WO2026017170A1 - 基于复变函数法的隧道衬砌设计方法及系统 - Google Patents

基于复变函数法的隧道衬砌设计方法及系统

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Publication number
WO2026017170A1
WO2026017170A1 PCT/CN2025/109445 CN2025109445W WO2026017170A1 WO 2026017170 A1 WO2026017170 A1 WO 2026017170A1 CN 2025109445 W CN2025109445 W CN 2025109445W WO 2026017170 A1 WO2026017170 A1 WO 2026017170A1
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Prior art keywords
lining
region
rock mass
tunnel
stress
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English (en)
French (fr)
Inventor
李利平
刘洪亮
蔡辉
陈昌源
屠文锋
孙尚渠
高鑫
王世成
张旭彤
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Shandong University
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Shandong University
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/13—Architectural design, e.g. computer-aided architectural design [CAAD] related to design of buildings, bridges, landscapes, production plants or roads
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2113/00—Details relating to the application field
    • G06F2113/14—Pipes

Definitions

  • This invention relates to the field of tunnel construction technology, and in particular to a tunnel lining design method and system based on the complex variable function method, as well as a tunnel lining construction method.
  • tunnel lining is usually used to support the tunnel rock mass.
  • the supporting effect of the lining is determined by parameters such as the elastic modulus and thickness of the lining. Therefore, the design of lining parameters affects the safety and economy of the tunnel.
  • Research on tunnel lining parameter design is one of the main research directions in tunnel engineering.
  • parameters such as the elastic modulus (EM) and thickness of tunnel linings are generally determined by engineering analogies based on multiple factors, including tunnel service life, geological conditions, and tunnel diameter.
  • the tunnel lining thickness and EEM determined by this method have poor adaptability to actual engineering projects.
  • the fabrication and support of linings have become increasingly faster.
  • Accurately calculating and optimizing parameters such as the lining's EEM and thickness has become crucial for scientific support. Designing linings that are compatible with geological conditions has significant economic and technical value.
  • this invention provides a tunnel lining design method and system based on the complex variable function method.
  • This method maps tunnels of different shapes using the complex variable function method and analyzes the lining design under different rock mass-lining contact conditions. It optimizes the elastic modulus and thickness of the lining design, improving the scientific rigor and accuracy of the lining design parameter calculations.
  • This method has broad applicability. When applied to actual tunnel lining construction projects, the optimized values of the output lining's elastic modulus and thickness can be used to construct the lining within the tunnel, achieving effective tunnel support.
  • the present invention provides a tunnel lining design method based on the complex variable function method.
  • a tunnel lining design method based on the complex function method includes:
  • the rock mass region and lining region of the tunnel are determined.
  • the rock mass region and lining region of the physical plane tunnel are mapped to the image plane region respectively;
  • the present invention provides a tunnel lining design system based on the complex function method.
  • a tunnel lining design system based on the complex function method includes:
  • the data acquisition module is used to select the tunnel excavation radius, tunnel burial depth, lining elastic modulus, and lining thickness to determine the rock mass area and lining area of the tunnel.
  • the mapping module is used to map the rock mass region and lining region of the tunnel to the image plane rock mass annular region and lining annular region, respectively, according to the mapping function;
  • the analytical function construction module is used to construct analytical functions for the rock mass annular region and the lining annular region under different contact conditions based on the rock mass annular region and the lining annular region in the mapped image plane.
  • the analytical function solving module is used to solve analytical functions based on the boundary conditions and continuity conditions of the rock mass region, the lining region, and their contact surfaces.
  • the stress and displacement calculation module is used to calculate the stress and displacement at any point in the lining area and the rock mass area based on the analytical function obtained by solving, combined with the stress and displacement expressions of the lining and the rock mass.
  • the lining design adjustment module is used to determine whether the tunnel design based on the selected lining elastic modulus and lining thickness meets the design requirements based on stress and displacement. If not, the lining elastic modulus and lining thickness are adjusted until the design requirements are met.
  • the present invention also provides an electronic device, including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when executed by the processor, perform the steps of the method described in the first aspect.
  • the present invention also provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, perform the steps of the method described in the first aspect.
  • a method for constructing tunnel lining comprising:
  • the initial values of the lining elastic modulus and the initial values of the lining thickness are set when the calculated stress and displacement meet the minimum allowable conditions of the predefined design specifications. These are used as the optimal values of the lining elastic modulus and the optimal values of the lining thickness.
  • the tunnel lining is constructed using these optimal values of the lining elastic modulus and the optimal values of the lining thickness within the lining area of the tunnel to be supported, thus supporting the tunnel to be supported.
  • This invention provides a tunnel lining design method and system based on the complex variable function method. It solves the problem that existing methods for determining tunnel lining thickness based on multiple factors such as tunnel service life, geological conditions, and tunnel diameter have poor accuracy.
  • the constructed tunnel lining is more suitable for tunnel construction.
  • the stress and displacement solutions of the tunnel rock mass and lining under lining support are calculated using the complex variable function method. Based on these solutions, the elastic modulus and thickness of the lining are optimized. Since the complex variable function method can map tunnels of various shapes, it is applicable to various types of tunnel chambers. Furthermore, it can calculate the elastic modulus of the lining, and combined with the lining thickness, it can guide the selection of lining material types, providing a foundation for flexible support with variable lining thickness, and has significant economic and technical value.
  • Figure 1 is a flowchart of the tunnel lining design method based on the complex variable function method according to an embodiment of the present invention
  • Figure 2 is a flowchart of the stress and displacement calculation in the rock mass region of the circular tunnel supported by the lining area in an embodiment of the present invention under different contact conditions.
  • Figure 3 is a schematic diagram of the lining area supporting the rock mass area of a circular tunnel in an embodiment of the present invention
  • Figure 4A is a schematic diagram of the rock mass region S1 being mapped to the rock mass annular region ⁇ 1 in an embodiment of the present invention
  • Figure 4B is a schematic diagram of the lining region S2 being mapped to the lining annular region ⁇ 2 in an embodiment of the present invention
  • Figure 5 is a flowchart of the tunnel lining construction method according to an embodiment of the present invention.
  • This embodiment provides a tunnel lining design method based on the complex function method, as shown in Figure 1, including:
  • the rock mass region and lining region of the tunnel in the physical plane are mapped to the rock mass annular region and lining annular region in the image plane, respectively.
  • Step S1 Select the tunnel excavation radius, tunnel burial depth, lining elastic modulus, and lining thickness to determine the rock mass region and lining region of the tunnel. Based on the mapping function, map the rock mass region and lining region of the tunnel in the physical plane to the rock mass annular region and lining annular region in the image plane, respectively.
  • the tunnel excavation radius, tunnel burial depth, lining elastic modulus, and lining thickness are selected first to determine the rock mass region and lining region of the tunnel. Based on this, it is analyzed whether the tunnel lining design meets the design requirements. Through a mapping function, a region with a complex boundary shape on the physical plane (z-plane) is transformed into a region with a simple boundary shape on the imaging plane ( ⁇ -plane). Then, the solution to the problem can be obtained under simple boundary conditions using the complex variable function method.
