A calculation method for the longitudinal deformation of an existing subway tunnel caused by a new tunnel passing under it

Abstract

The present invention discloses a method for calculating the longitudinal deformation of an existing subway tunnel caused by a new tunnel passing under the tunnel. The present invention takes into account the nonlinearity of the foundation reaction coefficient of the tunnel, and solves the longitudinal deformation and mechanical response of the existing subway tunnel lying adjacent to the new tunnel under the influence of the construction of the new tunnel passing under the tunnel by means of a superposition method based on beam theory. The obtained prediction result is relatively accurate, close to the actual situation, and has a small error.

Description

Technical Field

[0001] The invention specifically relates to a method for calculating the longitudinal deformation of an existing subway tunnel caused by a newly built tunnel passing under the tunnel. Background Art

[0002] Due to the limitation of urban underground space, new tunnels will inevitably approach and cross existing operating tunnels. The excavation of new tunnels will have adverse effects on existing tunnels, such as uneven longitudinal settlement, cracking of segments, ballast bed detachment, joint damage, water leakage, etc., which will seriously affect the safety of train operation. Reasonable estimation of the response of existing tunnels during tunnel underpass construction, control of tunnel construction risks, and protection of existing tunnel structures are one of the important issues facing urban rail transit projects.

[0003] Scholars at home and abroad mainly study the impact of underpasses on existing subway tunnels through on-site monitoring, finite element analysis and simplified analytical theory.

[0004] Most of the current methods for predicting the deformation of shield tunnels under unloading of foundation pits assume that the reaction force of the foundation increases linearly as the deformation of the foundation increases. As we all know, the deformation of foundation soil has nonlinear characteristics, that is, as the displacement of the soil increases, the foundation reaction force will not increase infinitely, but will approach a certain critical value. Therefore, it is inaccurate to use a linear foundation model to evaluate the stress and deformation characteristics of the tunnel, and there is a large error. Summary of the invention

[0005] In view of the above situation, in order to overcome the defects of the prior art, the present invention provides a method for calculating the longitudinal deformation of an existing subway tunnel when a new tunnel is adjacent to or under the existing subway tunnel.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] A method for calculating the longitudinal deformation of an existing subway tunnel caused by a newly built tunnel passing under the existing subway tunnel comprises the following steps:

[0008] (1) Using the elastic foundation model, calculate the tunnel deflection s1(y) and elastic reaction force F1(y) caused by the unloading pressure q(y), F1(y) = k·s1(y), where k is the roadbed reaction coefficient;

[0009] (2) Apply an external load F1(y) of equal magnitude and opposite direction to the elastic foundation model in step (1), with the load action range being y∈[-L,L], and recalculate the tunnel deflection s2(y) and elastic reaction force F2(y);

[0010] (3) If the elastic reaction force F2(y) is less than 0.1 kN, s(y) is obtained by superimposing s1(y) and s2(y); otherwise, similar calculations are repeated until the elastic reaction is less than 0.1 kN;

[0011] Each superimposed external pressure using the elastic foundation model is equal to the initially applied additional pressure q(y);

[0012] At the same time, when the superimposed reaction force from the range -L to L is close to zero, it means that the springs in this range have been "deleted", and it can be regarded as the lower tunnel "0 distance" passing through the existing tunnel above;

[0013] Therefore, the elastic reaction force F(y) and deflection s(y) of the model can be obtained by the following formula:

[0014] F(y)=F1(y)-F1(y)+F2(y)-F2(y)+…-F n-1 (y)+F n (y) (11)

[0015] s(y)=s1(y)+s2(y)+…+s n (y) (12).

[0016] Furthermore, in step (1), the Winkler foundation is used to simulate the interaction between the foundation soil and the tunnel, that is, the interaction between the foundation soil and the existing tunnel is simulated by a series of tight independent linear "soil springs". Due to the construction of the new tunnel underpass, the stiffness of the original "soil spring" changes along the direction of the existing tunnel.

[0017] Furthermore, in step (1), an Euler-Bernoulli beam is used to simulate an existing subway tunnel.

