Calculation method of foundation reaction coefficient applicable to interaction between shield tunnel and stratum

Abstract

The present invention discloses a method for calculating the foundation reaction coefficient applicable to the interaction between a shield tunnel and a stratum, and belongs to the technical field of shield tunnels. The method comprises the following steps: based on the Timoshenko beam and elastic continuous medium action model, a semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel is proposed, and the tunnel deformation is calculated. w 1( x ); Based on the deformation differential equation of the Timoshenko beam on the Winkler foundation, the deformation differential equation is solved to obtain the displacement solution and calculate the tunnel deformation w 2( x ); the semi-analytical and semi-numerical displacement solution of the tunnel in the elastic semi-infinite space and the peak value of the displacement solution of the Timoshenko beam on the Winkler foundation are equivalently fitted, and the foundation reaction coefficient is calculated by an iterative method. The beneficial effect is that the present invention determines a reasonable recommended value of the foundation reaction coefficient, which has guiding significance for actual engineering.

Description

Technical Field

[0001] The invention belongs to the technical field of shield tunnels, and in particular relates to a method for calculating a foundation reaction force coefficient applicable to the interaction between a shield tunnel and a stratum. Background Art

[0002] In order to ensure the safety of existing urban rail transit lines, it is necessary to reveal the mechanical response of shield tunnel segments in long-term service due to construction such as close proximity to foundation pit excavation, loading and tunnel crossing. Many scholars regard shield tunnels as straight beams on elastic foundations and calculate and analyze the deformation and internal forces of tunnels. Among them, the foundation reaction coefficient is an indispensable and important parameter in the calculation process, which is crucial to evaluating the deformation and internal forces of tunnel structures.

[0003] The foundation reaction coefficient is an important parameter in subway design and construction. The accuracy of the foundation reaction coefficient is related to the safety and economy of shield tunnels. At present, the foundation reaction coefficient is mainly determined by three methods. The first is to determine the value according to the recommended value given in the specification. This method is the simplest to operate, but the accuracy is not high because the range of recommended values ​​given in the specification is too wide; the second is to measure through field tests, which is too complicated to operate; the third is to calculate according to the existing foundation reaction coefficient calculation formula, but the selection of the formation conditions and the tunnel model itself is too simplified, making it difficult for the derived formula to fully and accurately meet the actual construction calculation.

[0004] Therefore, it is necessary to provide a calculation method for the foundation reaction coefficient suitable for the interaction between shield tunnel and stratum to solve the above problems. Summary of the invention

[0005] The purpose of an embodiment of the present invention is to provide a method for calculating the foundation reaction coefficient suitable for the interaction between a shield tunnel and a stratum, which determines a reasonable recommended value of the foundation reaction coefficient, has guiding significance for actual engineering, and can thus solve at least one technical problem involved in the background technology.

[0006] In order to solve the above-mentioned technical problems, the present invention is achieved as follows:

[0007] The embodiment of the present invention provides a method for calculating the foundation reaction force coefficient applicable to the interaction between a shield tunnel and a stratum, comprising the following steps:

[0008] Step S1: Based on the Timoshenko beam and elastic continuum model, a semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel is obtained to calculate the tunnel deformation. w 1 ( x );

[0009] Step S2, according to the deformation differential equation of the Timoshenko beam on the Winkler foundation, solve the deformation differential equation to obtain the displacement solution and calculate the tunnel deformation w 2 ( x );

[0010] Step S3, equivalently fitting the peak values ​​of the semi-analytical and semi-numerical displacement solution of the tunnel in the elastic semi-infinite space and the displacement solution of the Timoshenko beam on the Winkler foundation is performed, and the foundation reaction coefficient is calculated by an iterative method.

[0011] Optionally, in step S1, the calculation assumption for obtaining the semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel is that the beam is in full contact with the foundation under the action of the load, that is, the deformation of the beam is consistent with the deformation of the foundation soil surface.