  • the two regions of the z-plane and z -1 plane namely the rock mass region S1 and the lining region S2 are mapped to the two annular regions of the ⁇ -plane and ⁇ -1 plane, namely the rock mass annular region ⁇ 1 and the lining annular region ⁇ 2 , respectively, as shown in Figure 4.
  • Figure 4A shows the mapping of the rock mass region S1 to the rock mass annular region ⁇ 1
  • Figure 4B shows the mapping of the lining region S2 to the lining annular region ⁇ 2 .
  • L1 is the surface boundary of the physical plane
  • L2 is the interface between the physical plane regions S1 and S2
  • L3 is the inner boundary of the physical plane lining
  • L' 1 is the surface boundary of the image plane
  • L' 2 is the interface between the image plane regions S1 and S2
  • L' 3 is the inner boundary of the physical plane lining.
  • the z-plane refers to the plane containing the rock mass region S1
  • the z -1 plane refers to the plane containing the lining region S2
  • R0 is the tunnel excavation radius
  • a H(1- ⁇ 2 )/(1+ ⁇ 2 ).
  • H represents the tunnel depth.
  • Step S2 Based on the mapped rock mass annular region and lining annular region, construct analytical functions for the rock mass region and lining region under different contact conditions.
  • the different contact conditions include: the rock mass region and the lining region in complete contact and the rock mass region and the lining region in smooth contact.
  • the analytical function corresponding to the rock mass region S1 is derived from... And ⁇ 1 (z) represents; as shown in Figure 3, T is the lining thickness, R1 is the inner radius of the lining, L1 is the surface boundary of the physical plane, L2 is the interface between the physical planes S1 and S2 , and L3 is the inner boundary of the physical plane lining.
  • T is the lining thickness
  • R1 is the inner radius of the lining
  • L1 is the surface boundary of the physical plane
  • L2 is the interface between the physical planes S1 and S2
  • L3 is the inner boundary of the physical plane lining.
  • And ⁇ 2 (z) represents that the rock mass region S1 is mapped to region ⁇ 1 , and the analytic function corresponding to region ⁇ 1 is given by And ⁇ 2 ( ⁇ ) represents; after the lining is constructed, the analytical function corresponding to the lining region S2 under the action of the rock mass region is given by And ⁇ 3 ( z1 ) indicates that the lining region S2 is mapped to region ⁇ 2 , and the analytic function corresponding to region ⁇ 2 is given by And ⁇ 3 ( ⁇ 1 ) represents it.
  • the above analytic function can be specifically expressed as:
  • F ⁇ sub>x ⁇ /sub> is the horizontal force
  • ⁇ 1 is a coefficient
  • ⁇ 1 3-4 ⁇ 1
  • ⁇ 1 is the Poisson's ratio of the rock mass
  • a k , b k , c k , d k are the coefficients of the analytic function to be determined.
  • Step S3 Solve for the analytical functions based on the boundary conditions and continuity conditions of the rock mass region, the lining region, and their contact surfaces. That is, as shown in Figure 2, solve for the analytical functions of the rock mass region and the lining region under complete contact conditions and under smooth contact conditions, respectively, based on the boundary conditions and continuity conditions of the rock mass region, the lining region, and their contact surfaces.
  • Step S3.1 Solve for the analytical functions of the rock mass region and the lining region under the condition of complete contact, 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, and the displacement boundary conditions of the excavation boundary (all of which are common knowledge in the field), and obtaining 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.
  • the stress boundary condition of the inner boundary of the lining can be expressed as:
  • the displacement continuity condition of the contact surface when the lining region and the rock mass region are in complete contact is determined.
  • G1 is the shear modulus of the tunnel
  • G1 E1 / [2(1+ ⁇ 1 )]
  • E1 is the elastic modulus of the rock mass
  • ⁇ 1 is Poisson's ratio
  • u1 is R.
  • uL and vL 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+ ⁇ 2 )]
  • E2 and ⁇ 2 are the elastic modulus and Poisson's ratio of the lining, respectively
  • ⁇ 2 3-4 ⁇ 2 .
  • the coefficients of the analytic functions can be obtained by solving the system of linear equations.
  • 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.
  • ⁇ k is a coefficient
  • ⁇ 1 is a point on the outer boundary of the lining
  • c01 + ic02 represents a real parameter
  • ⁇ 2 is a parameter of the lining.
  • ⁇ is a point on the excavation boundary.
  • ⁇ ⁇ represents the shear stress on the rock mass and the lining surface.
  • the coefficients of the analytic functions can be obtained by solving the system of linear equations.
  • Step S4 Based on the analytical function obtained from the solution, and combined with the stress and displacement expressions of the lining and rock mass, calculate and obtain the stress and displacement at any point in the lining area and the rock mass area respectively.
  • Step S4.1 Calculate the stress at any point in the lining area and the rock mass area.
  • the stress component expression for any point within the lining is:
  • Step S4.2 Calculate and obtain the displacement of any point in the lining area and the rock mass area.
  • the expression for the displacement component of any point in the lining area is as follows:
  • uL and vL are the horizontal and vertical displacements of any point inside the lining, respectively.
  • uR and vR are the horizontal and vertical displacements of any point within the rock mass, respectively.
  • the stress and displacement of the lining area and rock mass area of the tunnel under the lining support conditions can be obtained.
  • Step S5 Based on stress and displacement, determine 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 stress and displacement, and make a judgment until the design requirements are met. If yes, optimize the lining elastic modulus and lining thickness to obtain the optimal lining parameter design.
  • the tunnel design is satisfied; if the lining design is too conservative, the elastic modulus and thickness of the lining should be reduced until the tunnel stress and displacement reach the minimum value required by the specifications; if the lining design does not meet the design specifications, the elastic modulus and thickness of the lining should be increased until the tunnel stress and displacement meet the design specifications.
  • the tunnel lining design method based on the complex variable function method proposed in this embodiment solves the problem that existing methods for determining tunnel lining thickness based on multiple factors such as tunnel service life, geological conditions, and tunnel diameter have poor accuracy.
  • the stress and displacement solutions of the tunnel rock mass and lining under lining support are calculated using the complex variable function method.
  • the elastic modulus and thickness of the lining are optimized. Since the complex variable function method can map tunnels of various shapes, it is applicable to various types of tunnel chambers. Moreover, it can calculate the elastic modulus of the lining, and combined with the lining thickness, it can provide guidance for the selection of lining material types, providing a basis for flexible support with variable lining thickness, and has significant economic and technical value.
  • This embodiment provides a tunnel lining design system based on the complex function method, including:
  • the data acquisition module is used to select the tunnel excavation radius, tunnel burial depth, lining elastic modulus, and lining thickness to determine the rock mass area and lining area of the tunnel.
  • the mapping module is used to map the rock mass region and lining region of the tunnel in the physical plane to the rock mass annular region and lining annular region in the image plane, respectively, according to the mapping function.
