Tunnel lining design method and system based on complex variable function method

Abstract

The present invention relates to the field of tunnel construction. Disclosed are a tunnel lining design method and system based on a complex variable function method. The method comprises: selecting a tunnel excavation radius, a tunnel burial depth, a lining elastic modulus and a lining thickness, and determining a rock mass region of a tunnel and a lining region thereof; on the basis of a mapping function, respectively mapping the rock mass region and lining region of the tunnel into a rock mass annular region and a lining annular region, so as to respectively construct analytical functions of the rock mass region and the lining region under different contact conditions; solving the analytical functions on the basis of boundary conditions and continuity conditions of the rock mass region, the lining region and the contact surface therebetween; respectively calculating and acquiring the stress and displacement of any point in the lining region and the rock mass region by combining stress and displacement expressions for lining and rock mass; and determining whether a tunnel design meets design requirements, and if the tunnel design does not meet the design requirements, adjusting the lining elastic modulus and the lining thickness until the design requirements are met. The present invention is applicable to lining design parameters of tunnels of different shapes, and can effectively improve the accuracy of calculating lining design parameters.

Description

Tunnel lining design method and system based on complex variable function method

[0001] Cross-reference to Related Applications

[0002] The present application claims priority to the Chinese patent application No. 202410966098.X filed on July 18, 2024, and entitled "Tunnel lining design method and system based on complex variable function method", the content of which is incorporated herein by reference in its entirety and made part of the present application for all purposes. TECHNICAL FIELD

[0003] The present application relates to the technical field of tunnel construction, in particular to a tunnel lining design method and system based on a complex variable function method, and a tunnel lining construction method. BACKGROUND

[0004] The statements herein are provided only to enhance understanding of the present application and are not necessarily intended to constitute the prior art.

[0005] With the rapid development of highways, railways, etc., the demand for tunnels is also increasing. In order to ensure the safe construction of tunnels, lining support is usually used to support the tunnel rock mass, and the supporting effect of the lining is determined by parameters such as the elastic modulus and thickness of the lining, and thus the lining parameter design affects the safety and economy of the tunnel, and the study of tunnel lining parameter design is one of the main research directions of current tunnel engineering.

[0006] At present, the elastic modulus (referred to as elastic modulus) and thickness of the lining are generally determined according to engineering analogy according to multiple factors such as the service life of the tunnel, the geological conditions, and the diameter of the tunnel. However, the tunnel lining thickness and elastic modulus determined by this method have poor adaptability to actual engineering. With the continuous development of mechanization and automation technology, the production and support of the lining are becoming faster and faster, and how to accurately calculate and optimize the elastic modulus and thickness of the lining has become the key to scientific support, and designing a lining that is adapted to the geological conditions has important economic and technical value. SUMMARY

[0007] To solve the above problems of the prior art, the present application provides a tunnel lining design method and system based on a complex variable function method, which uses the complex variable function method to map tunnels of different shapes, and also analyzes the lining design under the condition that different tunnel rock masses are in contact with the lining, optimizes the elastic modulus and thickness of the lining design, and improves the scientificity and accuracy of the calculation of the lining design parameters. This method has wide applicability. When this method is applied to actual tunnel lining construction engineering, the specific construction of the lining in the tunnel can be carried out according to the optimized values of the elastic modulus and thickness of the lining output, so as to achieve effective support of the tunnel.

[0008] The first aspect of the present application provides a tunnel lining design method based on complex variable function method.

[0009] The tunnel lining design method based on complex variable function method comprises:

[0010] The tunnel excavation radius, the tunnel buried depth, the lining elastic modulus and the lining thickness are selected to determine the rock mass region and the lining region of the tunnel.

[0011] According to the mapping function, the rock mass region and the lining region of the physical plane tunnel are respectively mapped into the image plane region.

[0012] Based on the mapped rock mass annular region and the lining annular region in the image plane, the analytical functions of the rock mass region and the lining region under different contact conditions are respectively constructed.

[0013] According to the boundary conditions and the continuity conditions of the rock mass region, the lining region and the contact surface therebetween, the analytical functions are solved.

[0014] Based on the solved analytical functions, the stress and the displacement of any point in the lining region and the rock mass region are respectively calculated and obtained in combination with the stress expression and the displacement expression of the lining and the rock mass.

[0015] Based on the stress and the displacement, it is determined whether the tunnel design based on the selected lining elastic modulus and the lining thickness meets the design requirements, and if not, the lining elastic modulus and the lining thickness are adjusted until the design requirements are met.

[0016] The second aspect of the present application provides a tunnel lining design system based on complex variable function method.

[0017] The tunnel lining design system based on complex variable function method comprises:

[0018] The data acquisition module is configured to select the tunnel excavation radius, the tunnel buried depth, the lining elastic modulus and the lining thickness, and determine the rock mass region and the lining region of the tunnel.

[0019] The mapping module is configured to map the rock mass region and the lining region of the tunnel into the image plane rock mass annular region and the lining annular region respectively according to the mapping function.

[0020] The analytical function construction module is configured to construct the analytical functions of the rock mass region and the lining region under different contact conditions based on the mapped rock mass annular region and the lining annular region in the image plane.

[0021] The analytical function solving module is configured to solve the analytical functions according to the boundary conditions and the continuity conditions of the rock mass region, the lining region and the contact surface therebetween.

