A tunnel pipe roof deflection prediction method and system based on elastic constraint boundary

Abstract

This invention relates to the field of tunnel pipe roof deformation calculation technology, and discloses a method and system for predicting tunnel pipe roof deflection based on elastic constraint boundaries. The method includes treating the pipe roof at the tunnel entrance as an equivalent elastic foundation beam with translational and rotational springs at both ends, establishing a longitudinal segmented force model, determining the applied loads and governing equations, and obtaining the deflection values ​​at each node of the pipe roof. The specific mechanism is as follows: by introducing translational and rotational springs at the ends of the pipe roof at the tunnel entrance, this invention can more realistically simulate the actual constraint effect at the pipe roof ends. This invention establishes three types of analysis models: an entrance segment model, a finite-length pipe roof model, and an exit segment model. These models are applicable to the force and deformation analysis at different construction stages, such as when the pipe roof enters the tunnel, when excavation approaches the end of the pipe roof, and when it exits the tunnel. This improves the completeness, computational efficiency, and applicability of the pipe roof deflection prediction method.

Description

Technical Field

[0001] This invention relates to the field of tunnel pipe roof deformation calculation technology, and in particular to a method and system for predicting tunnel pipe roof deflection based on elastic constraint boundaries. Background Technology

[0002] Pipe roof support, as an important advanced support measure, has been widely used in the construction of shallow tunnels, tunnels in weak and broken surrounding rock, tunnel entrance sections, and underground engineering projects under adverse geological conditions.

[0003] Pipe roofs, by pre-installing steel pipes along the excavation outline and combining them with grouting reinforcement, can effectively improve the stress state of the surrounding rock, control excavation disturbance and surface settlement, thereby ensuring construction safety. Pipe roofs play a crucial role in load transfer and deformation coordination during construction; their stress and deformation characteristics directly affect the support effect and project safety. Therefore, it is necessary to accurately analyze their mechanical response.

[0004] Existing analytical methods for the stress and deformation of pipe roofs mainly include numerical simulation methods and theoretical analysis methods, as follows: Numerical simulation methods, while capable of comprehensively reflecting the interaction between pipe roof and surrounding rock under complex working conditions, typically suffer from problems such as complex modeling processes, cumbersome parameter selection, long calculation times, and low analysis efficiency, making it difficult to meet the needs for rapid analysis and parameter comparison in engineering design and construction.

[0005] The theoretical analysis method has the advantages of simple modeling and high solution efficiency. However, most existing theoretical models adopt idealized boundary conditions, usually simplifying the pipe roof end to a fixed end, hinged end, or free end. This makes it difficult to truly reflect the actual elastic constraint effect of steel arch frame, guide wall, initial support, and surrounding reinforcement on the pipe roof end. At the same time, most methods do not adequately consider the continuous deformation characteristics of the surrounding rock and the longitudinal segmentation effect caused by step excavation during construction, resulting in certain deviations between the calculation results and the actual engineering situation.

[0006] Therefore, it is necessary to propose a new method for predicting tunnel pipe roof deflection to address the problems existing in the current technology. Summary of the Invention