[0018] Furthermore, the governing equation of the infinite Euler Bernoulli beam on the Winkler foundation is:

[0019]

[0020] Where: EI is the equivalent bending stiffness of the existing tunnel; s(y) is the longitudinal displacement of the existing tunnel; k is the roadbed reaction coefficient; b is the cross-sectional width of the beam; q(y) is the unloading pressure on the existing subway tunnel caused by the construction of the new tunnel;

[0021] The solution of formula (1) is written as:

[0022] s(y)=s0(y)+v(y) (2)

[0023] Where: s0(y) is the general solution of equation (1), v(y) is the special solution related to the unloading pressure q(y);

[0024] Solved to:

[0025] s0(y)=e βy(c1cosβy+c2cosβy)+e -βy (c3cosβy+c4cosβy) (3)

[0026] in: I is a material mechanics parameter, which represents the inertia moment of the tunnel; parameters c1, c2, c3 and c4 are four integral constants, which can be determined by boundary conditions;

[0027] Because s(y)=0, when y approaches infinity,

[0028] Formula (3) is simplified to:

[0029] s0(y)=e -βy (c3cosβy+c4cosβy) (4)

[0030] If a concentrated load p is applied at the origin, the rotation angle θ(y) is zero and the shear force Q(y) is -p / 2 at the origin:

[0031]

[0032] Q(y) represents the shear force generated by applying a concentrated load p at the origin;

[0033] Substituting equation (4) into equation (5), constants c3 and c4 can be obtained:

[0034]

[0035] Therefore, the vertical deflection of an infinitely long beam caused by a concentrated load p can be expressed as:

[0036]

[0037] When q(δ) is applied to an infinitesimal element length dδ at any position δ of the infinitely long beam, it can be assumed that the origin of the finite-length beam moves from O to O1; the resulting deflection ds(y) can be calculated by equation (7), where p is replaced by q(δ)bdδ and y is replaced by |y-δ|;

[0038] Right now:

[0039]

[0040] By integrating the equation, the total deflection s(y) of the infinitely long beam caused by the unloading distributed pressure can be obtained:

[0041]

[0042] The additional bending moment M(y) related to s(y) can be obtained through beam theory, that is:

[0043]

[0044] The tunnel deflection s1(y) and elastic reaction force F1(y) caused by the unloading pressure q(y) are calculated using the above equations (1) to (9).

[0045] Furthermore, in step (2), the tunnel deflection s2(y) and the elastic reaction force F2(y) are recalculated using equations (1) to (9), where F2(y) = k·s2(y), where k is the roadbed reaction coefficient.

[0046] The beneficial effects of the present invention are:

[0047] The present invention takes into account the nonlinearity of the foundation reaction coefficient of the tunnel, and solves the longitudinal deformation and mechanical response of the existing subway tunnel adjacent to the new tunnel under the influence of the adjacent underpass construction through the superposition method based on beam theory. The obtained prediction result is relatively accurate, with an accuracy rate of more than 96%, close to the actual situation, and with a small error. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of the process of solving the longitudinal deformation of existing subway tunnels by superposition method.

[0049] Figure 2 This is the cross-section of the new tunnel.

[0050] Figure 3 It is a schematic diagram of the external load q(δ)dδ acting on the Winkler foundation model. DETAILED DESCRIPTION

[0051] In order to make the purpose, technical solutions and advantages of the present application clearer, the present application is described and illustrated 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 application and are not intended to limit the present application. Based on the embodiments provided in the present application, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the present application.

[0052] Reference to "embodiments" in this application means that a particular feature, structure, or characteristic described in conjunction with the embodiments may be included in at least one embodiment of the present application. The appearance of the phrase in various locations in the specification does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment that is mutually exclusive with other embodiments. It is explicitly and implicitly understood by those of ordinary skill in the art that the embodiments described in this application may be combined with other embodiments without conflict.

[0053] Unless otherwise defined, the technical terms or scientific terms involved in this application should be understood by people with ordinary skills in the technical field to which this application belongs. The words "one", "a", "a", "the" and the like involved in this application do not indicate a quantitative limitation and may represent the singular or plural. The terms "include", "comprise", "have" and any of their variations involved in this application are intended to cover non-exclusive inclusions; the words "connect", "connected" and the like involved in this application are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. The "multiple" involved in this application means greater than or equal to two. "And / or" describes the association relationship of associated objects, indicating that there may be three relationships. For example, "A and / or B" can represent: A exists alone, A and B exist at the same time, and B exists alone. The terms "first", "second", "third" and the like involved in this application are only used to distinguish similar objects and do not represent a specific ordering of objects.

[0054] Example 1

[0055] A method for calculating the longitudinal deformation of an existing subway tunnel caused by a newly built tunnel passing under the existing subway tunnel comprises the following steps:

[0056] (1) When a new tunnel is built under an existing subway tunnel, the classic elastic foundation model is used to calculate the tunnel deflection s1(y) and elastic reaction force F1(y) caused by the unloading pressure q(y). F1(y) = k·s1(y), where k is the roadbed reaction coefficient.