[0012] Optionally, in step S1, the tunnel is deformed w 1 ( x ) is calculated as follows:

[0013] Step S11, solve the relationship between the displacement and stress of the foundation under the surface load in the Hankel transform domain, and then perform an inverse transform on it, and the functional of the Timoshenko beam can be obtained as follows:

[0014] ;

[0015] In the formula, E b I b is the flexural stiffness of the Timoshenko beam; θ , s Timoshenko Liang x The rotation angle and vertical displacement at ; P, M, p are the concentrated force, concentrated bending moment and distributed load acting on the beam respectively; ;

[0016] Step S12, using the Mindlin solution to calculate the load acting on the beam element, the specific calculation formula is as follows:

[0017] ;

[0018] ;

[0019] ;

[0020] In the formula, q To act on the ground ( x , y ) uniformly distributed load at the position; h is the buried depth of the tunnel top;D is the diameter of the beam; is the Poisson's ratio of the formation; A is the action range of the uniformly distributed load on the surface; x 0, y 0) represents the position of the beam element; R1 and R2 are intermediate variables; H is the depth of uniform load action;

[0021] Step S13, using the two-node Hermite beam unit interpolation function to calculate the shear force and bending moment at both ends of the beam unit and the finite element equilibrium equation of the Timoshenko beam, the specific calculation formula is as follows:

[0022] ;

[0023] ;

[0024] ;

[0025] ;

[0026] In the formula, is the node displacement and rotation matrix; is the node shear force and bending moment matrix; T is the symbol of matrix transposition; P is the intermediate variable; Q is the shear force of the Timoshenko beam; is the stiffness matrix of the formation; l 1 is the length of the beam element; is an intermediate variable; is the integration variable;

[0027] Step S14, balance equation According to the assumptions, it becomes the following form:

[0028] ;

[0029] In the formula, δ is the flexibility coefficient of the soil; Y , Z are the moment and shear force produced by the unit force respectively; w is the soil displacement at the beam-soil interface;

[0030] Step S15, solving the equilibrium equation to obtain the tunnel deformation w 1 ( x ).

[0031] Optionally, in step S2, the tunnel is deformed w 2 ( x ) is calculated as follows:

[0032] Step S21, use Mindlin solution to calculate the load acting on the beam element , the deformation differential equation of the Timoshenko beam on the Winkler foundation is as follows:

[0033] ;

[0034] In the formula, w is the displacement of the Timoshenko beam;

[0035] Step S22: Perform Fourier series expansion;

[0036] Step S23, solving the deformation differential equation of the Timoshenko beam on the Winkler foundation to obtain the tunnel deformation w 2 ( x ).

[0037] Optionally, in step S3, calculating the foundation reaction coefficient by an iterative method specifically includes the following calculation steps:

[0038] Step S31, using the semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel to calculate the tunnel deformation w 1 ( x ) max ;

[0039] Step S32: Input foundation reaction coefficient k= 1000+10 *i , the tunnel deformation is calculated using the displacement solution of the Timoshenko beam on the Winkler foundation w 2 ( x ) max ,in, i is the number of iterations;

[0040] Step S33, comparing tunnel deformation w 1 ( x ) max and tunnel deformation w 2 ( x ) max , determine whether the difference between the two is less than 0.001mm, if so, they are considered equivalent and output the current k value, and consider that k The value is the required foundation reaction coefficient; if not, return to step S32 to adjust k The value continues to be calculated.

[0041] The beneficial effects of the embodiments of the present invention are:

[0042] The method provided by the present invention regards the shield tunnel as a Timoshenko beam model and the stratum as an elastic semi-infinite space, performs discrete analysis on the Timoshenko beam, and uses the finite element method to derive the theoretical solution of the Timoshenko beam, which is beneficial to improving the calculation accuracy of tunnel deformation. In addition, the method can be used for the evaluation and calculation of the deformation of existing shield tunnels under the influence of different external operations, which has guiding significance for practical engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for describing the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative work, among which:

[0044] Figure 1 A diagram of the Timoshenko beam model in an elastic semi-infinite space provided by the present invention;

[0045] Figure 2 Schematic diagram of the Timoshenko beam model on the Winkler foundation provided by the present invention. DETAILED DESCRIPTION

[0046] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0047] The terms "first", "second", etc. in the specification and claims of the present invention are used to distinguish similar objects, and are not used to describe a specific order or sequence. It should be understood that the data used in this way can be interchanged where appropriate, so that the embodiments of the present invention can be implemented in an order other than those illustrated or described herein, and the objects distinguished by "first", "second", etc. are generally of the same type, and the number of objects is not limited. For example, the first object can be one or more. In addition, "and / or" in the specification and claims represents at least one of the connected objects, and the character " / " generally indicates that the objects associated with each other are in an "or" relationship.

