Methods and systems for calculating tunnel deformation during foundation pit excavation considering inter-tunnel interactions

Abstract

This invention relates to the field of foundation pit engineering technology, specifically to a method and system for calculating tunnel deformation during foundation pit excavation considering the interaction between tunnels. The method includes: obtaining the foundation parameters of the foundation pit and the tunnel; calculating the second initial direct deformation and initial bending moment of the second tunnel based on the foundation parameters; calculating the first initial direct deformation of the first tunnel based on the foundation parameters; calculating the influence deformation of the second tunnel on the first tunnel based on the second initial direct deformation and the initial bending moment; and superimposing the first initial direct deformation and the influence deformation to obtain the total deformation of the first tunnel. This solves the problem in existing technologies where the calculation of tunnel deformation during foundation pit excavation is inaccurate because the interaction between tunnels is not considered. This invention improves the accuracy of deformation calculation by calculating the initial direct deformation of adjacent tunnels affected by the foundation pit, inferring the influence deformation of the tunnel to be deformed based on this deformation, and finally superimposing the influence deformation and the initial direct deformation to obtain the final deformation.

Description

Technical Field

[0001] This invention relates to the field of foundation pit engineering technology, specifically to a method and system for calculating tunnel deformation during foundation pit excavation that considers the interaction between tunnels. Background Technology

[0002] With the acceleration of urbanization, the scale of underground space development and construction continues to expand, and the number of foundation pit projects adjacent to existing tunnels is increasing. Foundation pit excavation inevitably disrupts the soil's stress balance, causing deformation of the surrounding soil and adjacent tunnels. If tunnel deformation exceeds limits, problems such as segment cracking, water leakage, and track deformation can easily occur, directly threatening the structural stability and operational safety of the tunnel. Therefore, accurately analyzing the stress and deformation patterns of adjacent tunnels under foundation pit excavation is of significant engineering value for the safe construction of underground projects.

[0003] Currently, scholars both domestically and internationally primarily conduct research on the impact of foundation pit excavation on adjacent tunnels through numerical simulations, model tests, and theoretical analysis. Among these, theoretical analysis has become the mainstream research method in this field due to its clear physical meaning and simple, efficient calculations. Existing theoretical models mostly focus on the working conditions of single-track tunnels, enabling preliminary calculations of tunnel deformation under foundation pit excavation, but they do not consider the interaction effects between adjacent double-track or multi-track tunnels.

[0004] In practice, subway tunnels are often laid out in a double-track configuration with small intervals between them, resulting in significant interactions between them. Ignoring the mutual influence between adjacent tunnels can lead to discrepancies between calculated tunnel deformation and actual engineering results, making it difficult to accurately predict the true deformation of adjacent tunnels during excavation. Therefore, there is an urgent need for a method to calculate tunnel deformation during excavation that considers the interactions between tunnels, in order to overcome the shortcomings of existing technologies. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a method and system for calculating tunnel deformation in foundation pit excavation that considers the interaction between tunnels, thus solving the problem that the calculation of tunnel deformation in foundation pit excavation is not accurate enough due to the failure to consider the influence of the interaction between tunnels in existing technologies.

[0006] To achieve the above objectives, the present invention provides a method for calculating the deformation of a foundation pit excavation tunnel considering the interaction between tunnels. The method includes: obtaining the foundation parameters of the foundation pit and the tunnel; calculating the second initial direct deformation and initial bending moment of the second tunnel based on the foundation parameters; calculating the first initial direct deformation of the first tunnel based on the foundation parameters; calculating the influence deformation of the second tunnel on the first tunnel based on the second initial direct deformation and the initial bending moment; and superimposing the first initial direct deformation and the influence deformation to obtain the total deformation of the first tunnel.

[0007] This invention accurately reflects the direct effect of foundation pit excavation on the second tunnel by obtaining the foundation parameters of the foundation pit and the tunnel and calculating the second initial direct deformation and initial bending moment of the second tunnel. It calculates the first initial direct deformation of the first tunnel to characterize the direct impact of foundation pit excavation on the first tunnel. Based on the second initial direct deformation and the initial bending moment, it solves for the deformation of the second tunnel on the first tunnel to reflect the interaction effect between the tunnels. The first initial direct deformation and the deformation of the influence are superimposed to obtain the total deformation of the first tunnel, which fully restores the joint effect of foundation pit unloading and tunnel interaction, and improves the accuracy of tunnel deformation calculation.

[0008] Optionally, the calculation of the second initial direct deformation and initial bending moment of the second tunnel based on the basic parameters includes: calculating the vertical additional stress of the second tunnel on the axis using the Mindlin elastic solution based on the basic parameters; extracting the diameter of the second tunnel from the basic parameters; multiplying the vertical additional stress by the diameter of the second tunnel to obtain the additional load distribution of the second tunnel; and calculating the second initial direct deformation and initial bending moment of the second tunnel based on the additional load distribution.

[0009] This invention employs the Mindlin elastic solution to accurately calculate the vertical additional stress on the axis of the second tunnel, ensuring the theoretical rigor of the stress solution. The additional load distribution is obtained by multiplying the vertical additional stress by the diameter of the second tunnel, achieving a reasonable conversion from stress to load. Based on the additional load distribution, the initial direct deformation and initial bending moment of the second tunnel are calculated, fully characterizing the direct effect of the foundation pit excavation on the second tunnel, thus improving the accuracy and reliability of the calculation of the initial deformation and internal forces of the second tunnel.