[0022] a stress and displacement calculation module, configured to calculate the stress and displacement of any point in the lining region and the rock region based on the analytical function, the stress expression and the displacement expression of the lining and the rock;

[0023] a lining design adjustment module, configured to judge whether the tunnel design based on the selected lining elastic modulus and the lining thickness meets the design requirement based on the stress and the displacement, and adjust the lining elastic modulus and the lining thickness until the design requirement is met if not.

[0024] In a third aspect, the present application further provides an electronic device, comprising a memory and a processor, and computer instructions stored in the memory and running on the processor, when the computer instructions are run by the processor, the steps of the method in the first aspect are completed.

[0025] In a fourth aspect, the present application further provides a computer readable storage medium for storing computer instructions, when the computer instructions are executed by a processor, the steps of the method in the first aspect are completed.

[0026] In a fifth aspect, the present application further provides a tunnel lining construction method.

[0027] A tunnel lining construction method comprises:

[0028] obtaining a tunnel excavation radius and a tunnel burial depth of a tunnel to be supported, setting an initial value of a lining elastic modulus and a lining thickness, and determining a rock region and a lining region of the tunnel to be supported;

[0029] mapping the rock region and the lining region of the tunnel to be supported in a physical plane into image plane regions according to a mapping function;

[0030] constructing analytical functions of the rock region and the lining region under different contact conditions based on the mapped rock annular region and the lining annular region in the image plane;

[0031] solving the analytical functions according to boundary conditions and continuity conditions of the rock region, the lining region and a contact surface therebetween;

[0032] calculating the stress and the displacement of any point in the lining region and the rock region based on the solved analytical functions, the stress expression and the displacement expression of the lining and the rock;

[0033] resetting the initial value of the lining elastic modulus and the initial value of the lining thickness, and repeating the calculation of the stress and the displacement of any point in the lining region and the rock region until the calculated stress and displacement meet a predefined lowest allowable condition of a design specification, and then stopping the calculation;

[0034] When the calculated stress and displacement meet the initial value of the lining elastic modulus and the initial value of the lining thickness set when the lowest allowable condition of the predefined design specification is met, the initial value of the lining elastic modulus and the initial value of the lining thickness are taken as the optimal value of the lining elastic modulus and the optimal value of the lining thickness, and the construction of the tunnel lining is performed in the lining area of the tunnel to be supported by using the optimal value of the lining elastic modulus and the optimal value of the lining thickness, so as to support the tunnel to be supported.

[0035] The above one or more technical solutions have the following beneficial effects:

[0036] The tunnel lining design method and system based on the complex variable function method provided by the application solve the problem of poor precision of a method for determining the tunnel lining thickness by comprehensively considering multiple factors such as the service life of a tunnel, geological conditions, and the diameter of the tunnel to determine the tunnel lining thickness, and the constructed tunnel lining is more in line with the tunnel construction; in the embodiment, the stress and displacement solutions of the tunnel rock mass and the lining under the lining support are calculated based on the complex variable function method, and the elastic modulus and the thickness of the lining are optimized according to the stress and displacement solutions of the tunnel rock mass and the lining, the complex variable function method can map multiple shapes of tunnels, is applicable to various types of tunnel caverns, and can calculate the elastic modulus of the lining, thereby providing guidance for the selection of the type of lining material in combination with the thickness of the lining, providing a basis for flexible support with variable lining thickness, and having important economic and technical values. BRIEF DESCRIPTION OF DRAWINGS

[0037] The accompanying drawings, which form a part of this application, are included to provide a further understanding of the application and are incorporated in and constitute a part of this application, illustrate embodiments of the present application and serve to explain the present application, and do not limit the present application in any manner.

[0038] FIG. 1 is a flowchart of the tunnel lining design method based on the complex variable function method according to the embodiment of the application;

[0039] FIG. 2 is a flowchart of the solution of the stress and displacement of the rock mass area of the circular tunnel in the lining area under different contact conditions according to the embodiment of the application;

[0040] FIG. 3 is a schematic diagram of the rock mass area of the circular tunnel in the lining area according to the embodiment of the application;

[0041] FIG. 4A is a schematic diagram of the mapping of the rock mass area S1 into the rock mass circular ring area Ω1 according to the embodiment of the application;

[0042] FIG. 4B is a schematic diagram of the mapping of the lining area S2 into the lining circular ring area Ω2 according to the embodiment of the application;

[0043] FIG. 5 is a flowchart of the tunnel lining construction method according to the embodiment of the application. DETAILED DESCRIPTION

[0044] It should be noted that the following detailed description is exemplary only and is intended to provide further description of the exemplary embodiments according to the present application, and is not intended to limit the exemplary embodiments according to the present application. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. Furthermore, it should be understood that the use of the terms "including", "comprising", "having" and the like are meant to be inclusive, and are intended to mean that there are additions to the listed items.