[0007] The main objective of this invention is to provide a method for predicting tunnel pipe roof deflection based on elastically constrained boundaries. Specifically, this method is an analysis of the load transfer mechanism of pipe roofs based on an elastic foundation beam model, considering end elastic constraints, and reflecting the characteristics of the surrounding rock continuous medium. It improves computational efficiency while ensuring calculation accuracy, thereby providing a more accurate and efficient theoretical tool for the design and construction optimization of pipe roof support structures. The specific technical solution is as follows: A method for predicting tunnel pipe roof deflection based on elastically constrained boundaries includes the following steps: Step S1: Based on the influence of tunnel excavation on the deformation and internal forces of the arched pipe roof system at the tunnel entrance, the pipe roof is simulated as an elastic foundation beam with horizontal springs and rotational springs applied to both ends respectively. Step S2: Establish a mechanical analysis model for the pipe roof. Specifically, simplify a single pipe roof into an Euler-Bernoulli beam, establish a stress model for the pipe roof placed on a Pasternak two-parameter elastic foundation, and establish end elastic constraint boundary conditions. Step S3: Establish a longitudinal segmented stress model, specifically including: dividing the pipe roof into multiple functional sections along the axial direction, preferably into a closed support section, an unclosed support section, an unsupported section, a plastic disturbance section, an elastic disturbance section, and an undisturbed section, and assigning corresponding load forms, foundation reaction coefficients, and foundation shear moduli to each section; the longitudinal segmented stress model includes a pipe roof inlet section model, a pipe roof finite length model, and a pipe roof outlet section model; Step S4: Determine the applied loads and governing equations. Specifically, based on the tunnel burial depth and the distribution law of the surrounding rock load, determine the vertical loads acting on each segment of the pipe roof. Combining beam bending theory, Pasternak foundation reaction relationship, and differential segment equilibrium conditions, establish the pipe roof deflection control differential equations considering end elastic constraints. Discretize the pipe roof deflection control differential equations using the finite difference method to obtain a set of finite difference algebraic equations with the deflection of each node as the unknown quantity. Step S5: Solve for the deflection and mechanical response of the pipe shed. Specifically, solve the finite difference algebraic equations obtained in step S4 to obtain the deflection values ​​at each node of the pipe shed. Based on the deflection values, use the finite difference relationship expression of internal forces to calculate the rotation angle, bending moment and shear force at each node to obtain the deformation and internal force response of the pipe shed.

[0008] Preferably, in step S1, based on the impact of tunnel excavation on the deformation and internal forces of the arch-roof pipe roof system at the tunnel entrance, the following assumptions are made: ① Idealize the pipe shed as an Euler-Bernoulli beam and establish a pasternak type elastic foundation beam model; ② Ignore the frictional interaction between the surrounding foundation and the steel pipe, assume that the two are completely coordinated in deformation, and assume that the pipe roof has no outward tilt angle, and ignore the influence of curvature on the structural response; ③ Considering the actual end constraints provided by the transverse end beams of the tube curtain, the umbrella arch of the tube curtain, and the surrounding ground, the tube roof is simulated as an elastic foundation beam with horizontal springs and rotational springs applied to both ends, where: the stiffness of the horizontal springs at both ends is denoted as... and The stiffness of the rotational springs located at both ends is denoted as . and subscript 0 and These represent the beginning and end of the pipe shed, respectively.

[0009] Preferably, step S2 includes the following steps: The governing differential equation for the pipe roof deflection is as follows: ; in: The bending stiffness of the cross section; For the flexural deformation of the pipe shed; This is the equivalent foundation beam width; It is the elastic foundation reaction coefficient; The shear modulus of the surrounding foundation; This refers to the load-bearing width of the pipe shed; This refers to the vertical distributed load acting above the pipe roof; The coordinates are along the axis of the pipe shed; Let be the length of the selected differential beam element.

[0010] Preferably, by introducing end constraints, the boundary condition expression for the pipe roof is obtained as follows: ; ; in: , , and These are the vertical deflection at the starting end of the pipe roof, and their first, second, and third derivatives, respectively. , , and These are the vertical deflection at the end of the pipe roof, the first derivative, the second derivative, and the third derivative.

[0011] Preferably, in step S3: The pipe roof inlet section model consists of an unsupported section, a plastic disturbance section, an elastic disturbance section, and an undisturbed section; In the finite-length model of the pipe roof, when the excavation approaches the end of the first set of pipe umbrellas, a model for overlapping is required. At this point, a virtual extension segment is introduced into the calculation to account for the bending stiffness of the pipe umbrellas in the extension segment. Assign a negligible minimum value; The pipe roof outlet section model consists of a closed support section, an unclosed support section, and an unsupported section.

