Method for calculating influence of expanding excavation on coplanar and parallel adjacent tunnels

Abstract

The invention provides a method for calculating the influence of expanding excavation on coplanar and parallel adjacent tunnels, which comprises the following steps of: 1, constructing an analysis model of a target tunnel, namely the coplanar and parallel adjacent tunnels, under the action of expanding excavation, and randomly selecting a tunnel unit with the length of dx in the target tunnel to carry out stress analysis and deformation analysis; 2, a foundation reaction coefficient is designed and used for calculating the stress and deformation conditions of the tunnel unit; 3, listing a fourth-order differential equation according to the analysis condition in the step 1 and the coefficient designed in the step 2, and calculating the deflection, bending moment and shearing force of the target tunnel; and 4, constructing a difference equation and solving the fourth-order differential equation in the step 3 to obtain the deflection, the bending moment and the shearing force of the target tunnel so as to complete the calculation of the influence of expanding excavation on the adjacent tunnels which are coplanar and parallel.

Description

Technical Field

[0001] The present invention relates to the field of tunnel construction engineering, namely a calculation method for the influence of excavation on a tunnel, and particularly a calculation method for the influence of excavation on adjacent tunnels that are in the same plane and parallel. Background Art

[0002] The information provided in this part is only background information related to the present disclosure, and it does not necessarily represent the prior art.

[0003] Currently, there is no corresponding theoretical calculation method for the influence of displacement caused by excavation on adjacent lateral tunnels. Excavation often causes huge losses to industrial and agricultural production and people's lives and property, and in some cases, even catastrophic disasters. As an important transportation passage, tunnels are inevitably affected by nearby excavations, and there is a lack of effective methods in the prior art for calculating the deflection, bending moment, and shear force of tunnels affected by excavation.

[0004] It should be noted that the information disclosed in the above background art section is only used to enhance the understanding of the background of the present disclosure, and thus may include information that does not constitute the prior art known to those of ordinary skill in the art. Summary of the Invention

[0005] Object of the Invention: The technical problem to be solved by the present invention is to provide a calculation method for the influence of excavation on adjacent tunnels that are in the same plane and parallel, aiming at the deficiencies of the prior art.

[0006] To solve the above technical problem, the present invention discloses a calculation method for the influence of excavation on adjacent tunnels that are in the same plane and parallel, including the following steps:

[0007] Step 1, construct an analysis model of the target tunnel, i.e., adjacent tunnels that are in the same plane and parallel, under the action of excavation. Select a tunnel element with a length of dx in the target tunnel for force analysis and deformation analysis;

[0008] Step 2, design a foundation reaction coefficient for calculating the force and deformation conditions of the tunnel element;

[0009] Step 3, list a fourth-order differential equation according to the analysis situation in Step 1 and the coefficient designed in Step 2 for calculating the deflection, bending moment, and shear force of the target tunnel;

[0010] Step 4, construct a difference equation and solve the fourth-order differential equation described in Step 3 to obtain the deflection, bending moment, and shear force of the target tunnel, and complete the calculation of the influence of excavation on adjacent tunnels that are in the same plane and parallel.

[0011] Further, the construction of the analysis model of the adjacent tunnel under the action of excavation in Step 1 includes:

[0012] Step 1-1: Arbitrarily select a tunnel element with a length of dx in the target tunnel, i.e., the adjacent tunnel.

[0013] Step 1-2: Conduct a force analysis and deformation analysis on the tunnel element. According to the force balance, we get:

[0014] F Q + k1s(x)dx = k1u(x)dx + F Q + dF Q

[0015] where F Q is the section shear force of the tunnel element, dF Q is the shear force increment of the tunnel element, u(x) is the lateral displacement of the tunnel element, k1 is the coefficient of subgrade reaction, and s(x) is the displacement caused by the excavation.