[0045] Embodiment One

[0046] The embodiment provides a tunnel lining design method based on a complex variable function method, as shown in FIG. 1, comprising:

[0047] Setting a tunnel excavation radius, a tunnel burial depth, a lining elastic modulus and a lining thickness, and respectively determining a rock region and a lining region of the tunnel;

[0048] According to a mapping function, respectively mapping the rock region and the lining region of the tunnel in a physical plane into a rock annular region and a lining annular region in an image plane;

[0049] Based on the mapped rock annular region and the lining annular region, respectively constructing analytical functions of the rock region and the lining region under different contact conditions;

[0050] According to boundary conditions and continuity conditions of the rock region, the lining region and a contact surface thereof, solving the analytical functions;

[0051] Based on the solved analytical functions, combining stress expressions and displacement expressions of the lining and the rock, respectively calculating and obtaining stresses and displacements of any point in the lining region and the rock region;

[0052] Based on the stresses and the displacements, judging whether the tunnel design based on the selected lining elastic modulus and the lining thickness meets design requirements, if not, adjusting the lining elastic modulus and the lining thickness until the design requirements are met.

[0053] The tunnel lining design method based on the complex variable function method proposed in the embodiment is described in more detail through the following content.

[0054] Step S1, selecting a tunnel excavation radius, a tunnel burial depth, a lining elastic modulus and a lining thickness, determining a rock region and a lining region of the tunnel, and according to a mapping function, respectively mapping the rock region and the lining region of the tunnel in a physical plane into a rock annular region and a lining annular region in an image plane.

[0055] Specifically, first, the tunnel excavation radius, the tunnel buried depth, the lining elastic modulus and the lining thickness are selected to determine the rock mass region and the lining region of the tunnel, so as to analyze whether the tunnel lining design meets the design requirements. Through a mapping function, a region with a complex boundary shape on a physical plane (z plane) is transformed into a region with a simple boundary shape on an image plane (ζ plane), and then the solution of the problem can be obtained under the simple boundary by using the complex variable function method. According to the two mapping functions, the two regions of the z plane and the z1 plane, i.e. the rock mass region S1 and the lining region S2, are mapped to the two circular ring regions of the ζ plane and the ζ1 plane, i.e. the rock mass circular ring region Ω1 and the lining circular ring region Ω2, as shown in FIG. 4, FIG. 4A is the mapping of the rock mass region S1 to the rock mass circular ring region Ω1, and FIG. 4B is the mapping of the lining region S2 to the lining circular ring region Ω2. In the figures, L1 is the ground boundary of the physical plane, L2 is the interface of the S1 and S2 regions of the physical plane, L3 is the inner boundary of the lining of the physical plane, L'1 is the ground boundary of the image plane, L'2 is the interface of the S1 and S2 regions of the image plane, and L'3 is the inner boundary of the lining of the physical plane. The expressions of the mapping functions are as follows:

[0056] wherein the z plane refers to the plane where the rock mass region S1 is located, the z1 plane refers to the plane where the lining region S2 is located, R0 is the tunnel excavation radius, a = H(1-α 2 ) / (1+α 2 ), H is the tunnel buried depth.

[0057] In step S2, based on the mapped rock mass circular ring region and the lining circular ring region, the analytical functions of the rock mass region and the lining region under different contact conditions are respectively constructed. In the process of constructing the analytical functions, the analytical functions of the rock mass region under the action of no support, the analytical functions of the rock mass region under the action of the lining region, and the analytical functions of the lining region under the action of the rock mass region are respectively constructed, and the different contact conditions include the complete contact condition of the rock mass region and the lining region and the smooth contact condition of the rock mass region and the lining region.

[0058] Specifically, when the tunnel rock mass region is under the action of no support and only gravity, the analytical function corresponding to the rock mass region S1 is represented by and ψ1(z); as shown in FIG. 3, T is the lining thickness, R1 is the inner radius of the lining, L1 is the ground boundary of the physical plane, L2 is the interface of the S1 and S2 regions of the physical plane, and L3 is the inner boundary of the lining of the physical plane. After the lining is constructed, the rock mass and the lining interact, and the analytical function corresponding to the rock mass region S1 under the action of the lining region is represented by and ψ2(z), the rock mass region S1 is mapped to the region Ω1, and the analytical function corresponding to the region Ω1 is represented by and ψ2(ζ) represents; after the construction of the lining, the analytic function corresponding to the lining area S2 under the action of the rock mass area is represented by and ψ3(z1) represents, the lining area S2 is mapped to the area Ω2, and the analytic function corresponding to the area Ω2 is represented by and ψ3(ζ1) represents. The above analytic function can be specifically represented as:

[0059] wherein F x is a horizontal force, F x = 0, F y is a vertical force, κ1 is a coefficient, κ1 = 3-4μ1, μ1 is the Poisson's ratio of the rock mass, a k , b k , c k , d k are coefficients of the analytic function to be solved.

[0060] Step S3, according to the boundary conditions and the continuity conditions of the rock mass area, the lining area and the contact surface therebetween, the analytic functions are solved. That is, according to the boundary conditions and the continuity conditions of the rock mass area, the lining area and the contact surface therebetween, the analytic functions of the rock mass area and the lining area under the conditions of complete contact and smooth contact are solved, as shown in FIG. 2.

[0061] Step S3.1, the analytic functions of the rock mass area and the lining area under the condition of complete contact are solved, including: according to the stress boundary conditions of the inner boundary of the lining area, the stress continuity conditions of the excavation boundary and the displacement boundary conditions of the excavation boundary (all of which are well-known in the art), the analytic functions are solved to obtain the coefficients of the analytic functions; wherein the excavation boundary refers to the contact surface when the lining area and the rock mass area are in complete contact.

[0062] Firstly, since there is no external load acting on the inner boundary of the lining, the stress boundary condition of the inner boundary of the lining can be represented as:

[0063] wherein C1 is an unknown complex constant.

