A method for calculating equivalent thickness of waterproof single-layer lining

Abstract

The application belongs to the technical field of tunnel engineering support, and specifically discloses a waterproof single-layer lining equivalent thickness calculation method, which comprises the following steps: (1) determining a tunnel model, determining a first layer of sprayed concrete thickness h1, a first layer of sprayed concrete elastic modulus E1, a second layer of sprayed concrete thickness h2, a second layer of sprayed concrete elastic modulus E1, and calculating a tunnel plastic hinge length l through numerical simulation; (2) determining sprayed waterproof interface parameters, wherein the sprayed waterproof interface parameters comprise determining sprayed waterproof material performance parameters and determining superimposed structure sprayed waterproof interface parameters; (3) calculating equivalent thickness; (4) verifying by indoor test; and (5) verifying by numerical simulation. The application can provide a reliable waterproof single-layer lining equivalent thickness calculation method.

Description

Technical Field

[0001] This invention relates to the field of tunnel engineering support technology, specifically to a method for calculating the equivalent thickness of a waterproof single-layer lining. Background Technology

[0002] Due to existing problems with composite lining in terms of theory, efficiency, waterproofing, and maintenance, single-layer lining is adopted for hard rock tunnels. This means that, without removing the isolation layer, a layer of waterproof concrete is sprayed immediately after the tunnel is excavated, and necessary support components such as anchor bolts and steel arches are installed according to the surrounding rock grade. Then, according to the requirements of durability and flatness, a leveling layer is applied and waterproof material is sprayed, followed by one or more layers of concrete sprayed to form a support system with strong interlayer adhesion and sufficient shear force transmission.

[0003] Some scholars have made quantitative measurements on the integrity of structures with strong interlayer bonding, but the thickness of each layer of waterproof single-layer lining is difficult to determine. There are no standards or specifications to guide the design parameters, and no mechanical relationship between external load and each composite layer has been established. Summary of the Invention

[0004] This invention provides a method for calculating the equivalent thickness of a waterproof single-layer lining, with the aim of providing a reliable method for calculating the equivalent thickness of a waterproof single-layer lining.

[0005] This invention is achieved through the following technical solution: a method for calculating the equivalent thickness of a waterproof single-layer lining, characterized by comprising the following steps:

[0006] (1) Determine the tunnel model, determine the thickness h1 of the first layer of shotcrete, the elastic modulus E1 of the first layer of shotcrete, the thickness h2 of the second layer of shotcrete, and the elastic modulus E1 of the second layer of shotcrete. Calculate the length l of the tunnel plastic hinge through numerical simulation.

[0007] (2) Determine the interface parameters of the sprayed waterproofing, and conduct compressive and shear tests on the composite beam with sprayed waterproofing to obtain the thickness t, elastic modulus E2, shear modulus G, cohesion c, and internal friction angle φ of the sprayed waterproofing layer.

[0008] (3) Calculate the equivalent thickness;

[0009] (4) Verification was conducted using indoor tests;

[0010] (5) Verification was performed using numerical simulation.

[0011] Furthermore, compressive and shear tests were conducted on the composite beam with sprayed waterproofing to obtain the thickness t, elastic modulus E2, shear modulus G, cohesion c, and internal friction angle φ of the sprayed waterproofing layer.

[0012] Furthermore, in step (1), the equivalent thickness h is obtained by looking up a table using the Q-system method. 等效1-1 The equivalent thickness h was calculated using the block theory software Unwedge. 等效1-2 The equivalent thickness h is calculated using permanent loads. 等效1-3 Then, the maximum value is taken to obtain the shotcrete thickness h. 等效1 =max(h 等效1-1 h 等效1-2 h 等效1-3 );

[0013] Through numerical simulation methods and the thickness h of the sprayed concrete... 等效1 The length l of the tunnel plastic hinge was calculated.