[0016] Furthermore, the displacement s(x) caused by the excavation in Step 1-2 is calculated as follows:

[0017]

[0018] where s max is the maximum horizontal displacement caused by the excavation, x is the horizontal coordinate, and d is the distance from the inflection point of the lateral displacement to the midpoint. The above data are obtained through actual measurement.

[0019] Furthermore, the designed coefficient of subgrade reaction in Step 2 includes:

[0020] k1 = k0 * k

[0021] where k0 represents the coefficient of earth pressure at rest, and k represents the empirical estimation coefficient.

[0022] Furthermore, the empirical estimation coefficient k in Step 2 is expressed as follows:

[0023]

[0024] where B is the width of the target tunnel, E s and v s are the Young's modulus and Poisson's ratio of the soil mass respectively, and E p I p is the flexural stiffness of the target tunnel.

[0025] Furthermore, the listing of the fourth-order differential equation in Step 3 includes:

[0026]

[0027] Among them, α is the Timoshenko shear coefficient of the target tunnel, G is the shear modulus of the target tunnel, and A is the cross-sectional area of the target tunnel. After solving the above equation, the lateral displacement of the target tunnel, that is, the deflection u(x), is obtained.

[0028] Further, in step 3, calculating the shear force of the target tunnel includes:

[0029] Substitute the lateral displacement u(x) of the target tunnel into the following equation:

[0030]

[0031] Among them, M represents the shear force of the tunnel.

[0032] Further, in step 3, calculating the bending moment of the target tunnel includes:

[0033] Substitute the lateral displacement u(x) of the target tunnel into the following equation:

[0034]

[0035] Among them, Q represents the bending moment of the tunnel.

[0036] Further, the constructing the difference equation and solving the fourth-order differential equation described in step 4 includes:

[0037] Step 4-1: Using the finite difference method, discretize the target tunnel along the lateral direction into n tunnel elements with equal spacing, then the number of finite difference nodes is n + 1;

[0038] Step 4-2: Construct two virtual difference nodes at both ends of the target tunnel for constructing the difference equation at the endpoints, then there are a total of n + 5 finite difference nodes;

[0039] Step 4-3: According to the standard finite difference principle, the finite difference forms of the first-order differential term, second-order differential term, third-order differential term, and fourth-order differential term are obtained as follows:

[0040]

[0041] Among them, u i , u i-1 , u i+1 , u i+2 and u ii2 are the horizontal displacements of the target tunnel at the i, i - 1, i + 1, i + 2, and i - 1 nodes respectively, and L represents the length of each tunnel element;

[0042] Thus, a system of linear algebraic equations consisting of n + 5 equations is obtained, and its matrix form is expressed as:

[0043] (K1 + K2 + K3)U = 0

[0044] Among them, K1, K2, K3, and U are matrices. According to the above equations, the shear force M of the target tunnel and the bending moment Q of the tunnel are solved.

[0045] Furthermore, the expressions of the matrices K1, K2, K3, and U described in step 4 are as follows:

[0046]

[0047]

[0048] Beneficial effects:

[0049] The present invention considers the action of the lateral force on the tunnel (different from the previously commonly used vertical force), and uses the Timoshenko beam model to calculate the influence of the tunnel under excavation, and calculates the deflection, bending moment, and shear force of the tunnel under excavation, having the following beneficial effects:

[0050] (1) The lateral subgrade reaction coefficient is first introduced to consider the lateral influence of the excavation on the tunnel.

[0051] (2) The Timoshenko beam calculation model is applied to calculate the lateral influence of the excavation on the tunnel.

[0052] (3) The method of difference is used to solve the fourth-order differential equation, and the deflection, bending moment, and shear force of the tunnel can be solved in combination with the boundary conditions. Description of the drawings

[0053] The following further specifically describes the present invention in combination with the drawings and specific embodiments, and the above and / or other advantages of the present invention will become clearer.

[0054] Figure 1 It is a schematic diagram of the influence of the excavation of a double-track tunnel on the adjacent tunnel.

[0055] Figure 2 It is an analysis model of the adjacent tunnel under the excavation action.