[0064] Secondly, since the contact surface between the lining and the rock mass satisfies the condition of complete contact, the stress continuity condition needs to be satisfied on the contact surface, and the expression is:

[0065] wherein C2 is an unknown complex constant.

[0066] Finally, based on the displacement of the rock mass region under no support, the displacement of the rock mass region under the support of the lining region, and the displacement of the lining region under the action of the rock mass region, the displacement continuity condition of the contact surface when the lining region and the rock mass region are in complete contact is determined.

[0067] Specifically, under gravity alone, the displacement expression for any point within the rock mass of an unsupported, shallowly buried circular tunnel is:

[0068] In the formula, G1 is the shear modulus of the tunnel, G1=E1 / [2(1+μ1)], E1 is the elastic modulus of the rock mass, μ1 is Poisson's ratio, and u1 is the elastic modulus of the rock mass. R and These are the horizontal and vertical displacement components of the rock mass under the action of gravity, respectively.

[0069] The displacement expression for any point in the rock mass of a shallow-buried tunnel under lining support only is:

[0070] In the formula, and These are the horizontal and vertical displacement components of the rock mass under the action of gravity, respectively.

[0071] The displacement expression for any point of the lining under the action of the rock mass is:

[0072] In the formula, u L and v L G1 and G2 are the horizontal and vertical displacement components of the lining under the action of the rock mass, respectively. G2 is the shear modulus of the lining, G2=E2 / [2(1+μ2)], E2 and μ2 are the elastic modulus and Poisson's ratio of the lining, respectively, and κ2=3-4μ2.

[0073] Since the contact surface between the lining and the rock mass satisfies the perfect contact condition, the displacement continuity condition is also satisfied at the contact surface, and its expression is:

[0074] By combining the above analytic functions, the coefficients of the analytic functions can be obtained by solving the system of linear equations.

[0075] Step S3.2: Solve for the analytical functions of the rock mass region and the lining region under smooth contact conditions, including: solving the analytical functions simultaneously based on the stress boundary conditions of the boundary of the lining region, the stress continuity condition of the excavation boundary, the shear stress condition of the excavation boundary, and the normal displacement continuity condition of the excavation boundary, and obtaining the coefficients of the analytical functions; where the excavation boundary refers to the contact surface when the lining region and the rock mass region are in complete contact.

[0076] First, since there is no external load acting on the inner boundary of the lining, the expression for the stress boundary condition is:

[0077] where β k is a coefficient, σ1 is a point on the outer boundary of the lining, c 01 +ic 02 represents a real parameter, and κ2 is a parameter of the lining.

[0078] Secondly, since the contact surface between the lining and the rock mass satisfies the complete contact condition, the stress continuity condition on the contact surface needs to be satisfied, and its expression is:

[0079] where σ is a point on the excavation boundary.

[0080] Then, the shear stress on the excavation boundary is zero, and its formula is:

[0081] where τ ρθ is the shear stress on the rock mass and the lining surface.

[0082] Finally, the normal displacement continuity condition on the excavation boundary is:

[0083] where u is the normal displacement of the excavation boundary of the tunnel without lining support under the action of gravity, is the normal displacement of the tunnel excavation boundary under the action of the lining support only, is the normal displacement of the outer boundary of the lining under the action of the rock mass.

[0084] By solving the linear equations, the coefficients of the analytical functions can be obtained by combining the above analytical functions.

[0085] Step S4, based on the analytical functions obtained by solving, combining the stress expressions and displacement expressions of the lining and the rock mass, the stress and displacement of any point in the lining region and the rock mass region are calculated and obtained.

[0086] Step S4.1, the stress of any point in the lining region and the rock mass region is calculated and obtained. The stress component expression of any point in the lining is:

[0087] where σ are the normal stress, tangential stress, and shear stress in the lining in the orthogonal curvilinear coordinate system, respectively; are the normal stress, tangential stress, and shear stress in the lining in the orthogonal curvilinear coordinate system without considering the original stress, respectively. Therefore, the stress components are superimposed as:

[0088] where σ respectively, are the normal stress, tangential stress and shear stress of any point in the rock mass under the orthogonal curve. Through the above formula, the stress of any point in the lining area and the rock mass area is calculated and obtained.

[0089] Step S4.2, the displacement of any point in the lining area and the rock mass area is calculated and obtained. Wherein, the displacement component expression of any point in the lining area is:

[0090] In the formula, u L and v L are the horizontal displacement and vertical displacement of any point in the lining, respectively.

[0091] The displacement component expression of any point in the rock mass area is:

[0092] In the formula, u R and v R are the horizontal displacement and vertical displacement of any point in the rock mass, respectively.

[0093] Based on the selected parameters of the lining (lining elastic modulus and lining thickness), the corresponding stress and displacement of the lining area and the rock mass area of the tunnel under the lining support condition can be obtained.

[0094] Step S5, based on the stress and displacement, it is judged whether the tunnel design based on the selected lining elastic modulus and lining thickness meets the design requirements. If not, adjust the lining elastic modulus and lining thickness, recalculate the stress and displacement and judge until the design requirements are met; if yes, optimize the lining elastic modulus and lining thickness to obtain the optimal parameter design of the lining.