  • the analytical function construction module is used to construct analytical functions for 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 analytical function solving module is used to solve analytical functions based on the boundary conditions and continuity conditions of the rock mass region, the lining region, and their contact surfaces.
  • the stress and displacement calculation module is used to calculate the stress and displacement at any point in the lining area and the rock mass area based on the analytical function obtained by solving, combined with the stress and displacement expressions of the lining and the rock mass.
  • the lining design adjustment module is used to determine whether the tunnel design based on the selected lining elastic modulus and lining thickness meets the design requirements based on stress and displacement. If not, the lining elastic modulus and lining thickness are adjusted until the design requirements are met.
  • This embodiment provides an electronic device, including a memory and a processor, as well as computer instructions stored in the memory and running on the processor.
  • the processor executes the computer instructions, it completes the steps in the tunnel lining design method based on the complex function method as described above.
  • This 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 function method as described above.
  • This embodiment provides a tunnel lining construction method, as shown in Figure 5, including:
  • the rock mass region and lining region of the tunnel in the physical plane are mapped to the rock mass annular region and lining annular region in the image plane, respectively.
  • the initial values of the lining elastic modulus and the initial values of the lining thickness are set when the calculated stress and displacement meet the minimum allowable conditions of the predefined design specifications. These are used as the optimal values of the lining elastic modulus and the optimal values of the lining thickness.
  • the tunnel lining is constructed using these optimal values of the lining elastic modulus and the optimal values of the lining thickness within the lining area of the tunnel to be supported, thus supporting the tunnel to be supported.
  • Step S1 Obtain the tunnel excavation radius and tunnel burial depth of the tunnel to be supported, set the initial value of the lining elastic modulus and the lining thickness, determine the rock mass region and lining region of the tunnel to be supported, and map the rock mass region and lining region of the tunnel to be supported in the physical plane to the rock mass annular region and lining annular region in the image plane respectively according to the mapping function.
  • the tunnel excavation radius and tunnel burial depth of the tunnel to be supported are obtained, the initial value of the lining elastic modulus and the lining thickness are set, and the rock mass area and lining area of the tunnel to be supported are determined. Based on this, it is analyzed whether the tunnel lining design of the tunnel to be supported meets the design requirements. Through a mapping function, a region with a complex boundary shape on the physical plane (z-plane) is transformed into a region with a simple boundary shape on the imaging plane ( ⁇ -plane). Then, the solution to the problem can be obtained under the simple boundary using the complex variable function method.
  • the two regions of the z-plane and z -1 plane namely the rock mass region S1 and the lining region S2 are mapped to the two annular regions of the ⁇ -plane and ⁇ -1 plane, namely the rock mass annular region ⁇ 1 and the lining annular region ⁇ 2 , respectively, as shown in Figure 4.
  • Figure 4A shows the mapping of the rock mass region S1 to the rock mass annular region ⁇ 1
  • Figure 4B shows the mapping of the lining region S2 to the lining annular region ⁇ 2 .
  • L1 is the surface boundary of the physical plane
  • L2 is the interface between the physical plane regions S1 and S2
  • L3 is the inner boundary of the physical plane lining
  • L' 1 is the surface boundary of the image plane
  • L' 2 is the interface between the image plane regions S1 and S2
  • L' 3 is the inner boundary of the physical plane lining.
  • the z-plane refers to the plane containing the rock mass region S1
  • the z -1 plane refers to the plane containing the lining region S2
  • R0 is the tunnel excavation radius
  • a H(1- ⁇ 2 )/(1+ ⁇ 2 ).
  • H represents the tunnel depth.
  • Step S2 Based on the mapped rock mass annular region and lining annular region, construct analytical functions for the rock mass region and lining region under different contact conditions.
  • the different contact conditions include: the rock mass region and the lining region in complete contact and the rock mass region and the lining region in smooth contact.
  • the analytical function corresponding to the rock mass region S1 is derived from... And ⁇ 1 (z) represents; as shown in Figure 3, T is the lining thickness, R1 is the inner radius of the lining, L1 is the surface boundary of the physical plane, L2 is the interface between the physical planes S1 and S2 , and L3 is the inner boundary of the physical plane lining.
  • T is the lining thickness
  • R1 is the inner radius of the lining
  • L1 is the surface boundary of the physical plane
  • L2 is the interface between the physical planes S1 and S2
  • L3 is the inner boundary of the physical plane lining.
  • And ⁇ 2 (z) represents that the rock mass region S1 is mapped to region ⁇ 1 , and the analytic function corresponding to region ⁇ 1 is given by And ⁇ 2 ( ⁇ ) represents; after the lining is constructed, the analytical function corresponding to the lining region S2 under the action of the rock mass region is given by And ⁇ 3 ( z1 ) indicates that the lining region S2 is mapped to region ⁇ 2 , and the analytic function corresponding to region ⁇ 2 is given by And ⁇ 3 ( ⁇ 1 ) represents it.
  • the above analytic function can be specifically expressed as:
  • F ⁇ sub>x ⁇ /sub> is the horizontal force
  • ⁇ 1 is a coefficient
  • ⁇ 1 3-4 ⁇ 1
  • ⁇ 1 is the Poisson's ratio of the rock mass
  • a k , b k , c k , d k are the coefficients of the analytic function to be determined.
  • Step S3 Solve for the analytical functions based on the boundary conditions and continuity conditions of the rock mass region, the lining region, and their contact surfaces. That is, as shown in Figure 2, solve for the analytical functions of the rock mass region and the lining region under complete contact conditions and under smooth contact conditions, respectively, based on the boundary conditions and continuity conditions of the rock mass region, the lining region, and their contact surfaces.
  • Step S3.1 Solve for the analytical functions of the rock mass region and the lining region under the condition of complete contact, 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, and the displacement boundary conditions of the excavation boundary (all of which are common knowledge in the field), and obtaining 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.
  • the stress boundary condition of the inner boundary of the lining can be expressed as:
  • the displacement continuity condition of the contact surface when the lining region and the rock mass region are in complete contact is determined.
  • G1 is the shear modulus of the tunnel
  • G1 E1 / [2(1+ ⁇ 1 )]
  • E1 is the elastic modulus of the rock mass
  • ⁇ 1 is Poisson's ratio.
  • uL and vL 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+ ⁇ 2 )]
  • E2 and ⁇ 2 are the elastic modulus and Poisson's ratio of the lining, respectively
  • ⁇ 2 3-4 ⁇ 2 .
  • the coefficients of the analytic functions can be obtained by solving the system of linear equations.
  • 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.
  • ⁇ 1-T/R 0
  • ⁇ 1 is a point on the outer boundary of the lining
  • c 01 +ic 02 represents a real parameter
  • ⁇ 2 is a parameter of the lining.
  • ⁇ is a point on the excavation boundary.
  • ⁇ ⁇ represents the shear stress on the rock mass and the lining surface.
  • the coefficients of the analytic functions can be obtained by solving the system of linear equations.