[0056] Figure 3 It is the force analysis of the adjacent tunnel unit.

[0057] Figure 4 It is a schematic diagram of the discretization of the tunnel.

[0058] Figure 5a It is a schematic diagram of the influence of the maximum lateral displacement in the excavation area on the lateral displacement of the adjacent tunnel in the embodiment.

[0059] Figure 5b It is a schematic diagram of the influence of the maximum lateral displacement in the excavation area on the bending moment of the adjacent tunnel in the embodiment.

[0060] Figure 6a Schematic diagram of the influence of the distance from the inflection point to the midpoint in the excavation area on the lateral displacement of the adjacent tunnel in the embodiment.

[0061] Figure 6b Schematic diagram of the influence of the distance from the inflection point to the midpoint in the excavation area on the bending moment of the adjacent tunnel in the embodiment. Detailed implementation manners

[0062] As Figure 1 shown, during the excavation of two tunnels on the same horizontal plane, the supporting effect of the soil mass weakens or is removed. Due to the action of gravity and stress redistribution, the soil mass around the excavation area will move towards the excavation area. This horizontal displacement usually occurs in the middle of the excavation surface and gradually spreads to both sides. The displacement caused by excavation can be determined according to the displacement curve formula:

[0063]

[0064] In the formula: s max is the maximum horizontal displacement caused by excavation, x is the horizontal coordinate, and i is the distance from the inflection point of the lateral displacement to the midpoint (all these data can be obtained through on-site measurement).

[0065] The technical solution proposed by the present invention specifically includes the following steps:

[0066] 1. Derive according to the tunnel lateral deformation differential equation to obtain the lateral displacement u(x) caused by excavation. The overall displacement of the tunnel is also called the deflection. Then, calculate the bending moment M and shear force Q of the affected tunnel based on this.

[0067] Figure 2 is the analysis model of the adjacent tunnel under the excavation action. Take an element with a length of dx in the existing tunnel for force and deformation analysis. From the force balance of this element, we can get:

[0068] F Q + k1s(x)dx = k1u(x)dx + F Q + dF Q (2)

[0069] In the formula: F Q is the section shear force, dF Q is the shear force increment of the element, u(x) is the lateral displacement of the tunnel, k1 is the Winkler foundation coefficient, and s(x) is the displacement caused by tunnel excavation.

[0070] (1961) proposed an empirical formula to estimate the value of k, and the formula is expressed as:

[0071]

[0072] Among them, B is the tunnel width; E s and v s are the Young's modulus and Poisson's ratio of the soil mass respectively; E p I p is the bending stiffness of the tunnel. In the present invention, since the influence of the excavation is a horizontal acting force, the subgrade reaction coefficient adopted should be considered to be multiplied by the coefficient of earth pressure at rest k0, and finally the subgrade reaction coefficient in the present invention is obtained:

[0073] k1 = k0 * k

[0074] Simplifying Equation (2), we can get:

[0075]

[0076] From the moment balance of the element, we can get:

[0077]

[0078] In the formula: m is the bending moment, and dm is the increment of the bending moment of the element. Simplifying Equation (5) and omitting the high-order infinitesimals, we can get:

[0079]

[0080] According to the deformation mode of the Timoshenko beam model, under the action of shear force, the cross-section of the beam rotates relative to the normal direction of the neutral axis, and the rotation angle is θ.

[0081] The shear force F Q of the beam, the bending moment m, the deflection, the rotation angle θ and the shear angle have the following relationships:

[0082]

[0083] In the formula: is the shear stiffness of the tunnel, α is the Timoshenko shear coefficient of the tunnel, G is the shear modulus of the tunnel, and A is the cross-sectional area of the tunnel.