[0048] The embodiment of the present invention provides a method for calculating the foundation reaction force coefficient applicable to the interaction between a shield tunnel and a stratum, comprising the following steps:

[0049] Step S1: Based on the Timoshenko beam and elastic continuum model, a semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel is obtained to calculate the tunnel deformation. w 1 ( x );

[0050] Step S2, according to the deformation differential equation of the Timoshenko beam on the Winkler foundation, solve the deformation differential equation to obtain the displacement solution and calculate the tunnel deformation w 2 ( x );

[0051] Step S3, equivalently fitting the peak values ​​of the semi-analytical and semi-numerical displacement solution of the tunnel in the elastic semi-infinite space and the displacement solution of the Timoshenko beam on the Winkler foundation is performed, and the foundation reaction coefficient is calculated by an iterative method.

[0052] In step S1, the calculation assumption for obtaining the semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel is that the beam is in full contact with the foundation under the load, that is, the deformation of the beam is consistent with the surface deformation of the foundation soil.

[0053] Tunnel deformation w 1( x ) is calculated as follows:

[0054] Step S11, as Figure 1 As shown in the figure, the relationship between displacement and stress of the foundation under the surface load is solved in the Hankel transform domain, and then the inverse transform is performed, and the functional of the Timoshenko beam is obtained as follows:

[0055] ;

[0056] In the formula, E b I b is the flexural stiffness of the Timoshenko beam; θ , s Timoshenko Liang x The rotation angle and vertical displacement at ; P, M, p are the concentrated force, concentrated bending moment and distributed load acting on the beam respectively; ;

[0057] Step S12, using the Mindlin solution to calculate the load acting on the beam element, the specific calculation formula is as follows:

[0058] ;

[0059] ;

[0060] ;

[0061] In the formula, q To act on the ground ( x , y ) uniformly distributed load at the position; h is the buried depth of the tunnel top; D is the diameter of the beam; is the Poisson's ratio of the formation; A is the action range of the uniformly distributed load on the surface; x 0, y 0) represents the position of the beam element; R1 and R2 are intermediate variables; H is the depth of uniform load action;

[0062] Step S13, using the two-node Hermite beam unit interpolation function to calculate the shear force and bending moment at both ends of the beam unit and the finite element equilibrium equation of the Timoshenko beam, the specific calculation formula is as follows:

[0063] ;

[0064] ;

[0065] ;

[0066] ;

[0067] In the formula, is the node displacement and rotation matrix; is the node shear force and bending moment matrix; T is the symbol of matrix transposition; P is the intermediate variable; Q is the shear force of the Timoshenko beam; is the stiffness matrix of the formation; l 1 is the length of the beam element; is an intermediate variable; is the integration variable;

[0068] Step S14, balance equation According to the assumptions, it becomes the following form:

[0069] ;

[0070] In the formula, δ is the flexibility coefficient of the soil; Y , Z are the moment and shear force produced by the unit force respectively; w is the soil displacement at the beam-soil interface;

[0071] Step S15, solving the equilibrium equation to obtain the tunnel deformation w 1( x ).

[0072] In step S2, the tunnel is deformed w 2 ( x ) is calculated as follows:

[0073] Step S21, as Figure 2 As shown in Figure 2, the Mindlin solution is used to calculate the load acting on the beam element. , the deformation differential equation of the Timoshenko beam on the Winkler foundation is as follows:

[0074] ;

[0075] In the formula, w is the displacement of the Timoshenko beam;

[0076] Step S22: Perform Fourier series expansion;

[0077] Step S23, solving the deformation differential equation of the Timoshenko beam on the Winkler foundation to obtain the tunnel deformation w 2 ( x ).

[0078] In step S3, the foundation reaction coefficient is calculated by the iteration method, which specifically includes the following calculation steps:

[0079] Step S31, using the semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel to calculate the tunnel deformation w 1 ( x ) max ;

[0080] Step S32: Input foundation reaction coefficient k= 1000+10 *i , the tunnel deformation is calculated using the displacement solution of the Timoshenko beam on the Winkler foundation w 2 ( x ) max ,in, i is the number of iterations;

[0081] Step S33, comparing tunnel deformation w 1 ( x ) max and tunnel deformation w 2 ( x ) max, determine whether the difference between the two is less than 0.001mm, if so, they are considered equivalent and output the current k value, and consider that k The value is the required foundation reaction coefficient; if not, return to step S32 to adjust k The value continues to be calculated.

[0082] It should be noted that, in this article, the terms "include", "comprises" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, an element defined by the sentence "comprises a ..." does not exclude the existence of other identical elements in the process, method, article or device including the element.