[0057] (2) Apply external loads F1(y) of equal magnitude and opposite direction to the classical elastic foundation model in step (1), with the load action range being y∈[-L,L], and recalculate the tunnel deflection s2(y) and elastic reaction force F2(y);

[0058] (3) When the elastic reaction force F2(y) is close to 0 kN, for example, if the elastic reaction force F2(y) is less than 0.1 kN, s(y) can be obtained by superimposing s1(y) and s2(y); otherwise, similar calculations will be repeated until the elastic reaction force is less than 0.1.

[0059] Each superimposed external load using the elastic foundation model is equal to the initially applied additional pressure q(y); at the same time, when the superimposed reaction force from the range -L to L is close to zero, it means that the springs in this range have been "deleted", which means that the lower tunnel is considered to pass through the existing tunnel above at a "0 distance";

[0060] Therefore, the elastic reaction force F(y) and deflection s(y) of the model can be obtained by the following formula:

[0061] F(y)=F1(y)-F1(y)+F2(y)-F2(y)+…-F n-1 (y)+F n (y) (11)

[0062] s(y)=s1(y)+s2(y)+…+s n (y) (12)

[0063] In some preferred embodiments, in step (1), the newly built tunnel is closely connected to the existing subway tunnel model, and the Winkler foundation is used to simulate the interaction between the foundation soil and the tunnel, that is, the interaction between the foundation soil and the existing tunnel is simulated by a series of tight independent linear "soil springs". Due to the construction of the new tunnel under the existing subway tunnel, the stiffness of the original "soil spring" changes along the direction of the existing tunnel.

[0064] Euler-Bernoulli beam is used to simulate the existing subway tunnel.

[0065] The unloading pressure q(y) on the existing tunnel caused by the construction of the new tunnel can be obtained by the unloading load p related to the construction of the new tunnel. The cross section of the new tunnel is as follows: Figure 2 As shown, Figure 2 middle, represents the internal friction angle of the foundation soil, D represents the outer diameter of the new tunnel; the green box represents the outer contour of the new tunnel, the gray part represents the scope of the foundation soil affected by the construction of the new tunnel in close proximity, and L represents half of the scope of the foundation soil affected by the construction of the new tunnel in close proximity. Figure 2 In the figure, the distance between the new tunnel and the original tunnel is 0, that is, it passes under the existing subway tunnel.

[0066] The governing equations of the infinite Euler Bernoulli beam on the Winkler foundation are:

[0067]

[0068] Where: EI is the equivalent bending stiffness of the existing tunnel; s(y) is the longitudinal displacement of the existing tunnel; k is the roadbed reaction coefficient; b is the cross-sectional width of the beam; q(y) is the unloading pressure on the existing subway tunnel caused by the construction of the new tunnel.

[0069] The solution of formula (1) can be written as:

[0070] s(y)=s0(y)+v(y) (2)

[0071] Where: s0(y) is the general solution of equation (1), and v(y) is the special solution related to the unloading pressure q(y).

[0072] The solution can be written as:

[0073] s0(y)=e βy (c1cosβy+c2cosβy)+e -βy (c3cosβy+c4cosβy) (3)

[0074] in: I is a material mechanics parameter, representing the inertia moment of the tunnel. Parameters c1, c2, c3, and c4 are four integral constants that can be determined by boundary conditions.

[0075] Because s(y)=0, when y approaches infinity,

[0076] Formula (3) can be simplified as:

[0077] s0(y)=e -βy (c3cosβy+c4cosβy) (4)

[0078] If a concentrated load p is applied at the origin, the rotation angle θ(y) is zero and the shear force Q(y) is -p / 2 at the origin:

[0079]

[0080] Q(y) represents the shear force generated by applying a concentrated load p at the origin;

[0081] Substituting equation (4) into equation (5), constants c3 and c4 can be obtained:

[0082]

[0083] Therefore, the vertical deflection of an infinitely long beam caused by a concentrated load p can be expressed as:

[0084]

[0085] When q(δ) is applied to an infinitesimal element length dδ at any position δ of the infinitely long beam (e.g. Figure 3 As shown in Figure 1, it can be assumed that the origin of the finite length beam moves from O to O1. The resulting deflection ds(y) can be calculated using equation (7), where p is replaced by q(δ)bdδ and y is replaced by |y-δ|.

[0086] Right now:

[0087]

[0088] By integrating the equation, the total deflection s(y) of the infinitely long beam caused by the unloading distributed pressure can be obtained:

[0089]

[0090] The additional bending moment M(y) related to s(y) can be obtained through beam theory, that is:

[0091]

[0092] The tunnel deflection s1(y) and elastic reaction force F1(y) caused by the unloading pressure q(y) are calculated by the above equations (1) to (9);

[0093] In some preferred embodiments, equations (1) to (9) are used to recalculate the tunnel deflection s2(y) and the elastic reaction force F2(y).