[0084] From Equations (4) and (6), the relationship between the shear force F Q of the tunnel element, the bending moment m and the tunnel deflection u can be obtained:

[0085]

[0086] Among them, M represents the shear force of the tunnel, and Q represents the bending moment of the tunnel;

[0087] Taking the first derivative of Equation (10), we can get:

[0088]

[0089] Substituting Equation (11) into Equation (4) and after rearrangement, the differential equation related only to the lateral displacement u(x) is obtained as follows:

[0090]

[0091] By solving Equation (12), the lateral displacement u(x) of the tunnel can be obtained. Substituting u(x) into Formulas (9) and (10), the bending moment m and shear force F of the tunnel can be obtained respectively. Q .

[0092] However, Equation (12) is a fourth-order differential equation and is difficult to solve directly in mathematics. Therefore, the present invention proposes to use the finite element difference method for solution.

[0093] 2. Construction and solution of the difference equation

[0094] Solving Equation (12) can obtain the deformation of the tunnel under the action of tension. However, since the above equation is a fourth-order differential equation and is difficult to solve directly in mathematics, to simplify the calculation, the solution domain is divided into a finite number of grids to replace the continuous solution domain. Using the finite difference method, the tunnel is discretized into n elements with equal spacing along the lateral direction, then the number of finite difference nodes is n + 1. In addition, to construct the difference equations at the endpoints, two virtual difference nodes are constructed at both ends of the tunnel, with a total of n + 5 nodes, as Figure 4 shown in the schematic diagram of the discretization analysis of the tunnel.

[0095] According to the standard finite difference principle, the finite difference forms of the first, second, third, and fourth-order differential terms can be obtained respectively as follows:

[0096]

[0097] where: u i , u i-1 , u i+1 , u i+2 and u i-2 are the horizontal displacements of the tunnel at the i, i - 1, i + 1, i + 2, and i - 1 nodes respectively, and L represents the length of each tunnel element.

[0098] Substituting the difference form of the differential of u(x) into (12), the finite difference expression of the tunnel flexure differential equation is obtained as follows:

[0099]

[0100] Similarly, from Formulas (9) and (10), the bending moment and shear force at any node i can be expressed as:

[0101]

[0102] The boundary conditions are:

[0103]

[0104] Among them, \(m_0\) represents the tunnel bending moment at node 0, and \(m\) n represents the tunnel bending moment at node \(n\), represents the tunnel shear force at node 0, represents the tunnel shear force at node \(n\);

[0105] Combining equations (15)-(17), the equation related to the deflection of the virtual node can be obtained as follows:

[0106]

[0107] Substitute \(i = 0, 1, 2, \cdots, n - 2, n - 1, n\) into equation (14) and combine it with equation (18) to obtain a system of linear algebraic equations consisting of \(n + 5\) equations, and its matrix form can be expressed as:

[0108] (K1 + K2 + K3)U = 0 (19)

[0109] The expressions of each matrix are as follows:

[0110]

[0111] According to the above system of equations, the shear force \(M\) and bending moment \(Q\) of the tunnel can be solved.

[0112] Example:

[0113] In a specific example, according to the geological exploration results of the engineering project, the project area is located in a mountainous and hilly area with complex terrain changes. The main rock formations include soft rock, relatively soft rock, and medium-hard rock, and the bedrock is hard rock. The existing tunnel is completely in the medium-hard rock, and the specific parameters are listed in Table 1. For the maximum lateral displacement \(S\) of the excavation area max and the distance \(i\) from the inflection point to the midpoint, parameter analysis is carried out using the method proposed in the present invention to study the influence of these parameter changes on the tunnel deformation and bending moment.

[0114] Table 1 Material parameters

[0115]

[0116] I. The maximum lateral displacement \(S\) of the excavation area max

[0117] To study the influence of the maximum lateral displacement \(S\) of the excavation area max on the mechanical response of the tunnel, calculations are carried out for the cases where the maximum lateral displacement \(S\) of the excavation area max = 0.5m, 1.0m, 2.0m, and 3.0m, while keeping the distance \(i\) from the inflection point to the midpoint of the excavation area = 3m. According toFigure 5a and Figure 5b As shown, S max The change has a significant impact on both displacement and bending moment. S max A larger excavation area results in greater lateral displacement and greater bending moment of the tunnel. S max When S = 3.0m compared to S max = 0.5m, the maximum displacement of the tunnel increases by about 4.8 times, and the bending moment increases by about 5.1 times. Obviously, when the distance i from the inflection point of the excavation area to the midpoint is 3m, the increase in the maximum lateral displacement S max of the excavation area will cause a sharp increase in the lateral displacement and bending moment of the adjacent tunnel.