[0012] Preferably, in step S4, based on the finite difference method, the governing differential equation of the pipe roof deflection and the boundary conditions of the pipe roof can be expressed in the following form: The governing differential equation for the pipe roof deflection is expressed by the following formula:

[0013] in: , , , and To indicate Vertical deflection at the nodes adjacent to the left and right of each node; This represents the distance between adjacent discrete nodes, i.e., the finite difference step size; For the first Load at each node ; For the first Equivalent foundation reaction stiffness at each node , For the first Ground reaction coefficient at each node; For the first Equivalent foundation shear stiffness at each node , For the first The shear modulus of the foundation surrounding each node.

[0014] Preferably, the stiffness of the vertical springs at both ends and the stiffness of the corner springs are expressed by the following formula: ; ; ; ; and The deflection of the two virtual nodes used for finite difference calculations at the starting end node; and The deflection of the two virtual nodes used for finite difference calculations at the end node.

[0015] Preferably, in step S5... The expression for the finite difference relationship of the internal forces at each node is as follows: ; ; ; in: , , The first The rotation angle, bending moment, and shear force of each node.

[0016] The effect of applying the technical solution of this invention is: This invention provides a method for predicting tunnel pipe roof deflection based on elastic constraint boundaries. Targeting the stress and deformation characteristics of the tunnel arch pipe roof system during tunnel excavation, the pipe roof is equivalent to an elastic foundation beam with translational and rotational springs at both ends. A mechanical analysis model of the pipe roof considering the end elastic constraints is established. Furthermore, a longitudinal segmented stress model is constructed to determine the vertical loads acting on each segment of the pipe roof and establish the corresponding pipe roof deflection control equations. Based on this, the finite difference method is used to discretize and solve the control equations and elastic constraint boundary conditions to obtain the deflection values ​​at each node of the pipe roof. Based on the node deflection, the rotation angle, bending moment, and shear force at each node are further calculated, thereby obtaining the longitudinal deformation and internal force response of the pipe roof.

[0017] The specific mechanism of action is as follows: By introducing translational and rotational springs at the ends of the pipe roof at the tunnel entrance, this invention can more realistically characterize the actual constraint effects of structures such as steel arches, guide walls, initial supports, and reinforced bodies on the ends of the pipe roof. This overcomes the problem of overly idealized end boundary conditions in existing pipe roof mechanical analysis methods and improves the applicability of the model to complex support conditions at the tunnel entrance. Simultaneously, this invention establishes three types of analysis models: an entrance section model, a finite-length pipe roof model, and an exit section model. These models are applicable to stress and deformation analysis at different construction stages, such as when the pipe roof enters the tunnel, when excavation approaches the end of the pipe roof, and when it exits the tunnel. This solves the problem that a single model cannot simultaneously describe multiple construction states at the tunnel entrance, thus improving the completeness and applicability of the pipe roof deflection prediction method.

[0018] In addition, this invention establishes a finite-length pipe roof model by introducing a virtual extension section. This model can consider the impact of pipe roof overlap length and excavation disturbance range exceeding the length of a single pipe roof on structural deformation, thus better reflecting the actual situation of multi-cycle pipe roof overlap construction on site and improving the rationality of pipe roof deflection calculation results. This invention adopts a parameterized end constraint stiffness characterization method. By changing the vertical and rotational constraint stiffness at the ends, it can uniformly simulate various boundary constraint states such as free, hinged, semi-rigid, and approximately fixed connections. This allows for the analysis and comparison of pipe roof deflection, rotation angle, bending moment, and shear force responses under different end constraint conditions, providing a basis for the design of pipe roof support parameters at the opening section. Based on theoretical analysis, this invention can quickly calculate pipe roof deflection, offering advantages over numerical simulation methods such as higher computational efficiency, convenient parameter analysis, and suitability for rapid engineering comparison.

[0019] The present invention also provides a tunnel pipe roof deflection prediction system based on elastic constraint boundaries, including a controller; The controller includes one or more processors and a storage device for storing one or more programs; when the program is executed by the processor, it implements the tunnel pipe roof deflection prediction method based on elastic constraint boundaries as described above. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. 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 the structures shown in these drawings without creative effort.