[0010] Optionally, the step of calculating the vertical additional stress of the second tunnel on the axis using the Mindlin elastic solution based on the basic parameters includes: obtaining the angle between the axis of the second tunnel and the length direction of the foundation pit, the coordinate system of the foundation pit center, and the coordinate system of the tunnel axis; determining that when the angle is not equal to zero, constructing a coordinate transformation model; transforming the tunnel axis coordinate system to the coordinate system of the foundation pit center through the coordinate transformation model; and calculating the vertical additional stress of the second tunnel on the axis using the Mindlin elastic solution based on the coordinate system of the foundation pit center and the basic parameters.

[0011] This invention obtains the angle between the tunnel and the foundation pit and two sets of coordinate systems. When the angle is non-zero, a coordinate transformation model is constructed and coordinate system one is completed. Based on the unified coordinate system, the Mindlin elastic solution is used to solve the vertical additional stress, which is adapted to the stress calculation scenario under the skew condition. This ensures the rationality and applicability of the stress distribution solution and improves the accuracy and engineering adaptability of the calculation of the additional stress of the tunnel during foundation pit excavation.

[0012] Optionally, the coordinate transformation model satisfies the following formula: The coordinate transformation model satisfies the following formula: , in, The angle between the axis of the second tunnel and the length of the foundation pit. The distance between the center of the foundation pit and the origin of the tunnel coordinate system is denoted as . Let ξ be the angle between the ξ-axis of the foundation pit coordinate system and the line segment pointing from the center of the foundation pit to the origin of the tunnel coordinate system. These are the coordinates in the tunnel axis coordinate system. These are the coordinates in the coordinate system of the foundation pit center.

[0013] This invention constructs a coordinate transformation model that includes the angle between the tunnel and the foundation pit, the distance between the coordinate origins, and the azimuth angle. This model accurately achieves a unified transformation from the tunnel axis coordinate system to the foundation pit center coordinate system, clearly quantifies the coordinate mapping relationship under different relative positions, adapts to any oblique intersection of the tunnel and the foundation pit, and improves the accuracy of additional stress calculation and engineering applicability.

[0014] Optionally, the step of calculating the influence deformation of the second tunnel on the first tunnel based on the second initial direct deformation and the initial bending moment includes: calculating the resistance force exerted by the second tunnel on the surrounding soil based on the second initial direct deformation and the initial bending moment; calculating the additional stress generated by the resistance force on the axis of the first tunnel using the Mindlin elastic solution; extracting the first tunnel diameter from the foundation parameters, multiplying the additional stress and the first tunnel diameter to obtain the additional load distribution of the first tunnel; and calculating the influence deformation of the first tunnel based on the additional load distribution.

[0015] This invention calculates the resistance of the second tunnel to the soil based on the second initial direct deformation and initial bending moment, accurately quantifies the mechanical source terms of the interaction between tunnels, uses the Mindlin elastic solution to solve for the additional stress on the axis of the first tunnel caused by the resistance, multiplies the additional stress caused by the influence of the influence by the diameter of the first tunnel to obtain the distribution of the additional load caused by the influence, calculates the deformation caused by the influence of the first tunnel based on the distribution of the additional load caused by the influence, and fully restores the effect of the second tunnel on the first tunnel, thus improving the accuracy of deformation calculation under the interaction between tunnels.

[0016] Optionally, the resistance satisfies the following formula: , This is to provide resistance to the surrounding soil exerted by the second tunnel. The coordinates of the calculated points on the tunnel axis. The distribution of additional loads on the second tunnel. The diameter of the second tunnel. The real part of the characteristic root of the Pasternak foundation beam determined based on the second tunnel. This is the imaginary part of the characteristic root of the Pasternak foundation beam determined by the second tunnel. This represents the positional variable along the tunnel axis.

[0017] This invention establishes a formula for calculating the resistance of a second tunnel based on Pasternak's foundation beam theory. It organically combines parameters such as additional tunnel load, diameter, and characteristic roots of the foundation beam to accurately quantify the mechanical effects of the second tunnel on the surrounding soil, clearly characterize the mechanical transmission mechanism of the interaction between tunnels, and improve the accuracy and reliability of the analysis of the interaction between tunnels.

[0018] Optionally, the additional stress that influences the stress satisfies the following formula: , in, Additional stress is applied to the impact of the second tunnel on the first tunnel. The coordinates of the calculated points on the tunnel axis. The diameter of the second tunnel. This is the length of the second tunnel. For resistance, Let the coordinate axes be along the axis of the second tunnel. The horizontal coordinate axis is perpendicular to the axis of the second tunnel. Poisson's ratio of soil The burial depth of the first tunnel axis, The burial depth of the second tunnel axis, , Both are distance functions related to coordinate position. This refers to the depth of the foundation pit excavation.

[0019] This invention establishes a formula for calculating the additional stress on the first tunnel caused by the second tunnel, organically integrating the resistance of the second tunnel, tunnel geometric parameters, soil parameters, and spatial distance function. This accurately quantifies the mechanical transmission effect between tunnels, fully restores the distribution law of the additional stress on the first tunnel caused by the second tunnel, and improves the accuracy and reliability of the calculation of additional stress under the interaction between tunnels.

[0020] Optionally, the step of calculating the impact deformation of the first tunnel based on the distribution of the impact additional load includes: treating the first tunnel as an infinitely long beam on the Pasternak foundation model, constructing a vertical deformation control equation using the distribution of the impact additional load, and solving the vertical deformation control equation to obtain the impact deformation.