[0094] Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present invention.

Claims

1. A method for calculating the longitudinal deformation of an existing subway tunnel when a new tunnel is adjacent to or under the existing subway tunnel, characterized in that: The following steps are involved: (1) Using the elastic foundation model, calculate the tunnel deflection s1(y) and elastic reaction force F1(y) caused by the unloading pressure q(y), F1(y) = k·s1(y), where k is the roadbed reaction coefficient; (2) Apply an external load F1(y) of equal magnitude and opposite direction to the elastic foundation model in step (1), with the load action range being y∈[-L,L], and recalculate the tunnel deflection s2(y) and elastic reaction force F2(y); (3) If the elastic reaction force F2(y) is less than 0.1 kN, s(y) is obtained by superimposing s1(y) and s2(y); otherwise, similar calculations are repeated until the elastic reaction is less than 0.1 kN; Each superimposed external pressure using the elastic foundation model is equal to the initially applied additional pressure q(y); At the same time, when the superimposed reaction force from the range -L to L is close to zero, it means that the springs in this range have been "deleted", and it can be regarded as the lower tunnel "0 distance" passing through the existing tunnel above; Therefore, the elastic reaction force F(y) and deflection s(y) of the model can be obtained by the following formula: F(y)=F1(y)-F1(y)+F2(y)-F2(y)+…-F n-1 (y)+F n (y) (11) s(y)=s1(y)+s2(y)+…+s n (y) (12)。 2. The method for calculating the longitudinal deformation of an existing subway tunnel caused by a newly built tunnel passing under the existing subway tunnel according to claim 1 is characterized in that: In step (1), the Winkler foundation is used to simulate the interaction between the foundation soil and the tunnel, that is, the interaction between the foundation soil and the existing tunnel is simulated by a series of tight independent linear "soil springs". Due to the construction of the new tunnel underpass, the stiffness of the original "soil spring" changes along the direction of the existing tunnel.

3. The method for calculating the longitudinal deformation of an existing subway tunnel caused by a newly built tunnel passing under the existing subway tunnel according to claim 2 is characterized in that: In step (1), the Euler-Bernoulli beam is used to simulate the existing subway tunnel.

4. The method for calculating the longitudinal deformation of an existing subway tunnel caused by a newly built tunnel passing under the existing subway tunnel according to claim 3 is characterized by: The governing equations of the infinite Euler Bernoulli beam on the Winkler foundation are: Where: EI is the equivalent bending stiffness of the existing tunnel; s(y) is the longitudinal displacement of the existing tunnel; k is the roadbed reaction coefficient; b is the cross-sectional width of the beam; q(y) is the unloading pressure on the existing subway tunnel caused by the construction of the new tunnel; The solution of formula (1) is written as: s(y)=s0(y)+v(y) (2) Where: s0(y) is the general solution of equation (1), v(y) is the special solution related to the unloading pressure q(y); Solved to: s0(y)=e βy (c1cosβy+c2cosβy)+e -βy (c3cosβy+c4cosβy) (3) in: I is a material mechanics parameter, which represents the inertia moment of the tunnel; parameters c1, c2, c3 and c4 are four integral constants, which can be determined by boundary conditions; Because s(y)=0, when y approaches infinity, Formula (3) is simplified to: s0(y)=e -βy (c3cosβy+c4cosβy) (4) If a concentrated load p is applied at the origin, the rotation angle θ(y) is zero and the shear force Q(y) is -p / 2 at the origin: Q(y) represents the shear force generated by applying a concentrated load p at the origin; Substituting equation (4) into equation (5), constants c3 and c4 can be obtained: Therefore, the vertical deflection of an infinitely long beam caused by a concentrated load p can be expressed as: When q(δ) is applied to an infinitesimal element length dδ at any position δ of the infinitely long beam, it can be assumed that the origin of the finite-length beam moves from O to O1; the resulting deflection ds(y) can be calculated by equation (7), where p is replaced by q(δ)bdδ and y is replaced by |y-δ|; Right now: By integrating the equation, the total deflection s(y) of the infinitely long beam caused by the unloading distributed pressure can be obtained: The additional bending moment M(y) related to s(y) can be obtained through beam theory, that is: The tunnel deflection s1(y) and elastic reaction force F1(y) caused by the unloading pressure q(y) are calculated using the above equations (1) to (9).

5. The method for calculating the longitudinal deformation of an existing subway tunnel caused by a newly built tunnel passing under the existing subway tunnel according to claim 4 is characterized in that: In step (2), the tunnel deflection s2(y) and elastic reaction force F2(y) are recalculated using equations (1) to (9), where F2(y) = k·s2(y), where k is the roadbed reaction coefficient.

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