[0021] Figure 1 This is a flowchart of the tunnel pipe roof deflection prediction method based on elastic constraint boundaries in Embodiment 1 of the present invention; Figure 2 This is a schematic diagram of the force balance of the differential element beam element in Embodiment 1 of the present invention; Figure 3 This is a schematic diagram of the vertical spring and the rotary spring in Embodiment 1 of the present invention; Figure 4 This is a schematic diagram of the pipe roof inlet section model in Embodiment 1 of the present invention; Figure 5 This is a schematic diagram of the finite length model of the pipe shed in Embodiment 1 of the present invention; Figure 6 This is a schematic diagram of the pipe roof outlet section model in Embodiment 1 of the present invention; Figure 7 This is a schematic diagram of a virtual node using the finite element method in Embodiment 1 of the present invention; Figure 8 The deflection curves of the outlet section pipe roof under different constraint stiffnesses in Embodiment 1 of the present invention are shown. Figure 9 The curves show the deflection curves of the inlet section pipe roof under different constraint stiffnesses in Embodiment 1 of the present invention.

[0022] The realization of the objective, functional features and advantages of the present invention will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0023] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0024] Example 1: A method for predicting tunnel pipe roof deflection based on elastically constrained boundaries, see details. Figure 1 This includes the following steps: Step S1: Investigate the impact of tunnel excavation on the deformation and internal forces of the arched pipe roof system at the tunnel entrance. Make reasonable assumptions about the pipe roof structure, surrounding rock mechanical parameters, and stress characteristics. Specific assumptions are as follows: ① Idealize the pipe roof as an Euler-Bernoulli beam, establish a pasternak type elastic foundation beam model, and study the evolution of the pipe roof response and deformation during tunnel excavation. ② Ignore the frictional interaction between the surrounding foundation and the steel pipe, assuming complete deformation coordination between the two, and assuming the pipe roof has no outward tilt angle, ignoring the influence of curvature on the structural response. ③ Consider the actual end constraints provided by the transverse end beams at the pipe roof ends, the pipe roof umbrella arch, and the surrounding ground. Accordingly, simulate the pipe roof as an elastic foundation beam with horizontal springs and rotational springs applied to both ends respectively. In this embodiment, the stiffness of the horizontal springs at both ends is denoted as... and The stiffness of the rotational springs located at both ends is denoted as . and subscript 0 and These represent the beginning and end of the pipe shed, respectively.

[0025] Step S2: Establish a mechanical analysis model for the pipe roof. Specifically, simplify a single pipe roof into an Euler-Bernoulli beam and establish a stress model of the pipe roof placed on a Pasternak two-parameter elastic foundation. Establish elastic constraint boundary conditions at the ends, use the parametric method, take the end stiffness as the parameter to be analyzed, take several levels for sensitivity analysis, and then give a recommended interval.

[0026] In this preferred embodiment, the specific process includes the following: 1. Considering the Pasternak foundation, the ground reaction force at any point. The displacement at that point (i.e., the deflection of the pipe roof) The following relationship exists: 1); in: This is the ground reaction force, measured in N. It is the elastic foundation reaction coefficient; The shear modulus of the surrounding foundation is given in MPa / m. The bending deformation of the pipe roof is expressed in meters (m). The coordinates are along the axis of the pipe shed; Let be the length of the selected differential beam element.

[0027] Based on the relationship between tube roof deformation and internal forces according to Euler-Bernoulli beam theory: 2); in: , , and These represent rotation angle (in °), axial strain, bending moment (in N·m), and shear force (in N), respectively. It represents the perpendicular distance from a point on the cross section to the neutral axis; Let be the bending stiffness of the cross section. It is the elastic modulus. It is the moment of inertia of the cross section.

[0028] For pipe sheds The calculation is performed using expression 3): 3); in: , , and These are the elastic modulus of the filling concrete, the elastic modulus of the steel used for the pipe roof, the moment of inertia of the section of the filling concrete, and the moment of inertia of the section of the steel used for the pipe roof.