[0021] This invention equates the first tunnel to an infinitely long beam on a Pasternak foundation model, constructs a vertical deformation control equation by utilizing the distribution of the influencing additional load, accurately characterizes the interaction between the tunnel and the soil, solves the control equation to obtain the deformation of the first tunnel, and fully restores the deformation influence of the second tunnel on the first tunnel, thereby improving the accuracy and theoretical rigor of deformation calculation under the interaction between tunnels.

[0022] Optionally, the step of treating the first tunnel as an infinitely long beam on the Pasternak foundation model and constructing the vertical deformation control equation using the distribution of the influencing additional load includes: extracting the soil physical and mechanical parameters, the flexural stiffness of the first tunnel, and the diameter of the first tunnel from the foundation parameters; calculating the elastic coefficient and shear coefficient of the first tunnel foundation using the soil physical and mechanical parameters and the diameter of the first tunnel; constructing the vertical deformation control differential equation of the first tunnel using the flexural stiffness, elastic coefficient, and shear coefficient of the first tunnel foundation; and substituting the distribution of the influencing additional load into the vertical deformation control differential equation to obtain the vertical deformation control equation of the first tunnel.

[0023] This invention extracts the physical and mechanical parameters of the soil, the bending stiffness and diameter of the first tunnel, calculates the elastic coefficient and shear coefficient of the first tunnel foundation to accurately characterize the mechanical properties of the foundation, constructs the vertical deformation control differential equation of the first tunnel and substitutes it with the distribution of the influencing additional load, and establishes a tunnel deformation mechanics model that fits the reality, thereby improving the theoretical rigor and accuracy of the deformation calculation of the first tunnel.

[0024] Another aspect of the present invention provides a foundation pit excavation tunnel deformation calculation system considering the interaction between tunnels, comprising: a processor, an input device, an output device, and a memory, wherein the processor, the input device, the output device, and the memory are interconnected, wherein the memory is used to store a computer program, the computer program including program instructions, and the processor is configured to call the program instructions to execute the foundation pit excavation tunnel deformation calculation method considering the interaction between tunnels as described in any of the preceding aspects of the present invention.

[0025] The present invention provides a foundation pit excavation tunnel deformation calculation system that considers the interaction between tunnels. It has a compact structure, stable performance, high integration and simple configuration. It can stably execute the foundation pit excavation tunnel deformation calculation method considering the interaction between tunnels provided in the foregoing aspect of the present invention, further improving the overall applicability and practical application capability of the present invention. Attached Figure Description

[0026] Figure 1 This is a flowchart of a method for calculating tunnel deformation during foundation pit excavation that considers the interaction between tunnels, according to an embodiment of the present invention. Figure 2 This is a schematic diagram showing the planar positional relationship between the foundation pit and the underlying tunnel according to an embodiment of the present invention. Figure 3 This is a schematic diagram of the Pasternak foundation model structure according to an embodiment of the present invention; Figure 4 This is a schematic diagram of a tunnel interaction structure according to an embodiment of the present invention; Figure 5 This is a schematic diagram of the planar positional relationship of the twin-track tunnel according to an embodiment of the present invention; Figure 6 This is a comparison diagram of the vertical deformation of the left tunnel in an embodiment of the present invention; Figure 7 This is a comparison diagram of the vertical deformation of the right tunnel line in an embodiment of the present invention; Figure 8 This is a diagram showing the tunnel deformation curve caused by inter-tunnel interaction according to an embodiment of the present invention. Figure 9 This is a schematic diagram of the structure of a foundation pit excavation tunnel deformation calculation system that considers the interaction between tunnels, according to an embodiment of the present invention. Detailed Implementation

[0027] Specific embodiments of the present invention will now be described in detail. It should be noted that the embodiments described herein are for illustrative purposes only and are not intended to limit the invention. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the invention. However, it will be apparent to those skilled in the art that these specific details are not necessary to practice the invention. In other instances, well-known circuits, software, or methods have not been specifically described to avoid obscuring the invention.

[0028] Throughout this specification, references to "an embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in connection with that embodiment or example is included in at least one embodiment of the invention. Therefore, the phrases "in an embodiment," "in an embodiment," "an example," or "an example" appearing in various places throughout the specification do not necessarily refer to the same embodiment or example. Furthermore, specific features, structures, or characteristics can be combined in one or more embodiments or examples in any suitable combination and / or sub-combination. Moreover, those skilled in the art will understand that the illustrations provided herein are for illustrative purposes and are not necessarily drawn to scale.

[0029] Please see Figure 1 In order to solve the problems in the prior art, in an alternative embodiment, such as Figure 1 The method for calculating tunnel deformation during foundation pit excavation, considering the interaction between tunnels, includes the following steps: Step S1: Obtain the basic parameters of the foundation pit and tunnel, and calculate the second initial direct deformation and initial bending moment of the second tunnel based on the basic parameters.

[0030] In this embodiment, the basic parameters include the geometric parameters of the foundation pit, the physical and mechanical parameters of the soil, the structural parameters of the first and second tunnels, and the relative position parameters of the tunnels. The geometric parameters of the foundation pit include the excavation width, length, and depth. The physical and mechanical parameters of the soil include the actual unit weight of each soil layer, the thickness of each soil layer, the elastic modulus of the soil, the Poisson's ratio of the soil, and the thickness of the shear layer. The structural parameters of the tunnels include the diameter of the first tunnel, the diameter of the second tunnel, the flexural stiffness of the first tunnel, the flexural stiffness of the second tunnel, the burial depth of the first tunnel axis, the burial depth of the second tunnel axis, and the tunnel length. The relative position parameters of the tunnels include the angle between the second tunnel axis and the length direction of the foundation pit, the distance between the center of the foundation pit and the origin of the tunnel coordinate system, and the angle between the ξ-axis of the foundation pit coordinate system and the line segment pointing from the center of the foundation pit to the origin of the tunnel coordinate system.