  • Step S4 Based on the analytical function obtained from the solution, and combined with the stress and displacement expressions of the lining and rock mass, calculate and obtain the stress and displacement at any point in the lining area and the rock mass area respectively.
  • Step S4.1 Calculate the stress at any point in the lining area and the rock mass area.
  • the stress component expression for any point within the lining is:
  • Step S4.2 Calculate and obtain the displacement of any point in the lining area and the rock mass area.
  • the expression for the displacement component of any point in the lining area is as follows:
  • uL and vL are the horizontal and vertical displacements of any point inside the lining, respectively.
  • uR and vR are the horizontal and vertical displacements of any point within the rock mass, respectively.
  • the stress and displacement of the lining area and rock mass area of the tunnel under the lining support conditions can be obtained.
  • Step S5 Reset the initial values of the lining elastic modulus and the lining thickness, and repeat the calculation of stress and displacement at any point in the lining area and the rock mass area until the calculated stress and displacement meet the minimum allowable conditions of the predefined design specifications, then stop the calculation.
  • the tunnel design is satisfied; if the lining design is too conservative, the elastic modulus and thickness of the lining should be reduced until the tunnel stress and displacement reach the minimum value required by the specifications; if the lining design does not meet the design specifications, the elastic modulus and thickness of the lining should be increased until the tunnel stress and displacement meet the design specifications.
  • Step S6 Record the initial values of the lining elastic modulus and lining thickness when the calculated stress and displacement meet the minimum allowable conditions of the predefined design specifications. These initial values are used as the optimal values of the lining elastic modulus and lining thickness. Within the lining area of the tunnel to be supported, the optimal values of the lining elastic modulus and lining thickness are used to construct the tunnel lining and support the tunnel to be supported.
  • the tunnel lining construction method proposed in this embodiment can accurately select the type of lining material based on the optimal solution parameter values of the elastic modulus and thickness of the lining obtained by calculation and screening. This provides a basis for construction parameters for flexible support with variable lining thickness and has important economic and technical value.
  • Embodiments 2 to 4 correspond to those in Embodiment 1.
  • the term "computer-readable storage medium” should be understood as a single medium or multiple media including one or more instruction sets; it should also be understood as including any medium capable of storing, encoding, or carrying an instruction set for execution by a processor and enabling the processor to perform any of the methods in this invention.
  • modules or steps of the present invention described above can be implemented using general-purpose computer devices.
  • they can be implemented using computer-executable program code, thereby allowing them to be stored in a storage device for execution by a computer device, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module.
