shield tunneling direction is along the positive direction of the x axis, and the cutter head notch is located at x=0; S24, the uplink tunnel and the downlink tunnel are stacked, the downlink tunnel is excavated first, and the uplink tunnel is excavated later.

[0008] Further, in step S12, specifically comprising: S31, the stress component of any point in the soil in the vertical z direction is calculated: (1) (2) (3) Wherein, , , : the stress component caused by the horizontal force in the vertical z direction, respectively, x axis, y axis, z axis; , , : the stress component along the x axis, y axis, z axis; x, y, z: the coordinates of any point in the soil; , , : the coordinates of any point below the ground; μ is the Poisson's ratio of the soil; M is the relative displacement of any point; N is the superimposed displacement of any point; S32, based on formulas (1)-(3), the stress component of each load along the z axis at any point is calculated: (4) (5) (6) wherein, is the vertical stress component caused by the cutterhead additional thrust; is the vertical stress component caused by the shield shell friction; is the vertical stress component caused by the grouting additional stress; q is the cutterhead additional thrust, along the positive direction of the x-axis; f is the shield shell friction, along the positive direction of the x-axis; p is the grouting additional stress, along the radial direction outward; r is the radial distance related to the cutterhead additional thrust; θ is the angle variable; and is the parameter related to the distance from the calculation point to the force action point, used for calculating the stress component; s is the position variable along the tunnel axis direction; L is the length of the shield machine; m is the parameter related to the grouting pressure distribution; h is the burial depth of the new shield tunnel; z is the coordinate of the calculation point in the vertical direction; R is the radius of the new shield tunnel; μ is the Poisson's ratio of the soil; is the component of the radial distance R in the vertical direction in the polar coordinate system; S33, calculating the additional stress component at the axis of the existing tunnel caused by the soil loss: (7) wherein, is the additional vertical stress component at the axis of the existing tunnel caused by the soil loss; is the equivalent width in the axis direction; 、 、 is the coefficient of the virtual force theorem; is the base bed coefficient; is the radius of the new shield tunnel; is the burial depth of the new shield tunnel; is the coordinate of the calculation point in the vertical direction; is the coordinate of the calculation point in the transverse direction; is the distance from the soil movement focus point to the tunnel center; then the total additional vertical stress is obtained: (8) wherein, is the total additional vertical stress at the axis of the downward tunnel caused by the upward tunnel excavation.

[0009] Further, in step S13, the steps of calculating the longitudinal displacement amount of the shield tunnel are as follows: ​​​​S41. Determine the relationship between the work done by the total additional vertical stress on the tunnel, the work done by the ground resistance, the work done by the shear force between the segments, and the work done by the tension force between the segments: (9) in, is the work done on the tunnel by the total additional vertical stress; is the work done by the formation resistance; is the work done by the shear force between the segments; is the work done by the tension between the segments; Calculate formation resistance, inter-annular shear force, and inter-annular tension: (10) (11) (12) in, is the formation resistance; is the shear force between the segments; is the tension between the segments; is the vertical displacement value of the down tunnel; is the characteristic diameter of the tunnel; is the rigid body rotation effect coefficient; 、 are the inter-annular shear and tensile stiffness, respectively; is the relative displacement of adjacent segments; is the angle between adjacent segments; The total additional vertical stress, stratum resistance, inter-ring shear force, and inter-ring tension are integrated along the tunnel longitudinal direction to obtain their respective work amounts: (13) (14) (15) (16) Where N is the discretization of the tunnel longitudinally into multiple small segments for numerical calculation; is the segment width; m is the segment index; is the total additional vertical stress; S42. Calculate the ground resistance and work done by the ground resistance on the down tunnel within the range of the anti-floating anchor rods: (17) (18) in, is the formation resistance; is the work done by the formation resistance; is the soil resistance enhancement coefficient of the tunnel; is the segment index, used to identify the start and end positions in the anti-floating anchor setting range; is the segment ring width; correspondingly adjusts the total function conservation formula: (19) wherein, is the amount of work done by the total additional vertical stress on the tunnel; is the amount of work done by the stratum resistance; is the amount of work done by the inter-segment ring shear force; is the amount of work done by the inter-segment ring tension; S43, calculating the longitudinal displacement of the shield tunnel: (20) wherein, is the vertical displacement function of the tunnel; is the cosine term coefficient of the Fourier series; is the sine term coefficient of the Fourier series; n is the expansion order of the Fourier series; is the application of the first Chebyshev polynomial in the cosine term; is the application of the first Chebyshev polynomial in the sine term; is the undetermined coefficient matrix related to the Chebyshev polynomial; is the undetermined coefficient matrix related to the Chebyshev polynomial.