[0031] The calculation of the second initial direct deformation and initial bending moment of the second tunnel based on the aforementioned basic parameters specifically includes the following sub-steps: Step S101: Calculate the vertical additional stress on the axis of the second tunnel using the Mindlin elastic solution based on the basic parameters.

[0032] The calculation of the vertical additional stress on the axis of the second tunnel using the Mindlin elastic solution based on the aforementioned basic parameters specifically includes the following sub-steps: Step S10101: Obtain the angle between the axis of the second tunnel and the length direction of the foundation pit, the coordinate system of the foundation pit center, and the coordinate system of the tunnel axis.

[0033] In this embodiment, a coordinate system for the foundation pit center is established with the center of the foundation pit as the origin, the length direction of the foundation pit as the horizontal axis, and the width direction of the foundation pit as the vertical axis. A coordinate system for the tunnel axis is established with a point on the axis of the second tunnel as the origin, the direction of the second tunnel axis as the horizontal axis, and the horizontal direction perpendicular to the second tunnel axis as the vertical axis. The angle between the axis of the second tunnel and the length direction of the foundation pit is obtained according to the relative layout positions of the foundation pit and the second tunnel.

[0034] Step S10102: When the included angle is not equal to zero, construct a coordinate transformation model.

[0035] The coordinate transformation model satisfies the following formula (Formula 3): , in, The angle between the axis of the second tunnel and the length of the foundation pit. The distance between the center of the foundation pit and the origin of the tunnel coordinate system is denoted as . For the foundation pit coordinate system The angle between the axis and the line segment pointing from the center of the excavation pit to the origin of the tunnel coordinate system. These are the coordinates in the tunnel axis coordinate system, where Let be the longitudinal coordinates along the axis of the second tunnel. The horizontal coordinate is perpendicular to the axis of the second tunnel. These are the coordinates in the coordinate system of the foundation pit center.

[0036] like Figure 2 As shown, with the center of the foundation pit as the origin. Along the length and width of the foundation pit, set shaft and Axis, Establish Coordinate system. Furthermore, a point on the tunnel axis is used as... Establish the point with the tunnel axis as the x-axis and the perpendicular axis as the y-axis. coordinate system They are respectively axis and x-axis and line segment The included angle; For line segments The length.

[0037] Step S10103: Transform the tunnel axis coordinate system to the pit center coordinate system using a coordinate transformation model.

[0038] In this embodiment, based on the constructed coordinate transformation model, the coordinates of each calculation point on the second tunnel axis in the tunnel axis coordinate system are uniformly converted into the corresponding coordinates in the foundation pit center coordinate system, realizing the coordinate conversion between the two coordinate systems and providing a unified coordinate reference for the subsequent calculation of vertical additional stress using the Mindlin elastic solution.

[0039] Step S10104: Based on the coordinate system of the foundation pit center, calculate the vertical additional stress of the second tunnel on the axis using the Mindlin elastic solution according to the foundation parameters.

[0040] In this embodiment, based on the coordinate system of the foundation pit center, combined with the foundation pit geometric parameters (excavation width) in the foundation parameters... ,length ,depth ), soil physical and mechanical parameters (Poisson's ratio of soil) Actual unit weight of each soil layer With layer thickness The total unloading of the foundation pit excavation was calculated. ) and the axial depth of the second tunnel The Mindlin elastic solution was used to calculate the vertical additional stress generated on the second tunnel axis by the excavation unloading of the foundation pit. (Formula 1). Excavation of the foundation pit will cause stress unloading of the existing strata, resulting in additional stress on the underlying tunnel. When calculating the additional stress on the tunnel, it is assumed that the soil involved in the calculation is considered as an isotropic elastic body. The formula for calculating this vertical additional stress is: , In the formula, , These are the two axes of a coordinate system with the center of the foundation pit as the origin. The coordinates are the longitudinal coordinates along the tunnel axis; , , in Let be the coordinates of any point on the tunnel axis in the coordinate system at the center of the excavation pit after coordinate transformation. The distribution of vertical additional stress at each location along the second tunnel axis can be obtained using this integral formula.

[0041] Step S102: Extract the second tunnel diameter from the basic parameters, and multiply the vertical additional stress by the second tunnel diameter to obtain the additional load distribution of the second tunnel.

[0042] In this embodiment, the second tunnel diameter is extracted from the basic parameters. The vertical additional stress on the second tunnel axis With the diameter of the second tunnel Multiplying these yields the distribution of the additional load acting on the second tunnel. Its expression is: , In the formula, The additional vertical stress generated on the second tunnel axis to relieve the load during the excavation of the foundation pit. This is the diameter of the second tunnel. The additional load distribution... This is the load input required to calculate the initial direct deformation and initial bending moment of the second tunnel in subsequent steps.

[0043] Step S103: Calculate the second initial direct deformation and initial bending moment of the second tunnel based on the additional load distribution.