[0029] Force balance based on differential element beam elements, such as Figure 2 As shown, with vertically upward as positive, according to... Figure 2 The vertical force equilibrium yields the following: 4); in: and These are the shear forces at the left and right ends of the differential element, respectively. This represents the ground reaction force per unit length of the pipe shed; Let be the length of the selected differential beam element; This indicates the vertical external load per unit length of the pipe shed; Take the moment about the left end section, according to Figure 2 From the direction of the bending moment and the moments of each force about the left end, the bending moment equilibrium can be obtained as follows: 5); in: The coordinates are along the axis of the pipe shed; Let be the length of the selected differential beam element; For pipe shed in Vertical deflection at the location; This refers to the vertical distributed load acting above the pipe roof; The load-bearing width of the pipe roof is usually taken as the outer diameter of the pipe roof; The equivalent foundation beam width can be calculated using expression 6); and These represent the bending moments at the left and right ends of the differential element, respectively; where and They represent shear force and bending moment along the length, respectively. The increment within.

[0030] 6); Based on the vertical force balance and bending moment balance of the differential unit, the control relationship between shear force, bending moment and deflection of the pipe roof can be further established.

[0031] From expressions 4) and 5), the equilibrium equations are as follows: 7); Substituting expressions 1) and 2) into expression 7), the final solution yields the governing differential equation for the pipe roof deflection: 8).

[0032] 2. Introduce more realistic end constraints by adding vertical springs and rotary springs at both ends to simulate actual constraints, such as... Figure 3 As shown. At this point, the boundary condition expression for the pipe shed can be obtained as follows: 9); 10); in: and These represent the rotational constraint stiffness and vertical constraint stiffness at the starting end of the pipe roof, respectively. and These represent the rotational constraint stiffness and vertical constraint stiffness at the ends of the pipe shed, respectively. =0 indicates the starting point of the pipe shed. =n represents the end of the pipe shed; and These are the vertical deflections at the beginning and end of the pipe roof, respectively.

[0033] Step S3: Establish a longitudinal segmented force model, details of which are as follows: Based on the variations in the surrounding rock disturbance range, support closure status, and foundation parameters during the phased excavation of the tunnel, the pipe roof is divided into multiple functional sections along the axial direction. The preferred divisions are: supported closed section, supported unclosed section, unsupported section, plastic disturbance section, elastic disturbance section, and undisturbed section. Corresponding load forms, foundation reaction coefficients, and foundation shear moduli are assigned to each section to reflect the influence of construction spatiotemporal effects on the mechanical response of the pipe roof. This embodiment focuses on the stress mechanism of the pipe roof in the entrance and exit sections; therefore, three analytical models are established as follows: ① The pipe roof inlet section model consists of an unsupported section, a plastic disturbance section, an elastic disturbance section, and an undisturbed section, such as... Figure 4 As shown; ② Finite-length pipe roof model: When excavation approaches the end of the first set of pipe umbrellas, a model of overlapping is required. In this case, a virtual extension segment is introduced into the calculation, and the bending stiffness of the pipe umbrella in the extension segment, σL, is assigned a negligible minimum value to better represent the actual site conditions. For example... Figure 5 As shown; ③ The pipe roof outlet section model consists of a supported closed section, a supported unclosed section, and an unsupported section, as shown below. Figure 6 As shown.

[0034] The specific values ​​of the length of each zone, the foundation reaction coefficient, and the foundation shear modulus are shown in Table 1.

[0035] Table 1 Determination of parameters for each segment

[0036] In Table 1: , , , , and These are the lengths of the closed support section, the unclosed support section, the unsupported section, the plastic disturbance section, the elastic disturbance section, and the undisturbed section, respectively. The initial load strength is for the unexcavated or original state. This is the equivalent foundation reaction coefficient provided after the initial support is formed. is the initial foundation reaction coefficient of the surrounding rock. To reduce the initial foundation reaction coefficient of the surrounding rock, it is generally taken as 0.6× ; This is the stress release coefficient of the surrounding rock at the working face, which is generally taken as 0.5; This is the stress release coefficient of the surrounding rock in the disturbed zone, which is generally taken as 1.2; This provides the equivalent foundation shear coefficient after the initial support is formed. The shear coefficient of the foundation; The excavation height, The internal friction angle of the surrounding rock; Step S4: Determine the applied loads and governing equations.