[0014] Furthermore, in step (3), the equivalent thickness is calculated using the following formula:

[0015]

[0016]

[0017] Based on formulas (1) and (2), the following can be deduced:

[0018]

[0019] According to the formula for moment of inertia The derivation leads to:

[0020]

[0021] in,

[0022]

[0023] Simplified to:

[0024]

[0025] Substituting formulas (6) and (7) into formula (4), and simplifying them, we can obtain the overall equivalent thickness of the waterproof single-layer lining:

[0026]

[0027] in,

[0028]

[0029]

[0030] D = h1 + h2 (12)

[0031]

[0032] In the formula:

[0033] (EI) 叠合 Overall bending stiffness of composite structures

[0034] (AQ) 叠合 Overall shear stiffness of composite structures

[0035] (EI) 第一层砼 : Flexural stiffness of the first layer of shotcrete

[0036] (EI) 喷涂防水 : The flexural stiffness of the waterproof coating

[0037] (EI) 第二层砼 The flexural stiffness of the first layer of shotcrete.

[0038] I 等效 Moment of inertia of an equivalent homogeneous body

[0039] h 等效2 The overall equivalent thickness of a waterproof single-layer lining.

[0040] δ 叠合 Waterproof single-layer lining deflection,

[0041] δ 等效 Deflection of homogeneous lining

[0042] P: Concentrated load

[0043] a: The distance from the upper edge of the sprayed waterproof layer to the centroid of the single-layer waterproof lining.

[0044] l: The span of the beam (l is approximately equal to the arc length between the plastic points of the tunnel).

[0045] h1: Thickness of the first shotcrete application

[0046] h2: The thickness of the second layer of shotcrete.

[0047] t: Thickness of the waterproof coating.

[0048] E1: Elastic modulus of shotcrete.

[0049] E2: Elastic modulus of waterproof coating.

[0050] G: Shear modulus of waterproof coating.

[0051] Furthermore, in step (4), a four-point beam bending toughness test was used for verification, and the equivalent thickness calculation method was obtained. According to the specification, the deflection at the midpoint of the span is:

[0052]

[0053] From formula (14), it can be seen that, under the condition of equivalent deflection, the ratio of bending strength to thickness is:

[0054]

[0055] The equivalent thickness calculation method was derived through energy absorption testing, and the formula for the center deflection was obtained based on the calculation:

[0056]

[0057] Under the condition of equivalent deflection, the ratio of deflection to thickness is:

[0058]

[0059] in,

[0060] b: Width of the beam used in the bending toughness test

[0061] d: Height of the beam used in the bending toughness test.

[0062] r: radius of the energy absorption test disk

[0063] E: Elastic modulus of concrete in energy absorption test

[0064] q: Load for energy absorption test

[0065] q1: Load for energy absorption test of homogeneous body (without waterproof interlayer)

[0066] q2: Load for energy absorption test of composite structure (with waterproof interlayer)

[0067] y: Deflection of the beam in the bending toughness test.

[0068] y1: Deflection of a homogeneous (without waterproof interlayer) beam in a bending toughness test.

[0069] y2: Deflection of the composite beam (with waterproof interlayer) in the bending toughness test.

[0070] p: Load on the beam used in the bending toughness test.

[0071] p1: Load on a homogeneous (without waterproof interlayer) flexural toughness test beam.

[0072] p2: Load on the composite beam (with waterproof interlayer) for bending toughness test

[0073] δ: The central deflection of the energy test disk.

[0074] δ1: The central deflection of the homogeneous (without waterproof interlayer) energy testing disk.

[0075] δ2: The central deflection of the energy testing disk of the composite (with waterproof interlayer).

[0076] Furthermore, in step (5), when verifying using numerical simulation, the initial values ​​of the first layer of sprayed concrete thickness h1 and the second layer of sprayed concrete thickness h2 determined in step (1) are used to perform numerical simulation based on the thickness t, elastic modulus E2, shear modulus G, cohesion c, and internal friction angle φ of the sprayed waterproof layer determined in step (2). At the same time, it is compared with a homogeneous body without sprayed waterproof material to compare the thickness of the homogeneous body when the deflection is equal under the same load.