[0095] Specifically, by querying the tunnel support design specification, if the stress and displacement of the tunnel under the lining support meet the specification, the tunnel design can be met; if the lining design is too conservative, the elastic modulus and thickness of the lining are reduced until the tunnel stress and displacement reach the minimum value required by the specification; if the lining design does not meet the design specification, the elastic modulus and thickness of the lining are increased until the tunnel stress and displacement meet the design specification.

[0096] The tunnel lining design method based on the complex variable function method provided in the embodiment solves the problem of poor precision of the method for determining the tunnel lining thickness by comprehensively considering multiple factors such as the service life of the tunnel, the geological conditions, and the diameter of the tunnel, and determining the lining thickness according to the lining elastic modulus and the lining thickness, and the stress and displacement solution of the tunnel rock mass and the lining under the lining support. In the embodiment, the complex variable function method can be mapped to multiple shapes of tunnels, and is applicable to various tunnel caverns. Moreover, the elastic modulus of the lining can be calculated, and the selection of the lining material model can be guided in combination with the lining thickness, thereby providing a basis for flexible support with variable lining thickness, and having important economic and technical values.

[0097] Embodiment two

[0098] The embodiment provides a tunnel lining design system based on a complex variable function method, which comprises:

[0099] The data acquisition module is configured to select a tunnel excavation radius, a tunnel burial depth, and a lining elastic modulus and a lining thickness, and determine a rock mass region and a lining region of the tunnel.

[0100] The mapping module is configured to map the rock mass region and the lining region of the tunnel in the physical plane into a rock mass annular region and a lining annular region in the image plane, respectively, according to a mapping function.

[0101] The analytic function construction module is configured to construct analytic functions of the rock mass region and the lining region under different contact conditions based on the mapped rock mass annular region and the lining annular region.

[0102] The analytic function solving module is configured to solve the analytic functions according to boundary conditions and continuity conditions of the rock mass region, the lining region, and a contact surface therebetween.

[0103] The stress and displacement calculation module is configured to calculate and obtain stress and displacement of any point in the lining region and the rock mass region, respectively, based on the solved analytic functions and stress expressions and displacement expressions of the lining and the rock mass.

[0104] The lining design adjustment module is configured to determine whether the tunnel design based on the selected lining elastic modulus and lining thickness meets the design requirements based on the stress and displacement, and if not, adjust the lining elastic modulus and the lining thickness until the design requirements are met.

[0105] Embodiment three

[0106] The embodiment provides an electronic device, which comprises a memory and a processor, and computer instructions stored in the memory and running on the processor. When the computer instructions are run by the processor, the steps in the tunnel lining design method based on the complex variable function method are completed.

[0107] Embodiment Four

[0108] The embodiment also provides a computer readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps in the tunnel lining design method based on the complex variable function method as described above.

[0109] Embodiment Five

[0110] The embodiment provides a tunnel lining construction method, as shown in FIG. 5, which includes the following steps.

[0111] Obtaining a tunnel excavation radius to be supported, a tunnel burial depth, setting an initial value of a lining elastic modulus and an initial value of a lining thickness, and determining a rock region and a lining region of the tunnel to be supported, respectively;

[0112] According to a mapping function, the rock region and the lining region of the tunnel in a physical plane are mapped into a rock annular region and a lining annular region in an image plane, respectively;

[0113] Based on the mapped rock annular region and the lining annular region, an analytical function of the rock region and the lining region under different contact conditions is constructed, respectively;

[0114] According to boundary conditions and continuity conditions of the rock region, the lining region and a contact surface therebetween, the analytical function is solved;

[0115] Based on the solved analytical function, in combination with stress expressions and displacement expressions of the lining and the rock, stress and displacement of any point in the lining region and the rock region are calculated, respectively;

[0116] The initial value of the lining elastic modulus and the initial value of the lining thickness are reset, and the stress and the displacement of any point in the lining region and the rock region are repeatedly calculated until the calculated stress and displacement meet a predefined design specification minimum allowable condition, and then the calculation is stopped;

[0117] The initial value of the lining elastic modulus and the initial value of the lining thickness, which are set when the calculated stress and displacement meet the predefined design specification minimum allowable condition, are recorded as an optimal value of the lining elastic modulus and an optimal value of the lining thickness, and the tunnel lining is constructed in the lining region of the tunnel to be supported by using the optimal value of the lining elastic modulus and the optimal value of the lining thickness, so as to support the tunnel to be supported.

[0118] The tunnel lining construction method proposed in the embodiment is described in more detail through the following content.

[0119] Step S1, obtaining a tunnel excavation radius and a tunnel buried depth of a tunnel to be supported, setting an initial value of a lining elastic modulus and a lining thickness, determining a rock region and a lining region of the tunnel to be supported, and mapping the rock region and the lining region of the tunnel to be supported in a physical plane into a rock annular region and a lining annular region in an image plane respectively according to a mapping function.