[0037] In this embodiment, based on the tunnel burial depth and the distribution law of the surrounding rock load, the vertical load acting on each segment of the pipe roof is determined. Combining the beam bending theory, Pasternak foundation reaction relationship and micro-segment equilibrium conditions, a pipe roof deflection control differential equation considering segment characteristics and end elastic constraint conditions is established.

[0038] It can be obtained through any one of the following methods: railway tunnel design code method, Terzaghi theory, Ptolemy theory, or experimental results.

[0039] The finite difference method is used to calculate the pipe top deflection. The region is discretized into N nodes with a spacing of... Uniform units, L Let be the length of the pipe roof. For numerical implementation, two virtual nodes are introduced at each end of the pipe roof: nodes -2 and -1 on the left boundary, and nodes n+1 and n+2 on the right boundary, as shown below. Figure 7 As shown. To achieve high-order accuracy during the discretization process, a five-point central difference scheme is used for all derivatives: 11); in: Indicates the pipe shed in the first Vertical deflection at each node; , , and To indicate the first Vertical deflection at the left and right adjacent nodes of each node. This represents the distance between adjacent discrete nodes, i.e., the finite difference step size. , , and Representing the pipe roof deflection function respectively In the The first, second, third, and fourth derivatives at each node.

[0040] Therefore, substituting expression 11) into the governing differential equation of the pipe roof deflection, i.e., expression 8), we can obtain: 12); in: Indicates the first Vertical deflection at each node; and They represent the first Vertical deflection at the two adjacent nodes to the left of each node; and They represent the first Vertical deflection at the two adjacent nodes to the right of each node; This represents the distance between adjacent discrete nodes, i.e., the finite difference step size; For the first Load at each node ; For the first Equivalent foundation reaction stiffness at each node , For the first Ground reaction coefficient at each node; For the first Equivalent foundation shear stiffness at each node , For the first Shear modulus of the foundation surrounding each node; The vertical spring stiffness at both ends and the corner spring stiffness, i.e., the aforementioned expressions 9) and 10), are expressed by the following formula: 13); 14); 15); 16); and For the virtual node deflection used in finite difference calculations at the starting end node, and The virtual node deflection used for finite difference calculations at the end node.

[0041] Finally, by combining expressions 12), 13), 14), 15), and 16), the matrix equation Aw = P can be constructed. The deflection of each node of the pipe roof can be solved using Matlab programming.

[0042] The equations for matrices A, w, and P are as follows: 17); 18); 19); In matrix A, the expressions for each parameter are as follows: 20); twenty one); twenty two); Finally, the deflection of each node is obtained. Combining expressions 2) and 11) By taking the derivative, we can obtain the expressions for the rotation angle, bending moment, and shear force at each node: twenty three); twenty four); 25); in: , , The first The rotation angle, bending moment, and shear force of each node; Step S5: Solve for the deflection and mechanical response of the pipe roof. Solve the finite difference algebraic equations obtained in Step S4 to obtain the deflection values ​​at each node of the pipe roof. Based on these deflection values, calculate the rotation angle, bending moment, and shear force at each node using the finite difference relationship expression for internal forces, ultimately obtaining the deformation and internal force response of the pipe roof. The deflection curves of the outlet and inlet sections of the pipe roof under different elastic constraint stiffnesses are shown below. Figure 8 and Figure 9 As shown.

[0043] The effect of applying the technical solution of this invention is: 1. By introducing translational and rotational springs at the ends of the pipe shed at the opening section, this invention can more realistically simulate the actual constraint effect of steel arch frame, guide wall, initial support and reinforcement on the ends of the pipe shed, overcome the problem of overly idealized boundary conditions in existing methods, and improve the engineering applicability of the model.