[0010] Further, in step S13, calculating the longitudinal displacement of the shield tunnel further includes: Deriving the derivative of each undetermined coefficient on both sides of equation (17) can obtain: (21) In the formula: is a polynomial composed of elements ai, bi in the matrix , ; Substituting each work calculation formula into equation (21) can obtain the control equation as: (22) wherein, is the polynomial combination coefficient, is the soil stiffness matrix of the region without setting the anti-floating anchor, is the undetermined coefficient of the Chebyshev sine expansion term in the downward tunnel displacement function is the soil stiffness distribution function of the region without setting the anti-floating anchor at position x, is the soil stiffness distribution function of the region setting the anti-floating anchor at position x; Converting the above formula into a matrix form is: (23) In the formula: is the combined load vector of the soil stiffness effect, is the inter-segment interaction effect; is the soil resistance effect, wherein is the soil stiffness matrix of the anti-floating anchor segment, and is the soil stiffness matrix of the left and right anti-floating anchor segments; the matrix expressions are as follows: ; ; ; wherein, is the inter-segment shear interaction effect function matrix of the anti-floating anchor segment at position x; By formula (20), it can be obtained that is: (24) Substituting formula (24) into formula (20), the vertical displacement of the descending tunnel is: (25).

[0011] Further, the matrix in the process of calculating the longitudinal displacement of the shield tunnel is a ten-order matrix.

[0012] The second aspect of the present application provides a computer program product, comprising a computer program, which, when executed by a processor, realizes the steps of the tunnel displacement calculation method with an anti-floating anchor during the excavation of the overlapping tunnel according to any one of claims 1-6. The third aspect of the present application provides a computer-readable storage medium, which stores a computer program, which, when executed by a processor, realizes the steps of the tunnel displacement calculation method with an anti-floating anchor during the excavation of the overlapping tunnel according to any one of claims 1-6. Compared with the prior art, the beneficial effects of the present application are: The application provides a tunnel displacement deformation calculation method with anti-floating anchor rods during overlying tunnel excavation, and the method comprises the following steps: establishing a shield tunnel load mechanical model with anti-floating anchor rods, combining a Mindlin solution to calculate soil stress components caused by additional loads, accurately evaluating the stress state of the soil during tunnel excavation, and identifying potential unstable areas in advance to avoid safety accidents such as tunnel collapse caused by soil instability. The uplink and downlink tunnel overlying excavation sequence is determined to avoid soil stress redistribution caused by improper construction sequence, effectively reduce mutual interference during construction, and reduce the risk of tunnel deformation and damage. The longitudinal displacement of the shield tunnel is calculated based on the energy variation method, and a segment ring cooperative deformation model is considered to more accurately predict the deformation of the tunnel during construction. The anti-floating anchor rod can effectively resist the upward force of the tunnel structure and enhance the anti-floating stability of the tunnel. The application can be used for predicting actual overlying tunnel excavation projects at any angle, and can also be used as a reference for selecting anti-floating anchor rods as anti-floating measures for shield tunnels. BRIEF DESCRIPTION OF DRAWINGS