[0044] In this embodiment, the additional load distribution of the second tunnel is used as input. Based on the Pasternak foundation beam model, the second initial direct deformation and initial bending moment of the second tunnel under the unloading action of the foundation pit excavation are calculated. The calculation principle is as follows: like Figure 3As shown, the Pasternak foundation model adds a shear layer to the traditional Winkler foundation model, overcoming the Winkler foundation's inability to consider soil continuity, making it superior to the Winkler foundation. According to Tanahashi's research, the tunnel under additional stress... The governing differential equation under the action of (Equation 4) is: , In the formula: For the tunnel's bending stiffness; This is the vertical deformation function of the tunnel; This is the diameter of the tunnel. (Formula 5), (Formula 6) represents the elastic coefficient and shear coefficient of the foundation, respectively, and the calculation formula is as follows: , , In the formula: The elastic modulus of the soil; Let the thickness of the soil shear layer be taken as... Perform calculations. for The expansion of .

[0045] Tunnels under additional stress The governing differential equations under the influence of the action are quite complex, and in order to obtain The equation can be taken as formula 4 for tunnels under additional stress. The homogeneous equation of the governing differential equation under the action (Equation 7) is: , Based on the derivation process of Ou Xuefeng et al., the general solution of Formula 7 can be obtained as (Formula 8): , In the formula: , , , These are coefficients to be determined; ; .

[0046] To study the impact of foundation pit excavation on the underlying tunnel, it is necessary to first solve for the concentrated loads on the tunnel on the Pasternak foundation. The solution when applied. As the distance approaches infinity, the tunnel will no longer be affected by the foundation pit construction; therefore, the tunnel deformation will occur at infinity. Because the tunnel cross-section is at No rotation occurred at that point, and it remains perpendicular to the neutral axis; therefore, at... The boundary conditions at this point still satisfy the conditions of zero rotation angle and shear force equilibrium. The boundary conditions of the tunnel at this point can be summarized as (Equation 9): , Substituting Formula 9 into Formula 8, the concentrated load can be obtained. The control equation for tunnel deformation under the action is (Formula 10): , Therefore, concentrated loads can be further obtained. Tunnel bending moment under action Expression (Formula 11): , Assuming that under the influence of foundation pit excavation, at any point on the tunnel The additional load acting on the surface is Then, according to Formulas 10 and 11, the load at any point on the tunnel axis can be calculated. Vertical deformation and bending moment : , , Vertical deformation within the distribution range of the additional load on the tunnel (Formula 14) and bending moment By integrating (Formula 15), the vertical deformation and bending moment of the tunnel caused by the excavation of the foundation pit can be obtained: , , Step S2: Calculate the first initial direct deformation of the first tunnel based on the basic parameters.

[0047] In this embodiment, based on the aforementioned basic parameters, the vertical additional stress generated on the first tunnel axis by the excavation unloading is calculated using the Mindlin elastic solution. This vertical additional stress is multiplied by the diameter of the first tunnel to obtain the additional load distribution of the first tunnel. Then, using the Pasternak foundation beam model as the calculation basis, the additional load distribution is integrally solved by combining the bending stiffness of the first tunnel, the elastic coefficient of the foundation, and the shear coefficient to obtain the first initial direct deformation of the first tunnel under the unloading action of the excavation.

[0048] Step S3: Calculate the deformation of the second tunnel on the first tunnel based on the second initial direct deformation and the initial bending moment.

[0049] The calculation of the impact deformation of the second tunnel on the first tunnel based on the second initial direct deformation and the initial bending moment specifically includes the following sub-steps: Step S301: Calculate the resistance exerted by the second tunnel on the surrounding soil based on the second initial direct deformation and the initial bending moment.

[0050] like Figure 4 As shown, when multiple adjacent tunnels exist below the excavation pit, the excavation and unloading disturb the surrounding soil. To resist deformation, the tunnels also exert a resistance force on the surrounding soil, hindering soil movement and thus affecting the stress and deformation of the adjacent tunnels. Taking a twin-tunnel case as an example, the influence of the second tunnel on the vertical deformation of the first tunnel is analyzed: When the second tunnel deforms under the excavation and unloading action, it exerts a resistance force on the surrounding soil that is equal in magnitude and opposite in direction to the internal forces of the tunnel. This reaction force changes the stress field of the soil, causing the first tunnel to experience additional stress caused by the second tunnel, ultimately affecting its deformation response.

[0051] in, Figure 4 (a) in the figure is a schematic diagram of the overall calculation model of the double-track tunnel, which intuitively shows the relative position and geometric parameters of the lower tunnel 1 and tunnel 2 under the condition of excavation and unloading of the foundation pit, and provides a basic model for the analysis of the interaction between the tunnels; Figure 4 (b) shows the stress and deformation of the second tunnel (tunnel 2). Under the unloading action of the excavation, tunnel 2 undergoes upward deformation. To resist this deformation, tunnel 2 applies a downward reaction force to the surrounding soil. This reaction force is the core mechanical source term that triggers the interaction between tunnels; Figure 4 (c) in the figure is a schematic diagram of the additional stress distribution of the first tunnel (tunnel 1). The black curve is the additional stress considering only the unloading effect of the foundation pit excavation. The blue curve is the additional stress caused by the reaction force of the second tunnel. The red curve is the total additional stress after superposition. It clearly shows the influence law of the second tunnel on the stress state of the first tunnel.