  • the present invention is not limited to any particular combination of hardware and software.

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Abstract

本发明公开一种基于复变函数法的隧道衬砌设计方法及系统，涉及隧道施工领域，该方法为：选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域；根据映射函数，将隧道的岩体区域和衬砌区域分别映射到岩体和衬砌圆环区域中，以此分别构建两区域在不同接触条件下的解析函数；根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；结合衬砌和岩体的应力和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移；判断隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，直至满足设计要求。本发明适用于不同形状隧道的衬砌设计参数，能够有效提高衬砌设计参数计算的准确性。

Description

基于复变函数法的隧道衬砌设计方法及系统
相关申请的交叉引用
本发明要求于2024年07月18日提交中国国家知识产权局、申请号为202410966098.X、发明名称为“基于复变函数法的隧道衬砌设计方法及系统”的中国专利申请的优先权，其全部内容通过引用结合在本发明中并构成本发明的一部分，用于所有目的。
技术领域
本发明涉及隧道施工技术领域，尤其涉及一种基于复变函数法的隧道衬砌设计方法及系统，以及一种隧道衬砌建造方法。
背景技术
这里的陈述仅提供与本发明相关的背景技术，而不必然地构成现有技术。
随着公路、铁路等的快速发展，隧道的需求也越来越多。为了保证隧道的安全施工，通常使用衬砌支护隧道岩体，而衬砌的支护作用由衬砌的弹性模量、厚度等参数决定，进而衬砌参数设计影响隧道的安全性和经济性，研究隧道衬砌参数设计是当前隧道工程的主要研究方向之一。
目前，衬砌的弹性模量(简称弹模)、厚度等参数一般根据隧道的使用寿命、地质条件、隧道直径等多个因素根据工程类比确定，然而，这一方法所确定的隧道衬砌厚度和弹性模量与实际工程适配性较差。随着机械化、自动化技术的不断发展，衬砌的制作及其支护变得越来越快捷，如何准确的计算、优化衬砌弹性模量和厚度等参数，成为科学支护的关键，设计与地质条件相适配的衬砌具有重要的经济、技术价值。
发明内容
为解决上述现有技术的不足，本发明提供了一种基于复变函数法的隧道衬砌设计方法及系统，采用复变函数法映射不同形状的隧道，同时还分析了不同隧道岩体与衬砌接触情况下的衬砌设计，优化衬砌设计的弹模和厚度，提高衬砌设计参数计算的科学性与准确性，该方法具有广泛的普适性。当将该方法应用到实际的隧道衬砌建造工程中时，可根据输出的衬砌的弹性模量和厚度的优化值，进行所述隧道内衬砌的具体施工建造，以实现对隧道的有效支护。
第一方面，本发明提供了一种基于复变函数法的隧道衬砌设计方法。
一种基于复变函数法的隧道衬砌设计方法，包括：
选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域；
根据映射函数，将物理平面隧道的岩体区域和衬砌区域分别映射到像平面区域中；
基于映射后的像平面中的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移；
基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，直至满足设计要求。
第二方面，本发明提供了一种基于复变函数法的隧道衬砌设计系统。
一种基于复变函数法的隧道衬砌设计系统，包括：
数据获取模块，用于选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域；
映射模块，用于根据映射函数，将隧道的岩体区域和衬砌区域分别映射到像平面岩体圆环区域和衬砌圆环区域中；
解析函数构建模块，用于基于映射后的像平面中的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
解析函数求解模块，用于根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
应力和位移计算模块，用于基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移；
衬砌设计调整模块，用于基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，直至满足设计要求。
第三方面，本发明还提供了一种电子设备，包括存储器和处理器以及存储在存储器上并在处理器上运行的计算机指令，所述计算机指令被处理器运行时，完成第一方面所述方法的步骤。
第四方面，本发明还提供了一种计算机可读存储介质，用于存储计算机指令，所述计算机指令被处理器执行时，完成第一方面所述方法的步骤。
第五方面，本发明还提供了一种隧道衬砌建造方法。
一种隧道衬砌建造方法，包括：
获取待支护隧道的隧道开挖半径、隧道埋深，设置衬砌弹性模量初始值和衬砌厚度，确定待支护隧道的岩体区域和衬砌区域；
根据映射函数，将物理平面中的待支护隧道的岩体区域和衬砌区域分别映射到像平面区域中；
基于映射后的像平面中的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算衬砌区域和岩体区域中任一点的应力和位移；
重新设置衬砌弹性模量初始值和衬砌厚度初始值，重复计算衬砌区域和岩体区域中任一点的应力和位移，直至计算得到的应力和位移符合预定义的设计规范最低允许条件，则停止计算；
记录计算得到的应力和位移符合预定义的设计规范最低允许条件时所设置的衬砌弹性模量初始值和衬砌厚度初始值，作为衬砌弹性模量最优值和衬砌厚度最优值，在所述待支护隧道的衬砌区域内，采用所述衬砌弹性模量最优值和衬砌厚度最优值，进行隧道衬砌的建造，对待支护隧道进行支护。
以上一个或多个技术方案存在以下有益效果：
本发明提供了一种基于复变函数方法的隧道衬砌设计方法及系统，解决了现有的衬砌弹模、厚度等参数根据隧道的使用寿命、地质条件、隧道直径等多个因素综合考虑，以此确定隧道衬砌厚度的方法精准性较差的问题，建造的隧道衬砌更符合隧道施工；本实施例中，通过基于复变函数法计算出在衬砌支护作用下隧道岩体和衬砌的应力和位移解，根据隧道岩体和衬砌的应力和位移解优选衬砌的弹模和厚度，由于复变函数方法可映射多种形状隧道，针对各类的隧道洞室均可适用，而且可计算出衬砌的弹模，结合衬砌的厚度可对衬砌材料的型号选择提供指导，为变衬砌厚度灵活支护提供基础，具有重要的经济、技术价值。
附图说明
构成本发明的一部分的说明书附图用来提供对本发明的进一步理解，本发明的示意性实施例及其说明用于解释本发明，并不构成对本发明的不当限定。
图1为本发明实施例所述基于复变函数法的隧道衬砌设计方法的流程图；
图2为本发明实施例中衬砌区域支护圆形隧洞岩体区域在不同接触条件下应力、位移的求解流程图；
图3为本发明实施例中衬砌区域支护圆形隧洞岩体区域的示意图；
图4A为本发明实施例中岩体区域S1映射为岩体圆环区域Ω1的示意图；
图4B为本发明实施例中衬砌区域S2映射为衬砌圆环区域Ω2的示意图；
图5为本发明实施例所述的隧道衬砌建造方法的流程图。
具体实施方式
应该指出，以下详细说明都是示例性的，仅是为了描述具体实施方式，旨在对本发明提供进一步的说明，并非意图限制根据本发明的示例性实施方式。除非另有指明，本文使用的所有技术和科学术语具有与本发明所属技术领域的普通技术人员通常理解的相同含义。此外，还应当理解的是，当在本说明书中使用术语“包含”和/或“包括”时，其指明存在特征、步骤、操作、器件、组件和/或它们的组合。
实施例一
本实施例提供了一种基于复变函数法的隧道衬砌设计方法，如图1所示，包括：
设置隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，分别确定隧道的岩体区域和衬砌区域；