[0120] Specifically, first, a tunnel excavation radius and a tunnel buried depth of a tunnel to be supported are obtained, an initial value of a lining elastic modulus and a lining thickness are set, a rock region and a lining region of the tunnel to be supported are determined, and whether the tunnel lining design of the tunnel to be supported meets the design requirements is analyzed in this way. A mapping function is used to transform a region with a complex boundary shape in a physical plane (z plane) into a region with a simple boundary shape in an image plane (ζ plane), and then the solution of the problem can be obtained under the simple boundary by using the complex variable function method. According to two mapping functions, the two regions in the z plane and the z1 plane, i.e., the rock region S1 and the lining region S2, are mapped into the two annular regions in the ζ plane and the ζ1 plane, i.e., the rock annular region Ω1 and the lining annular region Ω2, as shown in FIG. 4. FIG. 4A shows that the rock region S1 is mapped into the rock annular region Ω1, and FIG. 4B shows that the lining region S2 is mapped into the lining annular region Ω2. In the figures, L1 is the ground boundary in the physical plane, L2 is the interface between the S1 and S2 regions in the physical plane, L3 is the inner boundary of the lining in the physical plane, L'1 is the ground boundary in the image plane, L'2 is the interface between the S1 and S2 regions in the image plane, and L'3 is the inner boundary of the lining in the image plane. The expressions of the mapping functions are as follows:

[0121] wherein the z plane refers to the plane where the rock region S1 is located, the z1 plane refers to the plane where the lining region S2 is located, R0 is the tunnel excavation radius, a = H (1-α 2 ) / (1+α 2 ), H is the tunnel buried depth.

[0122] Step S2, based on the mapped rock annular region and lining annular region, analytical functions of the rock region and the lining region under different contact conditions are respectively constructed. In the process of constructing the analytical functions, analytical functions of the rock region under no support, the rock region under the support of the lining region, and the lining region under the action of the rock region are respectively constructed, and the different contact conditions include complete contact condition and smooth contact condition of the rock region and the lining region.

[0123] Specifically, when the tunnel rock region is under no support and only under the action of gravity, the analytical function corresponding to the rock region S1 is and ψ1(z) represents; as shown in Figure 3, where T is the lining thickness, R1 is the inner radius of the lining, L1 is the ground boundary of the physical plane, L2 is the interface of the physical plane S1 and S2 area, and L3 is the inner boundary of the physical plane lining. After the construction of the lining, the rock mass and the lining interact, and the corresponding analytical function of the rock mass area S1 under the support of the lining area is and ψ2(z) represents, the rock mass area S1 is mapped to the area Ω1, and the corresponding analytical function of the area Ω1 is and ψ2(ζ) represents; after the construction of the lining, the corresponding analytical function of the lining area S2 under the action of the rock mass area is and ψ3(z1) represents, the lining area S2 is mapped to the area Ω2, and the corresponding analytical function of the area Ω2 is and ψ3(ζ1) represents. The above analytical function can be specifically represented as:

[0124] wherein F x is the horizontal force, F x = 0, F y is the vertical force, κ1 is a coefficient, κ1 = 3-4μ1, μ1 is the Poisson's ratio of the rock mass, a k , b k , c k , d k are the coefficients of the analytical function to be solved.

[0125] Step S3, according to the boundary conditions and continuity conditions of the rock mass area, the lining area and the contact surface thereof, the analytical function is solved. That is, as shown in Figure 2, according to the boundary conditions and continuity conditions of the rock mass area, the lining area and the contact surface thereof, the analytical functions of the rock mass area and the lining area under the complete contact condition and the smooth contact condition are solved respectively.

[0126] Step S3.1, solving the analytical functions of the rock mass area and the lining area under the complete contact condition, comprising: according to the stress boundary condition of the inner boundary of the lining area, the stress continuity condition of the excavation boundary and the displacement boundary condition of the excavation boundary (all of which are well-known in the art), the analytical functions are solved to obtain the coefficients of the analytical functions; wherein the excavation boundary refers to the contact surface when the lining area and the rock mass area are in complete contact.

[0127] Firstly, since there is no external load acting on the inner boundary of the lining, the stress boundary condition of the inner boundary of the lining can be represented as:

[0128] wherein C1 is an unknown complex constant.

[0129] Secondly, since the contact surface between the lining and the rock mass satisfies the complete contact condition, the stress continuity condition on the contact surface is needed to be satisfied, and its expression is:

[0130] where C2 is an unknown complex constant.

[0131] Finally, according to the displacement of the rock mass region under the action of no support, the displacement of the rock mass region under the action of the support of the lining region, and the displacement of the lining region under the action of the rock mass region, the displacement continuity condition of the contact surface when the lining region and the rock mass region are in complete contact is determined.

[0132] Specifically, the displacement expression of any point in the rock mass of the shallow-buried circular tunnel under the action of gravity without support is:

[0133] where G1 is the shear modulus of the tunnel, G1=E1 / [2(1+μ1)], E1 is the elastic modulus of the rock mass, μ1 is the Poisson's ratio, and are the displacement components of the rock mass in the horizontal direction and the vertical direction under the action of gravity, respectively.

[0134] The displacement expression of any point in the rock mass of the shallow-buried tunnel under the action of the lining support is:

[0135] where and are the displacement components of the rock mass in the horizontal direction and the vertical direction under the action of gravity, respectively.

[0136] The displacement expression of any point in the lining under the action of the rock mass is:

[0137] where u L and v L are the displacement components of the lining in the horizontal direction and the vertical direction under the action of the rock mass, respectively, G2 is the shear modulus of the lining, G2=E2 / [2(1+μ2)], E2 and μ2 are the elastic modulus and the Poisson's ratio of the lining, respectively, and κ2=3-4μ2.

[0138] Since the contact surface between the lining and the rock mass satisfies the complete contact condition, the displacement continuity condition on the contact surface is needed to be satisfied, and its expression is:

[0139] The above analytical functions are combined to obtain the coefficients of the analytical functions by solving the linear equations.