[0052] Combining formulas 4, 11, and 15, the resistance of the second tunnel to the soil under the influence of foundation pit excavation can be obtained. for: , In the formula: , These represent the vertical deformation and bending moment of the second tunnel under excavation unloading conditions only. For the bending stiffness of the second tunnel, and Please refer to formula 8. and The calculation method.

[0053] Therefore, resistance satisfies the following formula: , This is to provide resistance to the surrounding soil exerted by the second tunnel. Here is the longitudinal coordinate along the tunnel axis. The distribution of additional loads on the second tunnel. The diameter of the second tunnel. The real part of the characteristic root of the Pasternak foundation beam determined based on the second tunnel. This is the imaginary part of the characteristic root of the Pasternak foundation beam determined by the second tunnel. This represents the positional variable along the tunnel axis.

[0054] Step S302: The additional stress caused by the resistance force on the first tunnel axis is calculated using the Mindlin elastic solution.

[0055] The additional stress caused by the influence satisfies the following formula: , in, Additional stress is applied to the impact of the second tunnel on the first tunnel. The coordinates of the calculated points on the tunnel axis. The diameter of the second tunnel. This is the length of the second tunnel. For resistance, Let the coordinate axes be along the axis of the second tunnel. The horizontal coordinate axis is perpendicular to the axis of the second tunnel. Poisson's ratio of soil The burial depth of the first tunnel axis, The burial depth of the second tunnel axis, , Both are distance functions related to coordinate position. This refers to the depth of the foundation pit excavation.

[0056] , (Formula 18) is: , In the formula: , They are respectively The longitudinal and transverse coordinates of a point on the first tunnel in the coordinate system. For example... Figure 5 As shown, the origin is the center of the second tunnel. Along the axis of the second tunnel, perpendicular to the axis direction, shaft and Axis, Establish Coordinate system. Furthermore, a point on the first tunnel axis is used as... Point, the direction of the first tunnel axis is The axis, perpendicular to the axis direction is Axis establishment Coordinate system: From the positional relationship between the two coordinate systems, the coordinate transformation formula (Formula 19) can be obtained: , In the formula: , They are respectively shaft and Axis and line segment The included angle; For line segments The length.

[0057] Step S303: Extract the first tunnel diameter from the basic parameters, and multiply the influencing additional stress and the first tunnel diameter to obtain the influencing additional load distribution of the first tunnel.

[0058] In this embodiment, the diameter of the first tunnel is extracted from the obtained foundation parameters of the pit and the tunnel. The additional stress caused by the second tunnel on the first tunnel is multiplied by the diameter of the first tunnel to obtain the distribution of the additional load caused by the interaction between the tunnels and acting on the first tunnel, which provides load input for the subsequent solution of the deformation caused by the first tunnel.

[0059] Step S304: Calculate the impact deformation of the first tunnel based on the distribution of the impact additional load.

[0060] The calculation of the impact deformation of the first tunnel based on the distribution of the additional load specifically includes the following sub-steps: Step S30401: Treat the first tunnel as an infinitely long beam on the Pasternak foundation model, and construct the vertical deformation control equation using the influence of the additional load distribution.

[0061] Specifically, the first tunnel is considered as an infinitely long beam on the Pasternak foundation model, and the vertical deformation control equations are constructed using the aforementioned influence on the additional load distribution, including: Step S3040101: Extract the soil physical and mechanical parameters, the first tunnel bending stiffness, and the first tunnel diameter from the basic parameters.

[0062] In this embodiment, soil physical and mechanical parameters such as the elastic modulus and Poisson's ratio are extracted from the obtained foundation parameters of the pit and tunnel. At the same time, the bending stiffness and diameter of the first tunnel are extracted to provide parameter support for subsequent calculation of the subgrade coefficient and construction of deformation control equations.

[0063] Step S3040102: Calculate the elastic coefficient and shear coefficient of the first tunnel foundation using the soil physical and mechanical parameters and the diameter of the first tunnel.

[0064] In this embodiment, based on the extracted soil elastic modulus, Poisson's ratio and other soil physical and mechanical parameters and the diameter of the first tunnel, the standard calculation formula of the Pasternak foundation model is used to calculate the foundation elastic coefficient (Formula 5) and foundation shear coefficient (Formula 6) corresponding to the first tunnel.

[0065] Step S3040103: Construct the vertical deformation control differential equation of the first tunnel using the first tunnel bending stiffness, the first tunnel foundation elastic coefficient, and the first tunnel foundation shear coefficient.

[0066] In this embodiment, the first tunnel is regarded as an infinitely long beam on the Pasternak foundation model. Combining the determined bending stiffness of the first tunnel, the elastic coefficient of the first tunnel foundation, the shear coefficient of the first tunnel foundation, and the diameter of the first tunnel, the vertical deformation control differential equation (Formula 4) corresponding to the first tunnel is constructed based on the foundation beam mechanics control theory. This provides the core equation basis for the subsequent substitution of the influence of the additional load distribution to solve the tunnel deformation.

[0067] Step S3040104: Substitute the distribution of the influencing additional load into the vertical deformation control differential equation to obtain the vertical deformation control equation of the first tunnel.

[0068] In this embodiment, the distribution of the additional loads affecting the first tunnel is taken as a non-homogeneous load term and substituted into the already constructed differential equation for the vertical deformation control of the first tunnel. The load substitution and parameter matching of the equation are completed, and finally the control equation (Formula 4) suitable for solving the vertical deformation of the first tunnel under the interaction between tunnels is obtained.

[0069] Step S30402: Solve the vertical deformation control equation to obtain the deformation influencing factors.