[0013] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or the prior art description. Obviously, the drawings in the following description are only embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of the provided drawings. Figure 1 A circular shield tunnel model with anti-floating anchor rods in the tunnel displacement deformation calculation method with anti-floating anchor rods during overlying tunnel excavation provided by the embodiments of the present application is shown in the figure. Figure 2 A combined and simplified model of anti-floating anchor resistance and stratum resistance in the tunnel displacement deformation calculation method with anti-floating anchor rods during overlying tunnel excavation provided by the embodiments of the present application is shown in the figure. Figure 3 A schematic diagram of the influence of overlying tunnel excavation on existing shield tunnels with anti-floating anchor rods in the tunnel displacement deformation calculation method with anti-floating anchor rods during overlying tunnel excavation provided by the embodiments of the present application is shown in the figure. Figure 4 A position relationship diagram between a new tunnel and a shield tunnel in the tunnel displacement deformation calculation method with anti-floating anchor rods during overlying tunnel excavation provided by the embodiments of the present application is shown in the figure. Figure 5 A reliability verification diagram in the tunnel displacement deformation calculation method with anti-floating anchor rods during overlying tunnel excavation provided by the embodiments of the present application is shown in the figure. DETAILED DESCRIPTION

[0014] The present application is further described below in conjunction with the embodiments and drawings. The following description of the embodiments is intended to help understand the present application and verify the reliability of the prediction method proposed by the present application. It should be noted that for those skilled in the art, without departing from the principles of the present application, a number of improvements and modifications can be made to the present application, and these improvements and modifications also fall within the scope of protection of the claims of the present application.

[0015] The first aspect of the present embodiment proposes a tunnel displacement calculation method with anti-floating anchor during overlying tunnel excavation, which comprises: S11, based on the conditions related to the excavation face of the newly built tunnel, a mechanical model of the shield tunnel with anti-floating anchor is established, the coordinate system and the shield tunneling direction are determined, and the uplink and downlink tunnel overlying excavation sequence is determined; S12, based on the basic idea of Mindlin solution, the stress components in the soil caused by additional load are calculated, the stress components of each load along the z-axis at a specific position are obtained, and the additional stress components caused by soil loss are calculated, and then the total additional vertical stress is obtained; S13, based on energy variation method, the longitudinal displacement of shield tunnel is calculated, considering the pipe ring collaborative deformation model, the function conservation equation is established, the total additional vertical stress, ground resistance, inter-ring shear force, inter-ring tension along the longitudinal direction of the tunnel are integrated, the influence of anti-floating anchor is introduced, the displacement function is constructed and the vertical displacement is solved.

[0016] In the present embodiment, there is a section of about 155m up and down overlying section, both tunnels are constructed by shield + open cut method, and the construction sequence is first down and then up, taking the left line tunnel and the overlying section of the downlink right line tunnel as the research object, in order to avoid the excessive floating deformation influence of the uplink tunnel on the downlink tunnel during the excavation process, the anti-floating anchor is set at the bottom of the downlink tunnel to suppress the floating deformation, as shown in Figure 1 .

[0017] As a preferred embodiment, in step S11, specifically comprising: S21, the anti-floating anchor is uniformly arranged on the specific ring number of the tunnel lining along the tunnel axis direction at a preset interval.

[0018] S22, the coordinate system is established with the ground surface above the tunnel excavation face as the coordinate origin, the xoy plane is located at the ground level, the x axis is parallel to the direction of the existing tunnel, and the z axis is vertical downward.

[0019] S23, the shield tunneling direction is along the positive direction of x axis, and the cutter head cut is located at x=0.

[0020] S24, the uplink tunnel and the downlink tunnel are overlying, the downlink tunnel is excavated first, and the uplink tunnel is excavated later.

[0021] In the present embodiment, as shown in Figure 2As shown in the figure, the anti-floating anchor rod is used as early support in the tunnel section with weak surrounding rock and poor self-stability by reserving the anchor rod on the lower segment of the shield tunnel; combined with the anchor rod, steel mesh and sprayed concrete, a combined support is formed, and the space is connected with the shield tunnel. When the environment around the shield tunnel with the anti-floating anchor rod does not change, the anti-floating anchor rod does not work; when the tunnel above the shield tunnel with the anti-floating anchor rod is crossed, the unloading effect of soil excavation will be transmitted to the lower shield tunnel through the soil, causing additional load on the tunnel structure, destroying the stress balance of the segment structure, and thus generating deformation. At this time, the anti-floating anchor rod gives the shield tunnel a downward anti-floating force through the friction with the soil. Figure 3 As shown in the figure, the present application simplifies each segment ring of the shield tunnel as an elastic foundation short beam, at this time, the shield tunnel can be regarded as being subjected to the downward elastic force of the soil above and the anti-floating force generated by the anti-floating anchor rod, which can be combined and simplified as a larger vertical downward elastic force.