[0077] Furthermore, in step (2), tensile stress-strain curves corresponding to different waterproof membrane thicknesses are obtained through uniaxial tensile tests of the material. The slope of the curves is the elastic modulus E2, Poisson's ratio, yield stress, and yield strain.

[0078] Furthermore, the bond stress-displacement curve was obtained through bond strength test. The peak stress of the curve is the bond strength, the slope of the rising segment of the curve is the bond stiffness, and the area formed by the curve and the horizontal axis represents the bond failure energy.

[0079] Furthermore, shear stress-displacement curves were obtained through shear strength tests, with the peak stresses of the curves representing the shear strengths; the slope of the rising segment of the curve represents the shear stiffness, and the area formed by the curve and the horizontal axis represents the shear slip energy.

[0080] Furthermore, in the bond stress-displacement test of the composite structure, the bond stress-displacement curve is obtained through the bond strength test. The peak stress of the curve is the bond strength, the slope of the rising segment of the curve is the bond stiffness, and the area formed by the curve and the horizontal axis represents the bond failure energy.

[0081] Furthermore, in the shear stress-displacement test of the composite structure, the shear stress-displacement curve is obtained through the shear strength test. The peak stress of the curve is the shear strength. The slope of the rising segment of the curve is the shear stiffness, and the area formed by the curve and the horizontal axis represents the shear slip energy.

[0082] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0083] This invention relates to a method for calculating the equivalent thickness of waterproof single-layer lining based on laminated glass theory. Currently, there is no simple and reliable calculation method for the equivalent thickness of waterproof single-layer lining both domestically and internationally. Most methods rely on ANSYS or other finite element simulation software to establish a solid simulation model and obtain an approximate numerical simulation value of the equivalent thickness of the waterproof single-layer lining. Modeling is difficult and interface parameters are hard to determine, thus failing to meet the requirements of actual design and construction of waterproof single-layer lining.

[0084] This invention can determine the equivalent thickness h of a waterproof single-layer lining based on the thickness of the first and second layers of shotcrete, the thickness of the waterproof coating, the elastic modulus of the shotcrete, the elastic modulus of the waterproof coating, and the shear modulus of the waterproof coating, using theoretical analysis methods. 等效2 Perform the calculation.

[0085] Through case studies, indoor experiments, and numerical simulations, the theoretical calculation method for the equivalent thickness of waterproof single-layer linings has been verified. The results show that the method is correct and provides a reliable method for calculating the equivalent thickness of waterproof single-layer linings. This method can improve the design level and quality of single-layer linings, reduce design and testing costs, and ensure that the stress strength of waterproof single-layer linings meets requirements. Attached Figure Description

[0086] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly described below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort. In the drawings:

[0087] Figure 1 This is a flowchart illustrating an embodiment of a method for calculating the equivalent thickness of a waterproof single-layer lining according to the present invention.

[0088] Figure 2 This is a tensile stress-strain curve.

[0089] Figure 3 This is a bond stress-displacement curve.

[0090] Figure 4 This is a shear stress-displacement curve.

[0091] Figure 5 This is a schematic diagram of the equivalent moment of inertia of a homogeneous body. Detailed Implementation

[0092] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0093] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0094] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0095] In the description of this invention, it should be noted that the terms "first," "second," "third," etc., are used only for distinguishing descriptions and should not be construed as indicating or implying relative importance.

[0096] In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0097] like Figure 1 and Figure 5 As shown, this embodiment 1 provides a method for calculating the equivalent thickness of a waterproof single-layer lining, including the following steps:

[0098] (1) Determine the tunnel model, determine the thickness h1 of the first layer of shotcrete, the elastic modulus E1 of the first layer of shotcrete, the thickness h2 of the second layer of shotcrete, and the elastic modulus E1 of the second layer of shotcrete. Calculate the length l of the tunnel plastic hinge through numerical simulation.