[0070] In this embodiment, when solving the vertical deformation control equation (Formula 4) of the first tunnel, the corresponding homogeneous equation (Formula 7) is solved first to obtain the homogeneous general solution (Formula 8). Then, the non-homogeneous particular solution (Formula 10) is determined by combining the influence of the interaction between tunnels and the distribution of additional loads. The general solution and the particular solution are superimposed to obtain the complete solution of the equation. Subsequently, the boundary conditions of the infinitely long beam (Formula 9) are substituted to determine the undetermined coefficients. Finally, the vertical deformation of the first tunnel shown in Formula 14 is obtained by integral solution.

[0071] Step S4: The first initial direct deformation is superimposed with the influence deformation to obtain the first total tunnel deformation.

[0072] By superimposing the deformations of the first tunnel caused by the unloading effect of the foundation pit excavation and the influence of the adjacent second tunnel, the total vertical deformation of the first tunnel caused by the foundation pit excavation, considering the influence of the adjacent second tunnel, can be obtained. : , In the formula: This refers to the vertical deformation of the first tunnel under the unloading action of excavation only. The deformation is due to the influence of the second tunnel on the first tunnel.

[0073] Similarly, when analyzing the total vertical deformation of the second tunnel, the influence of the resistance force formed by the first tunnel during the process of resisting stratum deformation is also considered. The total vertical deformation of the second tunnel can be calculated using the same method as the calculation of the total vertical deformation of the first tunnel. Furthermore, when there are multiple underground tunnels and pipelines around the foundation pit project, the same method can be used to consider the interaction between multiple lines. Due to space limitations, this will not be elaborated further.

[0074] To verify the correctness of the proposed method, a case study of a subway station ventilation shaft and entrance / exit pit excavation project was conducted. The excavation depth of the ventilation shaft and entrance / exit pit is 9m, with a north-south length of 56.55m and an east-west length of 67.65m. A north-south oriented double-track shield tunnel exists directly beneath the pit, parallel to the short side of the pit, with the left tunnel located below the center of the pit. The distance between the left and right tunnel axes is approximately 22.3m, with an axial burial depth of 14.75m for both. The outer diameter is 6m, the wall thickness is 0.3m, and the tunnel segments are constructed of C50 concrete with an elastic modulus of 34.5GPa. The effective longitudinal stiffness is... Therefore, the bending stiffness of the tunnel is taken as... The soil surrounding the tunnel is mainly mudstone and silty clay. In the actual calculations, the elastic modulus of the soil was taken as [value missing]. Poisson's ratio .

[0075] Figure 6 , Figure 7 The figures show the vertical deformation values ​​of the left and right shield tunnels caused by the excavation of the foundation pit, calculated by the method proposed in this paper and considering the interaction between tunnels. These values ​​are compared with the results obtained by ignoring the interaction between tunnels and by on-site measurements. Figure 6 , Figure 7 It is evident that the calculated results, neglecting the interaction between tunnels, are significantly larger than the measured results, while the deformations of the left and right tunnels calculated by the method presented in this paper are closer to the measured values. This is because the method in this paper considers the mutual influence between the left and right tunnels, which better reflects the coordinated response of tunnels and soil in actual engineering. Figure 8 The tunnel deformation curve caused by the interaction between tunnels, from Figure 8It can be seen that within the excavation length of the foundation pit, the tunnel settlement values ​​caused by the interaction between the tunnels are both greater than 2 mm for both the left and right tunnels, which illustrates the necessity of considering the mutual influence between the tunnels. Overall, the calculation results of the method in this paper agree well with the measured values, proving the accuracy and rationality of the method. Furthermore, this method demonstrates that it can more accurately predict the vertical deformation of adjacent double-track tunnels caused by foundation pit excavation compared to methods that ignore the mutual influence between tunnels.

[0076] It should be noted that in all the attached diagrams, Tunnel 1 refers to the first tunnel and Tunnel 2 refers to the second tunnel.

[0077] like Figure 9 As shown, in another aspect, the present invention also provides a foundation pit excavation tunnel deformation calculation system considering the interaction between tunnels, including: a processor, an input device, an output device, and a memory, wherein the processor, the input device, the output device, and the memory are interconnected, wherein the memory is used to store a computer program, the computer program includes program instructions, and the processor is configured to call the program instructions to execute the relevant steps of the relevant embodiments of the foundation pit excavation tunnel deformation calculation method considering the interaction between tunnels of the present invention.

[0078] This invention provides a tunnel deformation calculation system for foundation pit excavation that considers the interactions between tunnels. The functional components can be integrated into a single processing unit, exist as separate physical entities, or be integrated into a single unit. The integrated components can be implemented in hardware or as software functions.

[0079] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features. These modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A method for calculating tunnel deformation during foundation pit excavation considering the interaction between tunnels, characterized in that, The method includes: Obtain the basic parameters of the foundation pit and tunnel, and calculate the second initial direct deformation and initial bending moment of the second tunnel based on the basic parameters; Calculate the first initial direct deformation of the first tunnel based on the aforementioned basic parameters; The deformation of the second tunnel on the first tunnel is calculated based on the second initial direct deformation and the initial bending moment. The first initial direct deformation is superimposed with the influence deformation to obtain the first total tunnel deformation.