[0022] The anti-floating anchor rod is uniformly arranged on the specific number of rings of the tunnel lining along the tunnel axis direction at a preset interval, the selection of the interval and the specific number of rings needs to comprehensively consider factors such as the geological conditions of the tunnel, the structural stress and the anti-floating design requirements, so as to ensure that the anchor rod can effectively provide anti-floating force and inhibit the upward deformation of the tunnel; The ground surface above the excavation face of the new tunnel is taken as the coordinate origin to establish a coordinate system, the xoy plane is located on the ground surface, the x axis is parallel to the direction of the existing tunnel, and the z axis is vertically downward; the shield tunneling direction is along the positive direction of the x axis, and the cutter head cut is located at x=0; the uplink tunnel and the downlink tunnel are stacked up and down, the downlink tunnel is excavated first, and the uplink tunnel is excavated later. The depth h of the new shield tunnel, the depth hs of the existing shield tunnel with the anti-floating anchor rod, the radius R of the new shield tunnel, the radius Rs of the existing shield tunnel with the anti-floating anchor rod, the cutter head thrust q, the shield shell friction force f acting on the circumferential side wall of the shield, the grouting additional stress p acting on the shield tail, and the number 2N1 of the shield tunnel lining rings where the anti-floating anchor rod is arranged are measured, and the positional relationship between the shield tunnel with the anti-floating anchor rod and the new tunnel is established, as shown in the figure. Figure 4 As shown in the figure.

[0023] As a preferred embodiment, in step S12, the step specifically comprises: S31, based on the basic idea of the Mindlin solution, within its original assumptions, if a horizontal force along the x axis, the y axis and the z axis acts on any point (x1, y1, z1) below the ground , σy, σz, the stress component σz in the vertical z direction of any point (x, y, z) in the soil can be calculated , σy-z, σz-z, respectively as: (1) (2) (3) wherein M is the relative displacement of any point, ; N is the superimposed displacement of any point, ; is the Poisson's ratio of soil, wherein, , , : are the stress components caused by the horizontal forces in the x-axis, y-axis, z-axis directions in the vertical z direction, respectively; , , : are the stress components in the x-axis, y-axis, z-axis directions, respectively; x, y, z: are the coordinates of any point in the soil body; , , : are the coordinates of any point below the ground.

[0024] S32, based on the formulas (1)-(3), the corresponding integral derivation of the loads q, f, p can be performed to obtain the stress components of each load along the z-axis at any point (x, y, z) , σz-f, σz-p: (4) (5) (6) wherein, is the vertical stress component caused by the additional thrust of the cutter head; is the vertical stress component caused by the friction of the shield shell; is the vertical stress component caused by the additional stress of grouting; q is the additional thrust of the cutter head, in the positive direction of the x-axis; f is the friction of the shield shell, in the positive direction of the x-axis; p is the additional stress of grouting, in the radial and circumferential outward direction; r is the radial distance related to the additional thrust of the cutter head; θ is the angle variable; , , , , and are parameters related to the distance from the calculation point to the force action point, for calculating the stress component; s is the position variable in the direction of the tunnel axis; L is the length of the shield machine; m is a parameter related to the distribution of grouting pressure; h is the burial depth of the newly-built shield tunnel; z is the coordinate of the calculation point in the vertical direction; R is the radius of the newly-built shield tunnel; μ is the Poisson's ratio of soil; is the component of the radial distance R in the vertical direction in the polar coordinate system; the point (x, y, z) is the derivation point of stress, which needs to be arranged at the axis of the downward tunnel, so that y=0, z=hs are satisfied; The intermediate variables are respectively: 、 、 、 、 、

[0025] The additional stress component of the existing tunnel axis caused by the soil loss is based on the basic idea of Mindlin solution, and the complex load is converted into the stress component of a specific point through integration derivation, thereby providing a basis for subsequent tunnel displacement calculation, and the core lies in the related theory and formula of Mindlin solution, and each type of load involved in the shield tunnel excavation process is analyzed and processed, so as to accurately calculate the stress distribution caused by the load in the soil.