[0099] In this embodiment, a tunnel underwent static compressive elastic modulus testing according to the "Standard for Test Methods of Physical and Mechanical Properties of Concrete" (GB50081). The elastic modulus of the two layers of shotcrete was found to be E1 = 31.5 × 10000 MPa. Based on on-site geological sketching, Q was calculated to be 4–10. Using the Q-system method, the minimum thickness of the shotcrete was determined to be h. 等效1-1 =9cm;

[0100] The equivalent thickness h was calculated using bulk theory and the Unwedge software. 等效1-2 =20cm; h was obtained through numerical simulation of permanent load. 等效1-3 =15cm.

[0101] Therefore, the maximum value is taken to obtain the total thickness h of the sprayed concrete.等效1 =max(h 等效1-1 h 等效1-2 h 等效1-3 =20cm, and through numerical simulation, the thickness of the sprayed concrete is h. 等效1 =20cm, and the analysis and calculation yielded a tunnel plastic hinge length l = 10m. Through numerical simulation of temporary loads, the initial thickness of the first layer of shotcrete was determined to be h1 = 15cm, combined with h 等效1 Given h1 = 15cm and h2 = 20cm, the assumed thickness of the second layer of shotcrete is h2 = 10cm.

[0102] (2) Determine the interface parameters of the sprayed waterproofing layer. The interface parameters of the sprayed waterproofing layer include determining the performance parameters of the sprayed waterproofing material and the interface parameters of the sprayed waterproofing layer of the composite structure. Compressive and shear tests were conducted on the composite beam with sprayed waterproofing to obtain the thickness t, elastic modulus E2, shear modulus G, cohesion c, and internal friction angle φ of the sprayed waterproofing layer.

[0103] In this embodiment, when determining the performance parameters of the sprayed waterproofing material, tensile stress-strain curves corresponding to different waterproof membrane thicknesses are obtained through tensile stress-strain tests on the sprayed waterproofing material. The slope of the curve represents the elastic modulus E2, Poisson's ratio, yield stress, yield strain, etc. Figure 2 As shown, the specific values ​​are detailed in Table 1.

[0104] Table 1 Mechanical parameters of sprayed waterproofing

[0105] 2 33 0.45 1.55 0.04

[0106] In this embodiment, the parameters of the waterproof interface for the composite structure are determined through shear stress-displacement tests and bond stress-displacement tests of the composite structure. Specifically, the bond stress-displacement test of the composite structure obtains the bond stress-displacement curve through bond strength tests. The peak stress of the curve represents the bond strength, the slope of the rising segment of the curve represents the bond stiffness, and the area formed by the curve and the horizontal axis represents the bond failure energy. Figure 3 As shown, the specific values ​​are detailed in Table 2.

[0107] Table 2 Adhesion performance parameters of sprayed waterproofing interface

[0108] 1 0.5 0.7 0.3

[0109] Shear stress-displacement tests were conducted on composite structures. Shear stress-displacement curves were obtained through shear strength tests, with peak stresses corresponding to shear strengths. The slope of the rising segment of the curve represents the shear stiffness, and the area formed by the curve and the horizontal axis represents the shear slip energy. Figure 4 As shown, the specific values ​​are detailed in Table 3.

[0110] Table 3 Shear resistance parameters of the waterproof coating interface

[0111] 1 0.2 0.1 2.0

[0112] (3) Calculate the equivalent thickness;

[0113] The equivalent thickness is calculated using the following formula:

[0114]

[0115]

[0116] Based on the principle of deflection equivalence, namely formulas (1) and (2), the following can be derived:

[0117]

[0118] According to the formula for moment of inertia The derivation leads to:

[0119]

[0120] in,

[0121]

[0122] Simplified to:

[0123]

[0124] Substituting formulas (6) and (7) into formula (4), and simplifying them, we can obtain the overall equivalent thickness of the waterproof single-layer lining:

[0125]

[0126] in,

[0127]

[0128]

[0129] D = h1 + h2 (12)

[0130]

[0131] In the formula:

[0132] (EI) 叠合 Overall bending stiffness of composite structures

[0133] (AQ) 叠合 Overall shear stiffness of composite structures

[0134] (EI)第一层砼 : Flexural stiffness of the first layer of shotcrete

[0135] (EI) 喷涂防水 : The flexural stiffness of the waterproof coating

[0136] (EI) 第二层砼 The flexural stiffness of the first layer of shotcrete.