2. The method for calculating tunnel deformation during foundation pit excavation considering inter-tunnel interaction as described in claim 1, characterized in that, The calculation of the second initial direct deformation and initial bending moment of the second tunnel based on the aforementioned basic parameters includes: The vertical additional stress on the axis of the second tunnel is calculated using the Mindlin elastic solution based on the aforementioned basic parameters. The second tunnel diameter is extracted from the basic parameters, and the additional vertical stress is multiplied by the second tunnel diameter to obtain the additional load distribution of the second tunnel; The second initial direct deformation and initial bending moment of the second tunnel are calculated based on the additional load distribution.

3. The method for calculating tunnel deformation during foundation pit excavation considering inter-tunnel interaction as described in claim 2, characterized in that, The calculation of the vertical additional stress on the axis of the second tunnel using the Mindlin elastic solution based on the aforementioned basic parameters includes: Obtain the angle between the axis of the second tunnel and the length direction of the foundation pit, the coordinate system of the foundation pit center, and the coordinate system of the tunnel axis; When the included angle is not equal to zero, construct a coordinate transformation model; The coordinate system of the tunnel axis is transformed to the coordinate system of the foundation pit center using a coordinate transformation model. Based on the coordinate system of the foundation pit center, the vertical additional stress of the second tunnel on the axis is calculated using the Mindlin elastic solution according to the foundation parameters.

4. The method for calculating tunnel deformation in foundation pit excavation considering inter-tunnel interaction as described in claim 3, characterized in that, The coordinate transformation model satisfies the following formula: The coordinate transformation model satisfies the following formula: ， in, The angle between the axis of the second tunnel and the length of the foundation pit. The distance between the center of the foundation pit and the origin of the tunnel coordinate system is denoted as . Let ξ be the angle between the ξ-axis of the foundation pit coordinate system and the line segment pointing from the center of the foundation pit to the origin of the tunnel coordinate system. These are the coordinates in the tunnel axis coordinate system. These are the coordinates in the coordinate system of the foundation pit center.

5. The method for calculating tunnel deformation during foundation pit excavation considering inter-tunnel interaction as described in claim 1, characterized in that, The deformation calculation based on the second initial direct deformation and the initial bending moment to determine the influence of the second tunnel on the first tunnel includes: The resistance exerted by the second tunnel on the surrounding soil is calculated based on the second initial direct deformation and the initial bending moment; The additional stress resulting from the resistance force on the first tunnel axis was calculated using the Mindlin elastic solution. The first tunnel diameter is extracted from the basic parameters, and the influence additional stress is multiplied by the first tunnel diameter to obtain the influence additional load distribution of the first tunnel; The deformation of the first tunnel is calculated based on the distribution of the additional load.

6. The method for calculating tunnel deformation in foundation pit excavation considering inter-tunnel interaction as described in claim 5, characterized in that, The resistance satisfies the following formula: ， This is to provide resistance to the surrounding soil exerted by the second tunnel. The coordinates of the calculated points on the tunnel axis. The distribution of additional loads on the second tunnel. The diameter of the second tunnel. The real part of the characteristic root of the Pasternak foundation beam determined based on the second tunnel. This is the imaginary part of the characteristic root of the Pasternak foundation beam determined by the second tunnel. This represents the positional variable along the tunnel axis.

7. The method for calculating tunnel deformation in foundation pit excavation considering inter-tunnel interaction as described in claim 5, characterized in that, The additional stress caused by the influence satisfies the following formula: ， in, Additional stress is applied to the impact of the second tunnel on the first tunnel. The coordinates of the calculated points on the tunnel axis. The diameter of the second tunnel. This is the length of the second tunnel. For resistance, Let the coordinate axes be along the axis of the second tunnel. The horizontal coordinate axis is perpendicular to the axis of the second tunnel. Poisson's ratio of soil The burial depth of the first tunnel axis, The burial depth of the second tunnel axis. , Both are distance functions related to coordinate position. This refers to the depth of the foundation pit excavation.

8. The method for calculating tunnel deformation in foundation pit excavation considering inter-tunnel interaction as described in claim 5, characterized in that, The calculation of the impact deformation of the first tunnel based on the distribution of the impacting additional load includes: The first tunnel is treated as an infinitely long beam on the Pasternak foundation model, and the vertical deformation control equation is constructed using the distribution of the influence additional load. The vertical deformation control equation is solved to obtain the factors affecting the deformation.

9. The method for calculating tunnel deformation in foundation pit excavation considering inter-tunnel interactions according to claim 8, characterized in that, The method of treating the first tunnel as an infinitely long beam on the Pasternak foundation model and constructing the vertical deformation control equation using the influence of the additional load distribution includes: The physical and mechanical parameters of the soil, the bending stiffness of the first tunnel, and the diameter of the first tunnel are extracted from the basic parameters. The elastic coefficient and shear coefficient of the first tunnel foundation are calculated using the soil physical and mechanical parameters and the diameter of the first tunnel. The vertical deformation control differential equation of the first tunnel is constructed using the bending stiffness of the first tunnel, the elastic coefficient of the foundation of the first tunnel, and the shear coefficient of the foundation of the first tunnel. Substituting the distribution of the influencing additional load into the vertical deformation control differential equation, the vertical deformation control equation of the first tunnel is obtained.

10. A system for calculating the deformation of excavated tunnels in a foundation pit, considering the interactions between tunnels, characterized in that, include: The system includes a processor, an input device, an output device, and a memory, all interconnected. The memory stores a computer program comprising program instructions. The processor is configured to invoke the program instructions to execute the method for calculating tunnel deformation in foundation pit excavation considering inter-tunnel interactions as described in any one of claims 1 to 9.

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