[0044] 2. This invention establishes three types of analysis models: the entrance section model, the pipe roof finite length model, and the exit section model. These models are applicable to the stress and deformation analysis of different construction stages, such as when the pipe roof enters the opening, when the excavation approaches the end of the pipe roof, and when it exits the opening. This solves the problem that a single model cannot simultaneously describe multiple construction states of the opening section, and improves the completeness and applicability of the pipe roof deflection prediction method.

[0045] 3. By introducing a virtual extension section to establish a finite-length pipe roof model, this invention can consider the impact of pipe roof overlap length and excavation disturbance range exceeding the length of a single pipe roof on structural deformation, thus better conforming to the actual situation of multi-cycle pipe roof overlap construction on site and improving the rationality of pipe roof deflection calculation results.

[0046] 4. This invention adopts a parameterized end constraint stiffness characterization method, which can uniformly simulate various boundary constraint states such as free, hinged, semi-rigid and approximately fixed by changing the vertical and rotational constraint stiffness at the ends. This enables the analysis and comparison of the deflection, rotation, bending moment and shear force response of the pipe roof under different end constraint conditions, providing a basis for the design of pipe roof support parameters at the opening section.

[0047] 5. Based on theoretical analysis, this invention can quickly calculate the deflection of the pipe roof. Compared with numerical simulation methods, it has the advantages of high calculation efficiency, convenient parameter analysis, and suitability for rapid engineering comparison.

[0048] Example 2: This embodiment provides a tunnel pipe roof deflection prediction system based on elastic constraint boundaries, including a controller; The controller includes one or more processors and a storage device for storing one or more programs; when the program is executed by the processor, it implements the tunnel pipe roof deflection prediction method based on elastic constraint boundaries as described above.

[0049] The above description is merely a preferred embodiment of the present invention and does not limit the patent scope of the present invention. Any equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A method for predicting the deflection of a tunnel pipe roof based on an elastic restraint boundary, characterized by, Includes the following steps: Step S1: Based on the influence of tunnel excavation on the deformation and internal forces of the arched pipe roof system at the tunnel entrance, the pipe roof is simulated as an elastic foundation beam with horizontal springs and rotational springs applied at both ends respectively. Step S2: Establish a mechanical analysis model for the pipe roof. Specifically, simplify a single pipe roof into an Euler-Bernoulli beam, establish a stress model for the pipe roof placed on a Pasternak two-parameter elastic foundation, and establish end elastic constraint boundary conditions. Step S3: Establish a longitudinal segmented stress model, specifically including: dividing the pipe roof into multiple functional sections along the axial direction, namely, a closed support section, an unclosed support section, an unsupported section, a plastic disturbance section, an elastic disturbance section, and an undisturbed section, and assigning corresponding load forms, foundation reaction coefficients, and foundation shear moduli to each section; establishing a longitudinal segmented stress model including a pipe roof inlet section model, a pipe roof finite length model, and a pipe roof outlet section model; Step S4: Determine the applied loads and governing equations. Specifically, based on the tunnel burial depth and the distribution law of the surrounding rock load, determine the vertical loads acting on each segment of the pipe roof. Combining beam bending theory, Pasternak foundation reaction relationship, and differential segment equilibrium conditions, establish the pipe roof deflection control differential equations considering end elastic constraints. Discretize the pipe roof deflection control differential equations using the finite difference method to obtain a set of finite difference algebraic equations with the deflection of each node as the unknown quantity. Step S5: Solve for the deflection and mechanical response of the pipe shed. Specifically, solve the finite difference algebraic equations obtained in step S4 to obtain the deflection value at each node of the pipe shed; calculate the rotation angle, bending moment and shear force at each node based on the deflection value using the finite difference relationship expression of internal forces to obtain the deformation and internal force response of the pipe shed.