[0026] S33, calculating the additional stress component of the existing tunnel axis caused by the soil loss , specifically: (7) In the formula: is the additional vertical stress component of the existing tunnel axis caused by the soil loss; is the maximum soil loss rate; is the radius of the new shield tunnel; is the burial depth of the new shield tunnel; is the coordinate of the calculation point in the vertical direction; is the coordinate of the calculation point in the horizontal direction; is the distance from the soil movement focus point to the tunnel center; B is the equivalent width in the axial direction, ; parameter delta, lambda is the coefficient of the force position theorem, ; ; ; L is the length of the shield machine; k is the base bed coefficient, which can be calculated by Vesic formula, E0 is the deformation modulus of the foundation soil, Es is the compression modulus of the soil, and Ds is the diameter of the downward tunnel; is the bending stiffness of the pile body.

[0027] Therefore, the total additional vertical stress σz generated by the upward tunnel excavation at the axis of the downward tunnel is: (8) wherein, Total additional vertical stress generated by up-tunnel excavation at the axis of down-tunnel.

[0028] As a preferred embodiment, in step S13, the step of calculating the longitudinal displacement amount of the shield tunnel is as follows: S41, considering the joint angle and the segment ring cooperative deformation model, from the perspective of functional conservation, the relationship between the work done by the total additional vertical stress on the tunnel , the work done by the ground resistance , the work done by the shear force between segment rings , the work done by the tension between segment rings is: (9) Ground resistance , the shear force between rings Fj, and the tension between rings Fl are respectively: (10) (11) (12) In the formula, where, is the ground resistance; is the shear force between segment rings; is the tension between segment rings; is the vertical displacement value of the down-tunnel; is the characteristic diameter of the tunnel; is the rigid body rotation effect coefficient; are the shear and tensile stiffness between rings respectively; is the relative displacement between adjacent segments, and ; is the rotation angle between adjacent segments, and ; is the segment ring width.

[0029] Integrating the total additional vertical stress, the ground resistance, the shear force between rings, and the tension between rings along the longitudinal direction of the tunnel respectively can obtain the work done by each of them , , , are respectively: (13) (14) (15) (16) Where N is the number of small segments obtained by discretizing the longitudinal direction of the tunnel for numerical calculation; is the segment ring width; and m is the segment index.​ Total additional vertical stress; S42, the anti-floating anchor is arranged at the bottom of the downward tunnel, which can inhibit deformation when the tunnel floats and deforms, and the upper soil resistance and the downward anti-floating force provided by the anti-floating anchor can be simplified and combined into a larger soil resistance, as shown in Figure 2 The soil resistance enhancement coefficient is introduced. Within the range of the anti-floating anchor, the stratum resistance and the stratum resistance work will increase to and The calculation formulae are respectively: (18) Wherein, is the stratum resistance; is the stratum resistance work; is the soil resistance enhancement coefficient; is the segment index, used to identify the starting and ending positions within the anti-floating anchor setting range; is the segment ring width; The calculation formulae of other forces and work remain unchanged, and the total energy conservation formula should be adjusted to: (19) Wherein, is the work done by the total additional vertical stress on the tunnel; is the stratum resistance work; is the segment ring shear force work; is the segment ring tensile force work.

[0030] S43, based on the energy variation method, the longitudinal displacement of the shield tunnel is calculated by establishing the energy conservation equation, and the displacement response of the tunnel under various forces is solved. The basic principle is to balance the energy change of the tunnel system under external load and the internal deformation energy, and through mathematical derivation and numerical solution, the displacement distribution law of the tunnel is obtained, thereby providing a theoretical basis for the structural design and construction control of the tunnel.