[0140] S3.2, solving the analytical function of the rock mass region and the lining region under the smooth contact condition, comprising: according to the stress boundary condition of the inner boundary of the lining region, the stress continuity condition of the excavation boundary, the shear stress condition of the excavation boundary and the normal displacement continuity condition of the excavation boundary, the analytical functions are solved to obtain the coefficients of the analytical functions; wherein the excavation boundary refers to the contact surface when the lining region and the rock mass region are in complete contact.

[0141] Firstly, since there is no external load acting on the inner boundary of the lining, the expression of the stress boundary condition is:

[0142] In the formula, β = 1-T / R0, σ1 is a point on the outer boundary of the lining, c 01 +ic 02 represents a real parameter, and κ2 is a parameter of the lining.

[0143] Secondly, since the contact surface of the lining and the rock mass satisfies the complete contact condition, the stress continuity condition needs to be satisfied on the contact surface, and the expression is:

[0144] In the formula, σ is a point on the excavation boundary.

[0145] Then, the shear stress on the excavation boundary is zero, and the formula is:

[0146] In the formula, τ ρθ is the shear stress on the rock mass and the lining surface.

[0147] Finally, the normal displacement continuity condition on the excavation boundary is:

[0148] In the formula, is the normal displacement of the excavation boundary of the tunnel without lining support under the action of gravity, is the normal displacement of the tunnel excavation boundary only under the action of lining support, is the normal displacement of the outer boundary of the lining under the action of the rock mass.

[0149] By solving the above analytical functions, the coefficients of the analytical functions can be obtained by solving the linear equation set.

[0150] Step S4, based on the analytical functions obtained by solving, combining the stress expressions and displacement expressions of the lining and the rock mass, the stress and displacement of any point in the lining region and the rock mass region are calculated and obtained.

[0151] Step S4.1, calculating and obtaining the stress of any point in the lining region and the rock mass region. The stress component expression of any point in the lining is:

[0152] wherein, are normal stress, tangential stress and shear stress in the lining under the orthogonal curvilinear coordinate system respectively; are normal stress, tangential stress and shear stress in the lining under the orthogonal curvilinear coordinate system respectively without considering the original stress. Therefore, the stress components are superimposed as:

[0153] wherein, are normal stress, tangential stress and shear stress in the lining under the orthogonal curvilinear coordinate system respectively without considering the original stress. Therefore, the stress components are superimposed as:

[0154] Step S4.2, the displacement of any point in the lining area and the rock mass area is calculated. Wherein, the displacement component expression of any point in the lining area is:

[0155] wherein, u L and v L are the horizontal displacement and vertical displacement of any point in the lining respectively.

[0156] The displacement component expression of any point in the rock mass area is:

[0157] wherein, u R and v R are the horizontal displacement and vertical displacement of any point in the rock mass respectively.

[0158] Based on the selected parameters of the lining (the elastic modulus of the lining and the thickness of the lining), the corresponding stress and displacement of the lining area and the rock mass area of the tunnel under the lining support condition can be obtained.

[0159] Step S5, the initial value of the elastic modulus of the lining and the initial value of the thickness of the lining are reset, and the stress and displacement of any point in the lining area and the rock mass area are repeatedly calculated until the calculated stress and displacement meet the minimum allowable conditions of the predefined design specification, and then the calculation is stopped.

[0160] Specifically, by querying the tunnel support design specification, if the stress and displacement of the tunnel under the lining support meet the specification, the tunnel design can be met; if the lining design is too conservative, the elastic modulus and thickness of the lining are reduced until the stress and displacement of the tunnel reach the minimum value required by the specification; if the lining design does not meet the design specification, the elastic modulus and thickness of the lining are increased until the stress and displacement of the tunnel meet the design specification.

[0161] Step S6, record the initial value of the lining elastic modulus and the initial value of the lining thickness set when the calculated stress and displacement meet the predefined design specification minimum allowable conditions as the optimal value of the lining elastic modulus and the optimal value of the lining thickness, and construct the tunnel lining in the lining area of the tunnel to be supported using the optimal value of the lining elastic modulus and the optimal value of the lining thickness to support the tunnel to be supported.

[0162] The tunnel lining construction method proposed in the embodiments can accurately select the model of the lining material according to the optimal solution parameter values of the elastic modulus and the thickness of the lining screened by calculation, and provides a construction parameter basis for flexible support with variable lining thickness, and has important economic and technical values.

[0163] The steps and methods involved in the above embodiments two to four correspond to the method embodiment one, and the specific implementation can refer to the relevant description part of the embodiment one. The term "computer readable storage medium" should be understood as including a single medium or multiple media of one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying instruction sets for execution by a processor and causing the processor to perform any method in the present application.

[0164] Those skilled in the art should understand that the above-mentioned modules or steps of the present application can be realized by a general computer device, and alternatively, they can be realized by program codes executable by a computing device, so that they can be stored in a storage device for execution by a computing device, or they can be respectively manufactured into individual integrated circuit modules, or a plurality of modules or steps among them can be manufactured into a single integrated circuit module to realize. The present application is not limited to any specific combination of hardware and software.