根据映射函数，将物理平面中的隧道的岩体区域和衬砌区域分别映射到像平面中的岩体圆环区域和衬砌圆环区域中；
基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移；
基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，直至满足设计要求。
通过下述内容对本实施例所提出的基于复变函数法的隧道衬砌设计方法进行更详细的介绍。
步骤S1、选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域，根据映射函数，将物理平面中的隧道的岩体区域和衬砌区域分别映射到像平面中的岩体圆环区域和衬砌圆环区域中。
具体的，首先选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域，以此分析该隧道衬砌设计是否符合设计要求，通过一个映射函数，将物理平面(z平面)上一个边界形状复杂的区域变换成像平面(ζ平面)上简单边界形状的区域，进而就可以利用复变函数方法在简单边界下求得问题的解。根据两个映射函数，将z平面和z1平面的两个区域即岩体区域S1和衬砌区域S2分别映射到ζ平面和ζ1平面的两个圆环区域即岩体圆环区域Ω1和衬砌圆环区域Ω2，如图4所示，图4A为岩体区域S1映射为岩体圆环区域Ω1，图4B为衬砌区域S2映射为衬砌圆环区域Ω2，图中L1为物理平面的地表边界，L2为物理平面S1和S2区域的交界面，L3为物理平面衬砌的内边界，L'1为像平面的地表边界，L'2为像平面S1和S2区域的交界面，L'3是物理平面衬砌的内边界，其中映射函数的表达式分别为：
其中，z平面是指岩体区域S1所在平面，z1平面是指衬砌区域S2所在平面，R0为隧道开挖半径，a＝H(1-α2)/(1+α2)，H是隧道埋深。
步骤S2、基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数，其中，在构建解析函数的过程中，分别构建岩体区域在无支护作用下、岩体区域在衬砌区域的支护作用下、衬砌区域在岩体区域的作用下的解析函数，且该不同接触条件包括：岩体区域和衬砌区域完全接触条件和岩体区域和衬砌区域光滑接触条件。
具体的，当隧洞岩体区域在无支护作用下且仅受重力作用时，岩体区域S1所对应的解析函数由和ψ1(z)表示；如图3所示，图中T为衬砌厚度，R1为衬砌内半径，L1为物理平面的地表边界，L2为物理平面S1和S2区域的交界面，L3是物理平面衬砌的内边界，在构筑衬砌后，岩体和衬砌相互作用，岩体区域S1仅在衬砌区域的支护作用下所对应的解析函数由和ψ2(z)表示，岩体区域S1映射到区域Ω1，区域Ω1所对应的解析函数由和ψ2(ζ)表示；构筑衬砌后，衬砌区域S2在岩体区域的作用下所对应的解析函数由和ψ3(z1)表示，衬砌区域S2映射到区域Ω2，区域Ω2所对应的解析函数由和ψ3(ζ1)表示。上述解析函数具体可表示为：

其中，Fx是水平作用力，Fx＝0，Fy是竖直作用力，κ1是系数，κ1＝3-4μ1，μ1是岩体的泊松比，ak、bk、ck、dk是待求的解析函数的系数。
步骤S3、根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数。即，如图2所示，根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解岩体区域和衬砌区域分别在完全接触条件下和在光滑接触条件下的解析函数。
步骤S3.1、求解岩体区域和衬砌区域在完全接触条件下的解析函数，包括：根据衬砌区域内边界的应力边界条件、开挖边界的应力连续条件和开挖边界的位移边界条件(该条件均为本领域公知常识)，联立解析函数进行求解，求解得到解析函数的系数；其中，开挖边界是指衬砌区域和岩体区域完全接触时接触面。
首先，由于衬砌内边界没有外荷载作用，衬砌内边界的应力边界条件可以表示为：
其中，C1是一个未知的复常数。
其次，由于衬砌与岩体的接触面满足完全接触条件，因此，接触面上需满足应力连续条件，其表达式为：
其中，C2是一个未知的复常数。
最后，根据岩体区域在无支护作用下的位移、岩体区域在衬砌区域的支护作用下的位移、衬砌区域在岩体区域的作用下的位移，确定衬砌区域和岩体区域完全接触时接触面的位移连续条件。
具体的，仅在重力作用下，无支护浅埋圆形隧洞的岩体内任意一点的位移表达式为：
式中，G1是隧洞的剪切模量，G1＝E1/[2(1+μ1)]，E1是岩体的弹性模量，μ1是泊松比，u1 R和分别是岩体在重力作用下水平方向和竖直方向的位移分量。
仅在衬砌支护作用下浅埋隧洞岩体任意一点的位移表达式为：
式中，和分别是岩体在重力作用下水平方向和竖直方向的位移分量。
衬砌在岩体作用下任意一点的位移表达式为：
式中，uL和vL分别是衬砌在岩体作用下水平方向和竖直方向的位移分量，G2是衬砌的剪切模量，G2＝E2/[2(1+μ2)]，E2和μ2分别是衬砌的弹性模量和泊松比，κ2＝3-4μ2。
由于衬砌和岩体的接触面满足完全接触条件，因此，在接触面上满足位移连续条件，其表达式为：
联立上述解析函数，通过求解线性方程组即可获得解析函数的系数。
步骤S3.2、求解岩体区域和衬砌区域在光滑接触条件下的解析函数，包括：根据衬砌区域内边界的应力边界条件、开挖边界的应力连续条件、开挖边界的剪应力条件和开挖边界的法向位移连续条件，联立解析函数进行求解，求解得到解析函数的系数；其中，开挖边界是指衬砌区域和岩体区域完全接触时接触面。
首先，由于衬砌内边界没有外荷载作用，应力边界条件的表达式为：
式中，βk为系数，σ1为衬砌外边界上的点，c01+ic02代表一个实参数，κ2为衬砌的参数。
其次，由于衬砌与岩体的接触面满足完全接触条件，因此，接触面上需满足应力连续条件，其表达式为：
式中，σ为开挖边界上的点。
然后，开挖边界上剪应力为零，其公式为：
式中，τρθ为岩体和衬砌面上的剪应力。
最后，开挖边界上的法向位移连续条件为：
式中，是重力作用下无衬砌支护隧洞开挖边界的法向位移，是仅在衬砌支护作用下隧洞开挖边界的法向位移，是衬砌外边界在岩体作用下的法向位移。
联立上述解析函数，通过求解线性方程组即可获得解析函数的系数。
步骤S4、基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移。
步骤S4.1、计算获取衬砌区域和岩体区域中任一点的应力。其中，衬砌内任意一点的应力分量表达式为：
其中，分别是正交曲线坐标系下衬砌内的法向应力、切向应力、剪应力；分别是不考虑原始地应力时，正交曲线坐标系下衬砌内的法向应力、切向应力、剪应力。因此，将各应力分量叠加为：
式中，分别是正交曲线下岩体内任意点的法向应力、切向应力和剪应力。通过上式，计算获取衬砌区域和岩体区域中任一点的应力。
步骤S4.2、计算获取衬砌区域和岩体区域中任一点的位移。其中，衬砌区域中任一点的位移分量表达式为：
式中，uL和vL分别是衬砌内任意点的水平位移和竖直位移。
岩体区域中任意一点的位移分量表达式为：
式中，uR和vR分别是岩体内任意点的水平位移和竖直位移。
基于所选取的衬砌的参数(衬砌弹性模量和衬砌厚度)，即可获得相应的隧道在该衬砌支护条件下的衬砌区域和岩体区域的应力和位移。
步骤S5、基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，重新计算应力和位移并判断，直至满足设计要求；若是，则优化衬砌弹性模量和衬砌厚度，得到衬砌最优参数设计。
具体的，通过查询隧道支护设计规范，若隧道在衬砌支护作用下的应力和位移符合规范，即可满足隧道设计；若衬砌设计过于保守，则减小衬砌的弹模和厚度，直至隧道应力和位移达到规范要求的最小值；若衬砌设计不满足设计规范，则增大衬砌弹模和厚度，直至隧道应力和位移以符合设计规范。
本实施例所提出的基于复变函数法的隧道衬砌设计方法，解决现有的衬砌弹模、厚度等参数根据隧道的使用寿命、地质条件、隧道直径等多个因素综合考虑，以此确定隧道衬砌厚度的方法精准性较差的问题，本实施例中，通过基于复变函数法计算出在衬砌支护作用下隧道岩体和衬砌的应力和位移解，根据隧道岩体和衬砌的应力和位移解优选衬砌的弹模和厚度，由于复变函数方法可映射多种形状隧道，针对各类的隧道洞室均可适用，而且可计算出衬砌的弹模，结合衬砌的厚度可对衬砌材料的型号选择提供指导，为变衬砌厚度灵活支护提供基础，具有重要的经济、技术价值。