[0031] The energy variation method needs to assume a suitable displacement function to represent the tunnel displacement. In the initial shear dislocation cooperative deformation model, since the assumed displacement function only contains cosine function, it is more suitable for the case that the tunnel displacement curve is symmetrically distributed about the center, such as the case of vertically passing through or underpassing an existing tunnel. However, the research condition in this paper is the overlapping tunnel, and the displacement curve of the existing tunnel is not symmetrically distributed about a point. Therefore, this paper assumes that the vertical displacement function of the shield contains both sine function and cosine function, and the expression is: (20) wherein: is the vertical displacement function of the tunnel; is the cosine term coefficient of the Fourier series; is the sine term coefficient of the Fourier series; n is the expansion order of the Fourier series; is the application of the first kind of Chebyshev polynomial in the cosine term; is the application of the first kind of Chebyshev polynomial in the sine term; , , is the undetermined coefficient matrix related to the Chebyshev polynomial; is the undetermined coefficient matrix related to the Chebyshev polynomial; has , .

[0032] As a preferred embodiment, in step S13, the calculation of the longitudinal displacement of the shield tunnel further comprises: The derivation of each undetermined coefficient on both sides of equation (17) can be obtained as follows: (21) wherein: ζi is a polynomial containing the elements ai, bi in the matrix and .

[0033] Substituting each work calculation formula into equation (21) can obtain the control equation as follows: (22) The above equation is converted into a matrix form as follows: (23) wherein: is the inter- segment ring action effect; is the soil resistance action effect, wherein is the soil stiffness matrix of the anti-floating anchor segment setting, and are the soil stiffness matrices of the left and right anti-floating anchor segments not setting; the expressions of each matrix are as follows: ; ; .

[0034] wherein, is the inter- segment ring shear action effect function matrix of the anti-floating anchor setting area at position x; Through equation (20), we can obtain is: (twenty four) Substituting equation (24) into equation (20), the vertical displacement of the down tunnel can be obtained as: (25) The above calculation can be performed through Matlab programming, and when the matrix is ​​a tenth-order matrix, the calculation accuracy requirement can be met.

[0035] The specific formation and tunnel parameters in Matlab software of this embodiment are as follows: mu =0.35, E s =20MPa, h =10m, h s =18.35m, R = R s =3.1m 、 D t =1.2m, j =0.3, q =200kPa, f =150kPa, p =120kPa, k s =7.45×105kN / m, k t =1.94×106kN / m, E I =1.1×108kN / m2, L =9m, m =1.2m, N 1=100, d =3m, eta =3%, k m =1.46, the length of the anti-floating anchor is 8m.

[0036] At this time, the displacement deformation calculation value of the existing tunnel under the upper tunnel caused by the excavation of the upper tunnel before and after the anti-floating anchor rod is set is obtained, and the calculated value is compared with the finite element simulation displacement value and the measured data. Figure 5 As shown. Figure 5It can be known that the tunnel displacement curves obtained by the finite element simulation and the actual monitoring are roughly consistent with the curve obtained by the method, and the change trends are consistent; wherein, the maximum value of the vertical displacement calculation value of the tunnel without setting the anti-floating anchor is 12.18mm, and the numerical simulation value is 12.79mm; the maximum value of the vertical displacement calculation value of the tunnel with setting the anti-floating anchor is 9.16mm, and the numerical simulation value is 9.59mm, and the final stable displacement value of the tunnel measured is about 8.88mm, and the relative error of the calculation value and the finite element simulation value is only 3.2% and 9%, respectively, and the calculation result meets the accuracy requirement. It is shown that the prediction method can better reflect the displacement deformation of the existing shield tunnel with the anti-floating anchor caused by the superimposed tunnel excavation.