[0137] I 等效 Moment of inertia of an equivalent homogeneous body

[0138] h 等效2 The overall equivalent thickness of a waterproof single-layer lining.

[0139] δ 叠合 Waterproof single-layer lining deflection,

[0140] δ 等效 Deflection of homogeneous lining

[0141] P: Concentrated load

[0142] a: The distance from the upper edge of the sprayed waterproof layer to the centroid of the single-layer waterproof lining.

[0143] l: The span of the beam (l is approximately equal to the arc length between the plastic points of the tunnel).

[0144] h1: Thickness of the first shotcrete application

[0145] h2: The thickness of the second layer of shotcrete.

[0146] t: Thickness of the waterproof coating.

[0147] E1: Elastic modulus of shotcrete.

[0148] E2: Elastic modulus of waterproof coating.

[0149] G: Shear modulus of waterproof coating.

[0150] The equivalent thickness under different working conditions is calculated according to formula (8):

[0151] Table 4. Calculation of Equivalent Thickness of Waterproof Single-Layer Lining

[0152]

[0153] (4) Verification was conducted using indoor tests;

[0154] In this step, a four-point beam bending toughness test is used for verification, and an equivalent thickness calculation method is derived. Based on mechanics of materials, the deflection at the midpoint of the span is:

[0155]

[0156] From formula (14), it can be seen that, under the condition of equivalent deflection, the ratio of bending strength to thickness is:

[0157]

[0158] The equivalent thickness calculation method was derived through energy absorption testing, and the formula for the center deflection was obtained based on the calculation:

[0159]

[0160] Under the condition of equivalent deflection, the ratio of deflection to thickness is:

[0161]

[0162] in,

[0163] b: Width of the beam used in the bending toughness test.

[0164] d: Height of the beam used in the bending toughness test.

[0165] r: radius of the energy absorption test disk

[0166] E: Elastic modulus of concrete in energy absorption test

[0167] q: Load for energy absorption test

[0168] q1: Load for energy absorption test of homogeneous body (without waterproof interlayer)

[0169] q2: Load for energy absorption test of composite structure (with waterproof interlayer)

[0170] y: Deflection of the beam in the bending toughness test.

[0171] y1: Deflection of a homogeneous (without waterproof interlayer) beam in a bending toughness test.

[0172] y2: Deflection of the composite beam (with waterproof interlayer) in the bending toughness test.

[0173] p: Load on the beam used in the bending toughness test.

[0174] p1: Load on a homogeneous (without waterproof interlayer) flexural toughness test beam.

[0175] p2: Load on the composite beam (with waterproof interlayer) for bending toughness test

[0176] δ: The central deflection of the energy test disk.

[0177] δ1: The central deflection of the homogeneous (without waterproof interlayer) energy testing disk.

[0178] δ2: The central deflection of the energy testing disk of the composite (with waterproof interlayer).

[0179] In this embodiment, a four-point beam bending toughness test is used for verification, and the equivalent thickness calculation method is obtained. According to formula (15):

[0180]

[0181] The equivalent thickness calculation method was derived by verifying the method through energy absorption experiments. According to formula (17):

[0182]

[0183] (5) Verification was performed using numerical simulation.

[0184] In this step, numerical simulation is used for verification. The initial values ​​of the first layer of sprayed concrete thickness h1 and the second layer of sprayed concrete thickness h2 in step (1) are used. The thickness t of the sprayed waterproof layer, elastic modulus E2, shear modulus G, cohesion c and internal friction angle φ determined in step (2) are used for numerical simulation. At the same time, it is compared with a homogeneous body without sprayed waterproof material to compare the thickness of the homogeneous body with the same deflection under the same load.