实施例二
本实施例提供了一种基于复变函数法的隧道衬砌设计系统，包括：
数据获取模块，用于选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域；
映射模块，用于根据映射函数，将物理平面中的隧道的岩体区域和衬砌区域分别映射到像平面中的岩体圆环区域和衬砌圆环区域中；
解析函数构建模块，用于基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
解析函数求解模块，用于根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
应力和位移计算模块，用于基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移；
衬砌设计调整模块，用于基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，直至满足设计要求。
实施例三
本实施例提供了一种电子设备，包括存储器和处理器以及存储在存储器上并在处理器上运行的计算机指令，所述计算机指令被处理器运行时，完成如上所述的基于复变函数法的隧道衬砌设计方法中的步骤。
实施例四
本实施例还提供了一种计算机可读存储介质，用于存储计算机指令，所述计算机指令被处理器执行时，完成如上所述的基于复变函数法的隧道衬砌设计方法中的步骤。
实施例五
本实施例提供了一种隧道衬砌建造方法，如图5所示，包括：
获取待支护隧道开挖半径、隧道埋深，设置衬砌弹性模量初始值和衬砌厚度初始值，分别确定待支护隧道的岩体区域和衬砌区域；
根据映射函数，将物理平面中的隧道的岩体区域和衬砌区域分别映射到像平面中的岩体圆环区域和衬砌圆环区域中；
基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算衬砌区域和岩体区域中任一点的应力和位移；
重新设置衬砌弹性模量初始值和衬砌厚度初始值，重复计算衬砌区域和岩体区域中任一点的应力和位移，直至计算得到的应力和位移符合预定义的设计规范最低允许条件，则停止计算；
记录计算得到的应力和位移符合预定义的设计规范最低允许条件时所设置的衬砌弹性模量初始值和衬砌厚度初始值，作为衬砌弹性模量最优值和衬砌厚度最优值，在所述待支护隧道的衬砌区域内，采用所述衬砌弹性模量最优值和衬砌厚度最优值，进行隧道衬砌的建造，对待支护隧道进行支护。
通过下述内容对本实施例所提出的隧道衬砌建造方法进行更详细的介绍。
步骤S1、获取待支护隧道的隧道开挖半径、隧道埋深，设置衬砌弹性模量初始值和衬砌厚度，确定待支护隧道的岩体区域和衬砌区域，根据映射函数，将物理平面中的待支护隧道的岩体区域和衬砌区域分别映射到像平面中的岩体圆环区域和衬砌圆环区域中。
具体的，首先获取待支护隧道的隧道开挖半径、隧道埋深，设置衬砌弹性模量初始值和衬砌厚度，确定待支护隧道的岩体区域和衬砌区域，以此分析所述待支护隧道的隧道衬砌设计是否符合设计要求，通过一个映射函数，将物理平面(z平面)上一个边界形状复杂的区域变换成像平面(ζ平面)上简单边界形状的区域，进而就可以利用复变函数方法在简单边界下求得问题的解。根据两个映射函数，将z平面和z1平面的两个区域即岩体区域S1和衬砌区域S2分别映射到ζ平面和ζ1平面的两个圆环区域即岩体圆环区域Ω1和衬砌圆环区域Ω2，如图4所示，图4A为岩体区域S1映射为岩体圆环区域Ω1，图4B为衬砌区域S2映射为衬砌圆环区域Ω2，图中L1为物理平面的地表边界，L2为物理平面S1和S2区域的交界面，L3为物理平面衬砌的内边界，L'1为像平面的地表边界，L'2为像平面S1和S2区域的交界面，L'3是物理平面衬砌的内边界，其中映射函数的表达式分别为：
其中，z平面是指岩体区域S1所在平面，z1平面是指衬砌区域S2所在平面，R0为隧道开挖半径，a＝H(1-α2)/(1+α2)，H是隧道埋深。
步骤S2、基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数，其中，在构建解析函数的过程中，分别构建岩体区域在无支护作用下、岩体区域在衬砌区域的支护作用下、衬砌区域在岩体区域的作用下的解析函数，且该不同接触条件包括：岩体区域和衬砌区域完全接触条件和岩体区域和衬砌区域光滑接触条件。
具体的，当隧洞岩体区域在无支护作用下且仅受重力作用时，岩体区域S1所对应的解析函数由和ψ1(z)表示；如图3所示，图中T为衬砌厚度，R1为衬砌内半径，L1为物理平面的地表边界，L2为物理平面S1和S2区域的交界面，L3是物理平面衬砌的内边界，在构筑衬砌后，岩体和衬砌相互作用，岩体区域S1仅在衬砌区域的支护作用下所对应的解析函数由和ψ2(z)表示，岩体区域S1映射到区域Ω1，区域Ω1所对应的解析函数由和ψ2(ζ)表示；构筑衬砌后，衬砌区域S2在岩体区域的作用下所对应的解析函数由和ψ3(z1)表示，衬砌区域S2映射到区域Ω2，区域Ω2所对应的解析函数由和ψ3(ζ1)表示。上述解析函数具体可表示为：

其中，Fx是水平作用力，Fx＝0，Fy是竖直作用力，κ1是系数，κ1＝3-4μ1，μ1是岩体的泊松比，ak、bk、ck、dk是待求的解析函数的系数。
步骤S3、根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数。即，如图2所示，根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解岩体区域和衬砌区域分别在完全接触条件下和在光滑接触条件下的解析函数。
步骤S3.1、求解岩体区域和衬砌区域在完全接触条件下的解析函数，包括：根据衬砌区域内边界的应力边界条件、开挖边界的应力连续条件和开挖边界的位移边界条件(该条件均为本领域公知常识)，联立解析函数进行求解，求解得到解析函数的系数；其中，开挖边界是指衬砌区域和岩体区域完全接触时接触面。
首先，由于衬砌内边界没有外荷载作用，衬砌内边界的应力边界条件可以表示为：
其中，C1是一个未知的复常数。
其次，由于衬砌与岩体的接触面满足完全接触条件，因此，接触面上需满足应力连续条件，其表达式为：
其中，C2是一个未知的复常数。
最后，根据岩体区域在无支护作用下的位移、岩体区域在衬砌区域的支护作用下的位移、衬砌区域在岩体区域的作用下的位移，确定衬砌区域和岩体区域完全接触时接触面的位移连续条件。
具体的，仅在重力作用下，无支护浅埋圆形隧洞的岩体内任意一点的位移表达式为：
式中，G1是隧洞的剪切模量，G1＝E1/[2(1+μ1)]，E1是岩体的弹性模量，μ1是泊松比，和分别是岩体在重力作用下水平方向和竖直方向的位移分量。
仅在衬砌支护作用下浅埋隧洞岩体任意一点的位移表达式为：
式中，和分别是岩体在重力作用下水平方向和竖直方向的位移分量。
衬砌在岩体作用下任意一点的位移表达式为：
式中，uL和vL分别是衬砌在岩体作用下水平方向和竖直方向的位移分量，G2是衬砌的剪切模量，G2＝E2/[2(1+μ2)]，E2和μ2分别是衬砌的弹性模量和泊松比，κ2＝3-4μ2。
由于衬砌和岩体的接触面满足完全接触条件，因此，在接触面上满足位移连续条件，其表达式为：
联立上述解析函数，通过求解线性方程组即可获得解析函数的系数。
步骤S3.2、求解岩体区域和衬砌区域在光滑接触条件下的解析函数，包括：根据衬砌区域内边界的应力边界条件、开挖边界的应力连续条件、开挖边界的剪应力条件和开挖边界的法向位移连续条件，联立解析函数进行求解，求解得到解析函数的系数；其中，开挖边界是指衬砌区域和岩体区域完全接触时接触面。
首先，由于衬砌内边界没有外荷载作用，应力边界条件的表达式为：
式中，β＝1-T/R0，σ1为衬砌外边界上的点，c01+ic02代表一个实参数，κ2为衬砌的参数。
其次，由于衬砌与岩体的接触面满足完全接触条件，因此，接触面上需满足应力连续条件，其表达式为：
式中，σ为开挖边界上的点。
然后，开挖边界上剪应力为零，其公式为：
式中，τρθ为岩体和衬砌面上的剪应力。
最后，开挖边界上的法向位移连续条件为：