[0037] As an optional embodiment, the present application further provides a computer program product, comprising a computer program which realizes the steps of the method described in the foregoing method embodiments when executed by a processor. As another optional embodiment, the present application further provides a computer readable storage medium, which stores a computer program, and the computer program realizes the steps of the method described in the foregoing method embodiments when executed by a processor. The above only describes the preferred embodiments of the present application and is not used to limit the present application, and any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for calculating displacement of tunnels with anti-floating anchors during overlapping tunnel excavation, characterized in that: The following steps are involved: S11. Based on the relevant conditions of the new tunnel excavation surface, establish a shield tunnel loading mechanical model with anti-floating anchors, clarify the coordinate system and shield tunneling direction, and determine the overlapping excavation sequence of the upward and downward tunnels; S12. Based on the basic idea of ​​Mindlin solution, calculate the stress components in the soil caused by the additional loads, obtain the stress components along the z-axis at specific locations of each load, and calculate the additional stress components caused by soil loss, thereby obtaining the total additional vertical stress; S13. Based on the energy variation method, the longitudinal displacement of the shield tunnel is calculated. The cooperative deformation model of the segment ring is considered, and the functional conservation equation is established. The total additional vertical stress, stratum resistance, inter-ring shear force, and inter-ring tension are integrated along the longitudinal direction of the tunnel. The influence of anti-floating anchors is introduced, and the displacement function is constructed to solve the vertical displacement.

2. The method for calculating tunnel displacement with anti-floating anchors during overlapping tunnel excavation according to claim 1 is characterized in that: Step S11 specifically includes: S21, anti-floating anchors are evenly arranged along the tunnel axis at a preset spacing on a specific number of rings of the tunnel lining; S22. Establish a coordinate system with the ground above the tunnel excavation surface as the coordinate origin, with the xoy plane located at the ground level, the x-axis parallel to the existing tunnel direction, and the z-axis pointing vertically downward; S23, the shield tunneling direction is along the positive x-axis, and the cutterhead is located at x=0; S24: The up tunnel and the down tunnel overlap each other, with the down tunnel being excavated first and the up tunnel later.

3. The method for calculating tunnel displacement with anti-floating anchors during overlapping tunnel excavation according to claim 1 is characterized in that: Step S12 specifically includes: S31. Calculate the stress component in the vertical z direction at any point in the soil: (1) (2) (3) in, 、 、 : are the stress components in the vertical z direction caused by the horizontal forces in the x-axis, y-axis, and z-axis directions respectively; 、 、 : are stress components along the x-axis, y-axis, and z-axis respectively; x, y, z: are coordinates of any point in the soil; 、 、 : is the coordinate of any point below the ground; μ is the Poisson's ratio of the soil; M is the relative displacement of any point; N is the superimposed displacement of any point; S32. Based on formulas (1) to (3), calculate the stress component of each load at any point along the z-axis: (4) (5) (6) in, is the vertical stress component caused by the additional thrust of the cutterhead; is the vertical stress component caused by the friction of the shield; is the vertical stress component caused by the additional stress of grouting; q is the additional thrust of the cutterhead, along the positive direction of the x-axis; f is the friction force of the shield, along the positive direction of the x-axis; p is the additional stress of grouting, along the radial annular direction outward; r is the radial distance related to the additional thrust of the cutterhead; θ is the angle variable; 、 、 、 、 and is a parameter related to the distance from the calculation point to the force application point, used to calculate the stress component; s is the position variable along the tunnel axis; L is the length of the shield machine; m is a parameter related to the grouting pressure distribution; h is the burial depth of the new shield tunnel; z is the vertical coordinate of the calculation point; R is the radius of the new shield tunnel; μ is the Poisson's ratio of the soil; is the vertical component of the radial distance R in the polar coordinate system; S33. Calculate the additional stress component at the existing tunnel axis caused by soil loss: (7) in, is the additional vertical stress component at the existing tunnel axis caused by soil loss; is the equivalent width in the axial direction; 、 、 is the coefficient of the virial theorem; is the base bed coefficient; is the radius of the new shield tunnel; is the burial depth of the new shield tunnel; To calculate the coordinates of the point in the vertical direction; To calculate the horizontal coordinates of the point; The distance from the soil moving focus to the tunnel center; The total additional vertical stress is obtained: (8) in, It is the total additional vertical stress generated at the axis of the down tunnel by the excavation of the up tunnel.