[0118] The ultimate tensile strain limit of concrete is 115×10 -6 . According to the calculation of material mechanics, it can be known that: assuming the tunnel height is 4m, the lining thickness is 0.4m, and the elastic modulus of concrete is 30GPa, the corresponding ultimate tensile bending moment M T-limit is calculated to be 63·10 6 N·m. Therefore, when S max = 3.0m, the tunnel is in a dangerous state at this time. During the construction process, it is necessary to control the maximum lateral displacement S max of the excavation area as much as possible to protect the adjacent tunnel from damage.

[0119] II. The distance i from the inflection point of the excavation area to the midpoint

[0120] To study the influence of the distance i from the inflection point of the excavation area to the midpoint on the mechanical response of the tunnel, calculations are carried out for the cases where the distance i from the inflection point of the excavation area to the midpoint is 1m, 3m, 5m, and 7m respectively, while keeping the maximum lateral displacement S max of the excavation area = 1m. The tunnel displacements and bending moments in the four cases are calculated respectively, and the results shown in Figure 6a and Figure 6b are obtained. The distance i from the inflection point to the midpoint has a certain impact on the displacements and bending moments of the adjacent tunnel. When the distance i from the inflection point to the midpoint increases from 1m to 7m, the maximum lateral displacement of the adjacent tunnel increases from 0.068·m to 0.413m, an increase of about 5.1 times; the maximum bending moment of the adjacent tunnel increases from 11.197·10 6 N·m to 48.318·10 6 N·m, an increase of about 3.4 times. For all the distances i from the inflection point to the midpoint in this section, the adjacent tunnel is in a safe state.

[0121] In a specific implementation, the present application provides a computer storage medium and a corresponding data processing unit. Among them, the computer storage medium can store a computer program, and when the computer program is executed by the data processing unit, it can run the inventive content of a calculation method for the influence of excavation on adjacent tunnels on the same plane and parallel, as well as some or all of the steps in each embodiment. The storage medium can be a magnetic disk, an optical disk, a read-only memory (ROM), a random access memory (RAM), or the like.

[0122] Those skilled in the art can clearly understand that the technical solutions in the embodiments of the present invention can be implemented by means of a computer program and its corresponding general hardware platform. Based on such an understanding, the technical solutions in the embodiments of the present invention, in essence, or the parts that contribute to the prior art can be embodied in the form of a computer program, that is, a software product. This computer program software product can be stored in a storage medium and includes several instructions for causing a device (which can be a personal computer, a server, a single-chip microcomputer, an MCU, or a network device, etc.) including a data processing unit to execute the methods described in each embodiment or some parts of the embodiments of the present invention.

[0123] The present invention provides an idea and method for a calculation method for the influence of excavation on adjacent tunnels on the same plane and parallel. There are many methods and ways to specifically implement this technical solution. The above is only the preferred implementation manner of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention. Each component not clearly defined in this embodiment can be implemented by the prior art.

Claims

1. A method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other, characterized in that: The following steps are involved: Step 1, constructing an analysis model of the target tunnel under the action of expansion and excavation, i.e., an adjacent tunnel in the same plane and parallel to the target tunnel, and randomly selecting a tunnel unit with a length of dx in the target tunnel for force analysis and deformation analysis; Step 2: Design the foundation reaction coefficient to calculate the stress and deformation of the tunnel unit; Step 3, based on the analysis of step 1 and the coefficients designed in step 2, a fourth-order differential equation is listed to calculate the deflection, bending moment and shear force of the target tunnel; Step 4, construct a differential equation and solve the fourth-order differential equation described in step 3 to obtain the deflection, bending moment and shear force of the target tunnel, and complete the calculation of the impact of the expansion on the adjacent tunnels in the same plane and parallel.