[0083] In addition, it should be noted that the scope of the methods and systems in the embodiments of the present invention is not limited to performing functions in the order shown or discussed, and may also include performing functions in a substantially simultaneous manner or in reverse order according to the functions involved. For example, the described methods may be performed in an order different from that described, and various steps may be added, omitted, or combined. In addition, features described with reference to certain examples may be combined in other examples.

[0084] The embodiments of the present invention are described above in conjunction with the accompanying drawings, but the present invention is not limited to the above-mentioned specific implementation modes, which are merely illustrative rather than restrictive. Under the guidance of the present invention, ordinary technicians in this field can also make many forms without departing from the scope of protection of the present invention and the claims, all of which are within the protection of the present invention.

Claims

1. A method for calculating the foundation reaction coefficient applicable to the interaction between a shield tunnel and a stratum, characterized in that: The steps include: Step S1, based on the Timoshenko beam and elastic continuum model, obtain the semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel. The calculation assumption is that the beam is in full contact with the foundation under the load, and the tunnel deformation is calculated. w 1( x ), the calculation steps are as follows: Step S11, solve the relationship between the displacement and stress of the foundation under the surface load in the Hankel transform domain, and then perform an inverse transform on it, and the functional of the Timoshenko beam can be obtained as follows: ; In the formula, E b I b is the flexural stiffness of the Timoshenko beam; θ , s Timoshenko Liang x The rotation angle and vertical displacement at ; P, M, p are the concentrated force, concentrated bending moment and distributed load acting on the beam respectively; is the shear stiffness of the beam; Step S12, using the Mindlin solution to calculate the load acting on the beam element, the specific calculation formula is as follows: ； ； ； In the formula, q To act on the ground ( x , y ) uniformly distributed load at the position; h is the buried depth of the tunnel top; D is the diameter of the beam; is the Poisson's ratio of the formation; A is the action range of the uniformly distributed load on the surface; x 0, y 0) represents the position of the beam element; R1 and R2 are intermediate variables; H is the depth of uniform load action; Step S13, using the two-node Hermite beam unit interpolation function to calculate the shear force and bending moment at both ends of the beam unit and the finite element equilibrium equation of the Timoshenko beam, the specific calculation formula is as follows: ； ； ； ； In the formula, is the node displacement and rotation matrix; is the node shear force and bending moment matrix; T is the symbol of matrix transposition; P is the intermediate variable; Q is the shear force of the Timoshenko beam; is the stiffness matrix of the formation; l 1 is the length of the beam element; is an intermediate variable; is the integration variable; Step S14, balance equation According to the assumptions, it becomes the following form: ; In the formula, δ is the flexibility coefficient of the soil; Y , Z are the moment and shear force produced by the unit force respectively; w is the soil displacement at the beam-soil interface; Step S15, solving the equilibrium equation to obtain the tunnel deformation w 1 ( x ); Step S2, according to the deformation differential equation of the Timoshenko beam on the Winkler foundation, solve the deformation differential equation to obtain the displacement solution and calculate the tunnel deformation w 2( x ), the calculation steps are as follows: Step S21, use Mindlin solution to calculate the load acting on the beam element , the deformation differential equation of the Timoshenko beam on the Winkler foundation is as follows: ; In the formula, w is the displacement of the Timoshenko beam; Step S22: Perform Fourier series expansion; Step S23, solving the deformation differential equation of the Timoshenko beam on the Winkler foundation to obtain the tunnel deformation w 2 ( x ); Step S3, equivalently fitting the peak values ​​of the semi-analytical and semi-numerical displacement solution of the tunnel in the elastic semi-infinite space and the displacement solution of the Timoshenko beam on the Winkler foundation, and calculating the foundation reaction coefficient by an iterative method, which specifically includes the following calculation steps: Step S31, using the semi-analytical and semi-numerical displacement solution of the longitudinal structural response of the shield tunnel to calculate the tunnel deformation w 1 ( x ) max ; Step S32: Input foundation reaction coefficient k= 1000+10 *i , the tunnel deformation is calculated using the displacement solution of the Timoshenko beam on the Winkler foundation w 2 ( x ) max ,in, i is the number of iterations; Step S33, comparing tunnel deformation w 1 ( x ) max and tunnel deformation w 2 ( x ) max , determine whether the difference between the two is less than 0.001mm, if so, they are considered equivalent and output the current k value, and consider that k The value is the required foundation reaction coefficient; if not, return to step S32 to adjust k The value continues to be calculated.

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