2. The method for predicting tunnel pipe roof deflection based on elastic constraint boundaries according to claim 1, characterized in that, In step S1, based on the impact of tunnel excavation on the deformation and internal forces of the arch-roof pipe roof system at the tunnel entrance, the following assumptions are made: ① Idealize the pipe shed as an Euler-Bernoulli beam and establish a pasternak type elastic foundation beam model; ② Ignore the frictional interaction between the surrounding foundation and the steel pipe, assume that the two are completely coordinated in deformation, and assume that the pipe roof has no outward tilt angle, and ignore the influence of curvature on the structural response; ③ Considering the actual end constraints provided by the transverse end beams of the tube curtain, the umbrella arch of the tube curtain, and the surrounding ground, the tube roof is simulated as an elastic foundation beam with horizontal springs and rotational springs applied to both ends, where: the stiffness of the horizontal springs at both ends is denoted as... and The stiffness of the rotational springs located at both ends is denoted as . and subscript 0 and These represent the beginning and end of the pipe shed, respectively.

3. The method for predicting tunnel pipe roof deflection based on elastic constraint boundaries according to claim 2, characterized in that, Step S2 includes the following steps: The governing differential equation for the pipe roof deflection is as follows: ； in: The bending stiffness of the cross section; For pipe shed in Vertical deflection at the location; This is the equivalent foundation beam width; It is the elastic foundation reaction coefficient; The shear modulus of the surrounding foundation; This refers to the load-bearing width of the pipe shed; This refers to the vertical distributed load acting above the pipe roof; The coordinates are along the axis of the pipe shed; is the length of the selected differential beam element.

4. The method for predicting tunnel pipe roof deflection based on elastic constraint boundaries according to claim 3, characterized in that, Introducing end constraints, the boundary condition expression for the pipe roof is obtained as follows: ； ； in: , , and These are the vertical deflection at the starting end of the pipe roof, and their first, second, and third derivatives, respectively. , , and These are the vertical deflection at the end of the pipe roof, the first derivative, the second derivative, and the third derivative.

5. The method for predicting tunnel pipe roof deflection based on elastic constraint boundaries according to any one of claims 1-4, characterized in that, In step S3: The pipe roof inlet section model consists of an unsupported section, a plastic disturbance section, an elastic disturbance section, and an undisturbed section; In the finite-length model of the pipe roof, when the excavation approaches the end of the first set of pipe umbrellas, a model for overlapping is required. At this point, a virtual extension segment is introduced into the calculation to account for the bending stiffness of the pipe umbrellas in the extension segment. And assign a negligible minimum value; The pipe roof outlet section model consists of a closed support section, an unclosed support section, and an unsupported section.

6. The method for predicting tunnel pipe roof deflection based on elastic constraint boundaries according to claim 5, characterized in that, The differential equation for controlling the deflection of the pipe roof is expressed as follows: ； in: Indicates the first Vertical deflection at each node; and They represent the first Vertical deflection at the two adjacent nodes to the left of each node; and They represent the first Vertical deflection at the two adjacent nodes to the right of each node; This represents the distance between adjacent discrete nodes, i.e., the finite difference step size; For the first Load at each node ; For the first Equivalent foundation reaction stiffness at each node , For the first Ground reaction coefficient at each node; For the first Equivalent foundation shear stiffness at each node , For the first The shear modulus of the foundation surrounding each node.

7. The method for predicting tunnel pipe roof deflection based on elastic constraint boundaries according to claim 6, characterized in that, The stiffness of the vertical springs at both ends and the stiffness of the corner springs are expressed by the following formula: ； ； ； ； in: and The deflection of two virtual nodes used for finite difference calculations at the starting end node; and The deflection of two virtual nodes used for finite difference calculations at the end node.

8. The method for predicting tunnel pipe roof deflection based on elastic constraint boundaries according to claim 7, characterized in that, No. The finite difference relationship expression of the internal forces at each node is as follows: ； ； ； in: , , The first The rotation angle, bending moment, and shear force of each node.

9. A tunnel pipe roof deflection prediction system based on elastic constraint boundaries, characterized in that, Including the controller; The controller includes one or more processors and a storage device for storing one or more programs; when the program is executed by the processor, it implements the tunnel pipe roof deflection prediction method based on elastic constraint boundaries as described in any one of claims 1-8.

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