式中，是重力作用下无衬砌支护隧洞开挖边界的法向位移，是仅在衬砌支护作用下隧洞开挖边界的法向位移，是衬砌外边界在岩体作用下的法向位移。
联立上述解析函数，通过求解线性方程组即可获得解析函数的系数。
步骤S4、基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移。
步骤S4.1、计算获取衬砌区域和岩体区域中任一点的应力。其中，衬砌内任意一点的应力分量表达式为：
其中，分别是正交曲线坐标系下衬砌内的法向应力、切向应力、剪应力；分别是不考虑原始地应力时，正交曲线坐标系下衬砌内的法向应力、切向应力、剪应力。因此，将各应力分量叠加为：
式中，分别是正交曲线下岩体内任意点的法向应力、切向应力和剪应力。通过上式，计算获取衬砌区域和岩体区域中任一点的应力。
步骤S4.2、计算获取衬砌区域和岩体区域中任一点的位移。其中，衬砌区域中任一点的位移分量表达式为：
式中，uL和vL分别是衬砌内任意点的水平位移和竖直位移。
岩体区域中任意一点的位移分量表达式为：
式中，uR和vR分别是岩体内任意点的水平位移和竖直位移。
基于所选取的衬砌的参数(衬砌弹性模量和衬砌厚度)，即可获得相应的隧道在该衬砌支护条件下的衬砌区域和岩体区域的应力和位移。
步骤S5、重新设置衬砌弹性模量初始值和衬砌厚度初始值，重复计算衬砌区域和岩体区域中任一点的应力和位移，直至计算得到的应力和位移符合预定义的设计规范最低允许条件，则停止计算。
具体的，通过查询隧道支护设计规范，若隧道在衬砌支护作用下的应力和位移符合规范，即可满足隧道设计；若衬砌设计过于保守，则减小衬砌的弹性模量和厚度，直至隧道应力和位移达到规范要求的最小值；若衬砌设计不满足设计规范，则增大衬砌弹性模量和厚度，直至隧道应力和位移以符合设计规范。
步骤S6、记录计算得到的应力和位移符合预定义的设计规范最低允许条件时所设置的衬砌弹性模量初始值和衬砌厚度初始值，作为衬砌弹性模量最优值和衬砌厚度最优值，在所述待支护隧道的衬砌区域内，采用所述衬砌弹性模量最优值和衬砌厚度最优值，进行隧道衬砌的建造，对待支护隧道进行支护。
本实施例所提出的隧道衬砌建造方法，可根据计算筛选出的衬砌的弹性模量和厚度的最优解参数值，可对衬砌材料的型号进行精确的选择，为变衬砌厚度灵活支护提供了施工参数基础，具有重要的经济、技术价值。
以上实施例二至四中涉及的各步骤与方法实施例一相对应，具体实施方式可参见实施例一的相关说明部分。术语“计算机可读存储介质”应该理解为包括一个或多个指令集的单个介质或多个介质；还应当被理解为包括任何介质，所述任何介质能够存储、编码或承载用于由处理器执行的指令集并使处理器执行本发明中的任一方法。
本领域技术人员应该明白，上述本发明的各模块或各步骤可以用通用的计算机装置来实现，可选地，它们可以用计算装置可执行的程序代码来实现，从而，可以将它们存储在存储装置中由计算装置来执行，或者将它们分别制作成各个集成电路模块，或者将它们中的多个模块或步骤制作成单个集成电路模块来实现。本发明不限制于任何特定的硬件和软件的结合。
以上所述仅为本发明的优选实施例，虽然结合附图对本发明的具体实施方式进行了描述，但并非对本发明保护范围的限制，所属领域技术人员应该明白，在本发明的技术方案的基础上，本领域技术人员不需要付出创造性劳动即可做出的各种修改或变形仍在本发明的保护范围以内。

Claims (10)

  1. 一种基于复变函数法的隧道衬砌设计方法，其特征是，包括：
    选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域；
    根据映射函数，将隧道的岩体区域和衬砌区域分别映射到岩体圆环区域和衬砌圆环区域中；
    基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
    根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
    基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移；
    基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，直至满足设计要求。
  2. 如权利要求1所述的基于复变函数法的隧道衬砌设计方法，其特征是，所述不同接触条件包括：岩体区域和衬砌区域完全接触条件和岩体区域和衬砌区域光滑接触条件。
  3. 如权利要求1所述的基于复变函数法的隧道衬砌设计方法，其特征是，求解解析函数的过程包括：
    基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域在无支护作用下、岩体区域在衬砌区域的支护作用下、衬砌区域在岩体区域的作用下的解析函数；
    根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解岩体区域和衬砌区域分别在完全接触条件下和在光滑接触条件下的解析函数。
  4. 如权利要求3所述的基于复变函数法的隧道衬砌设计方法，其特征是，求解岩体区域和衬砌区域在完全接触条件下的解析函数，包括：
    根据衬砌区域内边界的应力边界条件、开挖边界的应力连续条件和开挖边界的位移边界条件，联立解析函数进行求解，求解得到解析函数的系数；其中，开挖边界是指衬砌区域和岩体区域完全接触时接触面。
  5. 如权利要求4所述的基于复变函数法的隧道衬砌设计方法，其特征是，根据岩体区域在无支护作用下的位移、岩体区域在衬砌区域的支护作用下的位移、衬砌区域在岩体区域的作用下的位移，确定衬砌区域和岩体区域完全接触时接触面的位移连续条件。
  6. 如权利要求3所述的基于复变函数法的隧道衬砌设计方法，其特征是，求解岩体区域和衬砌区域在光滑接触条件下的解析函数，包括：
    根据衬砌区域内边界的应力边界条件、开挖边界的应力连续条件、开挖边界的剪应力条件和开挖边界的法向位移连续条件，联立解析函数进行求解，求解得到解析函数的系数；其中，开挖边界是指衬砌区域和岩体区域完全接触时接触面。
  7. 如权利要求1所述的基于复变函数法的隧道衬砌设计方法，其特征是，基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则增大衬砌弹性模量和衬砌厚度，重新计算应力和位移并判断，直至应力和位移满足隧道设计要求；若是，则减小衬砌弹性模量或衬砌厚度，得到衬砌最优参数设计。
  8. 一种基于复变函数法的隧道衬砌设计系统，其特征是，包括：
    数据获取模块，用于选取隧道开挖半径、隧道埋深和衬砌弹性模量和衬砌厚度，确定隧道的岩体区域和衬砌区域；
    映射模块，用于根据映射函数，将隧道的岩体区域和衬砌区域分别映射到岩体圆环区域和衬砌圆环区域中；
    解析函数构建模块，用于基于映射后的岩体圆环区域和衬砌圆环区域，分别构建岩体区域和衬砌区域在不同接触条件下的解析函数；
    解析函数求解模块，用于根据岩体区域、衬砌区域及其接触面的边界条件和连续条件，求解解析函数；
    应力和位移计算模块，用于基于求解得到的解析函数，结合衬砌和岩体的应力表达式和位移表达式，分别计算获取衬砌区域和岩体区域中任一点的应力和位移；
    衬砌设计调整模块，用于基于应力和位移，判断基于选取的衬砌弹性模量和衬砌厚度的隧道设计是否满足设计要求，若否，则调整衬砌弹性模量和衬砌厚度，直至满足设计要求。
  9. 一种电子设备，其特征是，包括存储器和处理器以及存储在存储器上并在处理器上运行的计算机指令，所述计算机指令被处理器运行时，完成如权利要求1-7中任一项所述的一种基于复变函数法的隧道衬砌设计方法的步骤。
  10. 一种计算机可读存储介质，其特征是，用于存储计算机指令，所述计算机指令被处理器执行时，完成如权利要求1-7中任一项所述的一种基于复变函数法的隧道衬砌设计方法的步骤。
PCT/CN2025/109445 2024-07-18 2025-07-18 基于复变函数法的隧道衬砌设计方法及系统 Pending WO2026017170A1 (zh)

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