[0165] The above only describes the preferred embodiments of the present application, and the specific implementation of the present application is described in conjunction with the drawings, but it is not a limitation on the protection scope of the present application. Those skilled in the art should understand that various modifications or variations made on the basis of the technical solutions of the present application without creative labor are still within the protection scope of the present application.

Claims

1. A tunnel lining design method based on complex variable function method, characterized in that, The method comprises the following steps: selecting a tunnel excavation radius, a tunnel buried depth, a lining elastic modulus and a lining thickness, and determining a rock mass region and a lining region of the tunnel; mapping the rock mass region and the lining region of the tunnel into a rock mass annular region and a lining annular region respectively according to a mapping function; constructing analytical functions of the rock mass region and the lining region under different contact conditions respectively based on the mapped rock mass annular region and the lining annular region; solving the analytical functions according to boundary conditions and continuity conditions of the rock mass region, the lining region and a contact surface therebetween; calculating and obtaining stress and displacement of any point in the lining region and the rock mass region respectively based on the solved analytical functions, and combining stress expressions and displacement expressions of the lining and the rock mass; judging whether the tunnel design based on the selected lining elastic modulus and lining thickness meets design requirements based on the stress and the displacement, and if not, adjusting the lining elastic modulus and the lining thickness until the design requirements are met.

2. The tunnel lining design method based on complex variable function method according to claim 1, characterized in that, The different contact conditions include a complete contact condition of the rock mass region and the lining region and a smooth contact condition of the rock mass region and the lining region.

3. The tunnel lining design method based on complex variable function method according to claim 1, characterized in that, The process of solving the analytical functions comprises the following steps: constructing analytical functions of the rock mass region under no support, the rock mass region under support of the lining region, and the lining region under action of the rock mass region respectively based on the mapped rock mass annular region and the lining annular region; solving the analytical functions of the rock mass region and the lining region under the complete contact condition and the smooth contact condition respectively according to the boundary conditions and the continuity conditions of the rock mass region, the lining region and the contact surface therebetween.

4. The tunnel lining design method based on complex variable function method according to claim 3, characterized in that, Solving the analytical functions of the rock mass region and the lining region under the complete contact condition comprises the following steps: solving the analytical functions by simultaneously solving the analytical functions according to stress boundary conditions of an inner boundary of the lining region, stress continuity conditions of an excavation boundary and displacement boundary conditions of the excavation boundary, and obtaining coefficients of the analytical functions; wherein the excavation boundary refers to a contact surface when the lining region and the rock mass region are in complete contact.

5. The tunnel lining design method based on complex variable function method according to claim 4, characterized in that, determining displacement continuity conditions of the contact surface when the lining region and the rock mass region are in complete contact according to displacement of the rock mass region under no support, displacement of the rock mass region under support of the lining region, and displacement of the lining region under action of the rock mass region.

6. The tunnel lining design method based on complex variable function method according to claim 3, characterized in that, Solving the analytical functions of the rock mass region and the lining region under the smooth contact condition comprises the following steps: solving the analytical functions by simultaneously solving the analytical functions according to stress boundary conditions of an inner boundary of the lining region, stress continuity conditions of an excavation boundary, shear stress conditions of the excavation boundary and normal displacement continuity conditions of the excavation boundary, and obtaining coefficients of the analytical functions; wherein the excavation boundary refers to a contact surface when the lining region and the rock mass region are in complete contact.

7. The tunnel lining design method based on complex variable function method according to claim 1, characterized in that, Judging whether the tunnel design based on the selected lining elastic modulus and lining thickness meets design requirements based on the stress and the displacement, and if not, increasing the lining elastic modulus and the lining thickness, recalculating the stress and the displacement and judging until the stress and the displacement meet the tunnel design requirements; if yes, reducing the lining elastic modulus or the lining thickness to obtain optimal parameter design of the lining.

8. A system for tunnel lining design based on complex function method, characterized in that, The method comprises the following steps: a data acquisition module is configured to select a tunnel excavation radius, a tunnel buried depth, a lining elastic modulus and a lining thickness, and determine a rock mass region and a lining region of the tunnel; The mapping module is configured to map the rock mass region and the lining region of the tunnel into a rock mass annular region and a lining annular region, respectively, according to a mapping function; The analytic function construction module is configured to construct analytic functions of the rock mass region and the lining region under different contact conditions based on the mapped rock mass annular region and the lining annular region; The analytic function solving module is configured to solve the analytic functions according to boundary conditions and continuity conditions of the rock mass region, the lining region and the contact surface therebetween; The stress and displacement calculation module is configured to calculate and obtain stresses and displacements of any point in the lining region and the rock mass region based on the solved analytic functions, and stress expressions and displacement expressions of the lining and the rock mass; The lining design adjustment module is configured to judge whether the tunnel design based on the selected elastic modulus of the lining and the thickness of the lining meets the design requirements based on the stresses and the displacements, and if not, to adjust the elastic modulus of the lining and the thickness of the lining until the design requirements are met.

9. An electronic device, characterized by comprising: The computer program product comprises a memory and a processor, and computer instructions stored in the memory and run on the processor, and when the computer instructions are run by the processor, the steps of the tunnel lining design method based on the complex variable function method in any one of claims 1-7 are completed.

10. A computer readable storage medium characterized by, The computer program product is configured to store computer instructions, and when the computer instructions are executed by the processor, the steps of the tunnel lining design method based on the complex variable function method in any one of claims 1-7 are completed.

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