4. The method for calculating tunnel displacement with anti-floating anchors during overlapping tunnel excavation according to claim 1 is characterized in that: In step S13, the steps for calculating the longitudinal displacement of the shield tunnel are as follows: S41. Determine the relationship between the work done by the total additional vertical stress on the tunnel, the work done by the ground resistance, the work done by the shear force between the segments, and the work done by the tension force between the segments: (9) in, is the work done on the tunnel by the total additional vertical stress; is the work done by the formation resistance; is the work done by the shear force between the segments; is the work done by the tension between the segments; Calculate formation resistance, inter-annular shear force, and inter-annular tension: (10) (11) (12) in, is the formation resistance; is the shear force between the segments; is the tension between the segments; is the vertical displacement value of the down tunnel; is the characteristic diameter of the tunnel; is the rigid body rotation effect coefficient; 、 are the inter-annular shear and tensile stiffness, respectively; is the relative displacement of adjacent segments; is the angle between adjacent segments; The total additional vertical stress, stratum resistance, inter-ring shear force, and inter-ring tension are integrated along the tunnel longitudinal direction to obtain their respective work amounts: (13) (14) (15) (16) Where N is the discretization of the tunnel longitudinally into multiple small segments for numerical calculation; is the segment width; m is the segment index; is the total additional vertical stress; S42. Calculate the ground resistance and work done by the ground resistance on the down tunnel within the range of the anti-floating anchor rods: (17) (18) in, is the formation resistance; is the work done by the formation resistance; is the soil resistance enhancement coefficient; It is a segment index used to identify the starting and ending positions within the anti-floating anchor setting range; is the segment ring width; The overall conservation of work formula is adjusted accordingly: (19) in, is the work done on the tunnel by the total additional vertical stress; is the work done by the formation resistance; is the work done by the shear force between the segments; is the work done by the tension between the segments; S43. Calculate the longitudinal displacement of the shield tunnel: (20) in, is the vertical displacement function of the tunnel; is the cosine term coefficient of the Fourier series; is the sine term coefficient of the Fourier series; n is the expansion order of the Fourier series; Application of the first kind of Chebyshev polynomials in cosine terms; Application of Chebyshev polynomials of the first kind in sine terms; is the matrix of unknown coefficients associated with the Chebyshev polynomials; The matrix of undetermined coefficients associated with the Chebyshev polynomials.

5. The method for calculating tunnel displacement with anti-floating anchors during overlapping tunnel excavation according to claim 4 is characterized in that: In step S13, calculating the longitudinal displacement of the shield tunnel further includes: By taking the derivative of each undetermined coefficient on both sides of equation (17), we can obtain: (21) Where: To include the matrix 、 The polynomial composed of the elements ai and bi; Substituting the work calculation formulas into equation (21), the control equation can be obtained as follows: (22) in, are the coefficients of the polynomial combination, is the soil stiffness matrix in the area where no anti-floating anchors are set, is the unknown coefficient of the Chebyshev sine expansion term in the displacement function of the downlink tunnel is the soil stiffness distribution function of the area without anti-floating anchor at position x, is the soil stiffness distribution function of the area where the anti-floating anchor is set at position x; Convert the above formula into matrix form: (23) Where: is the combined load vector acting on soil stiffness, is the effect between the segments; is the soil resistance effect, where To set the soil stiffness matrix of the anti-floating anchor section, and is the soil stiffness matrix without anti-floating anchor sections on the left and right sides; the matrix expressions are as follows: ； ； ； in, is the shear effect function matrix between the segments at position x in the anti-floating anchor area; According to formula (20), we can get for: (24) Substituting equation (24) into equation (20), the vertical displacement of the down tunnel can be obtained as: (25)。 6. The method for calculating displacement of a tunnel with anti-floating anchors during overlapping tunnel excavation according to claim 4, characterized in that: The matrix used in the calculation of the longitudinal displacement of the shield tunnel is a tenth-order matrix.

7. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method for calculating the displacement of a tunnel with anti-floating anchors during the excavation of overlapping tunnels as described in any one of claims 1 to 6 are implemented.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method for calculating the displacement of a tunnel with anti-floating anchors during the excavation of overlapping tunnels as described in any one of claims 1 to 6 are implemented.

Images