2. The method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other according to claim 1, characterized in that: The analytical model of the adjacent tunnel under the action of excavation described in step 1 includes: Step 1-1, in the target tunnel, i.e., the adjacent tunnel, randomly select a tunnel unit with a length of dx; Step 1-2, perform stress analysis and deformation analysis on the tunnel unit, and obtain the following according to the stress balance: F Q +k1s(x)dx=k1u(x)dx+F Q +dF Q Among them, F Q is the cross-sectional shear force of the tunnel unit, dF Q is the shear force increment of the tunnel unit, u(x) is the lateral displacement of the tunnel unit, k1 is the foundation reaction coefficient, and s(x) is the displacement caused by excavation.

3. The method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other according to claim 2, characterized in that: The displacement s(x) caused by the excavation described in step 1-2 is calculated as follows: Among them, s max is the maximum horizontal displacement caused by excavation, x is the horizontal coordinate, d is the distance from the inflection point of the lateral displacement to the midpoint, and the above data are obtained through actual measurement.

4. The method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other according to claim 3, characterized in that: The design foundation reaction coefficient described in step 2, include: k1=k0*k Where k0 represents the static earth pressure coefficient and k represents the empirical estimation coefficient.

5. The method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other according to claim 4, characterized in that: The empirically estimated coefficient k described in step 2 is expressed as follows: Where B is the target tunnel width, E s and v s are Young's modulus and Poisson's ratio of the soil, E p I p is the target tunnel bending stiffness.

6. The method for calculating the impact of excavation expansion on adjacent tunnels in the same plane and parallel to each other according to claim 5, characterized in that: The fourth-order differential equations listed in step 3 include: Among them, α is the Timoshenko shear coefficient of the target tunnel, G is the shear modulus of the target tunnel, and A is the cross-sectional area of ​​the target tunnel. After solving the above equation, the lateral displacement of the target tunnel, that is, the deflection u(x), is obtained.

7. The method for calculating the impact of excavation expansion on adjacent tunnels in the same plane and parallel to each other according to claim 6, characterized in that: In step 3, the shear force of the target tunnel is calculated, including: Substitute the lateral displacement u(x) of the target tunnel into the following equation: Where M represents the shear force of the tunnel.

8. The method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other according to claim 7, characterized in that: In step 3, the bending moment of the target tunnel is calculated, including: Substitute the lateral displacement u(x) of the target tunnel into the following equation: Where Q represents the bending moment of the tunnel.

9. The method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other according to claim 8, characterized in that: The step 4 of constructing the differential equation and solving the fourth-order differential equation in step 3 includes: Step 4-1, using the finite difference method, discretize the target tunnel into n tunnel units with equal spacing in the horizontal direction, and the number of finite difference nodes is n+1; Step 4-2, construct two virtual difference nodes at both ends of the target tunnel to construct the difference equations of the endpoints, so there are a total of n+5 finite difference nodes; Step 4-3, according to the standard finite difference principle, the finite difference forms of the first-order differential term, the second-order differential term, the third-order differential term and the fourth-order differential term are obtained as follows: Among them, u i 、u i-1 、u i+1 、u i+2 and u i-2 are the horizontal displacements of the target tunnel at the i-th, i-1, i+1, i+2 and i-1 nodes respectively, and L represents the length of each tunnel unit; This results in a linear algebraic equation system consisting of n+5 equations, which is expressed in matrix form as follows: (K1+K2+K3)U=0 Among them, K1, K2, K3 and U are matrices. According to the above equations, the shear force M of the target tunnel and the bending moment Q of the tunnel are solved.

10. The method for calculating the impact of excavation on adjacent tunnels in the same plane and parallel to each other according to claim 9, characterized in that: The expressions of matrices K1, K2, K3 and U described in step 4 are as follows:

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