[0185] Numerical simulations using ANSYS software revealed that under the same load, the deflection was equal, and the following conclusions were drawn:

[0186] Table 5. Calculation of Equivalent Thickness of Waterproof Single-Layer Lining

[0187]

[0188] Example 2, as Figure 1 As shown, the difference between this embodiment and Embodiment 1 is that this embodiment also includes verification of the calculation method proposed in this invention through numerical simulation and experiments. Specifically:

[0189] This invention proposes an analytical calculation method for waterproof single-layer lining composite structures, establishing the mechanical relationship between the equivalent thickness of the composite structure and key tunnel parameters. The equivalent thickness h needs to be calculated using the Q-system method by looking up tables or the block theory software Unwedge. 等效1 The thickness h1 of the first layer of shotcrete was determined through temporary load calculations, combined with the equivalent thickness h. 等效1 Assuming the second layer of shotcrete thickness h2 is assumed, the equivalent thickness h is then calculated using the method proposed in this invention. 等效2 The calculation method proposed in this patent is verified through numerical simulation and experiments, and finally h is determined. 等效1 and h 等效2 The difference, thus adjusting the thickness of h1 and h2 so that h 等效2≥ h等效1 .

[0190] The equivalent thickness h is calculated based on the assumed thickness of the second layer of shotcrete. 等效2 Determine the equivalent thickness h 等效2 Is it greater than the total thickness h of the shotcrete? 等效1 If yes, the conclusion is that the requirements are met; if not, the thickness of the second layer of sprayed concrete is adjusted.

[0191] When determining the performance parameters of the sprayed waterproofing material and the interface parameters of the sprayed waterproofing material in the composite structure, finite element analysis is performed to verify the parameters in the tensile stress-strain test of the sprayed waterproofing material, the shear stress-displacement test of the composite structure, and the bond stress-displacement test of the composite structure. The numerical results are then compared with the sandwich theory. If they match, the conclusions meet the requirements; otherwise, the parameters of the sandwich theory formula are adjusted.

[0192] The flexural toughness test and energy absorption test were conducted to verify the waterproof single-layer lining. The test results were then compared with the sandwich theory. If they matched, the conclusion met the requirements. If not, the parameters of the sandwich theory formula were adjusted.

[0193] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for calculating the equivalent thickness of a waterproof single-layer lining, characterized in that, Includes the following steps: (1) Determine the tunnel model, determine the thickness h1 of the first layer of shotcrete, the elastic modulus E1 of the first layer of shotcrete, the thickness h2 of the second layer of shotcrete, and the elastic modulus E1 of the second layer of shotcrete. Calculate the length l of the tunnel plastic hinge through numerical simulation. (2) Determine the parameters of the sprayed waterproof interface. The parameters of the sprayed waterproof interface include determining the performance parameters of the sprayed waterproof material and determining the parameters of the sprayed waterproof interface of the composite structure. (3) Calculate the equivalent thickness; (4) Verification was conducted using indoor tests; (5) Verification was performed using numerical simulation.

2. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 1, characterized in that, Compression and shear tests were conducted on the composite beam with sprayed waterproof coating to obtain the thickness t, elastic modulus E2, shear modulus G, cohesion c, and internal friction angle φ of the sprayed waterproof coating.

3. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 2, characterized in that, In step (1), the equivalent thickness h is obtained by looking up a table using the Q-system method. 等效1-1 The equivalent thickness h was calculated using the block theory software Unwedge. 等效1-2 The equivalent thickness h is calculated using permanent loads. 等效1-3 Then, the maximum value is taken to obtain the shotcrete thickness h. 等效1 =max(h 等效1-1 h 等效1-2 h 等效1-3 ); Through numerical simulation methods and the total thickness h of the sprayed concrete... 等效1 The length l of the tunnel plastic hinge was calculated.

4. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 3, characterized in that, In step (3), the equivalent thickness is calculated using the following formula: Based on formulas (1) and (2), the following can be deduced: According to the formula for moment of inertia The derivation leads to: in, Simplified to: Substituting formulas (6) and (7) into formula (4), and simplifying them, we can obtain the overall equivalent thickness of the waterproof single-layer lining: in, D = h1 + h2(12) In the formula: (EI) 叠合 Overall bending stiffness of composite structures (AQ) 叠合 Overall shear stiffness of composite structures (EI) 第一层砼 : Flexural stiffness of the first layer of shotcrete (EI) 喷涂防水 : The flexural stiffness of the waterproof coating (EI) 第二层砼 : Flexural stiffness of the first layer of shotcrete I 等效 Moment of inertia of an equivalent homogeneous body h 等效2 The overall equivalent thickness of a waterproof single-layer lining. δ 叠合 Waterproof single-layer lining deflection, δ 等效 Deflection of homogeneous lining P: Concentrated load a: The distance from the upper edge of the sprayed waterproof layer to the centroid of the single-layer waterproof lining. l: The span of the beam (l is approximately equal to the arc length between the plastic points of the tunnel). h1: Thickness of the first shotcrete application h2: The thickness of the second layer of shotcrete. t: Thickness of the waterproof coating. E1: Elastic modulus of shotcrete. E2: Elastic modulus of waterproof coating. G: Shear modulus of waterproof coating.

5. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 4, characterized in that, In step (4), a four-point beam bending toughness test is used for verification, and the equivalent thickness calculation method is obtained. According to the mechanics of materials, the deflection at the midpoint of the span is: From formula (14), it can be seen that, under the condition of equivalent deflection, the ratio of bending strength to thickness is: The equivalent thickness calculation method was derived through energy absorption testing, and the formula for the center deflection was obtained based on the calculation: Under the condition of equivalent deflection, the ratio of deflection to thickness is: in, b: Width of the beam used in the bending toughness test d: Height of the beam used in the bending toughness test. r: radius of the energy absorption test disk E: Elastic modulus of concrete in energy absorption test q: Load for energy absorption test q1: Load for energy absorption test of homogeneous body (without waterproof interlayer) q2: Load for energy absorption test of composite structure (with waterproof interlayer) y: Deflection of the beam in the bending toughness test. y1: Deflection of a homogeneous (without waterproof interlayer) beam in a bending toughness test. y2: Deflection of the composite beam (with waterproof interlayer) in the bending toughness test. p: Load on the beam used in the bending toughness test. p1: Load on a homogeneous (without waterproof interlayer) flexural toughness test beam. p2: Load on the composite beam (with waterproof interlayer) for bending toughness test δ: The central deflection of the energy test disk. δ1: The central deflection of the homogeneous (without waterproof interlayer) energy testing disk. δ2: The central deflection of the energy testing disk of the composite (with waterproof interlayer).

6. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 2, characterized in that, In step (5), when verifying using numerical simulation, the initial values ​​of the first layer of sprayed concrete thickness h1 and the second layer of sprayed concrete thickness h2 determined in step (1) are used to perform numerical simulation based on the thickness t of the sprayed waterproof layer, elastic modulus E2, shear modulus G, cohesion c, and internal friction angle φ determined in step (2). At the same time, it is compared with a homogeneous body without sprayed waterproof material to compare the thickness of the homogeneous body with the same deflection under the same load.

7. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 1, characterized in that, In step (2), when determining the performance parameters of the sprayed waterproof material, the tensile stress-strain curves corresponding to different waterproof membrane thicknesses are obtained by tensile stress-strain test of the sprayed waterproof material. The slope of the curve is the elastic modulus E2, Poisson's ratio, yield stress, and yield strain.

8. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 1, characterized in that, The parameters of the waterproofing interface for the composite structure are determined by the shear stress-displacement test and the bond stress-displacement test of the composite structure.

9. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 8, characterized in that, In the bond stress-displacement test of the composite structure, the bond stress-displacement curve is obtained through the bond strength test. The peak stress of the curve is the bond strength, the slope of the rising segment of the curve is the bond stiffness, and the area formed by the curve and the horizontal axis represents the bond failure energy.

10. The method for calculating the equivalent thickness of a waterproof single-layer lining according to claim 9, characterized in that, In the shear stress-displacement test of the composite structure, the shear stress-displacement curve is obtained through the shear strength test. The peak stress of the curve is the shear strength. The slope of the rising segment of the curve is the shear stiffness, and the area formed by the curve and the horizontal axis represents the shear slip energy.

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